Theory of the Integer and Fractional Quantum Hall Effects

Transcription

1 Theory of the Integer and Frational Quantum all Effets Shosuke SASAKI Center for Advaned igh Magneti Field Siene, Graduate Shool of Siene, Osaka University, - Mahikaneyama, Toyonaka, Osaka 56-, Jaan Prefae The all resistane in the lassial all effet hanges ontinuously with alied magneti field. So, the disovery of a stewise hange of the all resistane in an ultra-thin layer of a MOSFET was a big surrise. The all resistane, R was found to take the following disrete values, R R K ν (59) Ω R K (.a) (.b) where, R K is alled the von Klitzing onstant and ν is the filling fator that takes integer values. It was found later that ν is not only restrited to integers but also takes seifi frational numbers. After these disoveries, a great deal of efforts has been made to eluidate the origin of the integer and frational quantum all effets (IQE and FQE). The many eletrons inside an ultra-thin layer onstrut a quasi-two-dimensional (quasi-d) system beause the quantum state along the diretion erendiular to the thin layer is onfined to the single ground state at low temeratures. The IQE and FQE are observed in several materials namely Si, GaAs, grahene and so on. Thus the QE aears indeendently of materials and is aused only by many eletrons in a thin onduting layer. If the Coulomb interation between eletrons is negleted, the eigen-value roblem of the D amiltonian an be solved exatly and the eigen-states are desribed by the Landau states. Therein the energy gas aear only in integer filling fators and then the FQE doesn t aear. So the FQE should be derived from the total amiltonian with the Coulomb interation between eletrons.

2 In the lassial all devie (D system) both the alied magneti field and the eletri field due to the all voltage at on eletrons. The magneti fore is balaned with the eletri fore in a steady state. Aordingly the total fore beomes zero. This anelation yields the relation between the magneti field strength and the all voltage. Thereby the measurement of the all voltage determines the magneti field strength as well known. That is to say the average orbit of eletron is a straight line (not a irle). In the quantum all system (D system) many artiles exress the eletron orbital by a irle. This view yields some onfusion. The irular orbital is aused by missing of the eletri otential gradient along the diretion of all voltage. In this book we take the eletri otential gradient into onsideration. The quasi-d eletron roblem with the Coulomb interation between eletrons annot be solved exatly. So we are obliged to use some aroximations for larifying the FQE. Laughlin introdued a quasi-artile with frational harge and its wave funtion. e obtained the wave funtion (Laughlin funtion) by using the variational-method. Jain introdued a quasi-artile alled the omosite fermion (CF) whih is an eletron aturing even number of magneti flux quanta. The number of the flux quanta deends uon the filling fator. That is to say there are many tyes of the omosite fermion (aturing flux quanta, flux quanta and so on). There has, however, been no diret evidene for the existene of these quasi artiles. Some hysiists might argue that the disovery of FQE manifests itself the existene of the quasi artiles. It is merely reeating the same thing. At least every CF wave funtion should be aroximately exressed by the wave funtion of many eletrons. The CF theory is laking in this desrition. On the other hand Tao and Thouless investigated the FQE using normal eletrons without any quasi-artile. owever they were not able to identify the filling fators that give stable states. We extended the Tao-Thouless theory and have sueeded in finding the most uniform eletron onfiguration in the Landau orbitals. The onfiguration is uniquely determined at any filling fator so as to minimize the exetation value of the total amiltonian. The Coulomb interation ats between two eletrons, and rodues the quantum transitions. That is to say, an eletron air in Landau orbitals transfers to the other emty orbitals. Therein the Coulomb interation deends uon only the relative distane. Aordingly the two eletrons move in a air so as to satisfy the momentum onservation law along the urrent diretion. (Note: the onservation law doesn t hold in the diretions erendiular to the urrent, beause other otentials exist.) The number of the allowed transitions varies abrutly with hanging the filling fator

3 beause of Pauli s exlusion rinile. The number takes a maximum at seifi filling fators ν suh as, ν ( ), ( ), ( ) ( ), ( ) et. When the filling fator ν deviates from ν, the transition number dereases disontinuously. So the transition number takes a maximum at ν with a ga. eause the airing energy is negative, the airing energy has a lowest value with aomanying an energy ga at eah seifi filling fator ν. Aordingly the quasi-d eletron system is onfined to these filling fators. The disontinuous form of the airing energy versus filling fator rodues the all lateaus whih are observed on the all resistane urve. Then the all resistane beomes equal to R K ν at the seifi frational numbers ν. The differene between the exerimental value and R K ν is extremely small. For 8 examle the relative unertainty is ±. at the filling fator /. This auray at the seifi filling fators will be larified in this book. ) Thousands of the Landau wave funtions are overlaed with eah other. ) The number of the Coulomb transitions dereases disontinuously by an infinitesimal deviation from ν as larified in hater 8 with taking aount of the momentum deendene of the air energy. Thereby the all resistane is reisely onfined to R K ν at the seifi frational numbers ν. In a high magneti field, all the eletron sins are direted antiarallel to the alied magneti field. At lower magneti fields, the differene between the numbers of u and down sins deends uon the alied magneti field strength. Then the wide lateaus and small shoulders aear on the exerimental olarization urves versus magneti field strength. The most uniform eletron onfiguration in the Landau orbitals has sin degeneray ie., many different sin arrangements are ossible for a given onfiguration of eletrons. Therefore the many-eletron states with different sin arrangements have the same exetation value of the Coulomb interation. The off-diagonal arts of the Coulomb interation rodue quantum transitions between these degenerate ground states. We have sueeded to solve exatly the eigen-value roblem of the artial amiltonian whih is omosed of the strongest and seond strongest interations among the degenerate ground states. Then the eigen-solutions yield the wide lateaus on the olarization urve. Furthermore the Peierls instability exists in this system. The sin Peierls effet rodues the small shoulders. The theoretial results are in good agreement with the exerimental data. Thus the IQE and FQE are exlained by a standard treatment of interating quasi-d eletron gas without assuming any quasi-artile in this book.

4 The author exresses his heartfelt areiation for enouragement of Professor Koihi Katsumata, Professor idenobu ori, Professor Yasuyuki Kitano, Professor Masayuki agiwara and Professor Takei Kebukawa. Eseially Professor Katsumata has given me imortant suggestions for imroving my desrition in this book. The friendly o-oeration of the Editorial Offie of Nova Siene Publishers is gratefully aknowledged. Also I wish to exress areiate for suort of my wife Makiko Sasaki and my sons Takashi Sasaki and Takuma Sasaki. I annot omlete this book without their suort.

5 Contents Prefae Chater. Formulation of the roblem. Potential in a quantum all devie. Single-eletron states. Effets of eletron sin Chater. Integer quantum all effet. Calulation of the all resistane. Distribution of urrent. A rigorous derivation of the all resistane Chater Coulomb energy in the FQ states. Total amiltonian and omlete set of many-eletron states. Many-eletron state with minimum lassial Coulomb energy. Momentum set seifying the many-eletron state with the minimum lassial Coulomb energy. ν -deendene of the minimum lassial Coulomb energy Chater inding energy of eletron air. Momentum onservation in Coulomb interation. Perturbation energy of a nearest eletron air. inding energy at the filling fators ( ) ( ) and ( ) ( ). Eletron-ole symmetry of nearest airs.5 Charater of the nearest eletron air and the nearest hole air at the 5

6 seial filling fators ν ( ±), ν ( ±) and ν ( ± ) ( ± ) Chater 5 Valley, flat and eak strutures in the energy setrum 5. Valley struture at the filling fator of ν ( ±) 5. Comarison of the theory with exerimental data in the neighborhood of ν ( ±) 5. Flat struture at ν and eak struture at ν and ν 5. Valley struture at ν ( ) ( ) and ν ( ) 5.5 Seifi filling fators with even number denominator 5.6 The states with non-standard filling fators 5.7 Exitation-energy-ga in FQ states 5.8 Comarison between the theory and the exerimental data in a wider range of magneti field 5.9 Pair energy of eletrons laed in the seond neighbouring Landau orbitals (Exlanation of the all lateaus in the region <ν < ) 5. Further investigation of the total energy in the quasi-d eletron system (air-energies of eletrons laed in more distant Landau orbitals) 5. all lateaus in the region <ν < 5. Short omment Chater 6 Effet of higher order erturbation and ontribution from uer Landau levels 6. Third order erturbation energy 6. igher order erturbation energy 6. Contribution of uer Landau levels Chater 7 Plateaus of quantum all resistane 6

7 7. Funtion-form of the total energy 7. Setrum of the total energy versus filling fator 7. ehaviour of all resistane urve near ν /, /, / and so on 7. Exlanation for the aearane of the lateaus in the all resistane urve 7.5 Comarison of the resent theory with exerimental data Chater 8 Auray of all resistane onfinement 8. Auray of the all resistane onfinement in FQE 8. Effets of Imurities and Lattie defets on the all resistane 8. Shae and size effet for the all resistane Chater 9 Sin olarization in the frational quantum all states 9. Coulomb interation between u and down sin states 9. Isomorhi maing from FQ states to one-dimensional fermion states 9. Diagonalization of the most effetive amiltonian at filling fator of / 9. Magneti field deendene of the Sin-Polarization at filling fator of / 9.5 Parameter Deendene of Polarization 9.6 Sin-Polarization for other filling fators 9.7 Small shoulder in Polarization Curve 9.8 Phenomena similar to the sin-peierls effet in various filling fators Chater FQE under a tilted magneti field. Formulation for the tilted magneti field. Comarison between theory and exeriment Chater Further exeriments 7

8 . Diagonal resistane in the FQ state under a eriodially modulated magneti field or urrent. Tunneling through a narrow barrier in a new tye of quantum all devie Chater Disussions on Traditional theories. Eletri otential along all voltage in the quasi-d eletron system. Laughlin theory and aldane-alerin hierarhy theory. Comosite Fermion theory. Effetive magneti field and flux quantization.5 Sin-olarization of omosite fermion theory Chater Summary Referenes Aendix Nearest-air-energy of FQ states with ν ( s ± ) ( 6s ± ) 8

9 . Formulation of the roblem The integer quantum all effet (IQE) was disovered by K. von Klitzing in the inversion layer of a metal-oxide-semiondutor field-effet transistor (MOSFET) [-]. The quantum all devie is shown shematially in Fig... The length of the devie is µm, the width 5 µm and the thikness of the inversion layer is about - µm. A magneti field is alied erendiularly to the surfae of the devie and the eletri urrent flows between the soure and drain. The eletron density was ontrolled by adusting the gate voltage. The all voltage and the urrent were measured at low temeratures. Then the all resistane R is obtained as [all voltage]/[urrent]. K. von Klitzing disovered a lateau in the all resistane urve for a wide range of gate voltage. The exerimental value of R is in the lateau. The value is lose to h ( e ), where h is Plank s onstant and (- e) is the harge of eletron. Thus K. von Klitzing larified that the quantum all effet gives an aurate method for determining the fine-struture onstant. Soure z y x Magneti field all robe Drain Potential robes : length of DES Gate width d Fig.. Quantum all Devie Thereafter many lateaus in the all resistane urve have been observed at ( ν ) e. The all resistane for ν, R K π e (59)Ω is alled the von Klitzing onstant and is used for the π, where ν is an integer and h /( π ) international resistane standard. The henomena are named the integer quantum all effets (IQE). Quantum all devies have been imroved so as to have higher eletron mobility. D. C. Tsui,. L. Stormer et al. [-9] found many lateaus in the all resistane urve at ( ν ) π e with the values of ν 5,, 5, 7, 9, 7, 5,,. This henomenon is alled the frational quantum all effet (FQE). The exerimental 9

10 values are drawn by the blak urves in Fig.. (referene [9]:. L. Stormer, Nobel leture). The uer anel of Fig.. shows the all resistane R versus magneti field strength and the lower anel indiates the diagonal resistane R whih is the usual eletri resistane between two otential robes in Fig... Fig.. all resistane R and diagonal resistane R The IQE and FQE have been observed in quantum all devies omosed of several materials Si, GaAs and so on, at low temeratures.5 K,. K,.5 K et. Furthermore it is surrising that the quantum all effet has been observed in grahene at a room temerature []. Thus the QE is a henomenon indeendent of materials.

11 Any quantum all devie has an ultra-thin layer where eletrons flow. eause the thikness is ultra small, only one ground state is effetive along the diretion erendiular to the thin layer at low temeratures and all the other exited states have negligibly small robability. Aordingly the eletron system may be aroximated to a two-dimensional (D) one. Grahene, a sheet of bonded arbon atoms, rovides us with an ideal D eletron system. Thus we an take a D eletron system as a working model for the alulation of the quantum all effets. As shown by the dashed lines in Fig.., the diagonal resistane is almost zero on the wide lateaus of the all resistane namely at ν,, /, /5 and /. The diretion of all voltage is erendiular to the urrent diretion, and so the all resistane yields no ohmi heating. On the other hand the diagonal resistane yields ohmi heating. eause the diagonal resistane is almost zero at the all resistane onfinements, the QE is a ure quantum roess without thermal exitations, eletron satterings and so on. In the tyial exeriments of IQE the all voltage is larger than about - Volts and the diagonal (otential) voltage is less than - Volts. Also in the FQE the all voltage is extremely large omared with the diagonal (otential) voltage at the all resistane onfinement. So we annot ignore the gradient of the eletri otential along the y-diretion of Fig... Consequently the total amiltonian for many eletrons should be omosed of three kinds of interations namely the strong magneti field interation, the Coulomb interations between eletrons, and the eletri field rodued by the all voltage. We take the three kinds of interations into onsideration in this book. Many theories ignore the eletri field along the all voltage. Then the orbital of eletron is exressed as a irle. This iture leads some misunderstandings in the investigation of the FQE. (Note: D all system: The magneti and eletri fields work on the eletrons in a D all system. Therein the eletri fore due to the all voltage anels the fore by the magneti field. So the eletron moves on a straight line along the x-diretion. The aneling ondition gives the relation between the all voltage and the magneti field strength. The relation makes it ossible to determine the magneti field strength by measuring the all voltage. Thus the eletri fore lays an imortant role in D all henomenon.) In this hater we study the basi formulation of the quantum all system where we take aount of the eletri field along the all voltage and ignore the Coulomb interation between eletrons. Then the eigen energy roblem of eletron an be exatly

12 solved and the wave funtion is exressed by the Landau solution. The wave funtion is the rodut of three wave funtions of the x-, y- and z-diretions. The wave funtion along the x-diretion (urrent diretion) is the lain wave form. That is to say the ballisti movement of eletron aears in the urrent diretion.. Potential in a quantum all devie Let us take the oordinate axes as in Fig.., in whih the urrent flows along the x-axis, the all voltage emerges along the y-axis, and the magneti field is alied along the z-axis. Potential V Thin (z) V Thin (z) Fig.. Potential of z-diretion (Potential width is very narrow) z Any quantum all devie has an ultra-thin onduting layer. Aordingly the otential, V Thin ( z), along the z-axis has a very high barrier outside the thin layer as in Fig... This otential yields a very large exitation-energy from the ground state in the z-diretion. The quantum all effets are observed in suh a low temerature that the robability of the ground state is almost % for the z-diretion. Potential -ev U(y) -ev y Fig.. Potential U ( y) yd y

13 In Fig.. we draw shematially the otential, U ( y), along the y-axis. ere the otential urve turns steely uward near the both edges so that the eletrons are onfined to the onduting region. The voltage is denoted as V and V at y and y d, resetively. The differene, V V, is determined by measuring the voltage between the all and otential robes. At the all resistane onfinement the value V V is extremely larger than the otential (diagonal) voltage between otential robes. Therefore we examine the nonzero ase of V V in the investigation of QE. U is studied in more details in hater. The shae of the otential ( y) We first study the wave funtion of single eletron. In the Landau gauge the vetor otential, A, has the omonents as ( y,, ) A, (.) where is the strength of the magneti field. This vetor otential ertainly satisfies the following fundamental equations: A A A A A z y x z y A x rot A,, y z z x x y A A x y Az div A x y z (,, ) The amiltonian,, of a single eletron is obtained as follows: where ( ea) m U ( y) V ( z) Thin, (.) m is an effetive mass of eletron and ( x, y, z ) is the eletron momentum. The effetive mass m differs from material to material [] and the value in GaAs is about.67 times the free eletron mass.. Single-eletron states If the Coulomb interation between eletrons is negleted, the eigen-energy roblem for the eletron system is exatly solved under the uniform magneti field. The eigen-equation is given by, ψ Eψ. (.a) From Eqs.(.), (.) and (.a) we obtain the eigen-value roblem as,

14 m i ey U ( y) VThin ( z) ψ Eψ (.b) x y z This equation has no otential term deending on x and so the eigen-state along the x-axis is desribed by a lane wave. Also the term ontaining z is searated and so the wave funtion ψ is exressed as, ψ ( x, y, z) ex( ikx) ϕ( y) φ( z), (.) where is the length of the devie along the x-axis. Therein the wave number k satisfies the eriodi boundary ondition; ( ), J : integer k π J (.5a) Eq.(.5a) is rewritten in terms of the momentum as ( π ) J. (.5b) ereafter we use the symbol for the x-omonent of the eletron momentum throughout in this book (namely we abbreviate along the z-axis satisfies the following equation, x as ). The wave funtion φ m z V Thin ( z) φ( z) λφ( z) (.6) where λ is the eigen-energy of the wave funtion φ. The exitation energy along the z-axis is very large beause the otential V Thin ( z) is very narrow. Consequently, the robability of finding the exited states in the z-diretion is extremely small at a low temerature. In the ase of grahene, the robability is exeted to be still small even at room temerature. Therefore, we may take only the ground state for the z-diretion in this book. That is to say, φ ( z) indiates the ground state and λ is the ground state energy. Substituting Eqs.(.6) and (.) into Eq.(.b) we have, ( k ey) m m y U We define an effetive otential, G ( y), as, ( ) ( k ey) G y U ( y) m Then, we get the eigen-equation for the y-diretion as ( y) ex( ikx) ϕ( y) ( E λ) ex( ikx) ϕ( y) (.8) (.7)

15 m y G ( y) ϕ( y) ( E λ) ϕ( y) (.9) We will examine the shae of G ( y) below. When the magneti field is strong, the first term of the right-hand-side in Eq.(.8) dominates. Then the effetive otential G ( y) takes a minimum at y α J where α J is given by α J k ( e) ( e) [ π ( e) ]J. (.) Thus the quantity α J of the y-diretion is related to the momentum along the x-diretion as in Eq.(.). As will be larified below in Eq.(.7), α J is the entre osition of the wave funtion along the y-diretion. The sreading region of the wave funtion along the y-axis is about.5 nm at the magneti field strength of 6 T as will be estimated below in Eq.(.9). eause this distribution is very narrow in omarison with the devie width, the otential U ( y) may be aroximated by ( ) (Note: A more reise treatment of this term) U α. eause the wave funtion along the y-axis has a very narrow width (about.5 nm), the y deendene of U ( y) an be aroximated with the linear funtion in the neighborhood of U y α as follows; J ( y) U ( α ) β ( y α ) where β reresents the derivative at G ( y) e ( m ) J J y α. Then we obtain ( )( y α ) U ( α ) β ( y α ) (Note: we an take aount of the quadrati y-deendene in ( y) alying the same roedure mentioned above.) J J J J J U near y α J by The funtion G ( y) takes a minimum at α J whih deviates by a small value b from the original value α J as follows: α α b b β m ( e ) G J J ( )( y ) U ( α ) βb ( y) e ( m ) α J J Thus the minimum osition of the effetive otential is y α J whih deviates by b from α. The funtion shae is the same as for b beause the deviation b and J 5

16 the residual term βb are onstant. Therefore the eigenvalue-roblem in the reise otential is solved by the same roedure as in the aroximate form mentioned below. (End of the Note) For simliity we use the value α. Then ( y) J G is aroximated as where G ( y) m ω ( y α ) U ( α ) J J, (.) e ω (.) m (Note: We an easily obtain the more reise results by relaing α J with α J and also by adding the residual onstant energy in all the haters of this book.) Substitution of Eq.(.) into Eq.(.9) yields the eigen-value roblem in the y-diretion as, m y m ω ( y α ) ϕ( y) ( E λ U ( α )) ϕ( y) J J. (.) Equation (.) is rewritten as, m y m ω ( y α ) ϕ( y) εϕ( y) J, (.a) where ( E λ U ( α )) ε (.b) J This eigen-value roblem has the same form as that of a harmoni osillator. The eigen-state, ( y) ϕ L,J, and the eigen-value, L ε are given resetively, by, ϕ L, J ( ) ( ) y u y α ex ( y α ) L L m ω J m ω J (.5) ( L,,,, ) (.6) where L is the ermite olynomial of L -th degree and u L is the normalization onstant. We all L the Landau level number. Substitution of Eq.(.5) into Eq.(.) gives the single-eletron wave funtion ψ L, J as follows, 6

17 ψ L, J m ω m ω ( x y z) ( ikx) u ( y α ),, ex ex ( y α ) φ( z) L L The eigenenergy E is derived from Eqs. (.b) and (.6) as J J (.7) λ U ( α ) ω (,,,, ) EL, J J L L (.8) The wave funtion, Eq.(.7), has the same form as that of the Landau state exet for the z-diretion. The eigenenergy E, is a sum of the three terms, namely, the energy L J λ in the z-diretion, the otential energy U ( α J ) in the y-diretion and the Landau energy with the level number L. Eq.(.7) exresses that the wave funtion along the y-diretion. α J is the entre osition of The distribution of the wave funtion is estimated here: eause the ermite olynomial for L is equal to, the robability density ψ L, Jψ L, J is L roortional to the following Gaussian form in the y-diretion as m ω ex m ω ( y α ) ex ( y α ) ex ( y α ) J J m ω When the magneti field strength is 6 T, the width of the Gaussian, y in the y-diretion beomes y.5 nm (.9) m ω e Therefore this value.5 nm indiates the sreading width y of the wave funtion. Next we alulate the number of states for a given L. We note that the value of satisfies as in Fig..5. Then, using Eq.(.) we have, So, the number of states for a given L is,. (.a). (.b) In Eq.(.7) the ermite olynomial L with L is equal to and u L is a onstant. So, the wave funtion is a rodut of the lane wave funtion along the x-axis and the Gaussian funtion along the y-axis. Figure.5 shematially exresses the wave J 7

18 funtion-shae by searate lines in order to be visible in the figure, although the robes of the wave funtions with J, J and J atually overla with eah other. π e due to Eq.(.). These lines are equally saed with a distane given by [ ( )] The overla of the wave funtions will be studied in details in hater 8. z y x Magneti field all robe Drain y x length of DES all robe Soure width d Potential robes : length of DES Potential robes Fig..5 Shemati figure of Landau states J J J. Energy differenes between various states The single eletron states have eah eigen-energies resetively. We estimate the rough values of the energy differenes as follows:.. Energy differene between Landau levels The energy differene between the neighbouring two Landau levels is equal to e m whih is derived from Eq.(.8) using Eq.(.). ( ) ΓLandau ω e m (.) As already mentioned above, the effetive mass m for GaAs is.67 times the eletron mass m in vauum. We roughly estimate the energy differene Γ for 6[ T] as Γ Landau ( m k ).5[K] k e for m.67 m, 6[ T] Landau (.) where k is the oltzmann onstant... Energy differene between u and down sins The Zeeman energy, E Z, is given by E Z ( e ( m) ) z g σ i g i,, i,, µ σ (.) z i 8

19 where g is the effetive g-fator, ( z ) σ is the z-omonent of eletron sin oerator z ( σ : Pauli matrix) and µ is the ohr magneton. The effetive g-fator for GaAs is about. times the g-fator of eletron in vauum, namely, g.. The energy differene between the two Zeeman levels is equal to g ( e ( m) ) Γ Sin ( e ( mk )).8[ K] as k g for. g, 6[ T] (.) Comarison of Eq.(.) with Eq.(.) yields that the energy differene between L and L Landau levels is about 67 times the Zeeman slit energy for GaAs... Energy by Coulomb interation between eletrons The exetation value of the Coulomb interation is named the lassial Coulomb energy. The eletron onfiguration in the Landau orbitals is uniquely determined at any filling fator so as to have the minimum lassial Coulomb energy. This roerty is roven in hater. Aordingly the ground state with the eletron onfiguration has the lowest value in the Landau energy, the lowest value in Zeeman energy and the lowest value in the lassial Coulomb energy. The Coulomb interation ats on any eletron air and the strength is deendent on the relative distane only. Also there is no otential along the x-diretion as in Eqs.(.) and (.). So the two eletrons satisfy the momentum onservation law along the urrent diretion (x-diretion). The residual Coulomb interation yields quantum transitions of eletron airs. We investigate the transition number versus filling fator in details in haters -5. The number of allowed transitions takes loal maximum at seifi filling fators ν and the number dereases disontinuously by an infinitesimal deviation from ν. This disontinuous hange rodues the energy ga via the Coulomb interation whih is studied in haters -8. The ga value deends on quantum all devies. The exerimental value Γ ga is disussed in hater 5. The tyial value of the energy ga is [ ] Γ k. ~ K (.5) ga As in Eqs.(.) and (.5) the energy ga in the FQE has a magnitude of the same order as in the Zeeman slitting energy. Therefore the sin olarizations of FQ states deend on the magneti field strength in a quasi-d many-eletron system. The deendenes are very omliated whih are examined in hater 9... Energy levels of single eletron 9

20 We show the energy levels of the single eletron system in Fig..6 where the Landau energy levels are exressed by left lines and the Zeeman slitting is drawn by the arrow airs. Then all the energy levels are exressed as in right lines. Fig..6 Single eletron energy levels inluding sin searations Next we define the filling fator ν as, ν [total number of eletrons] / [total number of Landau states with L ] (.) Some examles are given below. (Case ) When eah of the J Landau states with L is ouied by an eletron with down sin (the sin ointing oosite to ), ν. (Case ) When eah of the J Landau states with L is ouied by two eletrons with u and down sins, ν. (Case ) Similarly, when eah of the J Landau states with L is ouied by two eletrons with u and down sins and eah of the J Landau states with L is ouied by an eletron with down sin, ν. The states with integer filling fator ν,, 5 have the exitation energy Γ sin and ones with ν,, 6 have the exitation energy ( Γ Landau Γ sin ) as seen in Fig..6. These exitation energies are very large in omarison with the thermal exitation energy under a temerature alied in the QE exeriments. Therefore the

21 states with integer filling fators are atually realized and are resonsible for the integer quantum all effet. If the filling fator ν is a frational number in the region of n < ν < n, eah of the Landau states with L n are ouied by two eletrons with u and down sins and Landau states of L n are artially filled with eletrons having down sins. (It is noteworthy that L is ounted from.) In this ase both emty and filled states have the same Landau energy. So the lowest energy states are degenerate in the absene of the Coulomb interations between eletrons. Similarly the ground states in the ase of n < ν < n are degenerate. Therefore we should take the Coulomb interations between eletrons into onsideration for investigating the frational quantum all effet. The Coulomb interation yields reulsive fores between eletrons, and so the eletrons are distributed as uniformly as ossible among many Landau states. Thereby only one eletron-onfiguration has a minimum exetation-value of Coulomb interation at a given ν. As an examle, a ν state is obtained by reeating the unit-onfiguration (emty, filled, emty) as in Fig..7a. A ν state is obtained by reeating the unit-onfiguration (filled, emty, filled) as in Fig..7b. Fig..7a: Configuration attern at ν Fig..7b: Configuration attern at ν old lines indiate the Landau ground states filled with eletron, and dashed lines emty states. In a similar way, we an onstrut the most uniform filling attern for any given frational number ν. The state with the most uniform eletron-onfiguration onstitutes a basis for the frational quantum all effet. Quantum transitions from the state via the Coulomb interations will be studied in Chaters -9.

22 Chater Integer quantum all effet Klaus von Klitzing disovered the integer quantum all effet (IQE). The quantum all resistane R is defined by [all voltage] / [urrent]. The exerimental value takes the following value reisely: R R K ν (.a) R K h e (59) Ω (.b) where R K is the von Klitzing onstant, h the Plank onstant, e the elementary harge and ν is the filling fator with any integer value. The D eletron system was studied theoretially by T. Ando, Y. Matsumoto and Y. Uemura in 975. They alulated the eletri ondutivity by emloying the self-onsistent orn aroximation and obtained the all ondutivity as, σ ( )( ω τ ( ω ) ne. XY τ ( ) Then the all ondutivity was redited to be σ e N ( π ) ( ) attrative satterers and σ ( N ) ( π ) XY XY in the ase of e in the ase of reulsive satterers at zero temerature [5], where, N is the Landau level number. They have ointed out, for the first time, that the all ondutivity is exressed by the natural onstants only. efore the disovery of IQE, a standard of eletrial resistane had some ambiguity. At that time measurements of resistane inluded some error whih was aused by the lak of long-term stability and world-wide uniformity in the resistane standard. The value of the quantum all resistane is extremely aurate as in Eq.(.a, b) and then the henomenon has been alied to a resistane standard from 99. This astonishing auray means that the IQE should be exlained rigorously. It was ointed out for examle, in the referene [6], that The many theoretial models exlain various asets of the QE, at least in a qualitative way. A omlete theory whih onlusively exlains e.g. the remarkable auray of the QR is still missing. In this hater the IQE is studied ste by ste. At an integer filling fator the exitation is ossible only to the higher levels. The minimum exitation energy is ΓLandau Γsin or sin Γ for a sin-fliing exitation in

23 Fig..6. For the ase of non sin-fliing, the minimum exitation energy is equal to Γ Landau. Then the energy differene between an IQ state and its exited state is very large, omared with the thermal exitations, under a strong magneti field. Aordingly the IQ state is extremely stable at a low temerature. Even if we take the Coulomb interations into onsideration, the erturbation energy for any IQ state is very small beause of the large denominator aearing in the erturbation alulation. (The denominator is the energy differene between the ground state and an exited state whih is equal to or larger than Γ Landau beause of non-fliing of sin in Coulomb transitions.). Consequently neglet of the Coulomb interations is a good aroximation for an integer filling fator.. Calulation of the all resistane There are two kinds of veloity namely the hase veloity and the grou veloity. It is well known in quantum hysis that the veloity of a harge is the grou veloity u G, not hase veloity. Eah eletron has a wave funtion with the length in the urrent diretion. So eah eletron arries an amount of harge er unit time given by, (harge er unit time) eu G (.) where an eletron harge is e. The number of Landau states inside the momentum region from to d is equal to (number of Landau states) π d (.) whih is derived from Eq.(.5b). We exress the urrent inside the momentum region ~ d by the symbol d I, whih is given by d I (harge er unit time) (number of eletrons) In the ase of ν, the number of eletrons is equal to the number of Landau states and therefore d I is the rodut of (.) and (.) as eug eug di d d (.a) π π Therein we reexress to the oordinate y by making use of Eq. (.) in order to know the urrent distribution along the y-diretion as follows;

24 di eug e u d G dα J (.b) π π where α J indiates the entre osition of the y-diretion for eah eletron wave funtion and is related to the momentum as k eα (.5) J We exress the eigen-energy (.8) by the symbol f ( ). E The grou veloity u ( ) λ U ( α ) ω (,,,, ) L, J f J L G ug along the x-diretion is defined as ( ) L (.6) del,j df (.7) d d The grou veloity is exressed with the otential from Eqs.(.5), (.6) and (.7) as df ( ) du ( α J ) du ( α J ) dα J du ( α J ) ug (.8) d d dα d e dα This equation is substituted into Eq.(.b) and the result is di e du J ( α ) e du ( α ) J J dα J dα J (.9a) π e dα J π dα J We obtain the total eletri urrent by integration of Eq.(.9) as du ( α ) d e e I dα [ ( )] d U y y π dα π The otential differene at and the all robe in Fig.. as J (.9b) y and y d is measured between the otential robe U ( y ) ev, U ( y d ) ev (.) Substitution of Eq.(.) into (.9b) yields the total urrent as e e I π π [ ev ev ] ( V V ) (.) In the above disussion, we have negleted the effets of eletron sattering by lattie vibrations, imurities, and so on. This neglet is allowed due to the exerimental fat mentioned in Chater, namely, the usual (diagonal) resistane is ratially zero at the IQE as in Fig... Equation (.) gives the all resistane R at ν as follows:

25 R ( V V ) π e (for Thus the all resistane I ν ) (.) R at ν is desribed by the natural onstants only. It is noteworthy that the total urrent does not deend uon the funtion-form of the otential U ( y), but deends on only the voltage between at both edges of the devie (see Fig..). Furthermore the total urrent does not deend uon the devie sizes and d. Thus the relation (.) has been obtained. A rigorous roof will be given in setion.. For the filling fator larger than, the alulation is done similarly. The number of eletrons inside the region from y to y dy is ν e (number of eletrons inside y ~ y dy ) d y (for any integer ν ) (.) π whih gives ( y) d y ν e du di (.) π dy ν e I π ( V V ) (.5) Thus the all resistane R for any integer value of ν is exressed as ( V V ) π R (for any integer ν ) (.6) I ν e. Distribution of urrent M. üttiker investigated the distribution of eletri urrent in a quantum all devie [7]. We review his investigation in the ase of ν. A urrent element, d I through a width d y is obtained from Eq.(.9a) at the filling fator ν as, e du di π dy ( y) d y Therefore eletri urrent density namely urrent er unit width is given by ( y) di e du (.7) dy π dy 5

26 We have shown a tyial otential urve along the y-diretion in Fig.., where the large otential barriers at both edges onfine eletrons inside a ondution layer. The derivative d U ( y) dy at y is negative and d U ( y) dy at y d is ositive as easily seen in Fig... The absolute values of these two derivatives are very large to onfine eletrons inside the devie. In the entral region of the devie, d U ( y) dy is aroximately equal to d y dy ev ev (.8) U ( ) ( ) d We shematially draw U ( y) dy d in Fig... du(y) / dy y y - e(v -V ) /d du(y) / dy yd Fig.. Derivative value d U ( y) dy The urrent distribution is roortional to du ( y) dy - as in Eq.(.7). y using Fig.. we obtain the urrent density as shown in Fig... The absolute value of eletri urrent density, d I dy beomes large at both edges of the devie. The urrent distribution has been theoretially redited by üttiker [7]. The large urrent densities near both edges are named the edge urrents. There are many exerimental studies on the edge urrents [8]. The edge urrent near y flows arallel to the x-axis and that near y d does anti-arallel to the x-axis. The urrent density near the entral region of the devie is direted arallel to the x-axis, the magnitude of whih is given aroximately as, di dy ( e ( π ) )( ev ev ) d (.9) The absolute value of d I dy at the entral region is smaller than that at the both edges as in Fig... 6

27 di / dy y di / dy yd y Fig.. Current density di/dy This distribution of the urrent is also illustrated on a quantum all devie in Fig... Therein the urrent distribution is drawn by bold arrows. X Width d Voltage V Voltage V Y Fig.. Distribution of urrent density Current density is exressed by bold arrows. A rigorous derivation of the all resistane The exerimental value of all resistane at ν is (59) Ω whih is quite aurate. This fat means that the relation (.6) should be derived rigorously. In an atual system, we do not know the funtion-form of the exat eigen-energy E ( x ) beause the funtion form of the otential U ( y) is unknown and the Coulomb interations between eletrons are not treated exatly, and so on. Nevertheless we an rigorously derive Eq.(.6) as follows: 7

28 We re-study the eletri urrent. The eletri urrent er eletron is eu G, where e and u G are the eletron harge and the grou veloity resetively. The eletron number is ( π ) d inside the momentum region ~ d. Therefore the urrent d I inside the momentum region ~ d is given by the rodut of eu G and ( ) d π as follows: eu di G d (.) π The relation between di and d is indeendent of the length. So, even if the value of varies in different oints of y, the relation (.) is orret. Aordingly we an aly the relation (.) to atual quantum all devies. We introdue the exat eigen-energy E ( ), although the exat form of ( ) unknown. Thereby the grou veloity u G ( ) u G is exressed as E is de (.) d Substitution of Eq.(.) into Eq.(.) yields ( ) d e de di (.) π d We obtain the total eletri urrent by integrating Eq.(.) as edge ( ) e de e edge I d [ E( ) ] (.) edge π d π edge Therein the integration is arried out from the momentum at the edge to at edge. Consequently the total urrent is desribed by the exat eletron energies at edge and edge as follows: e I π ( E( edge ) E( edge) ) Thus the total urrent is exressed by the differene ( edge ) E( edge) (.) E of the exat eletron energies at the edges and. The energy differene an be determined by measuring the voltage between the otential robe and the all robe. E ( edge ) E( edge) ( ev ) ( ev ) e( V V ) (.5) 8

29 where e is exatly equal to the eletron harge and V V is the voltage between the all robe and the otential robe. Substitution of this equation into (.) gives e I π ( V V ) (.6) Thus the total urrent is indeendent of the funtion-form of the otential and the sizes and d of the devie. Therefore the all resistane R at ν is exatly equal to R ( V V ) π e (for I ν ) (.7) As exlained above, when the filling fator is ν, the number of eletrons is ν times that for ν. Therefore the total urrent is equal to ν e I π ( E( edge) E( edge) ) The atual energy differene ( ( edge) E( edge) ) (.8) E is equal to the voltage between the all robe and the otential robe as in the ase ν : E edge E edge e V V (for any value of ν ) (.9) ( ) ( ) ( ) Substitution of Eq. (.9) into Eq. (.8) yields the all resistane as R ( V V ) π ν e (for any value of ν ) (.) I Thus the relation (.) between the all resistane and the filling fator ν has been derived rigorously. For any integer ν the energy ga is Γ sin or ( Γ Landau Γsin ) whih are large in a strong magneti field. When the sin doesn t fli, the exitation energy is Γ Landau whih is very large as in Eq.(.). Therefore the ground state with an integer filling fator an be realized at low temeratures by adusting the magneti field strength or gate voltage. Therefore the all resistane given by Eq.(.) aears atually for any integer ν. When the filling fator ν is a frational number, the Landau levels are artially filled with eletrons. If we ignore the Coulomb interation between eletrons, the energy ga disaears and then the all resistane at a frational filling fator ν is not onfined. So we should take the Coulomb interation between eletrons into onsideration. In the later haters we examine the quasi-d eletron system with the Coulomb interation. 9

30 Chater Coulomb energy in the FQ states In this hater we study the many-eletron system in a quantum all devie and find a many-eletron state with a minimum exetation value of the Coulomb interation at any filling fator of ν.. Total amiltonian and omlete set of many-eletron states The total amiltonian T N ( x, y, z ) T of the many-eletron system is given by N N ( x x ) ( y y ) ( z z ) i i i i i > i πε i i i where N is the total number of eletrons, ( x i y i, z i ) eletron, ε is the ermittivity of the devie. The amiltonian ( x, y, z ) e (.), is the osition of the i-th i i i is given by Eq.(.) for the single eletron system. In hater, we have already obtained the eigenstates and eigenenergies of ( x, y, z). The seond term in the right hand side of Eq.(.) indiates the Coulomb interations between eletrons. Arbitrary many-eletron state an be exressed by suerosing of the many-eletron wave-funtions whih are omosed of diret roduts of single-eletron wave funtions. The single eletron wave funtion is exressed by the symbol ( x, y z ) ψ whih is given by Eq.(.7). The -th wave funtion is seified by a L,, Landau level number L and a momentum. Aordingly the many-eletron state is seified by a set of Landau level numbers L, L,, L and momenta N,,,. N The omlete set of many-eletron states is omosed of the Slater determinant as ( x, y, z ) ψ ( x, y, z ) ψ L, L, N N N Ψ ( L,, LN ;,, N ) (.) N! ψ This state is the eigenstate of ( x y z ) amiltonian is denoted as W ( L, L ;,, ) W N ( x, y, z ) ψ ( x, y z ) LN, N LN, N N N, i i, i i, N N,. The exetation value of the total whih is given by: ( L, L ;,, ) Ψ( L,, L ;,, ) Ψ( L,, L ;,, ), N N N N T N N N

31 N, N N L i N i ( L, L ;,, ) E ( ) C( L,, L ;, ) W i, where E Li ( i ) is given by Eq.(.8) whih is the eigen-energy of ( x, y, z ) Landau level number L and the momentum i value of the Coulomb interation defined by C ( L,, L ;,, ) Ψ( L,, L ;,, ) N N N i > i N πε N (.) i i i for the. In Eq.(.), C is the exetation i ( x x ) ( y y ) ( z z ) Ψ( L,, LN ;,, N ) dxdydz dx N dy N dz N ereafter we all C ( L, L ;,, ) i, N N e i lassial Coulomb energy. N i N (.) In this hater we investigate the ase of the filling fator ν <. (The ase of ν > an be derived easily from the method in this hater and also is studied in hater 5.) We also restrit our investigation to the ase of a low temerature and a strong magneti field. Aordingly all the eletron sins turn to the oosite diretion of the magneti field. (In Chater 9, we will examine the ase of a weak magneti field where the many-eletron state is onstruted by a mixture of u and down sins.) The ground N states of ( x, y z ) is seified by the quantum numbers ( L,, ) (,,) i i i, i L N for ν <. All the Landau states with L are artially filled with eletrons. Various eletron-onfigurations in the Landau states are ossible. If the eletrons are distributed most uniformly in the Landau states, then the many-eletron state has the minimum exetation value of the Coulomb interation.. Many-eletron state with minimum lassial Coulomb energy In this hater we onsider the ase where all the eletrons have down sins and L beause the magneti field is strong and the filling fator is smaller than. So the Landau level numbers L,, L are not shown, for simliity as follows: (, ;,, N ) Ψ(,, N ) (, ;,, N ) W (, N ) (, ;,, ) C(, ) N Ψ (.5) W, (.6) C N, N (.7) Thus the many-eletron state at ν < is seified by the set of momenta,,. N

32 We an distinguish the many-eletron state with,, from the other one with N,,. This momentum set is related to the entre ositions N α,,α by the N relation (.). If the distribution ( α,,α ) of eletron-ositions is the most uniform one among N ossible distributions, then the state has the minimum lassial Coulomb energy. We exlain how to find the most uniform distribution ( α,,α ). First we examine two N examles as follows: the most uniform distributions for ν and ν are shematially shown resetively in Fig..a and b where the straight bold lines indiate the Landau states filled with an eletron and the dashed lines emty states. (Note: The line shematially shows the Landau wave funtion namely the lane wave along the x-axis. The y-oordinate is the eak osition of the Gaussian funtion along π e due to the y-axis. So the lines are equally saed with the interval [ ( )] Eq.(.).) Therein the urrent flows along the x-diretion and the all voltage aears along the y-diretion. (see Figs.. and.5.) The definition of these diretions, x and y, is used throughout this book. It is easily understood that the filling atterns of Figs..a and.b are the most uniform eletron-onfigurations at ν and ν resetively. So these eletron onfigurations have the minimum lassial Coulomb energies resetively. y y x x Fig..a: Filling fator / Fig..b: Filling fator / We next study how to find the eletron-onfiguration with the minimum lassial Coulomb energy for any filling fator. First, we examine an examle of ν 5. We lassify the eletron onfigurations into the following two ases: Case : In the whole region, three eletrons exist in every sequential 5 Landau states. The filling fator beomes /5 beause any region with sequential 5 Landau states is artially filled with eletrons. Case : The average filling fator is equal to /5 in the whole system. Two eletrons exist in some sequential 5 Landau states, and four eletrons exist in some

33 sequential 5 states and so on. The Coulomb energy of Case is smaller than that of Case beause the filling attern of Case is more uniform than that of Case. So we searh the filling attern of Case to find the ground state of the many-eletron system. The searhing roess is divided into two stes as follows: Ste : We examine filling atterns inside sequential 5 Landau orbitals at ν 5. Ste : We next onsider the onnetion of the filling atterns. (Ste ) There are filling atterns for sequential 5 Landau orbitals with ν 5. All the filling atterns are drawn in Fig... Eah attern is alled unit-onfiguration. a- a- a- a- a-5 b- b- b- b- b-5 Fig.. All unit-onfigurations for eletron ouation in five orbitals The five unit-onfigurations a-, a-, a-, a-, a-5 give the same eletron-onfiguration by reeating of themselves as shown in Fig..a exet for both ends. The differenes at the boundaries an be negleted in a quantum all system with a marosoi number of eletrons. Similarly the five unit-onfigurations b-, b-, b-, b- and b-5 in Fig.. give the onfiguration shown in Fig..b. If we omare these two onfigurations, the eletron-onfiguration of Fig..a has a lower Coulomb energy than that of Fig..b. Fig..a: Filling fator /5 Fig..b: Filling fator /5 (Ste ) We examine the onnetions between different unit-onfigurations in Fig... Figure. shows five onnetions of (a- and a-), (a- and a-), (a- and a-) and (a- and a-5). All the onnetions have green areas where eletrons or eletrons exist inside the sequential five states.

34 a- a- a- a- a- a- a- a-5 Fig.. Connetions between different unit-onfigurations Thus these onnetions belong to Case. Therefore the onnetions between different unit-onfigurations have a lassial Coulomb energy larger than that of Fig..a. Consequently we an obtain the most uniform onfiguration by reeating of the single unit-onfiguration in Fig..a. For any frational filling fator ν r q, r eletrons exist in sequential q Landau states everywhere. The number of filling atters is q! ( r! ( q r)!). We an write all the atterns. y omaring the atterns with eah other we find the filling attern with the most uniform distribution. Of ourse there are q equivalent unit-onfigurations as in Fig..a. We have done this roedure for the filling fators ν r q with q,,, 5, 6, 7 and 8. The results are shown in Fig..5 whih exresses the unit-onfigurations with the minimum lassial Coulomb energy. Therein only the reresentative unit-onfigurations are shown among equivalent unit-onfigurations. That is to say, the unit-onfiguration a- reresents all the equivalent unit-onfigurations a-, a-, a-, a- and a-5 for ν 5. ν / ν / ν / ν / ν / ν /5 ν /5 ν /5 ν /5 ν /6 ν 5/6 ν /7 ν /7 ν /7 ν /7 ν 5/7 ν 6/7 ν /8 ν /8 ν 5/8 ν 7/8 Fig..5 Most uniform unit-onfigurations of eletron ouation For onfirmation, we show some examles with higher Coulomb energy in Fig..6. Comarison of Fig..5 with Fig..6 reveals that the unit-onfigurations in Fig..5 have the lassial Coulomb energy lower than that in Fig..6

35 ν /7 ν /7 ν /7 ν 5/7 ν /8 ν 5/8 Fig..6 Unit-onfigurations with higher energy Thus only one unit-onfiguration has the minimum lassial Coulomb energy among all the ossible unit-onfigurations of eletrons. Then the whole onfiguration with the minimum lassial Coulomb energy is obtained by reeating the reresentative unit-onfiguration for a given ν. This onlusion is roven as follows; (Proof): We onsider the most uniform eletron-onfiguration with ν r q. Therein r eletrons should exist inside sequential q Landau states everywhere. First we take sequential q Landau states (orbitals) in the whole onfiguration. Next we onsider the following new region: one Landau state is removed from the left end of the original region and one Landau state is added to the right end. Then we get a new region. Therein r eletrons should exist inside the new region in order to have the minimum lassial Coulomb energy. This requirement yields the following two onditions: () If the removed state is emty, the added state should be emty. () If the removed state is filled with an eletron, the added state should be also filled with an eletron. Thus we get a new region from left to right, one after another. Thereby we an rerodue the whole region of the many-eletron onfiguration. Consequently the whole eletron-onfiguration with the minimum lassial Coulomb energy is obtained by reeating only one unit-onfiguration with the most uniform filling attern. (End of Proof). Momentum set seifying the many-eletron state with the minimum lassial Coulomb energy The entral osition α J along the y-diretion is related to the momentum of the x-diretion as in Eqs.(.) and (.5b). That is to say, both entral osition α and J momentum deend on the same integer J as follows; 5

36 π π α J J and J (.8) e Aordingly any eletron-onfiguration is given by a set of momenta (The orresondene is one to one maing). Let us onsider three examles of Figs..b,.a and b. The many-eletron state with the onfiguration of Fig..b is seified by the momentum set as π π, ( ), for,,,,, (.9) where indiates the minimum momentum among all the momenta of eletrons. We next onsider two onfigurations at ν 5. Figure.a show the first onfiguration seified by the momentum set (,, N ) as π π π 5, ( 5 ), ( 5 ) for,,,,. (.) The seond onfiguration is shown in Fig..b whih is seified by the momentum set (,, N ) as π π π 5, ( 5 ), ( 5 ) for,,,,. (.) Thus arbitrary eletron-onfiguration is desribed by the orresonding momentum set. W,, of the total The many-eletron state has the exetation value ( ) amiltonian < T. As exlained in setion., all the Landau level numbers are zero for W,, W,, ν. Use of Eq.(.) yields the differene between ( N ) and ( N ) to be (, ) W (,, ) C(,, ) C(,, ), N N N N N W (.) ere C (,, ) is smaller than C (,, ) N N beause the onfiguration of Fig..a is more uniform than that of Fig..b. Aordingly, we get (, ) < W (,, ) W (.), N N Consequently the momentum set of Eq.(.) has the minimum exetation value of at ν 5. In this way we an obtain the momentum set with the minimum T lassial Coulomb energy for arbitrary filling fator. The funtion-form of the lassial Coulomb energy versus filling fator ν will be investigated in the next setion. 6

37 . ν -deendene of the minimum lassial Coulomb energy Equation (.b) gives the number of states with a given Landau level number as; e d π Therefore, the marosoi harge density of eletrons at the filling fator ν is given ν e number of states d whih is by ( ) ( ) ( ) e marosoi harge density σ ν (.) π The blak urve in the uer anel of Fig.. exresses the exerimental data of all resistane R divided by the Klitzing onstant RK π e. Therein the red line shows the average deendene of R RK. The ratio R RK is equal to ν. Aordingly the red line indiates that ν is almost roortional to the magneti field strength exet the lateau regions. That is to say, ν is nearly equal to a onstant value and then the harge density σ is indeendent of. So we an aroximate the marosoi Coulomb energy ( σ ) C to be a onstant value in Marosoi the exeriment of Fig... When we examine the mirosoi harge-distribution, we find a new differene of the lassial Coulomb energy by the eletron onfiguration. Aordingly we need to study how the lassial Coulomb energy deends on the mirosoi eletron-onfiguration. As an examle, we first examine the many-eletron state at ν. The eletron-onfiguration with the minimum lassial Coulomb energy is illustrated in Fig..7. Therein the bold lines exress the ouied orbitals with eletron and the dashed lines indiate the emty orbitals. ξ η ξ η ξ η ξ η ξ η ξ η A ζ C D κ Fig..7 Classial Coulomb energies between eletrons at ν 7

38 The eletron air loated at the orbitals A is one examle of the nearest-eletron-airs. Two eletrons laed at and C show the seond nearest air. The eletron air at A and C is the third nearest air. The air at A and D is the fourth nearest air and so on. The lassial Coulomb energy between two eletrons is exressed by the symbols ξ, η, ς and κ resetively as in Fig..7. Therein ξ is the largest one of the lassial Coulomb air energy, η is the seond largest, ς is the third largest, κ is the fourth largest and so on. The lassial Coulomb energy between air (A, C) is weakened by the sreening (shielding) effet of eletron. Also the lassial Coulomb energy between air (A, D) is weakened by the sreening effet of eletrons and C. Aordingly the ν -deendene of the lassial Coulomb energy mainly omes from the first nearest and the seond nearest airs. The number of the more distant airs (third, fourth, fifth and so on) are enormous many. The total number of eletron airs is N ( N ). On the other hand the total number of the first and seond nearest airs is N. Aordingly the residual energies (namely the sum of all the more distant air energies) may be aroximated by the marosoi Coulomb energy ( σ ) C whih is a onstant Marosoi value mentioned above. On the other hand the sum of the first and seond nearest air-energies is strongly deendent uon the filling fator. The ν -deendene is examined for various filling fators as follows: (Case of ν ) We an ignore the boundary effet in both ends for a marosoi eletron-number N. Then the number of the nearest airs is N and the number of the seond nearest airs is N in the onfiguration of Fig..7. Then the sum of the lassial Coulomb energies between the nearest airs is equal to ξ N and the sum of the energies between the seond nearest airs is equal to η N. Aordingly the total lassial Coulomb energy C (,, ) is aroximated by N (,, ) [( ξ ) ( η )] N C ( σ ) C N at ν (.5) Marosoi Next we estimate the lassial Coulomb energy for other ases. (Case of ν 5) The eletron-onfiguration with the minimum lassial Coulomb energy at ν 5 is shown in Fig..8. 8

39 ξ η η ξ η η ξ η η ξ η η Fig..8 Classial Coulomb energies between eletrons at ν 5 The number of the nearest airs is equal to ( )N and the number of the seond nearest airs is equal to ( )N at ν 5. Then the total lassial Coulomb energy is (,, ) [( ξ ) ( η )] N C ( σ ) C N at ν 5 (.6) Marosoi (Case of ν 7) Figure.9 show the eletron-onfiguration with the minimum lassial Coulomb energy at ν 7. ξ η η η ξ η η η ξ η η η Fig..9 Classial Coulomb energies between eletrons at ν 7 The number of the nearest airs is equal to ( )N and the number of the seond nearest airs is equal to ( )N at ν 7. Then the total lassial Coulomb energy is (,, ) [( ξ ) ( η )] N C ( σ ) C N for ν 7 (.7) Marosoi (Case of ν 5 7 ) The eletron-onfiguration with the minimum lassial Coulomb energy for ν 5 7 is shown in Fig... 9

40 ξ ξ η ξ η ξ ξ η ξ η ξ ξ η ξ η ξ ξ η ξ η Fig.. Classial Coulomb energies between eletrons at ν 5 7 The number of the nearest airs is equal to ( 5)N and the number of the seond nearest airs is equal to ( 5)N at ν 5 7. Then the total lassial Coulomb energy is (,, ) [( ξ 5) ( η )] N C ( σ ) C N at ν 5 7 (.8) 5 (Any ase of <ν < ) Marosoi We alulate the lassial Coulomb energy for a general ase of ν r q ( <ν < ). As roven in Se.., the eletron-onfiguration with the minimum lassial Coulomb energy is onstruted by reeating the reresentative unit-onfiguration where r eletrons exist in sequential q Landau orbitals (states). The number of emty orbitals er unit-onfiguration is q r. All the emty orbitals are searated by one or more filled-orbitals at <ν <. That is to say, all the emty orbitals are isolated as seen in Figs Therefore the q r seond-nearest airs exist er unit-onfiguration. The total numbers of the first and the seond nearest airs is equal to the total number of eletrons as easily seen in Figs Therefore the number of nearest airs beomes r ( q r) r q er unit-onfiguration. The total number of nearest eletron airs is equal to ( r q)( N r) and the total number of seond nearest eletron airs is equal to ( q r)( N r) for the filling fator ν r q. Consequently the total lassial Coulomb energy is obtained as C (,, N ) [( ξ ( r q) r) ( η( q r) r) ] N CMarosoi ( σ ) [( ξ η) (( ξ η) q r) ] N C ( σ ) Marosoi This equation is rewritten by using ν and then the result beomes (,, ) [( ξ η) ( ξ η) ν ] N C ( σ ) Marosoi at ν r C N at ν r q (.9) Equations (.) and (.9) yield the exetation value of the total amiltonian as W N N i i (,, ) E ( ) [( ξ η) ( ξ η) ν ] N C ( σ ) Marosoi q

41 at ν r q (.) The single eletron eigenenergy E ( i ) is given by the following form as in Eq.(.8). ( ) U ( α ) e ( ) E λ m (.) i i Substitution of Eq.(.) into Eq.(.) yields the exetation value of the total amiltonian as: W N (,, ) ( λ U ( α ) e ( m ) N i N i ( ξ η) N ( ξ η) ν C ( σ ) Marosoi (.a) We ut together the onstant art as follows; (,, ) [ f e ( m ) ( ξ η) ν ] N C ( σ ) W N (.b) Marosoi Therein f is the onstant value as f λ U ξ η (.a) U N i ( ) U α N (.b) i where U is the average of the otential along the y-diretion. (Note: We rove that the value U is invariant via the Coulomb transition as follows: The eletron momenta before the transition are and i whih are related to the entre ositions α and i α as in Eq.(.). Also (, i ) and ( α, i α ) exress the momenta and the ositions after the transition. The total momentum onserves via the Coulomb transition. So we obtain the equation. This equation gives i i the relation α α α α. The width y of the wave funtion is very small i i namely satisfies y.5 nm at 6T as in Eq.(.9). So the effetive quantum transition α α y. Therefore the eletri otential U ( y) is aroximated to the i i linear form in the overlaed region of the eletron wave funtions. Aordingly we obtain the relation as; U ( α ) U ( α ) U ( α ) U ( α ) i (.) i

42 Thus the value U an be treated as a onstant in the quantum transitions.) Eq.(.b) gives the funtion form of W whih deends linearly uon ν. The roortional oeffiient ( ξ η)n is negative, beause the lassial Coulomb energy between the first nearest eletron air is larger than that between the seond nearest air. Aordingly the exetation value of the total amiltonian, W, hanges ontinuously with ν as shown in Fig... W / ν Fig.. Exetation value W of the total amiltonian Thus the lassial Coulomb energy has no energy-ga. Aordingly the lassial Coulomb energy annot rodue the lateaus of all resistane. In the resent theory, the onfinement of the all resistane omes from another reason, whih will be studied in Chaters -8. Sine the quasi-d eletron roblem with the Coulomb interation annot be solved exatly, we are obliged to use some aroximations in the investigation of the FQE. Laughlin introdued a quasi-artile with frational harge and obtained the wave funtion by using the variational method [9, ]. aldane and alerin extended the sheme [-]. Jain roosed a theory to exlain the exerimental data on FQE []. e introdued a quasi-artile alled the omosite fermion whih is an eletron bound to even number of magneti flux quanta. These theories have used the many different tyes of quasi-artiles. On the other hand, there is a theory used normal eletrons without any quasi-artile. The theory is investigated by Tao and Thouless. They alulated the erturbation energy via the Coulomb interation between eletrons [5, 6]. We develo the Tao-Thouless theory in Chaters -5. The investigations has been ublished in Ref.[7-]. The most uniform onfiguration has been examined in []. The resent theory [7-] is omared with the traditional theories [9-] in Chater.

43 Chater inding energy of eletron air This hater is devoted to alulate the erturbation energy via the Coulomb transition. The Coulomb interation deends uon only the relative oordinate between eletrons. The roerty derives the momentum onservation along the x-diretion. In this hater we examine FQ states under the strong magneti field and so all the sins of eletrons are antiarallel to the magneti field. (We will examine the other ase namely the ase of weak magneti field in Chater 9.). Momentum onservation in Coulomb interation The omlete set for the quasi-d eletron system is omosed of the wave funtion ( L, L ;,, ) Ψ defined by the Slater determinant (.). The total amiltonian, N N (.) is re-exressed by using this omlete set. We desribe the diagonal art of T by D whih is defined by Ψ L L ;,, W L L ;,, L L ;,, D ( ) ( ) ( ) N N N N Ψ L,,,, L N N where W ( L L ;,, ) N N N (.) is the diagonal matrix element of T as Eq. (.). The ground state in D is identified by the momentum set whih gives the most uniform eletron-onfiguration. The onfiguration is determined uniquely at any frational filling fator as verified in the revious hater. This uniqueness means that the ground state in D is not degenerate. The total amiltonian is divided into two arts D and I. The latter art I is defined as N I. (.) T D where I is onstruted by the off-diagonal elements only. Next the roerties of the matrix elements ( L L,, ) Ψ( L L ;,, ) N ; N I N Ψ will be examined below. The quantum numbers ( L, ) and ( ) N L, identify the final and the initial states of the -th eletron, resetively. When the Coulomb interation ats between two

44 eletrons i and, the other eletrons have the same momenta before and after the transition. That is to say the matrix elements inlude the rodut of the Kroneker delta funtions as δ s i, s ( L L ) δ ( ) where the symbol ( a b) s Aordingly the matrix element of Ψ s s s δ is used instead of the Kroneker delta funtion δ a, b. I is ( L L ;,, ) Ψ( L L ;,, ) N N I N N s i, s ψ δ ( L L ) δ ( ) L, i i s ψ s L, s πε s ( x x ) ( y y ) ( z z ) i e i i ψ L, i i ψ L, (.a) where ψ indiates the eletron-air wave funtion with the quantum L, ψ L, i i numbers ( ) i i L, and ( L, ) matrix element is zero as Ψ. When the final state is idential to the initial state, the ( L L,, ) Ψ( L L ;,, ) N ; N I N N for ( L L n,,, N ) (.b) The x omonent of the total momentum onserves in this system as i i n n (.) beause the Coulomb interation deends uon only the relative oordinate between eletrons. In haters -8, we study the FQE under a strong magneti field where all the eletrons have the same sin-diretion antiarallel to the magneti field for ν. When an eletron air transfers to other states via the Coulomb interation at ν, the n n final two momenta i and should be different from the momenta of all the other eletrons beause of the Pauli exlusion rinile. That is to say the Coulomb transition is allowed to emty states only. The number of transitions is deendent uon the filling

45 fator. We examine the roerty in details below.. Perturbation energy of a nearest eletron air The eletron-onfigurations with the minimum lassial Coulomb energy are illustrated for the filling fators ν,, 5, 7,5 9,6,7 in Fig.. where bold lines indiate the Landau states filled with eletron and dashed lines the emty states. The eletron-onfiguration at ν is rodued by reeating the unit-onfiguration of (filled, emty). This onfiguration is the most uniform distribution of eletrons at ν. Figure. also shows the eletron-onfigurations with the filling fators of ν ( ) for,,,5,6, 7. These onfigurations give the most uniform distribution of eletrons. ν/ ν7/ ν6/ ν5/9 ν/7 ν/5 ν/ Fig.. Most uniform eletron onfigurations Red lines indiate nearest neighbor eletron airs and dashed lines exress emty states. 5

46 The uer most eletron-onfiguration on Fig.. has no eletron air laed in the nearest Landau orbitals. On the other hand the onfigurations other than ν have nearest neighbouring eletron airs whih are shown by red lines. We all the eletron air laed in the nearest neighbouring orbitals nearest eletron air. All the onfigurations in Fig.. are the ground state of D at ν /, 7/, 6/, 5/9, /7, /5 and /, resetively. The eletron onfiguration with the minimum energy of D is only one attern as it has been roven in hater. Aordingly we an aly a erturbation method in non-degenerate ground state in order to alulate the binding energy. We alulate the erturbation energy of the nearest eletron air at ν 7 as an examle. The Landau states denoted by A,, C, D, E and F in Fig.. are haraterized by the momenta,,,, and resetively. A C D E F Y X E C A D F Fig.. Coulomb transitions from nearest neighbour eletron air A at ν 7 We show a quantum transition from the eletron air A via the Coulomb interation by blak arrow air in Fig... When the eletron at the site A transfers to the fifth orbital to the left, the momentum A dereases by 5 π due to Eq.(.). After the transition, the new momentum beomes C as 5 π. (.5) C A The eletron at the site transfers to the site D with D. eause the total momentum onserves during the quantum transition, the following relation is satisfied: C D A (.6) Equations (.5) and (.6) yield the momentum D as 5 π. (.7) D Consequently the eletron at transfers from its original orbital to the fifth orbital to the right as shown in Fig... This transition is allowed beause the fifth orbital is emty. Also the nearest eletron air A an transfer to the states E and F, and the momenta 6

47 after the transition are given by π (.8a) E A ( ) ( ) π (.8b) F We have examined two examles mentioned above. The eletron air A an transfer to all the emty orbitals as easily seen in Fig... The momenta after transition are denoted by A, and then the total momentum onservation is exressed by (.9) A A Aordingly the momenta after the transition are given by A A (.a) (.b) where the momentum transfer at ν 7 takes the values as ( ) π, ( ) π, ( 5) π, ( 7) π, ( 9) π and ( ) π for, ±, ±, ±, (.) beause the eletrons are allowed to transfer to all the emty orbitals. The seond order erturbation energy of the nearest eletron air A via the Coulomb interation is given by ς ν 7 all allowed momenta at ν 7 A, W G I W, A exite, (, ) A A A I A, (.) where W G is the ground state energy of at ν 7. It is noted that takes D the values of Eq.(.). In Eq.(.) the denominator is negative beause the ground state energy W G is the lowest of all. Then the seond order erturbation energy is negative. ς (.) ν 7 < We have disussed the quantum transitions from air A. Also all the nearest eletron airs are able to transfer to all the emty orbitals. Therefore the erturbation energy of any nearest air has the same erturbation energy as Eq.(.). We examine another examle ν. Figure. shows the eletron-onfiguration with the minimum lassial Coulomb energy. 7

48 Y A X Fig.. Coulomb transitions from nearest neighbour eletron air at ν The momenta after the transition are denoted by the momentum transfer takes the values as ( ) for, ±, ±, ±, A A where π at ν (.) beause of the Pauli exlusion rinile. The seond order erturbation energy of the nearest eletron air is given by,,, A I A A I A ς ν (.5) ( ) π for, ±, ±, W W (, ) G exite A A, In order to alulate the erturbation energy for various filling fators systematially, we introdue the following summation Z as:,,,, Z (.6) A I A A I A, π W G Wexite A A where the momentum transfer (, ) takes all values ( π ) integer exet and π. If the momentum transfer is equal to or π then the transferred state is the same state as the initial one. Aordingly the transition matrix element is zero beause the diagonal element of I is zero. Therefore the state is eliminated in the summation of (.6). The ground state energy W G is lower than W exite and so the denominator is negative. In the definition of Z, (-) is multilied in the right-hand side of Eq.(.6). Aordingly Z is ositive. [7-] Let us omare Z with ς ν. The summation over in ς ν is erformed in π ste. On the other hand the summation in Z is erformed with the interval π. Then we get, in a marosoi sale ( ) ς ν Z (.7) 8

49 Thus we an exress the erturbation energy of the nearest eletron air by Z. The erturbation energy deends uon and namely devie size and magneti field strength. These deendenes are inluded in the summation Z. The total number of nearest eletron airs is ( )N at ν. Aordingly the total erturbation energy E of all the nearest eletron airs is equal to nearest air E nearest air N ς ν ZN for ν (.8) 6 Similar alulation leads the seond order erturbation energy of the nearest eletron air ς ν 7 at ν 7 as follows; 6 ς ν 7 Z (.9) whih is derived from the roerty that the number of emty states is 6 er sequential Landau states. The total erturbation energy E nearest air of all the nearest eletron airs is N 6 6 Enearest air ς ν 7 ZN ZN for ν (.) beause the total number of the nearest eletron airs is ( 7)N at ν 7. Next we examine the ase of ( ) beomes large, ( ) ν for arbitrary integer. When ν aroahes /. So we hoose the energy at ν as a referene. The total energy at ν is equal to ( ν ) g( ) N [ f e ( m ) ( ξ η) ν ] N C ( σ ) E (.) Marosoi where the last two terms on the right hand side of Eq.(.) reresent the eigenenergy of D obtained by Eq.(.b). The residual term g( )N indiates the total erturbation energy via the interation at ν. eause the ν state has no nearest I eletron air, the term g( )N is the total erturbation energy from all the non-nearest airs. As is evident from Fig.., nearest eletron airs exist in the ase other than ν and so the energy differene between ν ( ) and ν omes from the nearest eletron airs. Aordingly the total energy at the filling fator ν is 9

50 aroximately equal to where ( ν ) E g( ) N [ f e ( m ) ( ξ η) ν ] N ( σ ) E (.) nearest air CMarosoi E nearest air is the nearest air energy. ereafter we alulate the erturbation energy of the nearest eletron airs at ( ) ν for arbitrary integer. As easily seen from Fig.., the most uniform onfiguration has one nearest eletron air (red air in Fig..) and ν. emty states in the unit-onfiguration at ( ) eause the nearest eletron air an transfer to all the emty states, the erturbation energy ς ν ( ) er nearest eletron air is equal to ς ν ( ) ( ) Z The total number of nearest eletron airs is times the total number of eletrons. Therefore the total erturbation energy of the nearest eletron airs is obtained as E N ( ) ( ) ZN nearest air ς ν for ν ( ) (.a) There is no nearest eletron air at ν and so E is enearest air E at ν (.b) nearest air The erturbation energy of nearest eletron airs er eletron is equal to Enearest air for ν ( ) (.) N ( ) Z where the boundary effets of both ends in a quantum all system have been ignored for a marosoi number of N. The value of E is listed on Table. for enearest air several filling fators. Table. Energy of nearest eletron airs er eletron at ν ( ) ν E nearest air N / / -(/6) Z 5

51 /5 -(/5) Z /7 -(/8) Z 5/9 -(/5) Z 6/ -(5/66) Z 7/ -(6/9) Z 8/5 -(7/) Z We omment briefly how the number of allowed quantum-transitions deends on the filling fator. (Short omment) As will be larified in the next hater, the erturbation energy is not a ontinuous funtion of ν. At the seifi filling fators of ν /, /5, /5, /7, 5/9,, the nearest eletron airs are allowed to transfer to all the emty orbitals. For almost all the other frational filling fators, the quantum transitions from nearest eletron airs are forbidden to some emty orbitals. The forbidden or allowed transitions are aused by the ombined effets of the most uniform eletron-onfiguration, Pauli s exlusion rinile and the total momentum onservation in the x-diretion. At the seifi filling fators, the nearest airs an transfer to all the emty orbitals. When the filling fator hanges from this seifi value, the number of allowed transitions abrutly dereases. This dereasing of the transitions rodues the inrease of the erturbation energy beause the seond order erturbation energy is negative for the ground state. Thus the erturbation energy is not a smooth funtion of ν. It will be shown in the latter hater that this disontinuous struture yields the lateaus in the all resistane urve. The reader who is interested in knowing the disontinuous struture of the erturbation energy may ski the latter setions and roeed to the next hater. The disontinuous struture aears in all orders of the erturbation energy and so exists in the exat solution. The higher order alulation will be examined in Chater 6.. inding energy at the filling fators ( ) ( ) and ( ) ( ) In this setion, we study the states with the filling fators ( ) ( ) ( ) ( ) ν ν and. The former has an odd number of the denominator and the latter has an even number of the denominator. We first examine an examle ν 5 whih 5

52 belongs to the former ase for eletron-onfiguration at ν 5.. Figure. shows the most uniform A C D Fig.. Allowed transitions of nearest eletron airs at ν 5 There are three nearest eletron airs A, C and CD in the unit-onfiguration. When the eletron at site A is transferred to the first nearest orbital to the left in Fig.., the eletron at should be transferred to the first orbital to the right beause of the momentum onservation. owever that orbital is already filled with eletron and the transition is forbidden by the Fermi-Dira statistis. Aordingly the nearest eletron airs A and CD annot transfer to any emty state. Only the eletron air C an transfer to all the emty states. The total number of emty states is /5 times the number of all orbitals. Aordingly the airs A, C and CD have the erturbation energy of the seond order as ς A, ς C ( 5)Z, ς CD The number of the nearest eletron airs like C is / times the total number of eletrons N. Then we obtain the erturbation energy of all the nearest eletron airs E as nearest air N Enearest air N ( 5) Z ( )Z for ν 5 (.) N Next, we disuss the ase of ν 6 7. We draw the most uniform eletron-onfiguration at ν 6 7 in Fig..5. A C D E F Fig..5 Allowed transitions of nearest eletron airs at ν 6 7 5

53 Only the eletron air CD in the unit-onfiguration an transfer to all the emty states. The total number of the emty states is /7 times the total number of orbitals. Aordingly the air CD has the erturbation energy of the seond order as whih gives ς CD ( 7)Z N Enearest air N ( 7) Z ( )Z for ν 6 7 (.5) 6 N For the ase of arbitrary integer, the erturbation energy of the seond order for ν ( ) is given by multilying the air energy ( ( ) )Z and the total number of the airs N ( ). Thereby E nearest air N N ( ( ) ) Z for N ( ) Z Next we alulate the erturbation energy with ν ( ) ( ) ν (.6), the denominator of whih is an even integer. Fig..6 shows the most uniform onfiguration of eletrons in the ase of. A C Fig..6 Nothing of transition from nearest eletron airs at ν In this ase, the nearest eletron airs A and C annot transfer to all emty orbitals. Aordingly the erturbation energy of the nearest eletron airs is zero: E N for ν (.7) nearest air We illustrate the most uniform onfigurations for and in Figs..7 and.8, resetively. A C D E Fig..7 Nothing of transition from nearest eletron airs at ν 5 6 5

54 A C D E F G Fig..8 Nothing of transition from nearest eletron airs at ν 7 8 These figures show that all the quantum transitions from the nearest eletron airs are forbidden. Aordingly the energies of all the nearest eletron airs are zero at the filling fators ν 5 6, and 7 8. E N for ν 5 6, and 7 8 (.8) nearest air For the ase of arbitrary integer, the erturbation energies of all the nearest eletron airs at ( ) ( ) ν are equal to zero: E N for ν ( ) ( ) (.9) nearest air Thus the nearest eletron airs at ( ) ( ) ν annot transfer to all the emty states due to the ombined effets of the most uniform eletron-onfiguration, the Fermi-Dira statistis of eletrons and the momentum onservation [7-]. Table. Seond order erturbation energy of nearest ν eletron airs er eletron for ν ( ) ( ) and ( ) ( ) ν E nearest air N / / -(/6) Z / /5 -(/) Z 5/6 6/7 -(/) Z 7/8 5

55 8/9 -(/7) Z Table. shows the seond order erturbation energy of the nearest eletron airs at the filling fators with ν ( ) ( ) or ( ) ( ) ν. It is noteworthy that the nearest air energies are zero in all order erturbation ν beause of no Coulomb transition. alulation for the filling fators ( ) ( ) That is to say the exat binding energy of the nearest eletron airs is equal to zero at ν. ( ) ( ). Eletron-ole symmetry of nearest airs In this setion we examine the ase of the filling fator ν < where the number of emty orbitals is greater than the number of orbitals ouied with eletron. ereafter we all emty orbital hole. The eletron-onfiguration at ν is shown in Fig..9. This onfiguration is rodued by reeating the unit-onfiguration (emty, filled, emty), whih has the minimum energy of the lassial Coulomb interation. A Fig..9 Coulomb transitions of nearest neighbor hole air at ν In this onfiguration, there is no eletron air laed in the nearest neighbour Landau orbitals. owever there are nearest hole airs. If we exhange eletron for hole and hole for eletron in Fig..9, we get the eletron-onfiguration of ν. It may be said that eletron-hole symmetry aears between the ν and ν states. We alulate the erturbation energy of the nearest hole air A whih is denoted by ς A. The hole air A is seified by the momenta A,. The eletron air A is also seified by the momenta,. Then, the eletron air A transfers to the A 55

56 vaant orbitals at A and as is illustrated by arrow airs in Fig..9. Therein the momentum onservation is exressed as A A (.a) The momentum transfer takes the following values as ( ) for, ±, ±, ±, π (.b) at ν. Then the seond order erturbation energy of the hole air A is obtained by ς A ( ) π, A W W, (, ) for, ±, ±, G exite A A I A, A, I A (.) Now we introdue the summation Z as A, I A, A, I A, Z (.) W W (, ), π G exite A A where the momentum transfer takes all the values ( π ) integer exet and π. The transferred states for and π are eliminated in the summation (.) beause the diagonal element of denominator in Eq.(.) is negative and so Z is ositive. I is absent. The The erturbation energy of the nearest hole air, ς A, an be desribed by Z. eause the interval of momentum transfer is very small for the marosoi size of the devie, we obtain for ν ς A Z (.) The total number of nearest hole airs is equal to ( ) N at ν where N is the number of holes in the Landau states with L. The erturbation energy E is nearest air E nearest air N ς A Z N for ν (.) 6 The erturbation energy of nearest hole airs er hole is obtained as E nearest air N Z for ν (.5) 6 Comarison of Eqs.(.6) and (.) reveals that Z is lose to Z : Z Z at the same strength of magneti field (.6) This relation is derived from the fat that the right hand side of Eq.(.) has the same 56

57 form as the right hand side of Eq. (.6). So the value of the same strength of magneti field. Z is almost equal to Z at Also there is a symmetry between the filling fators ν and ν as will be exlained below. One examle is shown at ν /5 and /5 as in Fig... A XY Fig.. Eletron hole symmetry at ν /5 and /5 Eletron orbitals are drawn by solid lines and hole orbitals are shown by dashed lines As easily seen in this figure, the eletron air A has two quantum transitions er unit-onfiguration and the hole air XY has also two quantum transitions er unit-onfiguration. Therefore the eletron air A has the erturbation energy ς A as ς A ( 5)Z The hole air XY has the erturbation energy ς XY as ς ( ) Z XY 5 Therein the oeffiient /5 omes from the ratio of the number of the allowed quantum transitions and the total number of Landau states. So the oeffiient (/5) at ν 5 is idential to the oeffiient (/5) at ν 5. Then the seond order erturbation energy of all the nearest hole airs at ν 5 is given by multilying ς XY and the total number of nearest hole airs, N, as E nearest air ς XY N Z N for ν 5 (.7) 5 Next, we examine the quantum transitions at ν 7 whih are drawn in Fig... All the nearest hole airs (nearest dashed lines) an transfer to all eletron states. That is to say all the eletrons an transfer to the nearest hole air A. Counting of the transition number gives the seond order erturbation energy of all the nearest hole 57

58 airs er hole as E nearest air N Z for ν 7 (.8) 8 A Fig.. Coulomb transitions of nearest neighbour hole air at ν 7 Consequently the number of the allowed transitions from the nearest eletron airs at ν ( / <ν < ) is the same as that to the nearest hole airs at the filling fator ν. This symmetri roerty yields the following relation between the erturbation energies of the nearest eletron airs and the nearest hole airs: E nearest hole air N at filling fator E nearest eletron air N ν at filling fator ( ν ) (.9) Then the erturbation energies of nearest hole airs are listed in Table. where the entre olumn lists the energies of the nearest hole airs er hole. We an onfirm the eletron-hole symmetry by omaring the entre olumn of Table. with Table.. ereafter nearest hole air at ν < and nearest eletron air at ν > are simly named nearest air. Table. Perturbation energy of nearest hole airs er hole and er eletron ν E N E N nearest air nearest air / / -(/6) Z -(/) Z / /5 -(/) Z -(/5) Z /5 -(/5) Z -(/5) Z /6 /7 -(/) Z -(/7) Z 58

59 /7 -(/8) Z -(/7) Z /8 /9 -(/7) Z -(/9) Z /9 -(/5) Z -(/9) Z / / -(/) Z -(/) Z 5/ -(5/66) Z -(/) Z / / -(/56) Z -(/) Z 6/ -(6/9) Z -(/) Z / The rightmost olumn of Table. exresses the seond order erturbation energies of nearest hole airs er eletron (not er hole). This energy indiates the energy of eletron. Therefore the binding energy of eletron is related to the value in the rightmost olumn of Table...5 Charater of the nearest eletron air and the nearest hole air at the seial ± ± ν ± ± filling fators ν ( ), ν ( ) and ( ) ( ) We alulate the nearest-air energies ν ( ±) and ( ± ) ( ± ) E for the filling fators ( ±) nearest air ν, ν. These filling fators aroah ν, and, resetively in the limit of. We study the energies in the six regions of the following subsetions..5. (Region of <ν ) In the region of <ν, the most uniform eletron-onfiguration is omosed of two subunits S and S only, where S is the onfiguration (filled, emty) and S is (filled, filled, emty). (Proof) We onsider a subunit S (filled, emty, emty) whih has the filling fator smaller 59

60 than /. Also a subunit S is defined by (filled, filled, filled, emty) whih has the filling fator larger than /. If S or S is inluded in an eletron-onfiguration, the onfiguration doesn t yield the most uniform distribution in <ν. Consequently the most uniform onfiguration inludes the subunits S and S only in the region <ν. Figure. shows the most uniform eletron onfiguration at ν 6. The quantum transitions from the nearest eletron air A are allowed to all the emty oribitals as in Fig... The eletron at A an transfer to the left emty-orbitals as shown in blak arrows ( A A, for > ) and also to the right emty-orbitals as in green arrows ( for < ). These transitions onserve A A, the total momentum in the x-diretion ertainly. Thus the eletron air A is allowed to all the emty orbitals via the Coulomb interation. This allowane to all the emty orbitals aears at any filling fator of ν ( ) resulting in a large binding energy. We rewrite Eq.(.a) by relaing to as follows: E nearest air ( ) ZN for ν ( ) ( ) (.) A Fig.. Allowed transitions from the nearest eletron air A at ν 6 Substitution of 5 into Eq.(.) gives the air energy ( 5 66)ZN at ν 6. When the filling fator hanges from ν ( ) ( ) by an infinitesimally small value, then nearest eletron airs annot transfer to some emty orbitals. Aordingly the number of the transitions dereases in the lose viinity of ν. This roerty rodues a disontinuous struture in the energy setrum whih will be investigated in the next hater..5. (Region of ν < ) 6

61 In this region, the most uniform eletron-onfiguration is omosed of the subunits S (filled, emty) and S (filled, emty, emty) only. Figure. shows the quantum transitions of the hole air A at ν 5. A Fig.. Allowed transitions of the nearest hole air A at ν 5 The nearest hole air A has the five allowed transitions for eah unit-onfiguration omosed of Landau orbitals. That is to say the number of the allowed transitions from A is 5/ times all the orbitals. Also the number of nearest hole airs is equal to /6 times the number of holes. Aordingly we obtain the erturbation energy as; 5 E nearest air ZN 6 for ν 5 (.) ν. The eletron-hole Next we alulate the nearest-hole air energies at ( ) symmetry (.9) exresses that the nearest-hole air energy at ( ) to the nearest-eletron air energy at ( ) ( ) the erturbation energy of the nearest hole airs is obtained as E nearest air ν is equal ν given by Eq.(.). Then Z N for ( ) ( ) The erturbation energy er hole is equal to Enearest air N Z for ( ) ( ) ν (.) ν (.) The right hand side of this equation is a negative and the absolute value is a large value ν beomes stable. for small. Therefore the state with ( ).5. (Region of <ν ) We ount the number of quantum transitions from nearest hole airs in the region of <ν. The most uniform eletron-onfiguration is omosed of subunits S (filled, emty, emty) and S (filled, emty, emty, emty) only. Figure. shows the most uniform eletron-onfiguration at ν whih is an examle of 6

62 ( ) ν for. All the eletrons an transfer to the orbital air A as in Fig.. where the arrows indiate the eletron transitions. The movement of holes has the oosite diretion against the arrows in Fig... A CDE FG Fig.. Allowed transitions of the nearest hole air A at ν Three eletron-airs an transfer to the emty air A er unit onfiguration with eleven orbitals. Therefore the erturbation energy of the hole air A is given by ς A Z The nearest hole airs DE and FG have the erturbation energies as follows; (.) ς DE Z and ς FG Z (.5) The nearest hole airs CD and G have no allowed transition and therefore ς and ς (.6) CD G The nearest hole airs are exressed with nearest dashed-line airs. The number of the airs is /8 times the total number of holes namely N. Then the total erturbation energy of all the nearest hole airs is equal to N 5 E nearest air ( ς A ς CD ς DE ς FG ς G ) ZN for ν 8 8 (.7) The erturbation energy er hole is 5 E nearest air N Z 8 for ν (.8) We next examine the ase of the filling fator ν ( ). For any odd integer, the nearest hole airs have the following erturbation energy as ( ) Enearest air Z N ( ) 6

63 E nearest air ( ) Z N for ( ) ( ) ν (.9a) For any even integer, the nearest hole airs have the following erturbation energy as E nearest air ( ) Z N ( ) E nearest air ZN for ( ) ( ) ν (.9b) Consequently the erturbation energy er hole is E nearest air N Z for ν ( ) ( ), odd integer (.5a) E nearest air N Z for ν ( ) ( ), even integer (.5b) The nearest hole airs shown by red dashed lines in Fig.. an transfer to all the eletron orbitals. This means that the holes at the sites A and are bound tightly at ν FQ states beome stable. ν ( ). So the ( ).5. (Region of 5 ν < ) We examine the erturbation energy of the nearest hole air with a filling fator in the region 5 ν <. The most uniform eletron-onfiguration is omosed of subunits S (filled, emty, emty, emty) and S (filled, emty, emty, emty, emty) only. Figure.5 shows the most uniform eletron-onfiguration at ν whih is an ν for. examle of ( ) ACD E FG I J Fig..5 Allowed transitions of nearest hole airs C, EF and IJ at ν 6

64 All the eletrons an transfer to the hole air C as illustrated by arrow airs in Fig..5. Also a smaller number of eletrons an transfer to the hole airs EF and IJ. The total erturbation energy of all the nearest hole airs is equal to Enearest air Z N for ν (.5) The erturbation energy of nearest hole airs er hole is 5 E nearest air N Z for ν (.5) For any odd integer, the nearest hole airs have the following erturbation energy as E nearest air Z N for ( ) ( ) ν odd integer (.5a) For any even integer, the nearest hole airs have the following erturbation energy as E nearest air Z N for ( ) ( ) ν even integer (.5b) The erturbation energy er hole is E nearest air N Z for ν ( ) ( ), odd integer (.5a) E nearest air N Z for ν ( ) ( ), even integer (.5b) In subsetions we have larified that the nearest eletron (or hole) air has ± ν ±. In the next subsetions large binding energy at ν ( ) and ( ) we will study the region ν (Region of ν < ) We examine the ase of ν ( ) ( ). Figure.6 shows the most uniform eletron-onfiguration at ν 8 whih is the ase of ( ) ( ) for. Therein the quantum transitions from the nearest eletron airs A, DE and FG are illustrated by arrows. The eletron air A are allowed to transfer to all the emty oribitals. The transition number is the three er unit-onfiguration. 6

65 A CDE FG Fig..6 Allowed transitions from nearest eletron airs at ν 8 The eletron airs DE and FG an transfer to one emty orbital er unit-onfiguration. Then all the nearest eletron airs have the following erturbation energy. E nearest air 5 ZN ZN for ν 8 (.55) 8 88 Next we alulate the erturbation energy at ( ) ( ) ν for arbitrary integer. For an odd integer, all the nearest eletron airs have the following erturbation energy: E nearest air ( ) ZN for ( ) ( ) ν (.56a) For an even integer, the nearest eletron airs have the following erturbation energy: E nearest air ν (.56b) ( ) ZN for ( ) ( ) The erturbation energy er eletron is E nearest air N ( ) Z for ( ) ( ) ν, odd integer (.57a) E nearest air N ( ) Z for ( ) ( ) ν, even integer (.57b) 65

66 These nearest air energies are negative and their absolute values are large. Aordingly ν are stable. the states with ( ) ( ).5.6 (Region of <ν 5) Figure.7 shows the quantum transitions from the nearest eletron airs C, EF and IJ at ν for. The nearest ν whih is the ase of ( ) ( ) eletron airs have the following erturbation energy. E nearest air 5 ZN ZN for ν (.58) ACD EFG I J Fig..7 Allowed transitions from nearest eletron airs at ν At ( ) ( ) ν with an odd integer the nearest eletron airs have the following erturbation energy: E nearest air ν (.59a) ( ) ZN for ( ) ( ) For an even integer, the nearest eletron airs have the following erturbation energy: E nearest air ( ) ZN for ( ) ( ) ν (.59b) The erturbation energy er eletron is E nearest air N ( ) Z for ( ) ( ) ν, odd integer (.6a) 66

67 E nearest air N ( ) Z for ( ) ( ) ν, even integer (.6b) These nearest air energies are negative and their absolute values are large. Therefore ν are stable. the states with ( ) ( ) In this hater we have alulated the erturbation energy for the nearest eletron (or ν (in hole) airs at the filling fators ν ( ) (in Se..), ( ) Se..), ν ( ±), ν ( ±), and ( ± ) ( ± ) ν (in Se..5). Therein we an find the nearest eletron (or hole) airs transferring to all the emty (or ouied) orbitals. This roerty rodues the large binding energy and then the states beome stable. At the filling fators ν ( ) and ν ( ) ( ) (see Ses.. and.), the nearest eletron (or hole) airs annot transfer to any emty (or ouied) orbitals. So the binding energy of the nearest eletron (or hole) airs is zero. This roerty yields no lateau of the all resistane at the filling fator of ν ( ) and ν ( ) ( ). In the next hater we will examine the funtion-form of the energy versus filling fator. Then the funtion shows a disontinuous behaviour at the seifi filling fators. 67

68 Chater 5 Valley, flat and eak strutures in the energy setrum It is verified below that the energy of nearest eletron (or hole) airs takes a lowest ν and so on. value disontinuously at ν ( ), ν ( ), ( ) ± That is to say the erturbation energy at the seifi filling fators ν is lower than both limiting values from the left and right sides of ν. We all this tye of the disontinuity valley struture. (Professor K. Katsumata advised me to use the name valley.) On the other hand the erturbation energy of nearest eletron (or hole) airs at ν is ontinuously equal to the limiting value from its neighborhood. We all this ase flat struture. The erturbation energy of nearest eletron (or hole) airs is zero at the filling fators of ν,, 6, whih is aused by the absene of the quantum transition as verified in Chater. When the filling fator deviates from ν,, 6,, some of the nearest eletron (or hole) airs an transfer to the other orbitals. These allowed transitions yield the negative energy. Therefore the erturbation energy of the nearest eletron (or hole) airs at ν,, 6, is higher than that in their neighbourhood. The ase is named eak struture. Thereby the robability of the state with ν,, 6, is very small at a low temerature in omarison with that of the neighbourhood. This ase yields the henomenon similar to the ase with the flat struture. The valley struture is aused by the ombined effets of the most uniform eletron-onfiguration, the Fermi-Dira statistis and the momentum onservation in the x-diretion [-]. As will be shown in Chater 7, the valley struture rodues the lateaus in the all resistane when the magneti field or gate voltage is varied. We examine FQ states with ν < in setions and ν > in setion 5.9. The FQ states with <ν < are studied in setion 5.. It is noteworthy that Chaters -8 treat only the ase of a strong magneti field, where all the eletron sins are direted oosite to the magneti field at ν. The mixing of u and down sins will be investigated in Chater Valley struture at the filling fator of ν ( ±) We examine the erturbation energy of the nearest eletron airs in the neighborhood ν. As an examle, we first study the ase of namely ν. of ( ) 68

69 5.. Valley struture at ν The frational number ( s ) ( 6s ) ν is smaller than / and aroahes / in the limit of infinitely large s. We alulate the erturbation energy at ν s 6s and find the value in the limit s. ( ) ( ) As an examle we onsider the ase of s that gives ν 9. We draw the eletron onfiguration with the minimum lassial Coulomb energy in Fig.5.. This onfiguration is obtained by reeating the unit-onfiguration of (filled, emty, filled, filled, emty, filled, filled, emty, filled, filled, emty, filled, filled, emty). There are four nearest eletron airs A, CD, EF and G in the unit-onfiguration as seen in Fig.5.. A C D E F G Fig.5. Coulomb transitions of nearest neighbour eletron airs at ν 9 We first examine the eletron airs A and G. oth eletron airs A and G an transfer to the two emty states er unit-onfiguration as in Fig.5.. In the quantum transition from A the momentum transfer is defined by whih takes the following values via the Coulomb transitions: ( ) π, ( ) π for, ±, ±, (5.) That is to say the number of allowed transitions is two among the fourteen orbitals. Denoting the seond order erturbation energy of the air A by the symbol ς, we A obtain the erturbation energy as ς A ( )Z where Z have been already defined in Eq.(.6). The eletron airs CD and EF an transfer to the four emty states er unit-onfiguration. Then the erturbation energies of the airs CD and EF, namely 69

70 ς CD,ς EF are given by ς ς ( )Z CD EF Similarly the erturbation energies of the air G, ς G, is given by ς G ( )Z Then the total erturbation energy of all the nearest eletron airs is E Z Z Z Z N 9 eletron air [ ( ) ( ) ( ) ( ) ] ( ) ( ( 9) ) ZN for ν 9 (5.) where N is the total number of eletrons. Next we examine the ase of s whih gives the filling fator ( s ) ( 6 ) ν s. The most uniform onfiguration of eletrons for the ν state is shown in Fig.5. A CD E F G I J KL Fig.5. Coulomb transitions of nearest neighbour eletron airs at ν In this eletron onfiguration the six nearest eletron airs A, CD, EF, G, IJ and KL exist in the unit-onfiguration. These airs have the erturbation energies as ς ( )Z, ς ( )Z, ς ( 6 )Z, ς ( 6 )Z, A CD ς ( )Z, ς ( )Z at ν IJ KL The sum of the erturbation energies of all the nearest eletron airs is equal to EF G E nearest air [ ( ) Z ( ) Z ( 6 ) Z ( 6 ) Z ( ) Z ( ) Z ] ( N ) ( ( ) ) ZN for ν (5.) Similarly we an alulate the sum of the erturbation energies at ν ( s ) ( 6s ) for arbitrary integer s. Therein the unit-onfiguration is omosed of 6 s Landau orbitals, s of whih are ouied by eletrons. Therefore the number of emty states is s er unit-onfiguration. Aordingly 7

71 E nearest air ( ( 6s ) ) Z ( ( 6s ) ) Z ( s ( 6s ) ) ( ( 6s ) ) Z ( ( 6s ) ) Z s( s ) ZN ( s )( 6s ) ( s )( 6s ) Z ( N ( s ) ) s s Enearest air ZN for ν ( s ) ( 6s ) (5.) This result leads the limiting values as follows: ν ( s ) ( 6s ) ( ) s (5.5a) E nearest air N ( ) ( ) ( )Z s 6s, s for ν ( ) ε ν (5.5b) where ε indiates a ositive infinitesimal and the symbol ( ) ε means the limiting roess from the left. The revious result (.8) gives E ( 6) Z at nearest air N ν (5.6) When the filling fator ν aroahes from the left, the energy of the nearest eletron airs er eletron aroahes ( )Z whih is half of the energy at ν. That is to say, the energy at ν is lower than the limiting value from the left with a ga. We next alulate the limiting value from the right. The filling fator of ν s 6s is larger than and aroahes / in the limit of s. ( ) ( ) The eletron onfiguration for the ase of s is illustrated in Fig.5.. A C D E F G I J K Fig.5. Coulomb transitions of nearest neighbour eletron airs at ν 6 The seond order erturbation energies of the nearest airs A, CD, EF, G, IJ and JK are exressed by ς, ς, ς, ς, ς and ς resetively whih are ς A CD EF G IJ JK A ς G 6 Z, ς CD ς EF 6 Z, ς IJ ς JK ( ) ( ) 7

72 The sum of all the energies of the nearest eletron airs beomes E 6 Z 6 Z 6 Z 6 Z N nearest air [ ( ) ( ) ( ) ( ) ] ( ) ( ( 6 ) ) ZN for ν 6 (5.7) Next we onsider the ase s. Figure 5. shows the eletron onfiguration with the minimum lassial Coulomb energy at ν 5. A CD E F G I J KL MNO Fig.5. Coulomb transitions of nearest neighbour eletron airs at ν 5 The unit-onfiguration is omosed of twenty-two sequential Landau states, fifteen of whih are filled with eletron as illustrated in Fig.5.. There are eight nearest eletron airs A, CD, EF, G, IJ, KL, MN and NO in the unit-onfiguration. These airs have the erturbation energy as ς ς ( )Z, ς ς ( )Z, A KL CD ς ς ( 6 )Z, ς ς, for ν 5 EF G IJ MN NO Then the total erturbation energy of all the nearest eletron airs is equal to E nearest air [ ( ) Z ( ) Z ( 6 ) Z ] ( 5) ( ( 5) ) ZN for ν 5 N (5.8) We alulate the erturbation energy at the filling fator ( s ) ( 6s ) integer s. The unit-onfiguration is omosed of 6 ν for any s Landau states, s of whih are ouied by eletron. Then the number of emty states is s er unit-onfiguration. Aordingly we get the energy for all the nearest airs as E nearest air [ ( ( 6s ) ) Z ( ( 6s ) ) Z ( ( s ) ( 6s ) ) Z ] ( N ( s ) ) s( s ) ZN ( s )( 6s ) 7

73 s s E ZN for ν s (5.9) nearest air The limiting values are ( s )( 6s ) ( s ) ( 6s ) ( ) s ( s ) ( 6 ) ν (5.a) E nearest air N ( ) ( ) ( )Z s 6s, s ν for ( ) ε ν (5.b) where the symbol ( ) ε indiates the limiting roess from the right. The energy of the nearest eletron airs er eletron at ν is equal to ( 6)Z whih is twie of the limiting value ( )Z from the right. Therefore the ν state has a lower energy than the limiting one from the right. Consequently Eqs.(5.5b), (5.b) and (5.6) indiate that the erturbation energy is disontinuous at ν and has a valley struture. This valley struture yields the stability of the state with ν. Some readers might onsider that the lowest roerty at ν is aused by an even denominator of the frational number ν ( s ± ) ( 6s ± ) state at the filling fator ( s ± ) ( 6s ± ). We also examine the ν with an odd denominator in Aendix. The results of the limiting values are the same as in this setion. (Funtion form of total energy) We exress the funtion form of the total energy E ( ν ) in the neighborhood of ν. There are many non-nearest eletron airs, in addition to the nearest-neighboring ones. The energy of all the non-nearest eletron airs is denoted by the symbol ( )N g in Eqs.(.) and (.). Then the total g ν. We already used ( ) energy E ( ν ) is a sum of four terms, namely, the energy of all the nearest airs, the energy of all the non-nearest airs, the Landau energy and the lassial Coulomb energy as follows: (Note: The lassial Coulomb energy has been obtained in Eq.(.b) ) ( ν ) E g( ν ) N [ f e ( m ) ( ξ η) ν ] N ( σ ) E (5.a) nearest air CMarosoi ereafter the total energy er eletron is desribed by the symbol ε ( ν ) as ( ν ) E( ν ) N ε (5.b) There are ontributions from the higher order erturbation whih will be studied in the next hater. The exat nearest-air energy er eletron, χ ( ν ), is a sum of all order erturbation energies as ( ν ) χ n ( ν ) χ (5.a) n,,, 7

74 where the symbol χ n ( ν ) indiates the n-th order erturbation energy of all the nearest eletron airs er eletron. The seond order term χ ( ν ) was already alulated in haters and 5. For onfirmation we write the following relation: nearest air ( ν ) χ ( ν ) E N (5.b) χ Eqs.(5.a,b) and (5.a,b) yield ( ν ) χ( ν ) g( ν ) [ f e ( m ) ( ξ η) ν ] C N ε (5.) Eqs.(5.5b) (5.6) and (5.b) are rewritten as Marosoi χ ( ) 6 Z ( ± ε ) Z (5.a) χ (5.b) Equations (5.a) and (5.b) exress a disontinuous roerty of the energy at ν [-]. 5.. Valley struture at ν 5 We examine the FQ states in the neighborhood of ν 5. The filling fator ν ( 6s ) ( s ) is larger than 5 and has the limiting value /5 for infinitely large integer s. An examle is ν 8 for s whose most uniform eletron onfiguration is illustrated in Fig.5.5. A C D E F G A CD E F G A C D Fig.5.5 Coulomb transitions of nearest neighbour eletron airs at ν 8 The most uniform eletron onfiguration is obtained by reeating the unit-onfiguration (filled, filled, emty, filled, emty, filled, filled, emty, filled, emty, filled, filled, emty, filled, emty, filled, filled, emty) in whih 8 sequential Landau states are artially filled with eletrons. There are four nearest eletron airs A, CD, EF and G in the 7

75 unit-onfiguration as in Fig.5.5. The eletron air A an be transferred to the four sites in a unit-onfiguration shown by four arrow-airs but annot be transferred to the other sites shown by dashed arrow-airs beause of the momentum onservation and the Pauli exlusion rinile. Also the air CD is transferred to the six sites in a unit-onfiguration indiated by six solid arrow-airs. Then the nearest airs A, CD, EF and G have the seond order erturbation energy as ς ( 8)Z, ς ( 6 8)Z, ς ( 6 8)Z, A CD EF ς ( 8)Z, for ν 8 (5.5) G The erturbation energy of all the nearest eletron airs is equal to E nearest air [ ( 8) Z ( 6 8) Z] ( ) N ( ( 8 ) ) ZN for ν 8 (5.6) We examine the ase of s, namely ν ( 6 s ) ( s ) 7 8. Figure 5.6 shows the most uniform eletron onfiguration. Therein the dashed arrow airs indiate forbidden transitions and the other arrow airs indiate allowed transitions. The nearest eletron airs A and KL are allowed the six transitions er unit-onfiguration, the airs CD and IJ eight and the airs EF and G ten transitions. Then the air energies are given by ς ς ( 6 8)Z, ς ς ( 8 8)Z, A KL ς ς ( 8)Z, for ν 7 8 EF G CD IJ A C D E F G I J K L A CD E F G I J KL A C D Fig.5.6 Coulomb transitions of nearest neighbour eletron airs at ν 7 8 Therefore the total transition energy of all the nearest eletron airs is equal to E 6 8 Z 8 8 Z 8 Z 7 N nearest air [ ( ) ( ) ( ) ] ( ) ( 8 ( 8 7) ) ZN for ν

76 We alulate the erturbation energy at the filling fator ( 6s ) ( s ) ν for any integer s. The unit-onfiguration is omosed of s Landau states, 6s of whih are filled with eletron. Therefore the number of emty states is s er unit-onfiguration. Aordingly E nearest air nearest ( s ( s ) ) Z (( s ) ( s ) ) Z ( ( 6s ) ) (( s ) ( s ) ) Z (( s ) ( s ) ) Z s( 6s ) ZN ( 6s )( s ) s( 6s ) air ZN for ( 6s ) ( s ) ( 6s )( s ) E ν (5.7) The limiting values are ( 6s ) ( s ) ( 5) s ν (5.8a) N E nearest air N ( ) ( ) ( )Z 6s s, s ν for ( ) ε ν 5 (5.8b) The revious result Eq.(.a) gives the erturbation energy at ν 5 as, E nearest air N Z Z at ν 5 5 ( ) (5.8) The limiting value of the erturbation energy ( )Z is (/) times the nearest air energy er eletron at ν 5 and so disontinuous. Therefore the energy ga aears at ν 5. Next we alulate the limiting value from the left. The filling fator ν ( 6 s ) ( s ) is smaller than /5. First, we onsider the ase of s. The most uniform eletron onfiguration at ν is shematially drawn in Fig.5.7. A C D E F G A CD E F G A C D Fig.5.7 Coulomb transitions of nearest neighbour eletron airs at ν 76

77 There are four nearest eletron airs A, CD, EF and G in the unit-onfiguration. The eletron air A and G transfer to the six emty states and the airs CD and EF also transfer to the eight emty states in a unit-onfiguration. These allowed transitions are illustrated by solid arrow airs in Fig.5.7. The dashed arrow airs indiate forbidden transitions. The erturbation energies are given by ς ς ( 6 )Z, ς ς ( 8 )Z for ν (5.9) A G CD EF Then the sum of the erturbation energies for all the nearest eletron airs is equal to E nearest air [ ( 6 ) Z ( 8 ) Z] ( ) N ( 8 ( ) ) ZN for ν (5.) We alulate the erturbation energy for arbitrary integer s. At the filling fator ν 6 s s the unit-onfiguration is omosed of s sequential ( ) ( ) Landau states, 6 s of whih are ouied by eletron. Therefore the number of emty states is s er unit-onfiguration. Aordingly E nearest air (( s ) ( s ) ) Z (( s ) ( s ) ) ( s ( s ) ) Z ( 6s ) s ZN ( 6s )( s ) Z ( 6s )( s ) ( ( 6s ) ) 6s s E ZN for ν s (5.) ( 6s ) ( ) nearest air The limiting values are ( 6s ) ( s ) 5 s ν (5.a) N E nearest air N ( ) ( ) ( )Z 6s s, s for ν ( ) ε ν 5 (5.b) The revious result (.a) gives E nearest air N Z Z Z at ν ( ) (5.) Thus Eqs.(5.8b), (5.b) and (5.) indiate a disontinuous struture at ν 5. The energy of all the nearest eletron airs er eletron is equal to Z 5 at ν 5 whih is lower than the limiting value ( Z ) from the both sides. 77

78 5.. Valley struture at ν ( ) In this subsetion we examine the erturbation energies in the neighbourhood of ν ( ) for any integer. The filling fator ν ( ( s) ) (( )( s) ) aroahes ν ( ) in the limit of infinitely large s. As an examle, we onsider the ase of, s, where the filling fator is equal to ( ( s) ) (( )( ) ) 5 6 ν s The most uniform eletron onfiguration is illustrated in Fig.5.8 where the allowed transitions are drawn with solid arrow airs and the forbidden transitions with dashed arrow airs. A C A C D E F G A CD E F G D Fig.5.8 Coulomb transitions of nearest neighbour eletron airs at ν 5 6 The number of the allowed transitions is eight for the airs A and G, and is ten for the airs CD and EF er unit-onfiguration: ς ς ( 8 6)Z, ς ς ( 6)Z at ν 5 6 E A G nearest air CD EF 6 [ ( 8 6) Z ( 6) Z ] ( 5) N ZN at ν The roedure given above an be extended to arbitrary integers and s as follows: Let us introdue three arameters, α, β and γ to reresent, resetively, the number of orbitals in the unit-onfiguration, the number of orbitals filled with eletron, and the number of emty orbitals in the unit-onfiguration. These arameters are exliitly given at the filling fator ν ( ( s) ) (( )( s) ) by, α (( )( s) ), β ( ( s) ), γ (( )( s) ) (5.) y extending the alulation at ν 5 6 to any integer s, the total energy for all the nearest eletron airs for is given by, N E nearest air [ (( s) α ) Z (( s ) α ) Z (( 6s ) α ) Z ] (5.) β s is equal to γ for s is equal to γ s. Therein ( 6 ) and also ( ) 78

79 Aordingly we obtain N E nearest air [ (( γ s) α ) Z (( γ s) α ) Z (( γ ) α ) Z ] β Z N E nearest air [ ( γ s) ( γ s) ( γ ) ] α β E nearest air Z N at α β [ ( γ s) s] ν β α Substitution of Eq.(5.) into Eq.(5.5) yields (( ) s ) s ZN (( )( s) ) ( ( s) ) ( ( s) ) (( )( s) ) (5.5) E at ν (5.6) nearest air Thus we have obtained the erturbation energy of the nearest eletron airs for arbitrary integers of and s. Eq.(5.6) gives the limiting value from the right (for s ). ( ( s) ) (( )( s) ) ( ) ν β α E From Eq.(.a) (( ) s ) s ZN (( )( s) ) ( ( s) ) nearest air s nearest air s ( ) ZN at ( ) ( ) ZN at ( ) ν ε (5.7a) (5.7b) E ν (5.7) χ ( ) ( ) Z ( ) where we have used Eq.(5.b). The energy E at ( ) nearest air (5.7d) ν is omared with the limiting value, Eq.(5.7b), and we find the former is lower than the latter. The differene is equal to E nearest air E ( ν ) lim E( λ) λ ν ε ZN for ν ( ) (5.8) Next we alulate the limiting value from the left. We onsider the filling fator s s ν. We introdue ν ( ( ) ) (( )( ) ) whih is smaller than ( ) three arameters α, β, γ defined by (( )( s) ), β ( ( s) ), γ (( )( ) ) α s (5.9) The arameter α reresents the number of orbitals in the unit-onfiguration, β the number of eletrons in the unit-onfiguration and γ the number of emty orbitals in the unit-onfiguration, resetively. The filling fator is given by ν β α : 79

80 ( ( s) ) (( )( ) ) ν β α s The sum of the erturbation energy for all the nearest eletron airs is equal to E nearest air Substitution of Eq.(5.9) yields Z N at α β [ ( γ s) s] ν β α (( ) s ) s ZN (( )( s) ) ( ( s) ) ( ( s) ) (( )( s) ) E at ν (5.) nearest air The limiting value of Eq.(5.) is ( ( s) ) (( )( s) ) ( ) ν s (5.a) (( ) s ) s ZN ( ) ZN Enearest air at ν ε (5.b) s (( )( s) ) ( ( s) ) ( ) This limiting value from the left is omared with the energy at ν ( ) and we find the former is higher than the latter. The differene is given by E nearest air E ( ν ) lim E( λ) λ ν ε ZN for ν ( ) (5.) Thus the disontinuity in the energy setrum is larified theoretially as in Eqs (5.8) and (5.). If readers feel that the roofs mentioned above are a little omliated, the limiting value an be aroximately evaluated by a omuter. We summarize the results: ) All the nearest eletron airs are able to transfer to all the emty states at ν. ( ) ) When the filling fator slightly deviates from ν ( ) ) Then the erturbation energy at ( ), the nearest eletron airs are unable to transfer to some of the emty states. ν is lower than those at ν ± ε as; E nearest air E ( ν ) lim E( λ) ZN [ ( ) ] λ ν ± ε at ( ) ν. Thus the valley struture is derived from the resent theory. The disontinuity is aused by the ombined effets of the most uniform eletron onfiguration, the Fermi-Dira statistis and the momentum onservation. As will be shown in Chater 6, the disontinuity aears in all orders of the erturbation alulation. 8

81 5.. Valley struture at ν ( ) In this subsetion, we examine the FQ state with ( ) The ν ( ) state is related to the ( ) ( ) ν smaller than /. ν state by the eletron-hole symmetry as in Eq.(.9). Equations (5.6) and (5.) are rewritten by relaing : (( ) s ) s ZN (( )( s) ) (( )( s) ) (( ) s ) s ZN (( )( s) ) (( )( s) ) (( )( s) ) (( )( s) ) (( )( s) ) (( )( s) ) E at ν (5.) nearest air E at ν (5.) nearest air The limiting values are obtained as E where ε (or nearest air N s for ν ± ε ( )( ) Z ε ) indiates the limiting roess from the right (or left). (5.5) From the eletron-hole symmetry, the erturbation energy E is obtained by nearest hole air relaing N and ν ( ν ) (( ) s ) s Z N (( )( s) ) (( )( s) ) (( ) s ) s ZN (( )( s) ) (( )( s) ) Z Z, N in Eqs. (5.) and (5.) as follows: (( )( s) ) (( )( s) ) (( )( s) ) (( )( s) ) E at ν (5.6) nearest hole air E at ν (5.7) nearest hole air In the limit of E nearest hole s, both equations (5.6) and (5.7) have the same limiting value as air N ε Z for ν ± ε (5.8) ( )( ) The erturbation energy of all the nearest hole airs at ( ) obtained in Eq. (.) as E N Z at ν 8 ν has been already nearest hole air ( )( ) Thus the energy setrum versus filling fator is disontinuous at ( ) The energy er eletron ( ν ) N ε E total has been given by Eq.(5.) as ( ν ) χ( ν ) g( ν ) [ f e ( m ) ( ξ η) ν ] C N ε Marosoi ν. (5.9)

82 where g ( ν ) reresents the energy of all the non-nearest airs er eletron via the Coulomb transitions. The first term χ ( ν ) in the right hand side reresents the energy of all the nearest airs er eletron, the seond order term of whih has been alulated χ ν, the energy as mentioned above. Using the seond order term ( ν ) er eletron at ν ( ) ( ) is aroximated by ε ( ν ) Z ( )( ) g ( ν ) e f m χ instead of ( ) ( ξ η) ν C at ( ) ( ) The energy ga (energy deth of the valley) is defined by ( ν ) ε ( ν ) ε ( λ) Marosoi ν (5.) ε lim (5.a) λ ν ε ( ν ) ε ( ν ) ε ( λ) ε lim (5.b) λ ν ε We will study the disontinuity of g ( ν ) in setions 5.9 and 5. whih is small. So we ignore the disontinuity of g ( ν ) and then obtain the aroximate form by substitute Eqs. (5.), (5.) and (5.) into Eqs.(5.a) and (5.b), and obtain the energy ga ν as (the energy deth of the valley) at ( ) ( ) ε ( ν ( ) ( ) ) (5.) ( )( ) Z ε ( ν ( ) ( ) ) (5.) In the ase of ν ( ) ( )( ) Z, the number of emty Landau orbitals is larger than that of filled Landau orbitals. The erturbation energy has been obtained as Eqs.(5.6) and (5.7). The energy er eletron (not er hole) is obtained by multilying N N to Eqs.(5.8) and (5.9) as follows; Enearest hole air N ε Z for ν ± ε (5.a) ( ) Z Enearest hole air N Z at ν (5.b) ( ) ( ) Then the energy ga (energy deth of the valley) ( ν ) at ( ) ε ± ν is equal to Z ε ( ν ( ) ) ( ) (5.5a) Z ε ( ν ( ) ) (5.5b) ( ) N 8

83 It is noteworthy that Eqs.(5.5a) and (5.5b) indiate the energy deth of the valley er eletron (not hole). When the energy deth is dee in omarison with the thermal ν ± beomes very stable. exitation energy, the state with the filling fator ( ) 5. Comarison of the theory with exerimental data in the Neighbourhood of ν ( ±) We list the energy gas (energy deths of the valleys) at ( ±) and 5. whih are derived from Eqs.(5.) (5.5). ν in Tables 5. Table 5. Energy gas of nearest eletron airs er eletron at ν ( ) ν E nearest air N ν lim ( E N ) nearest air ε ( ν ) ε ( ν ) / (/) ± ε / -(/6) Z (/) ± ε -(/) Z -(/) Z /5 -(/5) Z (/5) ± ε -(/) Z -(/) Z /7 -(/8) Z (/7) ± ε -(5/56) Z -(/56) Z 5/9 -(/5) Z (5/9) ± ε -(7/9) Z -(/9) Z 6/ -(5/66) Z (6/) ± ε -(9/) Z -(/) Z 7/ -(6/9) Z (7/) ± ε -(/8) Z -(/8) Z 8/5 -(7/) Z (8/5) ± ε -(/) Z -(/) Z Table 5. Energy gas of nearest hole airs er eletron at ν ( ) ν E nearest air N E air N ν nearest lim ( E N ) nearest air ε ( ν ) ε ( ν ) / -(/6) Z -(/) Z (/) ± ε -(/6) Z -(/6) Z /5 -(/5) Z -(/5) Z (/5) ± ε -(/) Z -(/) Z /7 -(/8) Z -(/7) Z (/7) ± ε -(5/) Z -(/) Z /9 -(/5) Z -(/9) Z (/9) ± ε -(7/7) Z -(/7) Z 5/ -(5/66) Z -(/) Z (5/) ± ε -(9/) Z -(/) Z 6/ -(6/9) Z -(/) Z (6/) ± ε -(/56) Z -(/56) Z 7/5 -(7/) Z -(/5) Z (7/5) ± ε -(/) Z -(/) Z 8

84 Table 5. indiates the disontinuous struture at the filling fators of ν ( ). In the third olumn ε indiates an infinitesimally small ositive value, then ε indiates the limiting roess from the right and ε indiates the limiting roess from the left. Table 5. shows the disontinuity in the energy setrum of the nearest hole airs at ν ( ). The rightmost olumns in these two tables reresent the energy gas (energy deths of the valleys) er eletron. The resene of the valley rodues the all lateau as will be studied in Chater 7. Fig.5.9 Many loal minima of the diagonal resistane in the region of ν The exerimental urve is given in referene [, 5] We omare the theoretial results with the exerimental data in Fig.5.9 [, 5]. The theoretial energy setrum has the valleys with dee energy deths as in Tables 5. and 5.. The dee valley yields the strong onfinement to the ground state with the seifi filling fator. Thereby the ground state annot be exited by the eletri urrent and so the diagonal resistane beomes nearly zero. The exerimental urve of the diagonal resistane is almost zero at the filling fators 5, 6 ν, 5, 7, and 9 ν 5,, 9, 7,. 5 Thus the exerimental data are in good agreement with the theoretial results. 6 Furthermore there is a small loal minimum at ν 7 8 5,,,, 8, shown by solid 7 6 arrows in Fig.5.9. We will investigate the FQ states with ν ,,,,,, in 7 7 setions 5.5 and

85 5. Flat struture at ν and eak struture at ν and ν We examine the frational quantum all states in the neighborhood of ν. The ν is larger than / and aroahes / for an infinitely large. The energy Eq.(.) has the limiting value from the right of ν as filling fator ( ) ν ( ) (5.6a) Enearest air N Z ( ) for ν ( ) ε ν, ( ) Equation (.b) indiates that the nearest air energy is zero at ν as (5.6b) E N for ν nearest air (5.6) Thus the limiting energy from the right is equal to that at ν. Similarly we alulate the limiting value from the left of ν whih is already obtained in Eq.(5.b). We write it again here: Z Enearest air N at ν ( ) The limiting value from the left of ν beomes zero as: Z Enearest air N ( ) for ν ( ) ε (5.6d) ν, ( ) Aordingly the energy of all the nearest eletron airs er eletron is ontinuous at ν. The ontinuous roerty aears in the higher order erturbation as will be shown in hater 6. That is to say the energy setrum of the nearest eletron airs is ontinuous at ν and we all it flat struture. We next examine the frational quantum all states in the neighborhood of ν. The filling fator ν ( 6 s ) ( 8s ) aroahes / in the limit of s. As an examle we onsider the ase of s. The most uniform onfiguration of eletrons is illustrated in Fig.5.. The unit-onfiguration has seventeen sequential Landau states, thirteen of whih are filled with eletron. There are four emty states inside eah unit-onfiguration. 85

86 ACD E FG I J KLM Fig.5. Allowed transitions of nearest eletron airs at ν 7 The nearest eletron air C transfers to all the emty orbitals as drawn by blue arrows in Fig.5.. The nearest eletron air EF (and LM) transfers to the two emty orbitals er unit-onfiguration as exressed by blak arrows. The air I (and IJ) annot transfer to any emty orbital as shown by dashed arrow airs. The other airs A, CD, FG, KL annot transfer to any emty orbital. Aordingly the erturbation energy of the nearest eletron airs at ν 7 is obtained as ς ( 7)Z, ς ( 7)Z, ς ( 7)Z C EF ς A ς CD ς FG ς I ς IJ ς KL Then the erturbation energy of all the nearest eletron airs is equal to Enearest air [ ( 7) Z ( 7) Z ( 7) Z ] ( ) N ( 8 ( 7 ) ) ZN for ν 7 We next alulate the erturbation energy at the filling fator ( 6 s ) ( 8s ) any integer s. The unit-onfiguration is omosed of 8 LM ν for s Landau states, 6 s of whih are ouied by eletron. Therefore the number of emty states is s er unit-onfiguration. Aordingly we obtain the total energy for the nearest airs as E E nearest air nearest air ( ( 8s ) ) Z ( ( 8s ) ) Z (( s) ( 8s ) ) ( ( 8s ) ) Z ( ( 8s ) ) Z s( s ) ( s ) s ZN ( 8s )( 6s ) ( 8s )( 6s ) Z ( ( 6s ) ) s N Z (5.7) The filling fator and the energy of the nearest airs aroah in the limit of ν ( 6s ) ( 8s ) s N s Enearest air N Z s (5.8) 86

87 Thus the erturbation energy of the nearest eletron airs at ( 6 s ) ( 8s ) negative and the limiting value is ( )Z as in Eq.(5.8). ν is At ν, all the transitions from nearest eletron airs are forbidden [7-]. Thereby the energy of all the nearest airs is zero as in Eq.(.7); E N at ν nearest air (5.9) Consequently the energy of all the nearest eletron airs at ν is higher than that of the neighbourhood. So the state with ν is unstable. Thus this ase has a eak (disontinuous eak) in the energy setrum. We all the energy setrum eak struture. At the eak no lateau aears in the all resistane urve as it will be studied in hater 7. For arbitrary integer we alulate the energy of the nearest eletron airs in the neighbourhood of ( ) ( ) written here. ν by using a omuter. The several results are (neighbourhood of ν 5 6 ) E N 5 5 nearest air 9 Z at ν 65 E E E nearest air nearest air nearest air N N N Z at ν Z at ν Z at ν 67 (neighbourhood of ν 7 8 ) E N 5 76 nearest air 5697 Z at ν 87 E E E nearest air nearest air nearest air N N N Z at ν Z at ν Z at ν 89 The numerial results may give the limiting value as ; E E nearest air nearest air 5 N Z for ν 6 ± ε 6 7 N Z for ν 8 ± ε Thus the energy of the nearest eletron airs at ( ) ( ) ν is larger than that of 87

88 the neighbourhood. So the eak struture aears at ν ( ) ( ) are listed in Table 5... Some examles Table 5. Comarison between nearest eletron air energy ν and in the neighbourhood ν E nearest air at ( ) ( ) N ν lim ( E N ) nearest air / (/) ± ε / (/) ± ε Z 5/6 (5/6) ± ε Z 7/8 (7/8) ± ε Z 6 Table 5. Comarison between nearest hole air energy ν E nearest air N ν / (/) ε /6 (/6) ε /8 (/8) ε at ( ) ν and in the neighbourhood lim ( E N ) nearest air ± Z ± 6 Z ± Z ν (/) ε (/6) ε (/8) ε lim ( E N ) nearest air Z ± 8 ± Z ± 6 Z From the eletron-hole symmetry, the erturbation energies of the nearest hole airs are easily alulated. The results are shown in Table 5. [7-]. We find a eak struture in the energy setrum. The nearest eletron airs (or hole airs) at ν ( ) ( ) (or ν ( ) ) have the erturbation energy higher than in their neighborhood. Aordingly the states with ν ( ) ( ) (or ν ( ) ) are unstable. Thus the theoretial results yield no lateau in the all resistane urve at the filling fator with the eak struture. The results of the resent theory are in good agreement with the exerimental data. 5. Valley struture at ν ( ) ( ) and ν ( ) We omare the erturbation energy at ( ) ( ) 88 ν with that of the

89 neighbourhood. The filling fator (( ) s ) (( ) s ) for the neighbourhood of ( ) ( ) ν is an aroriate value ν beause ν aroahes ν in the limit of infinitely large s. We examine an examle for ( s ) whih gives ν 7. The most uniform onfiguration at ν 7 is illustrated in Fig.5.. The total number of emty states is / times the total number of orbitals. A C D E F G I J K L MNO PQ Fig.5. Allowed transitions of nearest eletron airs at ν 7 The eletron air FG an transfer to all the emty states as shown by solid arrow airs. The quantum transitions from the air FG yield the erturbation energy as ς FG ( )Z The nearest eletron airs C and JK transfer to two emty states er unit-onfiguration. Aordingly the airs C and JK have the erturbation energies as ς C ( )Z, ς JK ( )Z The erturbation energies of the remaining nearest eletron airs A, CD, EF, G, IJ, KL, MN, NO, OP, and PQ are zero beause the transitions of these airs are forbidden. ς A ς CD ς EF ς G ς IJ ς KL ς MN ς NO ς OP ς PQ The sum of the erturbation energies of all the nearest eletron airs is N N Enearest air ( ς C ς FG ς JK ) ( 8 ) Z Enearest air N Z for ν 7 (5.5) 7 Next we alulate the erturbation energy at the filling fator ν ( 8 s ) ( s ) for any integer s. The unit-onfiguration is omosed of s Landau orbitals, 8 s of whih are ouied by eletron. Therefore the number of emty orbitals is s er unit-onfiguration. Aordingly we obtain ( ( s ) ) Z ( ( s ) ) Z (( s) ( s ) ) Z Enearest air ( ( 8s ) ) N ( ( s ) ) Z ( ( s ) ) Z s( s ) ( s ) s ZN s 8s ( )( ) 89

90 E nearest air N s Z (5.5) ( s )( 8s ) The limiting values of the filling fator and the energy of the nearest airs are equal to ν ( 8s ) ( s ) 5 s Enearest air N Z s in the limit from the right (5.5a) The erturbation energy at ν 5 is listed in Table. as E nearest air N Z at ν 5 (5.5b) Eqs. (5.5a) and (5.5b) reveals that a valley struture exists in the energy setrum at ν 5. The method mentioned above is generalized for arbitrary integer, the limiting value of the energy for all the nearest eletron airs er eletron is obtained as E nearest air N Z for ν ± ε (5.5a) ν ± ( ) ε ( )( ) Eq.(.6) gives the energy of all the nearest eletron airs er eletron at ν as ( ) E N ν ( ) Z nearest air at (5.5b) Thus the limiting value from the both sides is half of the value at ( ) ( ) Then the energy ga (deth of the valley) is obtained as [-] ε ( ν ) ε ( ) ν ( )( ) Z We exress these values in Table 5.5. for ν ν. (5.5) Table 5.5 Comarison of nearest eletron air energies er eletron at ν ( ) and in its neighbourhood ν ν ε ( ν ) ε ( ν ) E N lim E N eletron air ( ) nearest air / -(/6) Z (/) ± ε -(/) Z -(/) Z /5 -(/) Z (/5) ± ε -(/) Z -(/) Z 6/7 -(/) Z (6/7) ± ε -(/8) Z -(/8) Z 8/9 -(/7) Z (8/9) ± ε -(/) Z -(/) Z 9

91 The energy of all the nearest hole airs at ( ) ν is alulated on the basis of the eletron-hole symmetry relation given in Eq.(.9). The erturbation energy E nearest hole air is obtained by relaing (5.5a) and (5.5b) as follows: E E nearest air nearest air N N ( ) Z Z, N Z for ± ( ) ε ( )( ) in Eqs. N and ν ( ν ) ν (5.5a) Z for ν ± ε (5.5b) ν The energies er eletron (not hole) are given by multilying N N to Eqs.(5.5a) and (5.5b); E E nearest air nearest air N Z for N ( ) ( ) ε ( ) ν (5.55a) Z for ν ± ε ν ± Then the energy ga (deth of the valley) er eletron beomes ε ( ν ) ε ( ) ν Z ( ) These results are listed in Table 5.6. [-] for ν (5.55b) (5.55) Table 5.6 Comarison of nearest hole air energies er hole and er eletron at ν ( ) and in its neighbourhood ν Enearest air N Enearest air N ν lim ( Enearest air N ) ε ( ν ) ε ( ν ) / -(/6) Z -(/) Z (/) ± ε -(/6) Z -(/6) Z /5 -(/) Z -(/5) Z (/5) ± ε -(/) Z -(/) Z /7 -(/) Z -(/7) Z (/7) ± ε -(/) Z -(/) Z /9 -(/7) Z -(/9) Z (/9) ± ε -(/8) Z -(/8) Z It is noteworthy that the erturbation energy of all the nearest hole airs er eletron is more imortant quantity than that er hole. The values er eletron are written in the third, fifth and sixth olumns in Table 5.6. Tables 5.5 and 5.6 indiate a disontinuity (deth of the valley) in the eletron-energy setrum at the filling fators of ν ( ) and ( ) ν. Therein ε 9

92 indiates the limiting roess from the right and ε does from the left. The eletron-energy gas (deths of the valley) in the energy setrum are listed in the last olumns of Tables 5.5 and 5.6. Aordingly the energy er eletron at ν ( ) (or ν ( ) ) is lower than that in their neighbourhood. Consequently these states beome stable. 5.5 Seifi filling fators with even number denominator We examine the energies of the nearest eletron airs at the filling fators of ν 5 8 and 7 whih have eah denominators of even number. The most uniform onfiguration of the many eletron state at ν 5 8 is shown in Fig.5.. X Y Z A CD Fig.5. Allowed transitions of nearest eletron airs at ν 5 8 This onfiguration is omosed of reeating a unit-onfiguration (filled, filled, emty, filled, filled, emty, filled, emty). Therein two nearest eletron airs A and CD exists in the unit-onfiguration as shown in Fig.5.. We examine allowed transitions from eletron air A. When the eletron at transfers to the first orbital to the right, the eletron at A transfers to the first orbital to the left beause of the momentum onservation. When the eletron at transfers to the fourth orbital to the right, the eletron at A is forbidden to transfer to the fourth orbital to the left beause of the Pauli exlusion rinile. Similarly the eletron at A annot transfer to the X orbital in Fig.5.. That is to say the eletron air A an transfer to the two emty orbital airs er unit-onfiguration. The transitions of the nearest eletron air CD are also allowed to the th, 6 th, 9 th, th, 7 th orbitals. Aordingly the number of allowed transitions er unit-onfiguration is two. The number of allowed quantum transitions from the nearest eletron airs is smaller than the number of emty orbitals. This roerty is in ontrast to that disussed in setions 5., 5. and 5., where the number of allowed transitions for the nearest 9

93 eletron airs is equal to the total number of emty orbitals. Thus the airs A and CD have the erturbation energy ς and A ς CD as ς ( 8)Z ς ( 8)Z A CD (5.56) The total erturbation energy of all the nearest eletron airs is equal to ( ς )( N 5) ( )ZN E ς nearest air A CD E N ( )Z at ν 5 8 nearest air (5.57) A similar situation ours at ν 7 as illustrated in Fig.5.. A CD Fig.5. Allowed transitions of nearest eletron airs at ν 7 This onfiguration is omosed of reeating a unit-onfiguration (filled, filled, emty, filled, filled, emty, filled, filled, filled, emty). Therein eletron an transfer to the th, 8 th, th, 8 th, th orbitals to the right in the quantum transitions from the air A. Also the transitions of the air CD are allowed to the th, 8 th, th, 8 th, th orbitals. That is to say, the two transitions are allowed er unit-onfiguration. The airs A and CD have the erturbation energy as ς ( )Z ς ( )Z A CD (5.58) Aordingly the total erturbation energy of all the nearest eletron airs is equal to ( ς )( N 7) ( )ZN E ς 5 nearest air A CD E N ( 5)Z for ν 7 nearest air (5.59) Similarly we an alulate the energies for the nearest hole airs at ν 8 and ν. The results of the alulation are listed in Tables 5.7 and

94 Table 5.7 Energy of nearest eletron airs er eletron for the filling fators with an even number of denominator ν Enearest air N ε ( ν ) ε ( ν ) 5/8 -(/) Z 7/ -(/5) Z Table 5.8 Energy of nearest hole airs er hole and er eletron for the filling fators with an even number of denominator ν E N E N nearest air nearest air ε ( ν ) ε ( ν ) /8 -(/) Z / -(/5) Z -(/6) Z -(/5) Z Next, we study the energy setrum in the neighbourhood of ν 5 8. We have introdued the sub-units S (filled, filled, emty) and S (filled, emty) in Chater, whih are drawn again in Fig.5.. Then the unit-onfiguration is T 5/8 S S S as shown in the uer anel of Fig.5.. Then the most uniform onfiguration of ν 5 8 is rodued by reeating the unit-onfiguration T 5/8. In the neighbourhood of ν 5 8, the most uniform eletron onfiguration should be omosed of only the two tyes of sub-units S and S, beause the frational number 5 8 is larger than / and smaller than /. S S T 5/8 (S, S, S ) (S, T 5/8, T 5/8, T 5/8, T 5/8, T 5/8, T 5/8 ) Fig.5. Sub-units and their ombinations The filling fator ( s ) ( 6s ) ν aroahes 5 8 in the limit of infinitely large value of s. The ase of s gives the filling fator ν 5. The ν 5 state has the most uniform eletron-onfiguration omosed of one sub-unit S and six 9

95 sub-units T 5/8 namely S T 5/8 T 5/8 T 5/8 T 5/8 T 5/8 T 5/8 whih is shown in the lower anel of Fig.5.. The unit-onfiguration has fifty-one sequential orbitals, thirty-two of whih are filled with eletron. This onfiguration is symmetri with reset to refletion with mirror lane at the entre of the air CD as easily seen in Fig.5.5. Therefore the nearest air CD an transfer to all the emty states and the number of the allowed transitions is nineteen er unit-onfiguration. Similarly the air G an transfer to seventeen emty states er unit onfiguration as shown by blue arrow airs as in Fig.5.5. A CD EF G I J KL MN OP QR ST UV WX YZ Fig.5.5 Allowed transitions of nearest eletron airs at ν 5 The unit-onfiguration inludes thirteen nearest eletron airs A, CD, EF, G, IJ, KL, MN, OP, QR, ST, UV, WX and YZ. The thirteen airs have the following erturbation energies: ς A ς EF ( 7 5)Z, ς CD ( 9 5)Z, ς G ς YZ ( 7 5)Z ς ς ( 9 5)Z, ς ς ( 5 5)Z, ς ς ( 5)Z IJ WX KL UV MN ST (5.6a) (5.6b) ς OP ς QR ( 5)Z (5.6) The sum of the erturbation energies is E nearest air ( ς ς ς ς ς ς ) A CD G IJ KL MN OP ς (( 9 8 6) 5) N Z 6 Enearest air N Z for ν 5 (5.6) 5 Next we onsider the ase of arbitrary integer s. The unit-onfiguration at ν ( s ) ( 6s ) is omosed of ( 6 s ) Landau states whih are artially ouied by ( s ) eletrons. Therefore the number of emty states is ( 6 s ) er unit-onfiguration. Aordingly we obtain N 95

96 E nearest air E E E (( s ) ( 6s ) ) Z (( s ) ( 6s ) ) Z (( 6s ) ( 6s ) ) (( s ) ( 6s ) ) Z (( s ) ( 6s ) ) Z ( s )( s ) ( s)( s) ZN ( 6s )( s ) nearest air nearest air nearst air 6s 6s 6s ( 6s )( s ) 6s ZN 6s 6.s.6.s. ZN 6s 6s 6.s. N Z Z for 6s 6s 6 6s ZN 6s 6 Z ( ( s ) ).s. ZN ZN 6s 6s 6 s ν (5.6) 6s We omare the erturbation energy of the nearest airs at ( s ) ( 6s ) N ν with that at ν 5 8 given in Table 5.7. Then we obtain the following inequality: ( Enearest air N ) ( ) ( ) > Z ( Enearest air N ) ν s 6s ν 5 8 In the limit of infinitely large integer s we get.s. ( Enearest air N ) ( s ) Z Z ν s 6s 6s 6 ( 6s ) for s (5.6) Z (5.6) That is to say, the energy setrum of the nearest eletron airs is ontinuous at ν 5 8. ε 5 ( ) ε ( ) ( ) lim ε λ λ ( 5 8 ) ε (5.65) The roerty aears at ν 5 8, 8, 7, as shown in Table 5.7 and 5.8 [-]. y examining Fig.5. in details we find that the seond nearest air C an transfer to all the emty states as shown by arrows. Therefore the transition number of the seond nearest airs dereases disontinuously when ν deviates slightly from 5/8. A similar effet is seen at ν 7 as exressed with arrows in Fig.5.. In setion 5.9 we will study an effet indued from the eletron airs laed in the seond neighbouring Landau orbitals. Therein small valleys aear at few filling fators. It is noteworthy that the small valley is not effetive at the filling fators with the eak struture beause the eak value is larger than the absolute value of the small valley energy. 5.6 The states with non-standard filling fators 96

97 In this setion we examine the states with the filling fators ν 7/, /, /, 5/, 5/7, 6/7 in Figs These frational numbers are named non-standard filling fators. It is diffiult to exlain the stability of these states by using the traditional theories. Jain has originally onsidered the multiflavor omosite fermion model in whih the omosite fermions arrying different numbers of flux quanta oexist. Also many theorists have roosed their extended models, for examle, Wos et al. [6, 7], Smet [8], Peterson and Jain [9] and Pashitskii []. Pashitskii redited new exoti frations at ν 5/, 5/6, and / based on the oneture roosed by alerin. Therein free eletrons and bound eletron airs oexist in the model. They used the different kinds of quasi-artiles for non-standard filling fators. Thus the theoretial results are model deendent. The resent theory an alulate the erturbation energy of the nearest airs for any FQ state. Let us start to alulate the energy for the non-standard filling fator ν 7/ as an examle. The most uniform eletron onfiguration with ν 7/ is shematially shown in Fig.5.6. The orbitals filled with eletron are illustrated by solid lines and the emty orbitals by dashed lines. A C D E F Fig.5.6 Coulomb transitions of nearest neighbor eletron airs at ν 7/ There are three airs of nearest eletrons in unit-onfiguration (filled, emty, filled, filled, emty, filled, filled, emty, filled, filled, emty). Therein CD air an transfer to all the emty orbitals. The airs like CD are shown by red solid lines in Fig.5.6. The erturbation energy of the nearest air CD is equal to ς ( )Z (5.66) CD The nearest eletron airs A and EF transfer to the two emty states er unit-onfiguration and have the erturbation energies ς and A ς as follows: EF ς A ς EF ( )Z (5.67) The sum of the energies for all the nearest airs is given by 97

98 8 ( E ) Z Z Z N ZN nearest air ν ( E N ) Z.896 Z nearest air ν for ν 7/ (5.68a) for ν 7/ (5.68b) We next onsider the neighbourhood of ν 7/. The filling fators are 7s s ν 7 s s whih aroah 7/ in an infinitely ν ( ) ( ) and ( ) ( ) large integer s. The most uniform eletron onfiguration at ( 7 s ± ) ( s ± ) ν is very omliated and so the erturbation energy of the nearest eletron airs is alulated by using a omuter rogram. The results of the alulation for s 76 ( E N ) Z.9777 Z nearest air ν ( E N ) Z Z nearest air ν We also alulate the ase of s the results of whih are 796 ( E N ) Z. 975 Z are for ν (5.69a) for ν 7 (5.69b) for ν nearest air ( E N ) Z. 97 Z for ν 7 nearest air 776 (5.7a) (5.7b) We omare the four values in Eqs.(5.69a,b) and (5.7a,b). Then we obtain the aroximate value for the limiting values from both sides as ν lim ( E N ) nearest air.97 Z ( 7 ) ± ε for ν ± 7 ± ε (5.7a) (The limiting value is robably equal to Z. The value may be larified in the evaluation for s although we annot alulate it due to very long CPU-time.) The energy of the nearest eletron airs er eletron is ( Z ) at ν 7/ as in Eq.(5.68). Therefore a small valley aears, the deth of whih ε is given by ± 8 77 ( ) ( E N ) lim ( E N ). Z ( ) ε ± 7 nearest air nearest air 69 ν 7 ν 7 ± ε (5.7b) Next we examine the ase of ν /. We illustrate the most uniform eletron onfiguration at ν / in Fig.5.7. For a quantum all devie with a marosoi size, we an ignore the effet of the boundaries namely both ends. Therein the eletron onfiguration in Fig.5.7 has left-right symmetry at the entre of the air CD. The nearest hole airs like CD are exressed by red dashed lines. All the hole airs with the 98

99 red dashed lines an transfer to all the eletron states. A C D E F Fig.5.7 Coulomb transitions of nearest neighbor hole airs at ν / The air A has no left-right symmetry and so there are some forbidden transitions to the eletron states. Also the dashed air EF annot transfer to some of the eletron states beause of the momentum onservation and the Fermi-Dira statistis. The erturbation energy of the nearest hole airs is obtained by ounting the number of allowed transitions as follows: ς CD ( ) Z, ς A ( ) Z, ς EF ( ) Z The sum of the energies for all the nearest airs is given by 8 ( Enearest air ) Z N ν 77 for ν / (5.7a) 8 ( E nearest air N ) Z ν 77 for ν / (5.7b) ν lim E N.97 Z ( nearest air ) ( ) ± ε for ν ± ± ε (5.7) Equations (5.7b) and (5.7) are re-exressed to the energy er eletron (not hole) as 8 7 ( E nearest air N ) Z Z. 888 Z ν 77 for ν (5.7a) lim E N.755 Z ( nearest air ) ( ) ± ν ε for ν ± ± ε (5.7b) The differene between Eqs.(5.7a) and (5.7b) gives the small energy ga (deth of the valley): ε ± ( ) ( E nearest air N ) lim ( Enearest air N ). 6 Z ( ) ν ν ± ε (5.7) These energy gas (deths of the valley) at ν 7/ and / are smaller than that at ν /. Similarly we alulate the energies of the nearest airs at ν 8/ and its neighbours as follows: 99

100 ( E N ) Z nearest air ν 8 58 ( E N ) Z nearest air ν ( E N ) Z nearest air ν Also the small valley aears at ν 5/. Table 5.9 shows the deths of non standard filling fators. Table 5.9 Energy ga of nearest airs er eletron at the filling fators 7/, /, 8/ and 5/ ν Enearest air N ε ( ν ) ε ( ν ) 7/ ( 8 77)Z. 69 Z / -(/) Z.6 Z 8/ ( )Z. 88 Z 5/ ( 65)Z.769 Z There is a ommon roerty in these ground states. We exlain it. The most uniform onfigurations at ν /, 5/, 5/7 and 6/7 are shematially drawn in Figs.5.8-, as follows; A C D E F Fig.5.8 Coulomb transitions of nearest neighbor hole airs at ν / A C D E F Fig.5.9 Coulomb transitions of nearest neighbor hole airs at ν 5/

101 A CD EF Fig.5. Coulomb transitions of nearest neighbor hole airs at ν 5/7 A CD EF G I J Fig.5. Coulomb transitions of nearest neighbor hole airs at ν 6/7 The nearest hole airs CD with red olour an transfer to all the eletron states at ν /, 5/ and 5/7 as shown in Figs Also the nearest red hole air EF in Fig.5. an transfer to all the eletron states. When the filling fator deviates slightly from the original value, the left-right symmetry with reset to the entre of CD (or EF for ν 6/7) is lost. The left-right symmetry ensures that the hole air an transfer to all the eletron states. So the broken symmetry means that the number of allowed transitions beomes disontinuously small. Thus the small valleys aear in the energy setrum at ν 7/, /, /, 5/, 5/7 and 6/7. The theoretial results are in good agreement with the exerimental data []. 5.7 Exitation-energy-ga in FQ states We have used the term energy ga (deth of the valley) for the ga in the energy setrum versus filling fator as in the revious setions. This ga (deth) rodues a lateau in the all resistane urve whih will be shown in Chater 7. There is another ga whih is named exitation-energy-ga. The exitation-energy-ga is defined as the differene between the ground state energy and the first exited state energy. The exited state has the same filling fator as that of the ground state. We investigate the relation between energy ga in the setrum and exitation-energy-ga. We first examine an exited state at ν as an examle. The ground state with ν has the eletron onfiguration as in Fig.. where all the eletrons are aired. When an eletron in the ground state is exited, the eletron moves to another emty

102 orbital. So, one eletron air is lost and one emty orbital is ouied by the eletron. A DCE F Fig.5. Eletron onfiguration in exited state # at ν / We illustrate an examle of the exited state. The eletron in the orbital is transferred to the orbital C by the exitation as in Fig.5.. Then the landau orbital beomes emty (bold dashed green line) and the Landau orbital C is filled with eletron (bold solid green line) as exressed in Fig.5.. We all this onfiguration exited state #. The exited state # has the filling fator ν / exatly same as that of the ground state. Therein the nearest air A disaears via this exitation and two nearest-eletron-airs DC and CE are newly reated. The nearest air DC an transfer to the two emty states as illustrated by blak arrow airs in Fig.5.. The other transitions from the air DC are forbidden as shown by ink olor. All the transitions from the air CE are forbidden beause of the momentum onservation and the Pauli exlusion rinile. In the ase of the revious setions the transition number is enormously large and so the number two is negligibly small. Aordingly the exitation energy is aused by disaearane of the eletron air A. We examine the details below. The seond order erturbation energy of the nearest air DC is desribed by the symbol ς DC, where the symbol rime (ς ) indiates the erturbation energy in the exited state #. The energy ς DC is given by ς DC,,,, D C I D C D C I D C 6( π ), 7( π ) W ( state# Wexite D D, C C ) We introdue the following summation Z as: all D C I D C D C I D C, π W ( state# Wexite D D, C C ) (5.75),,,, Z (5.76) It is noteworthy that Eq. (5.76) is obtained by relaing W G by W state# into Eq.(.6). The summation in (5.75) is arried out only for the two momenta. On the other hand, in

103 Eq. (5.76), the summation is arried out for a large number of momenta. The Landau wave funtion sreads in the width y along the y diretion, the value of whih was already alulated in (.9) of hater as y.5 nm m ω e for the ase of 6[T] (5.77) From Eq.(.), the distane between adaent orbitals is, π α [nm] for [mm] [nm] (5.78) e Then a large number of single eletron states exists inside the width y. The number of the states is y α π Aordingly about e effetive in the region as; (5.79) wave funtions overla and so the Coulomb transitions are π C C (5.8) The sum in Eq.(5.75) is done only for the two transitions and so the ratio ς Z is about : ( ) DC Z DC ς (5.8) eause this value is negligibly small, we obtain the following result. ς DC Z (5.8a) All the transitions from the air CE are forbidden and then the erturbation energy of the nearest air CE is zero: ς (5.8b) CE Furthermore the nearest air A in the ground state disaears in the exited state # as seen in Fig.5.. From these onsiderations the exitation-energy E exitation # is given by E exitation # ς DC ς CE ς A (5.8) where ς indiates the summation (.5), the result of whih is equal to (.7) as A ς A ( )Z (5.8) Substitution of Eqs.(5.8a, b) and (5.8) into Eq.(5.8) yields the exitation-energy-ga at ν as E ς ς ς ( )Z at ν (5.85) exitation # DC CE A

104 This exitation-energy-ga ( )Z is ositive beause Z is a ositive number. The seond examle is the filling fator ν / shown in Fig.5.. Therein the eletron A moves to the orbital from the ristine orbital. This exited state is named exited state #. The exited state # has the two nearest eletron airs C and D. A C D E F Fig.5. Eletron onfiguration in exited state # at ν / The exitation energy E exitation # from the ground state to the state # is given by E exitation # ς C ς D where ς indiates the erturbation energy in the exited state #. From the same argument as in Eq.(5.8), ς C and ς are extremely small. Aordingly we obtain D E ς ς at ν / (5.86) exitation # C D The third examle is the ase of ν /. When the hole at transfers to the orbital C, the orbital is filled with eletron and the orbital C beomes emty. This transition yields the exited state # as in Fig.5.. A DCE F Fig.5. Eletron onfiguration in exited state # at ν / The nearest hole airs DC and CE are reated and the two nearest-hole-airs disaear from the ground state via this exitation roess. Aordingly the exitation-energy is given by E exitation #

105 E ς ς ς ''' hole ''' hole hole exitation # DC CE ν where ς means the energy for the nearest hole air in the ground state. The value hole ν of ς is zero as shown in Table 5.. All the quantum transitions from the hole-airs hole ν DC and CE are forbidden as is illustrated by ink arrow airs in Fig.5.. Then the exitation-energy E exitation # is equal to zero as ''' hole ''' hole hole E ς ς at ν / (5.87) exitation # DC CE ς ν Thus the exitation-energies at ν / and / are nearly equal to zero. From the similar investigation mentioned above, the exitation-energy is almost equal to zero in ν and the ases of the flat struture and the eak struture namely at ( ) ν ( ) ( ). On the other hand, a large exitation-energy-ga aears for the ase of the valley ν ± and so on. The large struture at ν ( ), ν ( ) ( ), ( ) exitation-energy-ga onfines eletrons to the ground state and suresses the satterings of eletron by imurities, lattie defets and lattie vibrations. Aordingly the diagonal resistane R is exeted to be very small at these filling fators. The XX result of the resent theory is omared with the exerimental urve of R in the next XX setion. 5.8 Comarison between the theory and the exerimental data in a wider region of magneti field We have alulated the erturbation energies of the nearest eletron (or hole) airs at ± ν ±, the filling fators of ν ( ), ν ( ) ( ), ν ( ), ( ) ν ( ± ) ( ± ) ν 5,,,,,,,, ν ( ), ( ) ( ) ν and so on. Then we have obtained the three tyes of the energy setrum, namely valley, flat and eak strutures. The valley struture has been studied in setions 5., 5., 5.. The small valley struture has been examined in setion 5.6. The other strutures (eak and flat strutures) have been also investigated in setions 5., and 5.5. We omare the theoretial results with the exerimental data in Fig

106 . (Valley struture): At ν ( ), ( ) ( ), ( ±), and ( ±), the nearest eletron (or hole) airs an transfer to all of the emty (or filled) orbitals. When the filling fator ν deviates from ν, the reeating of the unit-sequene at ν is violated by the deviation ν ν. So the nearest eletron (or hole) airs annot transfer to some emty (or filled) orbitals. Thereby a valley aears at these filling fators. The energy gas (deths of the valleys) are obtained in Tables 5., Z ε Z 5., 5.5 and 5.6. The following deths ε ± ( ) ( ), ± ( ) ( ) and ( ) ( )Z ± 6 5 ε are the most dee three in our alulation. Then the eletron satterings are suressed by the large exitation energies. Aordingly we an exet that the value of R XX is nearly equal to zero at ν,, 5. Exerimentally is almost zero in a wide range at the three filling fators as R XX shown by blue arrows (The ν 5 state is out of range in Fig.5.5). The other filling fators of this tye are shown by green and ink arrows for ν ( ± ) and ( ±) R exhibits loal, resetively. The exerimental urve of XX minima at these filling fators. Thus the theoretial results are in good agreement with the exerimental data.. (Small valley struture): The red airs of eletrons (or holes) in Figs.5.6- an 6 transfer to all the emty (or filled) orbitals at ν 7 5 5,,,, 7,. owever the 7 brown airs annot transfer to some emty (or filled) orbitals. This roerty yields a small valley as alulated in Eqs. (5.7b), (5.7) and so on. The energy deth of the valley is very small;. 69 Z at ν 7 and.6 Z at ν and so on. The exerimental value of R has small loal-minima at XX these filling fators whih are exressed by brown arrows in Fig.5.5. Thus our theoretial results are in agreement with the exerimental data. for all the nearest eletron (or hole) airs annot transfer to any emty (or filled) orbital. Then all the. (Peak struture): At ν ( ) and ( ) ( ) nearest airs have zero binding energy whih is the highest value among the seond order erturbation energies. In the neighbourhood of ν the nearest air energies have finite negative values as in Tables 5. and 5.. Aordingly the state at ν has a eak struture. The effet of more distant airs annot anel the eak beause the eak value is large. So the exitation-energy-ga is almost zero at ν as in the revious setion 5.7, and then the eletron satterings are not suressed. Consequently it is onluded that the diagonal resistane R XX is finite at these filling fators. The roerty is in agreement with the exerimental data shown by red arrows in Fig (Flat struture): At ν the allowed transition number from the nearest 6

107 eletron (or hole) airs is equal to that in the neighbourhood of ν. Then the energy setrum of the nearest airs is ontinuous namely flat struture. Also we will examine the FQ states with ν 5, 7 in setion 5.9 and 5.. Therein the seifi asets aear as in Figs.5.6-, Fig.5.6 and Fig.5.9. Also the allowed transition number from the nearest eletron (or hole) airs is equal to that in the neighbourhood of ν 5 5 8, 8,7,. So the more distant airs are imortant to know the detail struture at ν 5. When we add the erturbation energy of the seond nearest airs at ν 5 5 8, 8,7,, the small valley struture aears in the theoretial setrum. These exerimental data are shown by blak arrows in Fig.5.5. R xx [kω] 6/7 5/7 / /7 /55/9 / / / 5/ /7 /9 /5 / / / / /8 6/ /5 7/ 5/ Magneti Field [T] Fig.5.5 Diagonal resistane versus magneti field in the region of < <8[T] Many tyes of the filling fators are drawn by the arrows with different olours. The exerimental data are measured by W. Pan,.L. Stormer, D.C. Tsui, L.N. Pfeiffer, K.W. aldwin, and K.W. West, in Ref. []. 7

108 Thus the resent theory an exlain the many filling fators with a loal minimum in the diagonal resistane on the basis of the fundamental amiltonian of the normal eletrons without any quasi artiles. On the other hand the traditional theories have emloyed many quasi-artiles with many different tyes. We will omare the resent theory with the traditional theories in hater. Further investigation will be done for ν > in the next setion 5.9 and for <ν < in the setion 5.. Setion 5. is devoted to examine the effet of the more distant airs. The valley struture is aused by the drasti number-dereasing of the allowed-transitions from the nearest airs when the filling fator varies slightly from the original value. This drasti derease is indeendent of the erturbation order. Therefore the disontinuous struture aears for all higher orders of the erturbation alulation whih will be examined in Chater 6. We will quantitatively omare the theoretial values of the valley deths with the exerimental data in Chater Pair energy of eletrons laed in the seond neighbouring Landau orbitals (Exlanation of the all lateaus in the region <ν < ) In the revious setions we have examined the air energies of eletrons (or holes) laed in the nearest neighbouring Landau orbitals. In this setion, we investigate the effet of quantum transitions from eletron airs laed in the seond neighbouring Landau orbitals. The air energy of eletrons (or holes) laed in the seond nearest orbitals yields a small valley at the seifi filling fators, whih is less than about. times that of the nearest airs Exerimental data for ν > In this subsetion we shortly study the exerimental results for ν >. The lateau at ν 5 attrats a great deal of attention beause of a new FQ harater. First, we review the exerimental findings at <ν <. In the exeriment by Pan, Du, Stormer, Tsui, Pfeiffer, aldwin, and West [, ], the diagonal resistane exhibits a dee minimum at ν 5 / and 7/ as in Fig On the other hand, at ν 9/ and / the diagonal resistane exhibits a strongly 8

109 anisotroi behaviour. Therein R XX has a shar eak while R YY is muh smaller than R XX and has a minimum at ν 9 /, / as is seen in the dashed urve of Fig.5.6 where R XX and R YY mean the diagonal resistanes along the x and y diretions, resetively. Fig.5.6 Diagonal resistane in wide region of magneti field [, ] The dashed urve indiates R. YY In the region <ν < the lateaus of all resistane have been found at ν 5 /, 7 / (whih have even number denominator) and at ν 7 /, 8/, /5, /5, 6/5, 9/5 (whih have odd number denominator) as seen in Fig.5.7. The exerimental data are obtained by Eisenstein, Cooer, Pfeiffer, and West []. Fig.5.7 ehavior of all resistane in the region of <ν < [] 9

110 The lateaus have the reise all resistane value. For examle, the lateau at ν 7 / has the all resistane value h /( 7e ) within the deviation of.5% as mentioned in []. Fig.5.8 all resistane urve in the region <ν < The left anel is quoted from referene [5] and the right anel from [6] We show other exerimental data in Fig.5.8. The all resistane-urve in the left anel [5] of Fig.5.8 is different from that in the right anel [6]. This differene means that the shae of the all resistane versus magneti field urve deends on the samles and the exerimental onditions (magneti field strength, temerature et.). Eseially the differene is large at ν 6 / 7, / 5 and /5. When the magneti field is tilted from the diretion erendiular to the D eletron system, the all resistane lateau at ν 5 / disaears as seen in Fig.5.9. On the other hand, the ν 7 /, 8 / lateaus ersist also under tilting of the magneti field in Fig.5.9.

111 Fig.5.9 Tilt deendene of the all resistane and diagonal resistane [7] Fig.5. Temerature-deendene of the diagonal resistane [8]

112 The diagonal resistane urve deends on temerature as in Fig.5. where the red urve indiates the data at 6 mk and the blak ones at 6 mk. Some loal minima in the diagonal resistane urve disaear at 6 mk. Many reserhers have measured the temerature deendene of R xx whih gives the energy gas from Arrhenius lots. The energy gas are shown in Fig.5.. Fig.5. Energy gas for the FQ states. Oen irles are quoted from referene [5]. Solid squares and irles are quoted from referene [9] In Fig.5. the energy gas exressed by oen rles are obtained by Dean, Piot, ayden, Sarma, Gervais, Pfeiffer, and West [5]. Also the data with the symbols of blue solid squares and red irles are mesured by Choi, Kang, Sarma, Pfeiffer, and West [9]. The energy gas in the high mobility samle are drawn by the blue squares in Fig.5.. The energy gas at ν 5, 8, 7 et. are listed in Table 5. whih is measured by the exeriment [9]. These values of the energy ga hange from samle to samle at the same filling fator as easily seen in Fig. 5. and Table 5.. Table 5.: Energy ga for the filling fators of /5, 9/7, 8/, 5/, 7/, 6/7 and /5 in [9] ν ν /5 ν 9/7 ν 8/ ν 5/ ν 7/ ν 6/7 ν /5 Samle A 5 mk 8 mk 56 mk 5 mk 58 mk 9 mk 6 mk Samle <6 mk 5 mk 7 mk 6 mk < mk

113 The energy gas for ν < are lotted in Fig.5. whih is quoted from the artile [5]. Therein the ν energy ga is about. K. On the other hand the energy ga of ν 5 state is about.7 ~. 5 K as measured in the referenes [9]. Thus the energy ga in the region <ν < is about / times that in ν <. So we need to investigate the more detail struture in order to disuss the FQ states with ν >. Fig.5. Energy gas at ν, 5, 7, 5 9, 6, 5, 9, 7, 5 in [5] 5.9. Theoretial aroahes other than our theory for ν > The exerimental findings at ν > have stimulated many hysiists and then theoretial studies have been arried out. The theories assumed their own models to exlain the lateaus of the all resistane at ν > eseially at ν 5. Some of them are briefly reviewed below. (The strie states): Fogler, Koulakov and Shklovskii have studied the ground state of a artially filled uer Landau level in a weak magneti field. They have used the effetive interation [5] whih was derived by Aleiner and Glazman in the D-eletron system with high Landau

114 levels, taking into aount the sreening effet by the lower fully ouied levels. Then they have found that the ground state is a harge density wave (CDW) state with a large eriod [5]. Moessner and Chalker studied a D-eletron system with a fermion hardore interation and without disorder. They found a transition to both unidiretional and triangular harge-density wave states at finite temeratures [5]. Rezayi, aldane and Yang numerially studied a D-eletron system in magneti field with a high Landau level half filled by eletrons. In finite size systems with u to eletrons and torus geometry, they found a harge density wave ordering in the ground state. Their results show that the highest weight single Slater determinant has the ouation attern where and stand, resetively, for an ouied and an emty orbitals [5]. (The R and MR states): aldane and Resayi investigated the air state with sin-singlet [55]. They used a hollow ore amiltonian. In the Landau level number L, the hollow ore amiltonian has the first seudootential V > although the zeroth aldane seudootential V is zero. They found a ground state alled R state. Moore and Read are insired by the struture of the R state, and onstruted the air state, a -wave ( x i y ) olarized state. They have desribed the FQ state in terms of onformal-field-theory [56]. The state is alled the Moore-Read state (MR state). In the referene [57], Read wrote the wavefuntion ψ reresents CS [58] airing of omosite fermions. One tye MR are the harged vorties disussed above, with harge ( q) whih aording to MR are suosed to obey nonabelian statistis. Greiter, Wen, and Wilzek investigated the MR state from the viewoint of the omosite fermion air [59]. The statistis is an ordinary abelian frational statistis. (Numerial works) There are many numerial works. We will shortly review several works below: Morf argued the quantum all states at ν 5 by a numerial diagonalization [6]. e studied sin olarized and unolarized states with N 8 eletrons. is result indiates that the 5/ state is exeted to be the sin-olarized MR state. Razayi and aldane [6] onfirmed Morf s results. Their results are based on numerial studies for u to 6 eletrons in two geometries: shere and torus. They found a first order hase transition from a stried hase to a strongly-aired state. They examined eletrons in a retangular unit ell with the aset ratio.5. They found the strie state, the robability weight of whih is 58% for the single Slater determinant state with the

115 ouation attern. Also they found an evidene that the ν 5 state is derived from a aired state whih is losely related to the MR olarized state or, more reisely, to the state obtained artile-hole (P) symmetrisation of the MR state [6]. Thus there are many investigations for the ν 5 state. owever the resent author has some questions: () frational harge, () small number of eletrons in numerial alulations and so on. These roblems are disussed in Chater Exlanation of 5/ Plateau In the revious setions and haters we have ignored the disontinuous struture of the seond-nearest and more distant airs, beause the air-energies are exeted to be smaller than those of the airs laed in nearest neighboring Landau orbitals. The FQ state at ν > has an energy-ga smaller than that at ν as shown in the subsetion So we need to take aount of the ontribution from the eletron airs laed in the seond nearest and more distant Landau orbitals [6] in order to investigate FQ states with ν >. The energy differene between the Landau levels has been alulated in Eq.(.). The value Γ Landau k is about 8.[K] at [T] for GaAs whih is very large in omarison with the gas shown in Fig.5. and Table 5.. Aordingly the mixing of higher levels ( L ) in the ground state is negligibly small in the region <ν <. Then all the Landau states with L are filled with eletrons of u and down sins, and the Landau states with L are artially ouied by eletrons. The roerty of FQ states with <ν < is very different from that with ν < as: ) The robability density of the Landau wave funtion is zero at the entre osition for L but that is maximum for L. Thus the shae of the L Landau wave funtion is quite different from that of L. So the interations between eletrons in L are quite different from that in L. ) The ν deendene of the lassial Coulomb energy at ν > is very small as will be studied at the subsetion The strength of ν deendene at ν 5 is. times that at ν as will be estimated in Eqs.(5.8), (5.9a) and (5.9b). So the ontribution from the seond nearest airs at ν > is more effetive than that at <ν <. ) Furthermore the interations between eletrons in L are shielded by the many eletrons in the lowest Landau level. 5

116 Thus the wave funtion and the interation in a higher Landau level L are different from those in the lowest Landau level L. On the other hand, the momentum onservation law and the Fermi-Dira statistis of eletrons are satisfied at ν > same as at ν. So we an use the same logi for the Coulomb transitions in the region ν >. Now we examine the ase of ν 5. The most uniform eletron onfiguration at ν 5 ( ) is drawn in Fig.5. where only the orbitals with L are shown. All the orbitals with L are filled with eletrons and these orbitals are not drawn in Fig.5. for simliity. The eletron air CD in the Landau level L an transfer to all the emty states in L whih are illustrated by the solid arrows in Fig.5.. These transitions satisfy the momentum onservation law. Any eletron air laed in the seond neighboring orbitals an also transfer to all the emty orbitals at ν 5. y C D x Fig.5. Most uniform onfiguration at ν 5 Allowed transitions are shown by arrows. Dashed lines indiate emty orbitals in the seond Landau level L We alulate the erturbation energy for the eletron air laed in the seond neighboring Landau orbitals. Therein we introdue the following integral S for the Landau level L. S L,, L,, L,, C D I C D C D I, π W ( G Wexite C C, D D ) L, D C π C C, D D (5.88) where the momentum-hanges, and π, have been eliminated beause the diagonal matrix element of is zero. It should be noted here that the summation I S is ositive beause the denominator in Eq.(5.88) is negative. Then the erturbation energy ς CD of the air CD is exressed by using S as ς CD ( )S (5.89) C, D 6

117 Therein the fator / omes from the fat that the number of allowed transitions is equal to the number of the emty orbitals whih is half of the total Landau orbitals with L. Let us ount the number of eletron airs like CD. We define the total number of eletrons laed in L by the symbol N L. Then the total number of the airs like CD is equal to N L. Aordingly the total energy of these airs is given by ( ) SN air E ν 5 L (5.9) The summation S deends on the thikness, size, shae and material in the quasi-d eletron system. The z-omonent of the eletron wave funtion deends on the thikness and the x-omonent deends on the devie size and shae. Also the effetive mass of eletron and the ermittivity deend on the material of the devie. Aordingly the lassial Coulomb energy W and the transition matrix element in Eq.(5.88) vary with hanging the quantum all devie. That is to say the value of S varies from samle to samle. Furthermore the L eletrons yield the sreening effets for the lassial Coulomb energy. Thus there are many unknown effets in a detail alulation of S. Aordingly the value of S is treated as a arameter. In order to larify the stability of the ν 5 state, we examine the ν 78 state whih is lose to 5/ as ν (5.9) This filling fator is different from ν 5 by about.6%. The most uniform eletron onfiguration at ν 78 6 is illustrated in Fig.5. where the Landau orbitals with L are not shown, for simliity. The eletron air CD an transfer to any emty orbital as shown by green arrows. On the other hand the air IJ an transfer to three sites er unit onfiguration as shown by blue arrows in Fig.5.. Therefore the seond order energy is given by ς CD ( 5 )S for ν 78 (5.9) ς ( )S for ν 78 (5.9) IJ Similarly ς DE ( )S, ς EF ( )S, ς FG ( 9 )S, ς G ( 7 )S, ς ( 5 )S, ς JA ( )S, for ν 78 (5.9) I A C D E F G I J A A unit onfiguration Fig.5. Quantum transitions at ν 78 state 7

118 The eletron airs, A, A, A, et. are laed in the nearest Landau orbitals as seen in Fig.5.. In order to alulate the energy of these airs, we introdue the following summation: T L,, L,, L,, A I A A I, π W ( G Wexite A A, ) L, A, (5.95) where π (5.96) A This summation T is also treated as a arameter like S. The nearest air A an transfer to all the emty states as shown by the red arrows in Fig.5. and the number of the emty states is 5 er unit onfiguration. The air energy of A is given by ς A ( 5 )T, for ν 78 (5.97) We alulate the total energy from all the eletron airs laed in the first and seond neighboring Landau orbitals with L : E ( ζ ζ ζ ζ ζ ζ ζ ζ ζ )( N 6) air ν 78 A CD DE EF FG G I IJ JA L (( ) T ( ) S ) ( N 6) air E ν 78 5 L (5.98) If air E 78 E ν is equal to air ν 5, then the ν 78 state is mixing into the ν 5 state with equal robability aused by the thermal transitions. In this ase the all resistane value deviates from ( 5e ) h by about.% whih is half of the differene between ν 78 and 5/. So we examine the magnitude of S and T. The differene of the air energies is derived from Eqs.(5.9) and (5.98) as ( T 9 )( 5 96) air L air L Eν 5 Nν 5 Eν 78 Nν 78 S (5.99) We disuss the following two ases and. (Case ) 9 S >> T In this ase the erturbation energy of the first and seond airs er eletron is air L air L Eν 5 Nν 5 << Eν 78 Nν 78 for S >> T 9 (5.) Aordingly the air energy at ν 5 is suffiiently lower than that in the 8

119 neighborhood of ν 5. When the energy differene air L air L ( E N E N )>> oltzmann onstant Temerature ν 5 ν 5 ν 78 ν 78 then the ν 5 state is onfined and the all lateau aears at ν 5. (Case ) 9 S << T In this ase, the ν 5 all lateau does not aear beause the air energy at ν 5 is higher than that in its neighborhood. Thus the FQ state is sensitive to the relative value of S and T whih are deendent on the materials, thikness, devie shae, and so on. In the exeriments [-9] and [6-65], we also see that the ν 5 all lateau is the samle deendent henomena. So these henomena an be understood by the resent theory on the basis of the normal amiltonian without any quasi artile Aearane or disaearane of the lateau at ν 5 and For examle, the ν 5 and 7/ all lateaus do not exist on the red urve of all ondutane in the artile [6] as seen in Fig.5.5. This result has been obtained by Dean, Young, Cadden-Zimansky, Wang, Ren, Watanabe, Taniguhi, Kim, one & Sheard. On the other hand, the exerimental results in Figs.5.6- indiate the aearane of 5/ and 7/ all lateaus. Thus aearane or disaearane of the ν 5 and 7/ lateaus seems to deend uon the samles used in the exeriments. Fig.5.5 Plateaus of all ondutane and loal minima of diagonal resistane in the exerimental results of the referene [6] 9

120 Similar henomena have been found at ν in grahene samles as follows: Figure 5.6 shows the aearane of the all lateau at ν. Fig.5.6 was obtained by Dong-Keun Ki, Vladimir I. Fal ko & Alberto F. Morurgo. Also, Fig.5.7 was obtained by Kirill I. olotin, Fereshte Ghahari, Mihael D. Shulman, orst L. Stormer & Phili Kim where the ν all lateau aears at the oint of Fig.5.7. Fig.5.6 Plateaus of all resistane and loal minima of diagonal resistane in the exerimental results of the referene [6] Fig.5.7 Plateaus of all ondutane in the exerimental results of the referene [65] On the other hand, the lateau of the all resistane disaears at ν as already examined in Figs.5.9 and 5.5. Also the disaearane at ν an be seen in Fig.5.8 [6]. Thus different samles show different behaviours for the all lateau at the filling fator with an even number denominator.

121 Fig.5.8 Plateaus of all ondutane and loal minima of diagonal resistane in the exerimental results of the referene [6] The differene has been also found in the GaAs samles. When the quantum all devie has a wide quantum well, a ν all lateau aears as shown in the exeriments [66-68]. For examle Shabani, Yang Liu, Shayegan, Pfeiffer, West, and aldwin [67, 68] have observed the lateau of the all resistane at ν as seen on the uer urve in Fig.5.9. Fig.5.9 Plateaus of all resistane and loal minima of diagonal resistane in the exerimental results of the referene [67, 68]

122 Thus the all lateau at ν 5 and ν deends on the samles used in the exeriments. In the resent theory this roerty is aused by the values of S and T whih are also deendent uon the samles. Next we will alulate the air energies for the filling fators.5 <ν < and <ν <. 5 in the subsetions and 5.9.6, resetively [6]. Similar alulation for ν > an be erformed by using the method of this setion. The air energy setrum takes loally lowest at the filling fator ν 7/, /, /, 6/5 and so on. These theoretial results are in agreement with the exerimental data FQ states at filling fators.5 <ν < In the region of.5 <ν <, the lateaus of all resistane are observed at the frational filling fators ν 8/, /5, 9/7 and so on. We examine the states in this subsetion. Figure 5. shows the most uniform onfiguration at ν 8 ( ) where two tyes of eletron airs, A and CD exist (where Landau orbitals with L are not drawn for simliity). The air A reresents the first nearest eletron air and the air CD the seond nearest one. oth the airs A and CD an transfer to all the emty orbitals with L. The allowed transitions are shown by arrows in Fig.5.. A Fig.5. Quantum transitions at ν 8 state C D The number of emty orbitals with L is / of the Landau orbitals with L. Therefore the air energies are given by ς A ( )T for ν 8 (5.a) ς CD ( )S for ν 8 (5.b) Aordingly the total energy of the eletron airs laed in the first and seond

123 neighboring Landau orbitals with L is E ( N ) ς ( N ) air ν 8 ς A L CD L Substitution of Eqs.(5.a and b) yields ( T 6 S ) N E ν (5.) air 8 6 L We examine the air energy in the limit from the right and left to ν 8. Suh a limiting value has already been alulated in setion 5.. Using the same method we obtain the limiting values from the right and left side to ν 8 as ν ν E ( 8 ) ν ε ( T S ) N air lim E ( 8 ) ν ε L ( T S ) N air lim L (5.a) (5.b) Therefore an energy valley aears as follows: air air air Eν 8 Eν 8 lim Eν ( T S ) N L (5.) ν ( 8 ) ± ε Thus the FQ state has a dee valley at ν 8. The ν 5,8 7,9 7 states have the most uniform onfiguration as shown in Figs.5., 5. and 5., resetively. The allowed transitions are shematially drawn by arrows for the first and seond nearest eletron airs, resetively. A Fig.5. Quantum transitions at ν 5 state C D The air energy of A and CD is given, resetively by ς A ( 5)T for ν 5 (5.5a) ς CD ( 5)S for ν 5 (5.5b) Then we obtain the total air energy from the eletron airs laed in the first and seond neighboring orbitals with L as ( T S ) N E ν (5.6) air 5 L

124 Figure 5. shows the allowed transitions of the airs A and CD at the filling fator ν 8 7. A C Fig.5. Quantum transitions at ν 8 7 state D The number of the emty orbitals is /7 times that of the Landau orbitals with L. Aordingly the air energy of A and CD is given, resetively ς A ( 7)T for ν 8 7 (5.7a) ς ( 7)S for ν 8 7 (5.7b) CD Then we obtain ( T 8 S ) N E ν (5.8) air L Next we ount the number of allowed transitions of the airs A and CD at ν 9 7. The eletron airs A and CD in Fig.5. an transfer to all the emty Landau orbitals with L. A C D Fig.5. Quantum transitions at ν 9 7 state Sine the number of the allowed transitions is two er unit onfiguration, the air energy of A and CD is given, resetively, ς A ( 7)T for ν 9 7 (5.9a) ς ( 7)S for ν 9 7 (5.9b) CD whih gives ( T 5 S ) N E ν (5.) air L

125 Thus the eletron airs A and CD an transfer to all the emty orbitals at ν 5/, 8/, /5, 8/7, 9/7, and therefore the air energy beomes very low, resulting in a large binding-energy. Next we examine the values of S and T whih vary from samle to samle. Our alulations mentioned above have treated the ase for no imurity-effet and absolutely zero temerature. So we should omare the theoretial results with the exerimental data in a high mobility samle under an ultra low temerature as in Fig.5. (see referene [9]). Eqs.(5.9) and (5.) give the theoretial air energy er eletron: E ( ) air ν 5 N L S (5.a) E ( T 6 6) air ν 8 N L S (5.b) The exerimental value of the energy ga at ν 5/ is nearly equal to that at ν 8/ for the high mobility samle in the exeriment [58] as found in Fig.5. and Table 5.. This exerimental roerty is derived from the following ondition between S and T : At ν (ondition): T S (5.) 5/, 8/, /5 and 9/7 the theoretial ratio of the air energies is obtained by emloying Eqs. (5.9), (5.), (5.6) and (5.) as follows: E N air air air : E 8 N : E 5 N : E 9 7 N air ν 5 L ν L ν L ν ( S ) : ( T 6 S 6) : ( T S ) : ( T 5 S 5) L (5.) When the ondition (5.) is satisfied, the theoretial ratio of the air energies beomes ( ): ( T 6 S 6) : ( T S ) : ( T 5 S 5) :: ( /) : ( 5) S (5.) The exerimental ratios are obtained from the exerimental data of the energy ga for the high mobility samle in Fig.5. and Table :.5 :. :..9 : :.6 :. (5.5) y omaring these ratios the theoretial ratio (5.) is almost equal to the exerimental ratio (5.5). Thus the resent theory exlains reasonably well the exerimental data for.5 ν < FQ states at filling fators <ν <. 5 5

126 A ν 7/ C A D ν /5 A C D ν 7/7 A C D ν 6/7 Fig.5. Quantum transitions at ν 7/, /5, 7/7, 6/7 states Dashed lines indiate emty Landau orbitals with L and solid lines are orbitals filled with eletron. C D The most uniform onfigurations at ν 7/, /5, 7/7 and 6/7 are shematially drawn in Fig.5.. The hole-airs A and CD an transfer to all the eletron states in L as easily seen in Fig.5. (where the Landau orbitals with L are not drawn for simliity). Figure 5.5 shows one to one orresondene of the allowed transitions between ν 8/ and 7/. The one to one orresondene yields the symmetry of the allowed transition numbers. That is to say the number of allowed transitions of the hole-airs at ν 7 is equal to that of the eletron-airs at ν 8. In subsetion we have alulated the air energies at ν 8/, /5, 8/7 and 9/7 in Eqs. (5.-). So we an obtain the air energies at ν 7/, /5, 7/7 and 6/7 by using the symmetry between hole and eletron. The absolute values of the air energies is large and therefore these states beome stable. ν 8/ A ν 7/ C D Fig.5.5 Comarison of allowed transitions between at ν 8/ and at 7/ 6

127 Thus the resent theory gives the fat that the state with ν 5/, 7/, 8/, /5, /5, 6/7, 7/7, 8/7 and 9/7 is stable in the region <ν <. The orresonding exerimental data [5] shows the lateaus of the all resistane at ν 5/, 7/, 8/, /5, /5, and 6/7 as in the left anel of Fig.5.8. Thereby the theoretial results are in agreement with the exerimental data Comarison of lassial Coulomb energies between <ν < and <ν < Equation (5.) gives the energy er eletron for FQ states where χ ( ν ) is the air energy between eletrons laed in the nearest neighbouring Landau orbitals and g ( ν ) is the air energy between eletrons in the seond-nearest and more distant Landau orbitals. We write again the energy er eletron in the quasi D eletron system: ε ( ν ) χ( ν ) g ( ν ) ( e ( m ) a ν b (5.6) where a ξ η (5.7a) ( ) ( C N ) b f (5.7b) Marosoi The energy inrease ε ( ν ) by a small inrease of ν is given by ε ν χ ν g ν a ν (5.8) ( ) ( ) ( ) ( ) ν where the last term has the following oeffiients at ν. 5 and.5: ( a ν ). 6 a at ν. 5 (5.9a) ( a ν ) a at ν. 5 (5.9b) Therefore the ontribution of the last term at ν. 5 is. times that at ν. 5 for the same variation ν χ ν g ν are more effetive at <. Aordingly the remaining terms ( ) and ( ) <ν than at <ν <. So we have examined the energy valley of g( ν ) in this setion 5.9 and sueed to exlain the lateaus of the all resistane in the region <ν <. 5. Further investigation for the total energy of D eletron system (Energy of eletron air laed in more distant Landau orbitals) We examine the struture of the exat total energy in more details [6]. 7

128 The total energy E of the quasi-d eletron system is the sum of the eigen-energy T air W and the air-energy E via the interation I as follows: air E T W E (5.) where W is the eigen-energy of D given by Eq.(.) as N i ( ) C( L, L ;, ) W E, Li i N, Substitution of Eq.(.8) yields the following equation: W C N ( L,, L ;, ) N U ( ( e) ) ( e m )( L ) N, N i i N N i i (5.) λ (5.) The Landau energies between different levels are estimated for GaAs in Eq.(.): ( E ) k ( E E ) k 8.[K] E for T (5.) whih is extremely large in omarison with the exerimental values of the energy gas. (The values are smaller than [K] at <ν < as in Table 5..) In order to make our disussions lear, we onsider the FQ states with <ν < as an examle. In this ase the robability of higher Landau levels L an be ignored in the exat ground state. Therein all the Landau states with L are ouied by eletrons with u and down sins and the Landau states with L are artially ouied by eletrons. The number of eletrons in the Landau level L is exressed by the symbol L N ν and the number of Landau orbitals by orbital N. The ratio N L ν orbital N is given for <ν < as follows: N L orbital ν N in the ground state at <ν < (5.a) orbital N L N ν in the ground state at <ν < (5.b) ν N L orbital ν N in the ground state at <ν < (5.) N L > orbital ν N in the ground state at <ν < (5.d) orbital where N deends on the magneti field strength and the devie size but is indeendent of L. The total number N of eletrons is L L N Nν Nν in the ground state at <ν < (5.5) 8

129 Substitution of Eqs.(5.a,b,,d) and (5.5) into Eq.(5.) gives the eigen-energy W of D as W N C N, N i ( L,, L ;, ) Nλ U ( e) L L ( ) ( e m ) N ( e m ) N i in the ground state at <ν < (5.6) ν ν Next we investigate the air energy whih is aused by the quantum transitions via. The eletron airs in the ground state with <ν < have been lassified into the I following three tyes: (First tye) oth eletrons in the air are laed in the orbitals with L only. (Seond tye) One eletron is laed in L and the other in L. (Third tye) oth eletrons in the air are laed in L only. These air energies are desribed by the symbols E air L, resetively. The total energy of all the eletron airs is air E L and and air E L, air air air air EL EL and EL E in the ground state with <ν < (5.7) Therein the air energies E air L and air E L and are negligibly small beause of the following reason: As in Eqs.(5.88) an d (5.95) the erturbation energy is omosed of the terms with the denominator W W G exite. The denominator is very large in the first and seond tyes beause the quantum transition inludes the exitation from L to L. (Note: Wexite WG inludes the Landau exitation energy e m.) On the other hand the denominator is very small in the third tye beause the transition arises between the eletron airs with L only. (Note: Wexite WG doesn t inlude the Landau exitation energy e m.) Therefore we may ignore the air energy belonging to the first and seond tyes. air E and E in the ground state with ν > (5.8) air L L and In the third tye the eletron air in L transfers to emty orbitals in L at <ν <. Aordingly the energy differene of W between the ground and the intermediate state is derived from the differene of the lassial Coulomb energies. So 9

130 the differene (denominator) is very small and the erturbation energy (whih is negative) beomes very low. We onsider any eletron (or hole) air laed in Landau orbitals with L. As an examle, we examine the ase of ν 8. Figure 5.6 shematially shows the eletron airs at ν 8. The eletron airs IL, M and GN osses the total momentum same as that of the air JK. So these airs an transfer to all the emty orbitals as easily seen in Fig.5.6. A C D E F G I J K L M N O P Q R S T JK air IL air M air GN air Fig.5.6 Various eletron airs with the same total momentum at ν 8 Dashed lines indiate emty orbitals and solid lines indiate filled orbitals in the Landau level L. Allowed transitions from the eletrons J and K are shown by blak arrow airs, from IL by blue, from M by brown and from GN by green. The momenta of the eletrons at G,, I, J, K, L, M and N are desribed by G,, I, J, K, L, M and N, resetively. Then the total momenta of the eletron airs take the same value due to Eq.(.) as (5.9) total G N M I The energies of the airs GN, M, IL and JK are exressed systematially by using a L symbol (, ) ν total L ς where total and indiate the total momentum and the distane between the air. (, ) ς (5.a) L JK ς ν 8 total (, 5) L IL ς ν 8 total ς (5.b) (, 7) L M ς ν 8 total ς (5.) (, ) L GN ς ν 8 total ς (5.d) Therein the momentum of eah eletron is given as π, π (5.a) ( ) ( ) J total K total J K

131 I ( total 5 π ), L ( total 5 π ) ( total 7 π ), M ( total 7 π ) G ( total π ), N ( total π ) Thus any momentum-air ( V, W ) is related to total V ( total π ), W ( total π ) ( ) ( π ) (5.b) (5.) (5.d) and as (5.a) total (5.b) V W, W V eause both momenta V and W should be equal to ( π ) integer, the values of and are lassified to the following two ases: total Case: total ( π ) ( odd integer) for ( odd integer) Case: ( π ) ( even integer) for ( even integer) total (5.a) (5.b) We have already shown the ase of odd integer in Fig.5.6. Next we examine the ase of even integer. Figure 5.7 shows quantum transitions with even integers given by Eq.(5.b). A C D E F G I J K L M N O P Q R S T KL air JM air IN air O air Fig.5.7 Various eletron airs with the same total momentum at ν 8 Dashed lines indiate emty orbitals and solid lines indiate filled orbitals in the Landau level L. Allowed transitions from the eletrons K and L are shown by blak arrow airs, from JM by blue, from IN by brown and from O by dark green. The eletron airs KL, JM, IN and O indiate the ases of,, 8 and resetively. All the eletron airs an transfer to all the emty orbitals as in Fig.5.7. The air energies are desribed as (, ) ς (5.a) L KL ς ν 8 total (, ) ς (5.b) L JM ς ν 8 total (, 8) ς (5.) L IN ς ν 8 total

132 (, ) L O ς ν 8 total ς (5.d) The total energy of all the eletron airs is desribed by the symbol Eq.(5.). Use of Eqs.(5.7) and (5.8) gives air air L ( ν ) air E defined by E E in the ground state with <ν < (5.5) air The energy ( ν ) E E is the sum of all the air energies as: L L ( ) (, ) air L ς ν total, ν in the ground state with <ν < (5.6) total where the sum is urried out for all the values total and. Equations (5.), (5.5) and (5.6) yield the total energy of the quasi-d eletron system: E T W L ν, total (, ) ς in the ground state with <ν < (5.7) total Substitution of Eq.(5.6) into Eq.(5.7) gives E T C Nλ N i U L L L ( i ( e) ) ( e m ) Nν ( e m ) Nν ςν ( total, ) total, in the ground state with <ν < (5.8) L We exress the air energy er eletron by the symbol ( ) L L ( ) (, ) N L ν ς ν total total ν ξ whih is defined by ξ in the ground state with <ν < (5.9) The exat air energy is the sum of all order terms in the erturbation alulation as L ν L where ( ; n) L ( ) ξ ( ; n) ξ (5.) ν ν n,,, ξ indiates the n-th order of the erturbation energy. Substitution of Eqs.(5.9) and (5.) into Eq.(5.8) yields E T W E air C ( L,, L ;,, ) Nλ U ( ( e) ) N N L L L L ( e m ) N ( ) ν e m Nν Nν ξν ( ; n) ν N i,,, n,,, in the ground state with <ν < (5.) Therein the funtion form of W is ontinuous with the hange in ν. On the other hand the air energy E air has a disontinuous form for ν at the seifi filling fators ν ν, beause the number of the allowed transitions deends disontinuously i

133 uon ν at ν ν. This disontinuous roerty rodues the lateaus of the all resistane at the seifi filling fators. We have already alulated the seond order erturbation energies for and. We list the results in Tables 5. and 5.. Table 5. Seond order of the erturbation energy er eletron for the eletron airs laed in the seond neighboring Landau orbitals ν 5/ 8/9 78/ 8/ /5 8/7 9/7 ξ L ν ( ;) ( )S ( 9)S ( 96)S S 6 S S 8 S 5 Table 5. Seond order of the erturbation energy er eletron for the eletron airs laed in the nearest Landau orbitals ν 5/ 8/9 78/ 8/ /5 8/7 9/7 ( ; ) L ξ ν ( 9 9)T ( 5 96)T T 6 T T 8 T 5 L Next we study the energies ( ) ξ of the eletron-airs laed in more distant ν 5 neighbouring Landau orbitals with. Figure 5.8 shows the most uniform onfiguration at ν 5. The entre osition between the nearest air A is equal to that of the eletron air A n n for n,,,. Then the total momentum of the air A n n is equal to that of the air A. Therefore the eletron air A n n ( n,,,, ) an transfer to all the emty states as shown by the arrow airs in Fig.5.8. A air A air A air A air A 8 A 7 A 6 A 5 A A A A Fig.5.8 Various eletron airs with the same total momentum at ν 5 Dashed lines indiate emty orbitals and solid lines indiate filled orbitals in the Landau level L. Allowed transitions from the eletrons A and are shown by blak arrow airs, from A by blue, from A by brown and from A by dark green.

134 Also the total momentum of the eletron air C D in Fig.5.9 is equal to that of the airs C n D n (n,,,...) and therefore the air C n D n an transfer to all the emty states exet the orbital Y. This only one forbidden transition may be ignored in omarison with the enormously many allowed transitions. C D air C 8 C 7 C 6 C 5 C C C C Y D D D D D 5 D 6 D 7 D 8 C D air C D air C D air Fig.5.9 Various eletron airs with the same total momentum at ν 5 Dashed lines indiate emty orbitals and solid lines indiate filled orbitals in the Landau level L. Allowed transitions from the eletrons C and D are shown by blak arrow airs, from C D by blue, from C D by brown and from C D by dark green. Thus the further neighbouring eletron (or hole) airs with an transfer to all the emty (or filled) orbitals at ν 8/, /5, 7/ and /5 as shown in Figs The energies of these airs with are negative in the seond order erturbation. Therefore the energies are aumulated to give a stronger binding energy and so the states beome more stable. 5. all lateaus in the region <ν < When the diretion of the magneti field is uward, the Zeeman energy of down-sin is lower than that of u-sin for eletrons. So, in the region <ν < all the Landau orbitals in the lowest level are ouied by the eletrons with down-sin under a strong magneti field. On the other hand the Landau orbitals are artially ouied by u-sins. (Note: We will examine the ase of a weak magneti field in Chater 9. The FQ state with any filling fator ν is realized in a weak magneti field by adusting the gate voltage. In the ase both down- and u-sin-eletrons artially ouy the lowest Landau orbitals.)

135 The u-sin eletron air laed in the nearest orbitals an transfer to all the emty orbitals of u-sin at ν ( ) ( ), ( ) ( ) ν and so on. When the filling fator ν deviates from ν, the number of allowed transitions derease abrutly in omarison with that at ν. This mehanism yields the energy gas at and so on. ν ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ) The energy gas rodue the frational quantum all effet in the region <ν < as examined in Ref.[69]. (Note: We have studied the energy gas for the seifi filling fators in the regions ν < and ν > in the revious setions.) We introdue the total number M of the L orbitals and also exress the number of eletrons with down-sin and u-sin by N and N resetively. Then we obtain the following relations for <ν < as N M (5.a) N < M (5.b) N N N (5.) ( N N ) M ν (5.d) 5.. Eletron onfigurations and energy gas for <ν < We examine the FQ states with <ν < in this sub-setion [69]. As an examle we onsider the filling fator ν 5. Equation (5.d) beomes ν N M 5 N M for ν 5 (5.) The most uniform onfiguration of u-sin eletrons is the reeat of (filled, emty, filled). Figure 5.5 shows the eletron onfiguration in a D view. Therein the tilted lines with the x-diretion exress the Landau orbitals of the lowest level shematially. All the orbitals are filled with down-sin eletrons under a strong magneti field beause of the Zeeman energy. The emty orbitals of u-sin are drawn by dashed blue lines in Fig.5.5. The u-sin eletrons ouy the red-oloured orbitals. This eletron onfiguration of u-sin has the minimum value of the lassial Coulomb energy. 5

136 X Z Y A Fig.5.5 Eletron onfiguration at ν 5 / We examine the quantum transitions via the Coulomb interation I. All the Coulomb transitions satisfy the momentum onservation along the x-axis. Figure 5.5 shows the quantum transitions from the eletron air A with u-sin. The momenta at A and are exressed by A and resetively. The eletron air A transfers to the orbitals with A and after the transition. Then the momentum onservation gives the following relations: A A (5.a) (5.b) where is the momentum transfer. All the allowed transitions from A are illustrated by the blue allow-airs in Fig.5.5. Aordingly the transfer momentum takes the following value: ( π ) ( n ) n, ±, ±, ±, ±, (5.5) We introdue the following summation Z in order to obtain the erturbation energy of the eletron air A. L,, L,, L,, I L, A, Z (5.6a) A I A A, π W G Wexite A A (, ) A π (5.6b) A A, (5.6) Therein the summation is arried out for all the values ( π ) integer exet and π. The elimination omes from disaearane of the diagonal matrix element of I. The summation Z is ositive, beause the denominator of Eq.( 5.6a) is negative. As shown in Fig.5.5, the transfer-momenta from A (u-sin eletron-air) satisfies Eq.(5.5). The number of the transfer-momenta is / of the total orbitals. Aordingly the erturbation energy ς A of the air A is exressed by Z as ς A ( )Z at ν 5 / (5.7) beause the momentum-interval, π, is extremely small in a marosoi size of a 6

137 quantum all devie. The total number of the nearest eletron airs with u-sin is / of N. Thereby we obtain the nearest air energy er u-sin-eletron ε as ( ς N ) N ( 6)Z ε at ν 5 / (5.8) A When the filling fator deviates from ν 5 /, the eletron onfiguration hanges from the regular reeating of (filled, emty, filled). Then the number of the Coulomb transitions dereases abrutly beause the hanging disturbs the Coulomb transitions. As an examle, the ν / 5 state is illustrated in Fig.5.5 where the nearest orbitals with u-sin are illustrated by red and brown olours. C D A CD E F G Unit length A I J KL MN OPQ C D A Fig.5.5 Eletron onfiguration at ν / 5 There are 9 nearest eletron airs, namely, A, CD, EF, G, IJ, KL, MN, OP and PQ in every unit sequene. The air CD an transfer to two orbital airs er unit length as shown by blak arrow airs. Aordingly the erturbation energy ς CD is equal to ς ( 5)Z at ν / 5 (5.9) CD The air A an transfer to all the emty orbitals of u-sin and then the number of allowed transitions is eight er unit length. Therefore the erturbation energy ς A is ς A ( 8 5)Z at ν / 5 (5.5) The other airs have the erturbation energies as ς ς ( 5)Z, ς ς ( 6 5)Z, ς ( 5)Z EF KL G IJ ς ς at ν / 5 (5.5) OP PQ The sum of the nearest eletron airs with u-sin is F ς ς ς ς ς ς ς ς ς ( 5)Z (5.5) A CD EF G IJ KL The number of eletrons with u-sin is seventeen in a unit sequene. Therefore the nearest air energy er u-sin eletron is MN OP MN PQ 7

138 ( ( 5 7) )Z ε at ν / 5 (5.5) When the filling fator ν is ( ) /( 6s ) nearest-air-energies inside the unit sequene is s (s is a ositive integer), the sum of the F Z( ( s ) s ( s ) ) /( 6s ) ( s ( 6s ) )Z (5.5) The filling fator ( s ) /( 6s ) is larger than 5/. The number of u-sin-eletrons inside a unit length is equal to ( s ) and therefore the air energy er u-sin-eletron is given by ε ( s (( 6s )( s ) ))Z at ( s ) /( 6s ) ν (5.55) When s beomes infinitely large, ε aroahes lim ε ( ) ν 5 ( )Z (5.56) Next we onsider the filing fator 8/ whih is smaller than 5/. The most uniform onfiguration is illustrated in Fig.5.5. C D A Unit length CD E F G A I J KL MN O C D A Fig.5.5 Eletron onfiguration at ν 8/ In this ase, the sum of the nearest-air-energies inside the unit sequene is F ς ς ς ς ς ς ς ( )Z (5.57) A CD At ν ( s ) /( 6s ) EF G IJ KL MN, the sum of the nearest-air-energies inside the unit sequene is F [ ( ( s ) s ( s ) ) /( 6s ) ] Z ( s ( 6s ) )Z (5.58) Aordingly ε ( s ( 6s )( s ) )Z at ( s ) /( 6s ) ν (5.59) ν lim ( 5 ) ε ( )Z (5.6) 8

139 Thus the energy ga aears between the energy at ν 5 / and the limiting energy from the left and right sides: ε ( ν 5 ) lim ε ν ( 5 ) ± ( 6) Z ( ) Z ( )Z Table 5. shows the energy gas in the fourth olumn at ( ) /( ) ν. Table 5. Energy gas for ν ( ) /( ) ν ( ) ε lim ε ( ν ) ε ( ν ) lim ε ( ν ) ν ν ν ± ν ν ± 5/ -(/6) Z -(/) Z -(/) Z 9/5 -(/) Z -(/) Z -(/) Z /7 -(/) Z -(/8) Z -(/8) Z 7/9 -(/7) Z -(/) Z -(/) Z (5.6) We onsider the other ases. Figure 5.5 shows the most uniform onfiguration of eletrons at ν 8/ 5. Therein the x-, y-, z-diretions are exressed at the uer-left of Fig.5.5. X Z Y A Fig.5.5 Eletron onfiguration at ν 8 / 5 The unit onfiguration is omosed of five Landau orbitals and three eletrons with u-sin. The number of the allowed transitions is two er unit length. Aordingly the erturbation energy ς A of the air A with u-sin is obtained as ς ( 5)Z at ν 8 / 5 (5.6) A The total number of the nearest eletron airs with u-sin is / times N. Therefore the nearest air energy er u-sin-eletron is ( ς N ) N ( 5)Z ε at ν 8 / 5 (5.6) A When the filling fator deviates from ν 8/ 5, the number of the Coulomb transitions dereases abrutly beause the eletron onfiguration at ν 8/ 5 disturbs the 9

140 Coulomb transitions as seen in Fig.5.5 for ν / 7. C D E F A Unit length CD E F A G I J C D E F A Fig.5.5 Eletron onfiguration at ν / 7 There are five u-sin-eletron airs laed in the nearest orbitals inside a unit length as in Fig.5.5. The number of allowed transitions is eleven for the air A, nine for EF and seven for CD in a unit length. Therefore the erturbation energies are obtained as follows: ς A ( 7)Z, ς EF ς G ( 9 7)Z, ς CD ς IJ ( 7 7)Z (5.6) The sum of these air energies is F ς ς ς ς ς ( 7)Z (5.65) A CD EF G IJ The number of eletrons with u-sin is sixteen in a unit length and then the nearest air energy er u-sin-eletron is ( ( 7 ))Z ε F 6 6 at ν / 7 (5.66) We examine more general ases of ν ( 6s 5) /( s ). At the filling fator, the sum of the nearest-air-energies inside a unit length is F Z( ( s ) ( s ) ( s ) ( s ) ( s ) )/( s ) (5.67) ( 6s s ) ( s ) )Z Aordingly ε [ ( 6s s ) (( s )( 6s ) )]Z at ν ( 6s 5) /( s ) (5.68) ν lim ( 8 5) ε ( )Z (5.69) Next we study the FQ state with ν / 7. The most uniform onfiguration is illustrated in Fig.5.55.

141 A Fig.5.55 Eletron onfiguration at ν / 7 The erturbation energy of the air A is ς ( 7)Z at ν / 7 (5.7) A The number of the nearest eletron airs with u-sin is / of N. Therefore the nearest air energy er u-sin-eletron is ( ς N ) N ( 8)Z A ε at ν / 7 (5.7) At ( ) /( ) ν the erturbation energy ς A for the nearest air A is obtained as ς [ ( ) /( ) ]Z at ( ) /( ) A ν (5.7) The total number of the nearest eletron airs with u-sin is / times N. Therefore the nearest air energy er u-sin-eletron is ( ς N ) N ( ) / ( ) ε [ ( )]Z at ( ) /( ) A ν (5.7) The energies are listed in the seond olumn of Table 5.. Next we alulate the ν /. Then number of the allowed transitions in the neighbourhood of ( ) ( ) the energy gas are shown in the fourth olumn of Table 5.. Table 5. Energy gas for ν ( ) /( ) ν ( ) ε lim ε ( ν ) ε ( ν ) lim ε ( ν ) ν ν ν ± ν ν ± 5/ -(/6) Z -(/) Z -(/) Z 8/5 -(/5) Z -(/) Z -(/) Z /7 -(/8) Z -(5/56) Z -(/56) Z /9 -(/5) Z -(7/9) Z -(/9) Z Thus the resent theory yields the energy gas at the seifi filling fators as in Tables

142 5. and Eletron onfigurations and energy gas for <ν < We examine the FQ states with <ν < in this sub-setion [69]. Four examles are shown in Figs where the filling fators are ν /, 6/5, 7/5 and /7, resetively. The eletron onfigurations are illustrated in a D view where the down-sin eletrons ouy all the Landau orbitals with the lowest level. X Z Y A unit length Fig.5.56 Eletron onfiguration at ν / The most uniform eletron onfiguration with u-sin is the reeat of the sequene (emty, filled, emty) at ν / as in Fig The emty orbitals for u-sin are shown by red dashed lines and the filled orbitals with u-sin by blue lines. The blue arrows exress the quantum transitions to the emty orbitals A (nearest hole air). The symbol ς A means the erturbation energy via all the quantum transitions shown by blue arrow airs. The nearest vaant-orbital-air A is seified by the momenta,. The eletron air A is also seified by the momenta A A,. The eletron air A transfers to the vaant orbitals at A and. Therein the total momentum of the air onserves in the Coulomb transition as A A (5.7) where the momentum transfer takes the following values as ( ) π for, ±, ±, ±, (5.75) at ν. Then the seond order erturbation energy of the hole air A is given by A, I A, A, I A, ς A (5.76) W W ( ) π (, ) for, ±, ±, G exite A A In order to evaluate the energy ς A we introdue the summation Z as

143 A, I A, A, I A, Z (5.77) W W (, ), π G exite A A where the momentum transfer takes all the values ( ) integer π exet and π. The transferred states for and π are eliminated in the summation (5.77) beause the diagonal element of I is absent. The denominator in Eq.( 5.77) is negative and so Z is ositive. (The value of Z is nearly equal to Z for the same magneti field strength.) The interval of momentum transfer is very small for a marosoi size of a devie and therefore the erturbation energy, ς A, an be exressed by Z as ς A Z for ν (5.78) The eletron onfigurations at ν 6/5, 7/5 and /7 are illustrated in Fig.5.57, 5.58 and 5.59, resetively. unit length A Fig.5.57 Eletron onfiguration at ν 6 / 5 unit length A Fig.5.58 Eletron onfiguration at ν 7 / 5 unit length A Fig.5.59 Eletron onfiguration at ν / 7

144 The erturbation energies, ς A, are also obtained by making use of Z as follows: ς A Z for ν 6 5 (5.79a) 5 ς A Z for ν 7 5 (5.79b) 5 ς A Z for ν 7 (5.79) 7 There are,, or eletrons in the unit length for the filling fator ν 6 5, 7/5, or 5 Z 5 Z and /7, resetively. Therefore the energy er eletron beomes ( ), ( ) ( 7) Z. We exress the erturbation energy er eletron by the symbol ( ν ) ε at ν ν whih is listed in the seond olumn of Table 5.5. Table 5.5 Energy gas for ν ( ) /( ) and ( ) /( ) ν ( ) ε lim ε ( ν ) ε ( ν ) lim ε ( ν ) ν ν ν ± ν ν ± / -(/) Z -(/6) Z -(/6) Z 6/5 -(/5) Z -(/) Z -(/) Z 7/5 -(/5) Z -(/) Z -(/) Z 8/7 -(/7) Z -(/) Z -(/) Z /7 -(/7) Z -(5/) Z -(/) Z /9 -(/9) Z -(/8) Z -(/8) Z /9 -(/9) Z -(7/7) Z -(/7) Z The limiting values from both sides are alulated and written in the third olumn of Table 5.5. Subtrations of the limiting value from ε ( ν ) give the energy gas whih are listed in the fourth olumn of Table 5.5. Tables 5., 5. and 5.5 show the energy gas at the filling fators. Thus the resent theory an exlain the onfinement of the all resistane in the region of <ν <.

145 5.. Peak struture at the filling fators ν ( ) ( ) and ( ) ( ) We examine the ν 7 state. The filling fator has the denominator of even number. Figure 5.6 shows the most uniform onfiguration at ν 7. X Z Y AC Fig.5.6 Eletron onfiguration at ν 7 / There are many eletron airs in Fig.5.6. The air A is an examle of the nearest eletron air. The quantum transition via the Coulomb interation onserves the total momentum. Aordingly the eletron should transfer to one orbital to the right when the eletron A transfers to one orbital to the left. owever the transformation to the right-diretion is forbidden beause the Landau orbital is already ouied by u-sin eletron as in Fig.5.6. When the eletron A transfers to the fifth orbital to the left, the eletron annot transfer to the fifth orbital to the right beause of the Pauli exlusion rinile. Thus all the transitions from the nearest eletron airs are forbidden via the Coulomb interation. Therefore the eletron air A has no binding energy. Also all the quantum transition from the eletron air C are forbidden. Aordingly all the nearest eletron airs have no binding energy. Similarly all the nearest eletron airs have no binding energy at the filling fator ν. ( ) ( ) ( ) ν ε at ( ) ( ) ν (5.8) The energies are listed in the seond olumn of Table 5.6. Table 5.6 Comarison of nearest eletron air energies ν and in its neighbourhood ν ( ) ν at ( ) ( ) ε lim ε ( ν ) ε ( ν ) lim ε ( ν ) ν ν ± ν ν ± / 5

146 7/ Z /6 Z Z 5/8 Z Z 6 6 Z We examine the energies of the nearest eletron airs in the neighbourhood of ν. As an examle for the neighbourhood of, we have alulated ( ) ( ) the number of allowed transitions by using a omuter, then we obtain ( ) 5 Z ε ν 76 at ( ) 5 Z ε ν 76 at 5 ( ) Z ν ν (5.8a) ν (5.8b) ε at ν 5 (5.8) 5 ( ) Z ε at ν 5 (5.8d) ν The limiting value of the air energy for ( ) ± ν is obtained as follows; lim ν ± ε ( ν ) Z (5.8) Next we examine the ase of ν 5 6. ( ) 5 Z ε ν 9 at 5 65 ( ) 5 Z ε ν 9 at 5 65 ( ) 5 Z ε ν 7 at ( ) 5 Z ε ν 7 at ν (5.8a) ν (5.8b) ν (5.8) ν (5.8d) For the limiting of ( 5 6) ± lim ν 5 6± ε ν the nearest air energy aroahes ( ν ) Z 6 (5.8) As already alulated in Eq.(5.8), the nearest air energy at ( ) ( ) Therefore the air energy at ( ) ( ) ν is zero. ν is higher than the energy in the 6

147 neighbourhood of ν. The eak values are listed in the fourth olumn of Table 5.6. Similarly we alulate the air energy of quantum transitions to the nearest emty ν. The values are listed in Table 5.7. It is orbitals at the filling fator ( ) ( ) noteworthy that the values indiate the air-energy er eletron (not hole). That is to say ν with an even number the eak struture aears at the filling fator ( ) ( ) denominator. Table 5.7 Comarison of nearest hole air energies er eletron ν and in its neighbourhood ν ( ) ν at ( ) ( ) ε lim ε ( ν ) ε ( ν ) lim ε ( ν ) ν ν ± 5/ Z 7/6 9/8 ν ν ± 8 8 Z Z Z 6 Z 6 Z Aordingly the FQ state is not stable at ν ( ) ( ) and ν ( ) ( ) for, beause the energy is higher than that of the neighbourhood. So the all resistane onfinement doesn t aear at ν ( ) ( ) and ( ) ( ) ν for. This logi an be extended to the FQE at the filling fator larger than. Then the all resistane lateau doesn t aear at the filling fator ( integer ( ) ( ) ) and ( integer ( ) ( ) ) for. This roerty is in agreement with the exerimental results. 5.Short omment ) The resent theory takes the eletri otential along the all voltage into onsideration (Traditional theories ignore it). So both all voltage and eletri urrent are not zero. The ratio between them yields the all resistane roerly. ) The total momentum of eletrons along the urrent diretion onserves via the Coulomb interation in FQE. The onservation is satisfied in the resent theory. ) The FQE at ν > have been exlained by the disontinuous energy setrum of the eletron airs as in setions 5.9 and 5.. On the other hand the traditional FQ 7

148 theories exlained the FQE at ν > by suerosing the IQ state of usual eletrons with that of quasi-artiles as will be disussed in Chater. That is to say, all the eletrons in a D system are lassified into the following two grous: ) the normal eletrons and ) the quasi-artiles. This lassifiation violates the anti-symmetri relation between eletrons in the wave funtion. The lak of anti-symmetry may disturb the Fermi-Dira statistis for all the eletrons existing in the same thin layer. ) The ν 5,7 FQE and the nonstandard FQE at ν 7/, /, /, 5/, 5/7, 6/7 have been exlained on the same logi as the FQE at ν /, /, /5, /5, /5, /5 and so on. The traditional theories have exlained the three tyes of the FQE by emloying the different models resetively. 5) The Coulomb interation ats between eletrons in the D-eletron system. So the binding energy belongs to eletron airs (not single eletron). The resent theory deals the air energy roerly. Thereby the henomena of FQE have been aused by the disontinuous deendene of the air energy uon the filling fator ν. 8

149 Chater 6 Effet of higher order erturbation and ontribution from uer Landau levels We will study the higher order erturbation energy due to I in this hater. The third order terms are examined in setion 6., the higher order term in setion 6. and the ontributions from the uer Landau levels in setion Third order erturbation energy As an examle we study the third order erturbation at the filling fator of ν. One of the third order transitions is illustrated in Fig.6.. The momenta of eletrons A and are denoted by A and. The momenta after the first transition are desribed by A and. The total momentum onserves as (6.) A A The seond transition yields the momenta A and whih satisfy the momentum onservation as (6.) A A A A A A Fig. 6. Third order erturbation of eletron air A at ν We introdue the following new summation Z as A, I A, A, I A, A, I A, Z ( W W (, ))( W W (, )), π, π G exite A A G exite A A 9 (6.) where W is the ground state energy of the amiltonian G D. Therein A,, A, indiate the momenta in the two intermediate states. These four momenta are exressed by using the momentum transfers, as follows: A A, (6.), (6.5) A A

150 whih are derived from the momentum onservation. In the summation Z, the momentum transfers, are able to take all of the values ( π ) integer exet,, π and π. The values,, π and π are eliminated in the summation of (6.), beause the diagonal matrix elements of I are zero. Therefore the denominator in Eq.(6.) is not zero. The third order erturbation energy an be exressed by Z systematially. As an examle, we alulate the third order erturbation energy ς A of the nearest eletron air A in Fig.6.. Therein the momentum transfers, take the values as 5 π, π, π, π, 7 π, (6.6) 5 π, π, π, π, 7 π, (6.7) The interval of the momentum transfers is π for the alulation of the third order erturbation energy ς A as in Eqs.(6.6) and (6.7). On the other hand the interval is π in Z. The interval value π is extremely small for a marosoi size of and then we get ς A Z at ν (6.8) It is noteworthy that the fator ( ) omes from double summations of and. eause the number of nearest eletron airs is (/) times the total number N of eletrons, the third order erturbation energy E of all the nearest eletron nearest air airs is exressed as E nearest air ( ) ( Z )N N ς A at ν (6.9) The energy er eletron is E nearest air N Z at ν (6.) Similarly the nearest air energy er eletron in the third order at ( ) given by ν is 5

151 5 Z N E air nearest at ( ) ν (6.) Also the energy of the nearest airs er eletron in the third order at ( ) ν is given by Z N E air nearest at ( ) ν (6.) The eletron-hole symmetry gives the following air energies er hole air nearest Z N E at ( ) ν (6.) air nearest Z N E at ( ) ( ) ν (6.) whih are derived from Eqs.(6.) and (6.) resetively. Then the energies er eletron (not hole) beomes air nearest Z N E at ( ) ν (6.5) air nearest Z N E at ( ) ( ) ν (6.6) 6. igher order erturbation energy We define the following summation Z n for the higher order alulations as ( ) ( ) ( ) ( ) π π,, A A exite G A A exite G A I A A I A A I A,,,,,,,, n n n n n n W W W W Z (6.7) We an obtain the n-th order erturbation energy of the nearest eletron airs by making use of Z n :

152 n E nearest air N n n Z at ν (6.8) It is noteworthy that we have used the symbol Z instead of Z and also E nearest air instead of E in haters and 5. nearest air At ( ) ν with arbitrary integer the n-th order erturbation energy er eletron is given by n E nearest air n n N Z at ν ( ) (6.9) At the filling fator of ν ( ) obtained as, the n-th order erturbation energy er eletron is n E nearest air N n n Z at ν ( ) (6.) The eletron-hole symmetry gives the following air energy er hole: n E nearest air N n n Z at ν ( ) (6.) n E nearest air N n n Z at ν ( ) ( ) (6.) The sum of the erturbation energies for all orders is given by n all order n E nearest air N Z n,, at ν ( ) (6.) n all order n E nearest air N Z n,, at ν ( ) (6.) Also we get the sum of the erturbation energies in all orders for the hole air 5

153 n all order n E nearest air N Z at ν ( ) (6.5) n,, n all order n E nearest air N Z at ν ( ) ( ) (6.6) n,, These energies of the nearest hole airs er hole are re-exressed to the energies er eletron as n all order n E nearest air N Z at ν ( ) (6.7) n,, n all order n E nearest air N Z at ν ( ) ( ) (6.8) n,, Next we examine the ase of the filling fator ν ( ) ( ), the denominator of whih is an even integer. There is no quantum transition from any nearest eletron (or hole) air at the filling fators of ν ( ) ( ) ( or ν ( ) sum of the all order terms is equal to zero: ). Therefore the all order E N at ν ( ) ( ) for nearest eletron airs (6.9) nearest air all order E N at ν ( ) nearest air for nearest hole airs (6.) We have already examined the number of the allowed transitions for various filling fators in Chaters and 5. The number of quantum transitions varies ontinuously at ν ± ν /, 5/8. 7/, disontinuously at ν ( ), ν ( ), ( ) and so on. The ontinuous (or disontinuous) struture is indeendent of the order of erturbation and is deendent on the filling fator only. Therefore the ontinuous or disontinuous struture is also maintained in the exat energy setrum. It is additionally noted here that (6.-8) indiate that the higher order terms are n multilied by ( ( ) ) n and (( ) ( ) ) the ase of, n and as follows: n ( ( ) ) ( 5).. These multiliers are evaluated in for, n (6.a) n (( ) ( ) ) ( ). for, n (6.b) 5

154 n ( ( ) ) ( 5). 8 for, n (6.) n (( ) ( ) ) ( ).7 for, n (6.d) These multiliers are small values and so the higher order ontribution beomes very small. Consequently the seond order ontribution may be a main art of E. all order nearest air 6. Contribution of uer Landau levels We examine the ontributions from uer Landau levels. The ground state of D at ν < is omosed of the Landau states with Landau level number L only. Aordingly all the orbitals with L are emty for the ground state at ν <. ere we examine the ontribution of the quantum transitions to the uer Landau levels. The transitions are lassified into the following two ases: (Case ) One eletron transfers to an orbital with L and the other eletron transfers to an orbital with L. Therein the one eletron transfers from the momentum to with L and then the other eletron should transfer from to beause the momentum onservation. All the orbitals in the uer Landau level L are emty. So the transition number is equal to the number of emty orbitals with L. The number of emty orbitals with L varies ontinuously with hanging of the filling fator ν. Therefore the erturbation energies deend ontinuously uon the filling fator. (Case ) oth eletrons in the eletron air transfer to the uer Landau levels. In this ase the transition number is equal to the number of Landau orbitals with the uer level. So the transition number is onstant. Consequently the erturbation energy via the uer Landau levels ( L ) does not rodue a disontinuous struture in the energy setrum. Therefore the energy ga is not derived from the ontribution of the uer Landau levels. (Although the disussion was arried out for ν <, the ontinuous roerty via the uer Landau levels in ν > is also the same as in ν <.) Furthermore the energy in the erturbation alulation is exressed by the Eq.(6.7). Therein the denominator of Eq.(6.7) is omosed of the differene Wexite W. The G differene value for the uer Landau levels is extremely large in omarison with the ase of the same Landau level. So the ontribution from the uer Landau levels is negligibly small omared with that in the same level. 5

155 Chater 7 Plateaus of quantum all resistane The exetation value of the total amiltonian is examined in hater and the air energy via the Coulomb interation is investigated in haters -6. We obtain the total energy by adding these results. Then the total energy er eletron ε ( ν ) on the quasi-d eletron system deends uon the filling fator ν disontinuously at the seifi filling fators ν ν but ontinuously in other filling fators. This ν -deendene yields the all resistane urve as will be larified below [7]. 7. Funtion-form of the total energy The total energy ( ) ν E of the FQ state for ν < has been exressed as ( ν ) χ( ν ) N g( ν ) N [ f e ( m ) ( ξ η) ν ] N C ( σ ) E Marosoi whih was obtained by Eq.(5.a) and Eq.(5.b). Therein χ ( ν )N indiates the energy via the Coulomb transitions from all of the nearest airs and g( ν )N indiates that from the rest of the airs. The onstant term f was defined by Eq.(.a, b) as f λ U ξ η where U is the mean value of the otential along the y-diretion. We reexress the total energy E ( ν ) as a sum of the following five terms: E ( ν ) ( χ( ν ) g( ν ) ( e m ) a ν b)n (7.a) a ( ξ η), b U ξ η ( C N ) λ (7.b) Marosoi When the gate voltage is hanged, the value of U varies. So the value of b an be ontrolled by adusting the gate voltage. We desribe the energy er eletron by the symbol ε ( ν ): ε ( ν ) χ( ν ) g ( ν ) ( e m ) a ν b (7.) Therein χ ( ν ) is defined in Eq.(5.a) as a sum of all order erturbation energies of the nearest eletron (or hole) airs as χ ( ν ) χ n ( ν ) n,, at the filling fator of ν (7.a) 55

156 Also g ( ν ) indiates the air energy laed in more distant Landau orbital air as given by the summation of all order erturbation energies as g ( ν ) ( ν ) g n n,, at the filling fator of ν (7.b) In haters and 5, we have alulated the seond order erturbation energy of the nearest airs er eletron namely, χ ( ν ). Also we have alulated the higher order erturbation energies as Eqs.(6.,, 7 and 8) in hater 6. The results are n χ( ν ) Z at ( ) n,, n n χ( ν ) Z at ( ) n,, n ν (7.a) ν (7.b) n χ( ν ) Z at ν ( ) n,, n n χ( ν ) Z at ν ( ) n,, n (7.) (7.d) At ν ( ) ( ) and ( ) ν with the denominator of even number, all order erturbation energies of the nearest eletron (or hole) air are zero where all the quantum transitions are forbidden. The results Eqs.(6.9) and (6.) are exressed as χ ( ) ( n,,,) at ν ( ) ( ) and ν ( ) n ν (7.5) χ( ν ) χ n ( ν ) at ν ( ) ( ) and ν ( ) n,,, (7.6) n The n-th order terms in Eqs.(7.a-d) have the multiliers ( ( ) ) n (( ) ( ) ) n, ( ( ) ) n and ( ( ) ) beome small for large n as,, resetively. These multiliers n ( ( ) ) n., ( ( ) ) n. 6, (( ) ( ) ). for n n ( ( ) ) n. 8, ( ( ) ) n. 6, (( ) ( ) ).7 for n 56

157 These results indiate that the seond order term may be a main art of χ ( ν ) beause the multiliers are small in the higher order terms. We summarize the seond order term χ ( ν ) for various filling fators: χ ( ) Z at ( ) ν χ ( ) Z at ( ) ν ν (7.7a) ν (7.7b) χ( ν ) Z at ν ( ) χ( ν ) Z at ν ( ) χ ( ν ) at ν ( ) ( ) and ν ( ) (7.7) (7.7d) (7.8) Equations (5.), (5.), (5.6) and (5.7) give the seond order energies in the ν as neighbourhood of ν ( ) ( ) and ( ) Enearest air ( ) (( ) s ) s Z χ ν at (( )( s) ) ν N (( )( s) ) (( )( s) ) (( )( s) ) (7.9a) Enearest air ( ) (( ) s ) s Z χ ν at (( )( s) ) ν N (( )( s) ) (( )( s) ) (( )( s) ) (7.9b) Enearest air ( ) (( ) s ) s Z χ ν at (( )( s) ) ν N (( )( s) ) (( )( s) ) (( )( s) ) (7.9) Enearest air ( ) (( ) s ) s Z χ ν at (( )( s) ) ν N (( )( s) ) (( )( s) ) (( )( s) ) (7.9d) Therein we have used the following ratio N N in the derivation of Eqs.(7.9, d). N N (( )( s) ) (( )( s) ) at ν (( )( s) ) (( )( s) ) N s s ν s s N (( )( ) ) (( )( ) ) at (( )( ) ) (( )( ) ) For arbitrary frational number ν we an alulate χ ( ν ) by using the same roedure as given in haters -5. When the denominator of the frational number is large, the total number of the quantum transitions is alulated by using omuter. eause the higher order erturbation energy inludes the small multiliers as mentioned above, ε ( ν ) is given aroximately by 57

158 ε ( ν ) χ ( ν ) g ( e m ) a ν b (7.) As examined in setions 5.9 and 5. the ontribution of the non-nearest airs is small and the ν -deendene is also small at ν <. Aordingly we may ignore the ν -deendene of g in Eq.(7.) at < ε ν is examined in the next setion 7.. [-] ν. The funtion-form ( ) 7. Setrum of the total energy versus filling fator Figure 7. shows four grahs of ε ( ν ) versus ν in the neighbourhood of ν /, /, / 5, / 7 where the green lines indiate the funtion γ ( ν ) as γ ( ν ) g ( e m ) a ν b (7.) Fig.7. Energy setra near ν,, 5, 7 We draw the value of χ ( ν ) with vertial bars in Fig.7., where the lengths of the bars are the absolute values of χ ( ν ). The uer end of eah vertial bar is laed on the 58

159 green line namely γ ( ν ). Then the lower ends of eah vertial bars indiate the values of ε ( ν ) beause χ ( ν ) is negative. ( ν ) χ ( ν ) γ ( ν ) ε (7.) lue vertial bars in the neighbourhood of χ ν of Eq.(7.9a) and (7.9b). Also blue vertial bars in the neighbourhood of ν indiate Eqs.(7.7b) and (7.7d), and those for ν 5 indiate Eqs.(7.9) and (7.9d). ν and /7 exress ( ) Figure 7. shows the following two features:. The lower end of the red bar in the resetive figure is lower than that of the blue bars in the neighbourhood of ν, 5, 7. The energy setrum has a valley as shown by Energy Ga in the resetive anel of Fig.7... There is no valley at ν. The blue vertial bar aroahes the base line (green line) near ν. (Note: The blue bars have the filling fators with even number denominators as in Eq.(7.9a,b,,d). So some readers might wonder whether the air energies would be different from that with odd denominator. In order to rely this asking, we have ν s 6s and alulated the energy for the nearest airs at ( ) ( ) ν ( s ) ( 6s ).) The nearest air energies are obtained in Aendix (A.6) and (A.) as follows: s Enearest air ZN for ν s ( s )( 6s ) ( s )( 6s ) ( s ) ( 6 ) s E ZN for ν s nearest air Then the energies er eletron are given as follows; χ s ( ) ν Z for ν s ( s )( 6s ) ( s )( 6s ) ( s ) ( 6 ) ( s ) ( 6 ) s ( ) χ ν Z for ν s ( s ) ( 6 ) (7.a) (7.b) These air energies are shown in Fig.7. together with the energy of Fig.7.. The green vertial bars indiate χ ( ν ) at ν ( s ) ( 6s ) and ν ( s ) ( 6s ) with odd number denominator. The blue bars indiate χ ( ν ) at ν ( s ) ( 6s ) and ν s 6s with even number denominator. ( ) ( ) 59

160 Fig.7. Energy setrum near ν The green vertial bars are drawn after showing the blue bars. So we an see only green bars in the viinity of ν beause the blue bars are hidden by the dense green bars. We an see both blue and green bars when we move from ν. The lengths of the green and blue bars are almost the same and ontinuously vary with ν. So the differene of χ ( ν ) between even and odd integer of the denominators is negligibly small in the neighbourhood of ν. We find many ranges of absent vertial-bar in Figs.7. and 7., beause we don t alulate χ ( ν ) yet inside the ranges. That is to say, many vertial bars are abbreviated in Figs.7. and 7.. Of ourse we an alulate χ ( ν ) inside these ranges by using a omuter. Then we an get the more dense vertial bars. itherto, a few theorists have alulated the energy setrum of FQ states. As an examle alerin s result is shown in Fig.7. whih has many uss in the energy setrum []. 6

161 Fig.7. Energy setrum of alerin s result [] The energy setra of the resent theory have been shown in Figs.7. and 7. in the neighbourhood of the four filling fators. These five grahs are drawn searately. Some readers may want to know the setrum in a wider region of the filling fator. We will show it. The ν -deendene of ε ( ν ) (see Eq.(7.)) is derived from only two arameters Z and a whih deend uon the size, thikness and shae of a quantum all devie. We draw the energy setrum in Fig.7. for the arameter ratio Z a. 5 as an examle. Fig.7. Energy setrum of the resent theory The funtion-shae of the energy setrum in the resent theory [7] is quite different from alerin s result [] in Fig.7.. 6

162 7. ehaviour of all resistane urve near ν We draw three grahs of ( ) ν /, /, / and so on ε in the neighbourhood of ν in Fig.7.5. The left anel shows the energy setrum at the magneti field, the middle one < < and the right one. The energy ε ( ν ) inreases with inrement of due to the term ( e m ) in Eq. (7.). We draw the hemial otential µ by the dark green line in Fig.7.5. The ink urve shows the lower end of eah blue bar whih indiates the value of ε ( ν ). Fig.7.5 Magneti field deendene of Energy setrum near ν The Fermi-Dira distribution funtion n ( ε ) is given by n( ε ) (7.) ex( ( ε µ ) k T ) where k is the oltzmann s onstant and T is the temerature. At a low temerature all the states with ε < µ are ouied by eletrons and the states with ε > µ are emty. At the magneti field (the left anel of Fig.7.5), ε ( ν ) is higher than µ for any filling fator ν > ν and ε ( ν ) is lower than µ for any filling fator ν < ν. (It is noteworthy that the horizontal axis is ν and so the filling fator in the left is larger than one in the right.) All the eletron states of ν ν are filled with eletron, and any state of ν > ν is emty at low temeratures. That is to say, the filling fator beomes 6

163 ν at. Also the filling fator beomes ν at as in the right anel of Fig.7.5. In the range of < <, the inverse of the filling fator ν deends uon the magneti field under the fixed value of the hemial otential µ. When the filling fator is nearly equal to /, the nearest air energy χ ( ν ) is very small as in Fig.7.5. Aordingly ε ( ν ) is almost equal to ( ν ) the filling fator is determined by the relation ( ν ) µ ε ( ν ) γ ( ν ) g ( e m ) a ν b µ a ν g ( e m ) b µ γ due to Eq.(7.). So the value of γ at a low temerature: (7.5) ( e ( m a ) ( b g µ ) a ν (7.6) Eq.(7.6) means that the inverse of the filling fator, the magneti field in the neighbourhood of ν. ν, is linearly deendent uon Next we study the magneti field deendene of the all resistane. We have already investigated the eletri urrent of the IQ states in hater. Equation (.8) is also satisfied at any frational filling fator ν beause the total eletri urrent is ν times that with the filling fator. Equation (.9) holds for any frational filling fator beause an eletron has the elementary harge e. Substitution of Eq.(.9) into Eq.(.8) yields the following relation: νe I ( V V ) π (7.7a) Eq. (7.7a) gives the all resistane R as ( V V ) π R for any frational filling fator (7.7b) I e ν Substitution of Eq.(7.6) into Eq.(7.7b) yields the following relation: π R ( e m a) ( b g µ ) a) near ν (7.8) e This theoretial result means that the all resistane deends linearly uon the magneti field in the neighbourhood of ν. One of the exerimental data is shown in Fig.7.6. The exerimental value of the all resistane also deends uon the magneti field linearly near ν as easily seen on the uer anel. Aordingly the result of the resent theory is in good agreement with the exerimental data near ν. 6

164 Fig.7.6 Exerimental results of all resistane R and diagonal resistane R xx R K is the von Klitzing onstant The states near ν have no valley struture as shown in Fig.7.5, and has almost zero exitation energy as disussed in setion 5.7. Therefore the exitations yield by eletron satterings due to the eletri urrent. Consequently the diagonal resistane R xx is redited to be finite and almost onstant in the neighbourhood of ν. The theoretial result aears in the exerimental data as in the lower anel of Fig.7.6. Next we disuss the ase with the eak struture. For an examle ν, the robability of the ν state is smaller than the robabilities with ν ( ) ± ε beause of the energy eak. Aordingly the state with ν ( ) ε varies to the state with ν ( ) ε skiing the ν state (ε is an infinitesimally small number) when the magneti field is inreased. Therefore the all resistane ontinuously hanges and linearly deends on the magneti field in the neighbourhood of ν. Thus the FQ state near the eak struture has a roerty similar to one in the neighbourhood of ν. 7. Exlanation for the aearane of the lateaus in the all resistane urve 6

165 The energy setrum ε ( ν ) yields the theoretial funtion-form of the all resistane versus magneti field. We draw figures of ε ( ν ) in the neighbourhood of ν at three magneti fields,, < < and, resetively in Fig.7.7. When the magneti field beomes stronger, the value of ε ( ν ) beomes higher beause of the term ( e m ) in Eq.(7.). We show the hemial otential µ by the green line. The ink urve indiates the value of ε ( ν ) in the neighbourhood of ν exet ν. The valley at ν is shown by the lower end of the red vertial bar. Fig.7.7 Magneti field deendene of Energy setrum near ν ε is lower than the hemial otential µ in the range of < < in Fig Therefore the state with ν is filled with eletrons at a low temerature. y ontrast ε ( ν > ) is higher than the hemial otential µ in the range of < <. Therefore all the states with ν > are emty in a low temerature. Consequently the filling fator is onfined to in the range of < < [-]. The all resistane is given by Eq.(7.7b) as follows; ( V V ) π π R in the range of < < (7.9) I νe e That is to say, the all resistane π e in the We an find that the energy er eletron ( ) R takes a onstant value ( ) ( ) range of < <. We examine below two more examles ν 5 and /5. From Tables 5. and 5.5, the two states with ν 5 and /5 have also large valleys in their energy setra, resetively. The exerimental data are shown in Fig.7.8 where three 65

166 lateaus aear at ν, /5 and /5. The results of the resent theory are in good agreement with the exerimental data. Thus we onlude that the aearane of the lateaus in the all resistane urve originates from the valley struture. Fig.7.8 Exerimental data of all resistane near ν 5,, 5 We examine the relation between the energy deth of the valley and the width. The left anel of Fig.7.7 exresses that the limiting value ε ( ν ) is equal to the hemial otential µ : ( ν, ) lim χ( ν ) ( ) g( ) ( e m ) a ( ) b µ lim ν lim ε (7.) ν ν Also the right anel of Fig.7.7 gives the following equation: ε ( ν, ) χ( ) g( ) ( e m ) a ( ) b µ (7.) Subtration of Eq.(7.) from Eq. (7.) yields the following relation: ( ) χ( ) ( e m )( ) lim χ ν ν The deth of the valley is given by Valley deth ε χ( ν ) ( ) (7.) lim χ ν (7.) Equations (7.) and (7.) yield the following relation: Valley deth e( ) ( ) ε. (7.a) m Thus the deth of the valley is related to the width of the lateau on the all resistane urve versus magneti field strength. at 66

167 Next we disuss a vanishing of the diagonal resistane as in Fig.7.9. This vanishing region ours simultaneously in the all lateau region. The roerty omes from the following reason: The ground state with ν has an exitation-energy-ga E ( )Z as in Eq.(5.85). The exitation energy is very large and therefore exitation # the eletron satterings are suressed. This absene of satterings yields a vanishing of the diagonal resistane R xx under the onfinement of the filling fator /. That is to say the vanishing region of R xx is equal to < < due to the resent theory. Thus the exerimental width for vanishing of R xx an be estimated by using Eq.(7.a) as follows: ( ) ( m e) ε (7.b) Fig.7.9 Exerimental data of diagonal resistane R near ν xx 7.5 Comarison of the resent theory with exerimental data In haters 5-7, we have found a valley struture at ν /, /, /5, /5, /5, /5, /7, /7, a flat struture at ν / and a eak struture at / /,. The flat and eak strutures rodue a linear deendene of the all resistane uon the magneti field as larified in setion 7.. The energy valley rodues a onfinement of the all resistane beause only one FQ state is realized in a range of the magneti field at low temeratures as larified in setion 7.. We omare the deths of the valleys with the exerimental data of the diagonal resistane [, 5]. If an exeriment is done using an ideal devie without imurity and 67

168 lattie defet at zero temerature, then R XX is zero in several ranges of magneti field. ut the atual exeriments use the devies with imurities and lattie defets. Also the exeriments are arried out at a finite temerature. Therefore the diagonal resistane of exeriments beomes very small but not zero. So we roughly estimate the width W ( ν ) from the exerimental data as follows; we take the width where green line in Fig.7.. These are W ( ).5[T], W ( 5).6[T], ( 7).9[T] W ( 9).7 [T], W ( 7).[T], ( 5).9[T] ( ).9[T] R XX is lower than the W, W, W (7.5) Fig.7. Widths of the vanishing regions on R XX urve Our seond order alulations give the energy gas (deths of the valleys) whih are obtained as in Tables 5. and 5.: ε ( ) ( )Z ε ± ( 5) ( )Z ε ± ( 7) ( 56)Z ε ( 9) ( 7)Z ε ± ( 7) ( ) Z ε ± ( 5) ( ) Z ε ( ) ( 6) Z ± ± ± (7.6) Negleting the magneti field deendene of Z and using the aroximate relation Z, Eqs.(7.6) give the theoretial ratio of the energy gas (deths of the valleys) Z as follows: 68

169 : : : : : :.5:. :.8 :.6 :.:.:.7 (7.7) The ratio of the exerimental values is derived from Eq.(7.5) as W ( ) W ( ): W ( ): W ( ): W ( ): W ( ): W ( ) : :.6 :.9 :.7 :. :.9 :.9 (7.8) The theoretial result, namely ratio (7.7), is in good agreement with the exerimental result (7.8) in site of a rough estimation. Thus the FQ states beome extremely stable at the filling fators ν /, /, /5, /5, /5 due to the large deths of valleys. Therein the deviation of the exerimental ex value ( ν ) R from the theoretial value π ( e ν ) ex ex fators. The deviation ( ( ν ) h ( e ν ) R ( ν ) is quite small at these filling R is measured to be about 5 at 8 ν /, about. at ν / as in [7] and about. at ν /5 as in [7-75]. These deviations will be investigated in details in the next hater. 69

170 Chater 8 Auray of all resistane onfinement As exlained in the revious haters, the lateaus of the all resistane are aused by the valleys in the disontinuous energy setrum. The valley struture is rodued by the drasti hange in the number of quantum transitions via the Coulomb interations. Thereby the theoretial auray of all resistane onfinement is rigorous at the FQE in the infinitely large D system without lattie defet. In this hater we study more detail roerties of the quasi D eletron system with a finite size. If the devie quality is bad, the Coulomb transitions are hindered by imurities and lattie defets. Therefore the quality of the quantum all devie is very imortant for us to observe the reise onfinement of the all resistane at the FQE. When the exeriment emloys a quantum all devie with an ultra high eletron mobility, the exerimental value of the all resistane is extremely lose to ( e ν ) h at the frational filling fators ν /, /, /5, /5, /5, and so on [7-75]. For examle the relative unertainty is differene between the theoretial value ( e ν ) 8 ±. at the filling fator / as in Ref. [7]. The h and the exerimental value deends uon imurities, lattie defets, size and shae of the devie. The auray of the all resistane onfinement is investigated in details for the FQE in this hater [76]. 8. Auray of the all resistane onfinement in FQE (air energy deending uon the air momentum in a finite size devie) We first examine the distribution of the single eletron wave funtion whih is given by Eq.(.7) as ψ L, J m ω m ω ( x y z) ( ikx) u ( y α ),, ex ex ( y α ) φ( z) L L J J (8.) The robability density in the y-diretion has the following form for the lowest Landau level L. m ω ex The distribution width m ω ( y α ) ex ( y α ) ex ( y α ) J y along the y-diretion is J m ω y m ω e where we have alied Eq.(.) to angular frequeny ω for deriving the seond J 7

171 equality. The value of y beomes about.5 nm at 6 T (Note: the exeriment [, 5] used the magneti field strength 6 T at ν ): J s e.6 C ( e).5[nm] y for 6[T] (8.) Next we estimate the interval between two eaks of the nearest Landau orbitals. The interval value α is equal to α J α J whih are derived from Eq.(.) as: π α α J α J [nm] for [ µ m] [nm] (8.) e where [ µ m] is the length of the devie. There are many single-eletron states inside the distribution width y α π e.5 y whose number is at 6[T] and [ µ m] (8.) Seondary we examine the transfer-momentum deendene of the transition matrix element, : A I A, A, I A, ψ, ψ A, ψ, ψ A, πε ( xa x ) ( ya y ) ( za z ) e whih is alulated as: e onstant ex ex m ω y ex m ω y ( i( A A ) xa / ) ex( i( ) x / ) φ ( za ) φ( za ) φ ( z ) φ( z ) πε ( x x ) ( y y ) ( z z ) A A e e A m A ex ω y A e ex m ω y e There are four Gaussian funtions whose eaks are at ya A e, y A e, y e, and y e. The differene A A is equal to e e e e A A dx A dy A dz A dx dy dz A beause of the momentum onservation. So the transition matrix element, deends uon the momentum differene A A only. When the A I A, 7

172 value of e A A e is larger than beomes small owing to the large searation of the two eaks. On the other hand, when e A A e y, the matrix element of the Coulomb interation is smaller than y, the overlaing of the wave funtions is large. Then the transitions are effetive. The momentum differene exists in the following region..5 That is to say, about ( π ).5 ( π ). (8.5) A A states ontribute to the Coulomb transitions. When there are many imurities in the devie, the number of the effetive transitions dereases beause of the disturbane by the imurities. This imurity effet is examined in the next setion. In this setion, we investigate the ideal ase without imurity and lattie defet. We shematially draw the effetive and non-effetive transitions for various filling fators in Figs whih are lassified into three ases (A-), (A-) and (A-). Case (A-) ( ν ) Figure 8. shows the quantum transitions at the filling fator ν. Therein eletrons ouy in states. A A A states Fig.8. Allowed transitions for ν The blak dashed arrows indiate a non-effetive transition exressed by the symbol. The effetive transitions are drawn by blue arrows beause the momentum transfer satisfies the ondition (8.5). The non-effetive transition is drawn by blak dashed arrows in Fig.8. where has a value larger than.5 ( π ) A A. The matrix element of this transition is negligibly small beause the two wave funtions don t 7

173 overla with eah other. The non-effetive transition is exressed by the symbol Fig.8.. Consequently the number of the effetive transition beomes equal to the number of emty states in the effetive region). in (whih is We onsider the following ritial ase where adaent three filled orbitals exist in every states and eletrons ouy in states. The filling fator is (.66 ). This ritial filling fator is exressed by the symbol ν. Also the symbol ν is defined by ν (.66 ), (.66 ) ν (8.6) Next we onsider another Case (A-) where the filling fator is larger than ν. Case (A-): (for the filling fator larger than ν ) In this ase, adaent three filled orbitals shown by brown bold lines aear inside every states beause the filling fator is larger than (.66 ). A A A states Fig.8. Allowed transitions for a filling fator ν larger than ν The symbol means non-effetive and indiates a forbidden transition. Figure 8. shematially exresses the Coulomb transitions. The adaent three filled orbitals disturb the sequene of (filled, filled, emty). The transition from ( A, ) to (, A ) satisfies the total momentum onservation A A. As in Fig.8. the state with momentum A is already filled with an eletron before the Coulomb transition, although the state of is emty. Therefore the quantum transition from ( A, ) to ( A, ) is forbidden. Thus the number of effetive transitions beomes smaller than. Consequently the absolute value of the air-energy ζ ( ν A ) for 7

174 is smaller than ζ ( ). That is to say, ζ ( ν ) > ( ) ν >ν air-energies are negative. A beause the A ζ A Case (A-): for the filling fator ν ( < ν < ν ) The most uniform onfiguration of eletrons is shown in Fig.8. at the filling fator ν for <.66. In this ase the adaent three brown lines with ( ) aear outside the region of states beause of < ν < ν. So there are effetive transitions from ( A, ). On the other hand the number of allowed transition from ( ) is only two as shown by blue arrows in Fig.8.. Thus the number of C, D effetive transitions deends on the total momentum of the eletron air. q α A A A q β CD EF states Fig.8. Allowed transitions for a filling fator ν ( < ν < ν ) The symbol indiates a forbidden transition. We have examined the average energy of eletron airs laed in the nearest Landau orbitals in the revious haters. The average of air energy is the ratio between the sum of all the air energies with different air momenta and the total number of air energies. So the momentum deendene doesn t exist in the average energy. Then the maor struture of FQE has been larified by using the average energy. When we examine more detail roerties of FQE (for examles: auray of the all resistane onfinement of FQE, the imurity effet and so on), we need to take the momentum deendene of the air energy into onsideration. The eletron air-energy deends uon the air momentum. For examle the nearest eletron air A has the total momentum Q as A 7

175 Q A A A π (8.7a) where we have emloyed the relation π in the nearest eletron air. A This air momentum rerodues the two momenta A and as π, ( ) π (8.7b) A ( Q A ) Q A ereafter the nearest air-energy ζ A ( ν ) is rewritten by ( ) ζ ν ;Q A to exress the deendene on the total momentum of the eletron air. Figure 8. shows that the number of effetive transitions from the air ( A, ) at the filling fator ν is almost equal to that at ν : ζ ( ν ; A) ζ ( ) Q for < (.66 ) It is noteworthy here that the air-energy (,Q) ν (8.8) ζ is indeendent of the air momentum Q beause any nearest air has the same number of effetive transitions at ν as easily seen in Fig.8.. So we an abbreviate the argument Q as ζ, Q ζ. ( ) ( ) At the filling fator ν the number of effetive transitions from the air CD is only two as in Fig.8.. Therefore ζ ( ν ; ) ( ) ζ ( ) Q (8.9a) CD Similarly we examine the transitions from the air EF shown by green air lines in Fig.8.. The number of effetive transitions from the eletron air EF is six whih give the air energy as ζ ( ν ; ) ( 6 ) ζ ( ) Q (8.9b) EF Thus the nearest air in the neighborhood of adaent three orbitals filled with eletron has very small air-energy beause the Coulomb transitions are disturbed by the adaent three filled orbitals. A similar situation aears for the filling fators smaller than /. At the filling fator ν ( ) ζ ζ ζ ( ν ; ) ζ ( ) Q (8.a) A ( ν ; ) ( ) ζ ( ) CD Q (8.b) ( ν ; Q ) ( 6 ) ζ ( ) for > (.66 ) EF ν (8.) Aordingly it is imortant to investigate the momentum deendene of the air-energy. We examine the following ase where the adaent three filled orbitals have the entral 75

176 momentum q α. The air momenta of the nearest eletron air are given by Eq.(8.7b) as ( Q ) ± π. When the momenta ( Q ) ± π ± momentum α are near the entral ± q, the absolute value of air-energy ζ ( ν ;Q) the absolute value of the differene as follows; ζ ( ν ; Q) onstant Q qα for is almost roortional to π ± qα <<.5 (8.a) beause the effetive transition number is almost roortional to the absolute value of ± q α. When the absolute value of qα ± aroahes.5 ( π ) for [ µ m] and 6[T], the effetive transition number aroahes the maximum value. Then the air energy ζ ( ν ;Q) saturates and aroahes ( ) ζ as in Fig.8.. Therein the air energies are shown at the four filling fators by the blue urve for ν, green for (.66 ) blak for ν (.7 ) ν, red for (. ) a b ν and. eause the effetive transition number is onstant at ν for any air momentum, the urve of the air-energy is onstant whih is exressed by blue straight line in Fig.8.. ζ( ) ζ(ν ; Q) Q Pair momentum q α q a q b q a q Fig.8. Pair momentum deendene of ζ ( ν ;Q) lue urve at ν Green for ν (.66 ) Red for ν (. ) lak for ν (.7 ) b a 76

177 The entral momentum in the adaent three filled orbitals is exressed by the symbols q α q, q, q,, q α q, q, q, and q α q, q, q, for the filling fators, a a a, b b b, ν a, ν b and ν, resetively. The momentum interval namely q a qα beomes larger when the filling fator ν a aroahes /. The total number of the adaent three filled orbitals is roortional to ν. Aordingly the momentum interval q a qα is related as a ( ν ) qa qα onstant a (8.) ν as: In Fig.8. the ratio of the momentum intervals is equal to the ratio of ( ) ( q q ) ( q q ): ( q q ) (.66) : (.) : (.7) a α : b α α (8.) The air-energy er eletron χ ( ν ) has been already defined by (total energy of nearest airs) / (total eletron number N ). So χ ( ν ) is the average value of ζ ( ν ;Q) in Figs.8. for a finite size devie with [ µ m]. The average value χ ( ν ) for a finite size devie is ontinuously deendent uon ν as in Fig.8.5. Fig.8.5 Average air-energy er eletron versus filling fator in the neighborhood of ν (This deendene is shematially shown for a quantum all devie with a finite size [ µ m]. orizontal axis indiates the filling fator ν and vertial axis is drawn by arbitrary sale. ) If we ignore the momentum deendene in Fig.8. the onfinement of the FQ 77

178 resistane has the theoretial value of the relative unertainty about for [ µ m] as in Fig.8.5. The exerimental result [7] has larified that the auray of the all resistane onfinement is the relative unertainty 8 ±. at ν. Thus we annot exlain the extremely reise onfinement of all resistane in an atual finite system. Aordingly we need to take the momentum deendene of the air energies in Fig.8.. The energy setrum is disontinuously deendent on the filling fator for the infinitely large D eletron system as in the revious haters -7. The total energy er eletron has been obtained in Eq.(5.) as ( ν ) χ( ν ) g( ν ) [ f e ( m ) ( ξ η) ν ] C N ε Marosoi In this hater we have examined the quasi D eletron system with a finite size and alulated the number of the effetive transitions. The air-energy ζ ( ν ;Q A ) belongs to the eletrons A and. Aordingly the eletron A has half of ζ ( ν ;Q A ). Therefore Eq.(5.) should be relaed with the following relation for a finite size devie: ε ( ν ) ζ ( Q ), ε ( ν ) ζ ( Q ) (8.a) A ν ; A ν ; A ( ) [ f e ( m ) ( ξ η) ] C N g ν ν (8.b) Marosoi The momentum deendene is exliitly desribed as follows: ( ν ; q ) ζ ( ν ;q ± ( π ) ) ε (8.5) where the sign is used for the smaller momentum in nearest two momenta and for the larger momentum. The energy setrum is drawn for four filling fators in Fig.8.6 where the hemial otential µ is exressed by the dashed ink line. eletron energy ε (ν,q) µ momentum q Fig.8.6 Energy setra for four filling fators lue urve at ν, Green urve at ν a, Red urve at ν b, lak urve at ν 78

179 There are many eletron states with ( ν ) µ the other hand, at the filling fator ε ; q > on the green, red and blak urves. On ν all the energies ( ;q) ε are smaller than the hemial otential µ as in Fig.8.6. The eletron states with ε ( ν ; q) > µ annot be realized at an ultra low temerature. Therefore the resent theory onludes that the FQ state with ν is realized under a suffiiently low temerature. Thus the funtion-form of ε ( ν ;q) rodues the rigorous onfinement of all resistane observed in the exeriment [7]. 8. Effets of Imurities and Lattie defets on the all resistane Imurities and lattie defets rodue an irregular otential in a quantum all devie. The irregular otential hinders the quantum transitions from the nearest airs to the orbitals near the imurity and/or lattie defet. The situation is illustrated in Fig.8.7 for a devie with very few imurities and Fig.8.8 for a devie with many imurities. Case (-) (very few imurities) We shematially draw the Coulomb transitions for the Case (-) in Fig.8.7. In this ase, the distane between the imurities is very large. Therefore the imurities aear outside the Landau states. That is to say the distane between the imurities is larger than the robe of the wave funtion. A A A imurity states imurity Fig.8.7 Disturbane of Coulomb transitions in very few imurities at ν The number of effetive Coulomb transitions is nearly equal to that in the ideal ase as 79

180 easily seen in Fig.8.7. The air-energy er eletron is almost the same as in the ideal ases (A-) and (A-). Case (-) (many imurities) Next we study the ase with many imurities and lattie defets. Figure 8.8 shematially shows the Coulomb transitions in a quantum all devie with many imurities. The number of Landau states between imurities beomes small beause of many imurities. Thereby many transitions are forbidden by the Pauli rinile. Therefore the absolute value of air-energy χ ( ν ) beomes small. The forbidden transitions are shown in red ross in Fig.8.8. A A A imurity imurity states imurity imurity Fig.8.8 Disturbane of Coulomb transitions by many imurities at ν Next we study the state with a filling fator ν larger than /. In this ase there are many adaent three orbitals filled with eletrons as shown in blue lines in Fig.8.9. When the differene ν beomes large, the adaent three filled orbitals aear more often in the most uniform eletron-onfiguration. A A A imurity imurity states imurity imurity Fig.8.9 Disturbane of Coulomb transitions by many imurities at ν > Many Coulomb transitions are forbidden by the following auses: () imurities, () χ ν for the Case lattie defets, () deviation of ν from /. The funtion shae of ( ) 8

181 (-) is shown by red urve where the valley width is ν in Fig.8.. The width is the same as the width in Fig.8.5 for the ideal ase. The funtion shae of χ ( ν ) ν for Case (-) is drawn by the blue urve in Fig.8. and the valley width is exressed by ν. Fig.8. Nearest air-energy χ ( ν ) versus filling fator ν Red urve indiates χ ( ν ) for Case (-). lue urve indiates ( ν ) χ for Case (-) The width ν is very wide and the binding energy beomes very small beause of the resene of the many imurities (Case (-)). When the distane between imurities is smaller than that between the adaent three orbitals, the imurities mainly hinder the Coulomb transitions. So the width ν in Fig.8. deends uon the quality of the devie. When the quality of the devie beomes worse, the lateau beomes unlear or disaears. ex The exerimental value of the all resistane ( ν ) R is lose to ( e ν ) h for the filling fators ν /, /, /5, /5, /5, and so on. The exerimental deviation ex ex ( R ( ν ) π ( e ν ) R ( ν ) has been measured and the results are ex ex 5 ( R ( ν ) π ( e ν ) R ( ν ) about for ν / (8.a) ex ex ( R ( ν ) π ( e ν ) R ( ν ) about. for ν /5 (8.b) as desribed in the literature [7-75]. These values are onsistent with the theoretial 8

182 result disussed in this hater. 8. Shae and size effet for the all resistane 8.. Case (C-): Small sized devie When the size of the devie is extremely small, the number of states in the robe of the wave funtion beomes small. For examle, when nm y α π e for 6[T] and [ nm] (8.) Therein the number of effetive Landau orbitals is three in the left side and three in the right side. At ν the number of the allowed transitions is / times that of Landau orbitals and therefore the number of effetive quantum transitions is only two. When the devie size and the magneti field strength are more and more small, the lateau of the all resistane disaears in the FQE. y ontrast the IQE is observed in a devie with suh a small size beause the ga energy is rodued in the single eletron system. 8.. Case (C-): Shae effet for the all resistane Usually quantum all devies have a retangular shae. Therein all the Landau wave funtions have the same length along the x-diretion (diretion of the eletri urrent). In this sub-setion, we examine how the shae of a devie affets the FQE. Three tyes of quantum all devie are illustrated in Fig.8.a, b and. Therein the boundary onditions are different from eah others in Figs.8.a, b and. y x all robe Soure Drain Fig.8.a: Right triangular shae Potential robes Fig.8.b: Isoseles triangular shae 8

183 y x all robe Soure Drain Potential robes Fig.8.: Ellitial irular shae The length of the wave funtion along the x-diretion is deendent uon the y-osition for the devies with the shaes as in Fig.8.a, b and : y (8.5) ( ) The momentum of the Landau state is related to the length as given by Eq.(.5b). [ π ( y) ] J J So the differene between adaent two momenta is given by [ π ( y) ] J J (8.6) The momentum-interval J J takes different values at the different y-ositions. Thus the shae of a quantum all devie affets the momentum onservation law. So the lateaus in the FQE may be smeared or disaear for quantum all devies with the shaes as Figs.8.a, b, and. It is interesting to examine how the all resistane urve deends uon the shae and the size of a quantum all devie. 8

184 Chater 9 Sin olarization in the frational quantum all states V. Kukushkin, K. von Klitzing, and K. Eberl [77] have measured the eletron sin olarization of the quantum all states. The olarization urves are shown in Figs.9.a-d. Their exerimental results reveal very imortant roerties of the FQ sates. The six filling fators /, /7, /5, /9, 8/5 and / have numerators with even integers resetively. Therein the sin olarization is nearly equal to zero u to eah ritial fields, above whih it begins to inrease as shown in Fig.9.. On the other hand, the six filling fators /, /, /5, /7, 7/5 and / have the numerators,,,, 7 and resetively whih are odd integers. At these filling fators, the olarization urve deends linearly uon the magneti field for a small magneti field region. Thus the sin olarization urves are qualitatively deendent on whether the numerator of the filling fator is even or odd. This feature is in ontrast to the eletron-hole symmetry seen in the all resistane. For examle a lateau in the all resistane aears at both ν /5 and ν /5. The binding energies at ν /5 and ν /5 have the eletron-hole symmetry. owever the sin olarization urve at ν /5 is quite different from that at ν /5. Similar differene aears between ν /7 and ν /7. Furthermore the ν /5 olarization urve behaves like the ν /7 olarization urve where the numerator of the filling fator is the same as eah other. Fig.9.a Fig9.b 8

185 Fig.9. Fig.9.d Fig.9. Magneti field deendene of sin olarizations at filling fators The numerator means the eletron number er unit-onfiguration. The fat indiates that the sin olarization belongs to only eletrons (not holes). ereafter we exress the eletron sin olarization by the symbol γ. Then, γ e e means a fully olarized state. The olarization urve measured in the exeriment has a wide lateau at γ for ν /7 and ν /9, at γ for ν /5 and ν /7 e and so on. The small shoulders aear at for ν /5 and ν /7, at e e e γ for ν / and ν /5, at γ γ and γ for ν /7 and ν /9, and at γ for ν /. In this hater, we exlain these interesting behaviors of the sin e olarization by solving the eigen-energy roblems of the Coulomb interations between the first and seond nearest eletron airs with u and down sins [78-8]. The solutions are obtained in this hater and the theoretial results are in good agreement with the exerimental data. In the revious haters -8 we have investigated a high magneti field region where eletron sins are fully olarized. In this hater we study the ase where some eletrons have u-sin and the others have down-sin. There are many ossible e e 85

186 sin-arrangements for a given eletron-onfiguration in the Landau orbitals. When we take the eletron onfiguration to be the most uniform one, all the sin-arrangements in the onfiguration have the same minimum eigen-energy of D. Therefore we annot aly a usual erturbation method of non-degenerate ase to solve the roblem. The interation amiltonian I yields quantum transitions between the ground states with different sin arrangements. Aordingly we must exatly solve the eigenvalue roblem of the Coulomb transitions among the degenerate ground states. These Coulomb transitions are equivalent to the sin exhange transition as will be verified below. We sueed to solve the eigen-value roblem of the first and seond nearest sin exhange interations. The sin olarization is determined by using the exat solutions [78-8]. The theoretial results of the sin olarization are in good agreement with the exerimental data. We examine in details the exerimental results of Kukushkin et al whih exhibit wide lateaus and small shoulders in the olarization urves. The small shoulders are studied in Ses.9.7 and 9.8 [8, 8]. 9. Coulomb interation between u and down sin states The degenerate ground-states have the same momentum set orresonding to the most uniform eletron-onfiguration and have various sin arrangements different from eah others. We study the matrix elements of I among the ground states of D. The interation amiltonian I given by Eq.(.) ats between two eletrons where the initial states are desribed by the momentum air,. We indiate the sin states by and for u and down sins, resetively. Then all the initial sin-states are desribed as,,,,,,, (9.) When these states transfer via I, their final states are desribed as follows:,,,,,,, (9.) where and indiate the final momenta via the Coulomb interation. We onsider only the transitions between the degenerate ground states so that the final momentum set should have the minimum energy of D. Aordingly the final momentum set is equivalent to the initial momentum set, and then, (9.) where the ase of (, ) is removed beause the diagonal matrix 86

187 87 elements of I are zero. We aly Eq.(9.) to the final state,. Then the final state beomes, whih is the same as the initial state. Also, beomes,. These two ases give the final state idential to the initial state and so the matrix elements of I are zero as defined in hater. Aordingly non-zero matrix elements are I,, (9.a) I,, (9.b) We will alulate these two matrix elements as in Eqs.(9.5-). ut these alulations are slightly long. If the reader is not interested in the alulation, she/he may ski Eqs.(9.5-). The matrix element (9.a) is obtained by using Eq.(.a) as follows: ( ) ( ) ( ) ( ) ( ) ( ) ( ) I d d d d d d,,,,,,,,,, z y x z y x z y x z y x z z y y x x e z y x z y x ψ ψ πε ψ ψ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) d d d d d d ex ex ex ex ex ex z y x z y x z z e y e e y e x i x i z z y y x x e z z e y e e y e x i x i u φ φ πε φ φ (9.5) Therein we have used Eq.(.7). Another matrix element I,, is the same as in Eq.(9.5). I I,,,, (9.6) The wave funtion ( ) z φ may be different from devie to devie beause the otential shae in the z-diretion is samle deendent. Then the wave funtion may be

188 88 aroximated by a Gaussian form as, ( ) z fe z β φ (9.7) where f and β are onstant arameters deending on the samle. Substituting Eq.(9.7) into Eq.(9.5), we obtain the following relation: ( ) ( )( ) ( ) ( ) ( ) ( ) I d d d d d d ex ex ex ex ex,, z y x z y x e e e y e e y e z z y y x x e e y e e y e x x i uf βz βz πε ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) d d d d d d ex ex ex ex ex z y x z y x e y e e y e e e z z y y x x e e y e e y e x x i uf z z z z β β πε (9.8) ( ) ex ex ex ex ex ex y y e e y y e e y e y e y e y e e y e e y e e y e e y e (9.9) We rewrite Eq.(9.8) by using Eq.(9.9) as follows: ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) I d d d d d d ex ex ex,, z y x z y x e e e e y y e y y e e e z z y y x x e x x i uf z z z z β β πε

189 89 (9.) The onstant f is determined from the normalization ondition as ( ) ( ) ( ) d d d z - z - z - β π ϕ ϕ β β β f z e f z fe fe z z z π β f (9.) Also the onstant u is determined as π e u (9.) whih is derived from d ex d ex ex e u y e y e u y e y e e y e u π Then, we have, ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) I d d d d d d ex ex ex,, z z z z y y y y x x e e e e y y e y y e e e z z y y x x e x x i uf z z z z β β πε The variables for the integration are transformed as, ( ) ( ) { } y y y y y, ( ) ( ) { } y y y y y ( ) ( ) { } z z z z z, and ( ) ( ) { } z z z z z. The integration variables are hanged as follows: ( ) ( ) d d d d y y y y y y and ( ) ( ) d d d d z z z z z z. Integrating the funtion by y y and z z we obtain the following equation; ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) I d d d d ex ex ex,, z z y y x x e e e e y y e e z z y y x x e x x i uf z z π β π πε β

190 9 ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) I d d d d ex ex ex,, z z y y x x e e e y y e e z z y y x x e x x i uf z z β πε (9.) Therein we have used Eqs.(9.) and (9.). We relae the two variables y y and z z by Y and Z, resetively as ( )( ) y y e Y, ( ) z z Z β (9.) Then the matrix element of Eq.(9.) is re-exressed as follows; ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) Z Y e x x e e e Y e Z e Y x x e x x i uf Z d d d d ex ex ex,, I β β πε ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) Z Y x x e e Y e Z e Y x x e x x i Z d d d d ex ex ex,, I π π β πε (9.5) Thus the momentum deendene of the matrix element has been exressed by the integration (9.5). We examine the matrix element among the degenerate ground states for the three momentum differenes π π π 6 and,, ± ± ±, whih is denoted by and,, ς η ξ, resetively: Case A : π ± ( ) ( ) ( ) ( ) ( ) ( ) ( ) π π π π β πε π ξ for d d d d ex ex ex,, I ± ± Z Y x x e e Y e Z e Y x x e x x i Z (9.6a)

191 In order to distinguish the Case from Case A, momentum suffixes and are relaed with and as follows: Case : π η πε ±, ( x x ) ( e) Y ( β ) π ex e e I e, dxdx dy π ( ) ex( ± iπ ( x x ) ) Z e Z ex dz for π ( Y ) (9.6b) ± π In the ase C, the two momentum suffixes are exressed by 5 and 6 as: Case C : 6π ς πε 5 6 ± 6, ( x x ) ( e) Y ( β ) 5 6π ex e e 5 6 I e 5, 6 dx5dx 6 dy π ( ) ex( ± i6π ( x x ) ) Z e Z ex dz for π ( Y ) ± 6π (9.6) The Coulomb transition in the ase A is illustrated in Fig. 9.a. The transition is shown by arrow airs in the left anel. The oen irle indiates the u sin state and the filled one the down sin state. The momenta after the transition are exressed by the symbols and whih are given by π π It is noteworthy that the final sin is unhanged via the Coulomb transition. Only the final momenta are hanged. Coulomb transition Initial state Final state u sin state down sin state Fig.9.a: Equivalene of seifi Coulomb transition and sin exhange interation for ase A 9

192 The final state is illustrated in the right anel of Fig. 9.a. This Coulomb transition is equivalent to the following roess: The sin at site flis from u to down and the sin at site flis from down to u simultaneously without hanging the momenta. That is to say the Coulomb transition of the ase A is equivalent to a sin exhange roess whih is desribed by the interation σ ξσ. Therein σ is the sin transformation oerator from down to u-sin state and σ is the adoint oerator of σ. There is another Coulomb transition given by Eq.(9.b) whih is equivalent to σ ξσ where the ouling onstant takes the same value as in σ ξσ beause of Eq.(9.6). Aordingly the Coulomb transition between the two eletrons at sites and is exressed as ξ ( σ σ σ ) σ (9.7) where ξ was already defined by Eq. (9.6a). In this Coulomb transition, the lassial Coulomb energy of the initial state is exatly equal to that of the final state. Next we examine the Coulomb transition of the ase whih is shown by Fig.9.b. Coulomb transition Initial state Final state Fig.9.b: Equivalene of seifi Coulomb transition and sin exhange interation for ase The momenta after the transition are desribed by the symbols,, the values of whih are given by π π 9

193 The Coulomb interation of this ase is equivalent to the following sin exhange interation between eletrons laed in the seond nearest-neighboring orbital air: ( σ σ ) η σ σ where η is the ouling onstant defined by Eq.(9.6b). (9.8) The Coulomb transition of the ase C is illustrated in Fig.9.. The momenta after the transition are desribed by the symbols 5, 6 : π π Coulomb transition Initial state Final state Fig.9.: Equivalene of seifi Coulomb transition and sin exhange interation for ase C Fig.9. shows that the Coulomb interation of the ase C is equivalent to the following sin exhange interation between eletrons laed in the sites 5 and 6: ς ( σ σ σ ) σ 6 These tyes of interations give the artial amiltonian of system. (9.9) I in the quasi-d eletron Figures.9.a and 9.b exress the most uniform onfigurations of eletrons at ν and ν 5, resetively. Therein the sin-states are numbered sequentially from the left to the right as indiated in eah figure. In the ν / state the nearest and seond nearest eletron airs have the ouling onstants ξ and η, resetively. At ν /5, the nearest eletron air is laed in the seond neighboring orbitals. The ouling onstant is η. The seond nearest eletron air is laed in third neighboring orbitals and so the ouling onstant is ζ as in Fig.9.b. The sin-site number is written 9

194 sequentially from the left to the right in the lowest art of eah figure. The sin-site number is different from the orbital number. x y ξ η ξ η ξ η ξ η ξ Fig.9.a Coulomb transitions for first and seond nearest eletron airs at ν / η ζ η ζ η ζ 5 6 Fig.9.b Coulomb transitions for first and seond nearest eletron airs at ν /5 7 Note here the word nearest eletron air. In Chater -8, we have used nearest eletron air only for <ν <, and so the nearest eletron air is laed in the nearest Landau orbital air. At <ν < we have examined nearest hole airs (not eletron air). In this hater we need to onsider the nearest eletron air at <ν <. For examle the nearest eletron air at ν 5 is laed in the seond nearest neighboring orbital air as in Fig.9.b. We onsider the interation between more distant airs. For examle the interation between eletron and eletron beomes weak due to the sreening effet of the interosing eletron as in Figs. 9.a and 9.b. Thus the interation between the third nearest eletrons is weak and so may be ignored. Aordingly the most effetive interation at ν / is obtained as follows; effeive [ ( σ σ σ σ ) η( σ σ σ σ )],, ξ (9.) where the oerator σ indiates the transformation from down to u-sin state of the eletron at the ( ) -th site. This amiltonian, Eq.(9.) yields the quantum 9

195 95 transition between the degenerate ground states. eause the external magneti field yields Zeeman energy in the z-diretion, the amiltonian beomes ( ) ( ) [ ] ( ),,,, i z i g σ µ σ σ σ η σ σ σ σ σ ξ (9.) where g is the effetive g-fator, is the magneti field, ( ) z σ is the eletron sin oerator in the z-diretion and µ is the ohr magneton [8, 8]. The matries (the Pauli sin matries) are exliitly indiated below:,, z σ σ σ (9.) We an obtain the amiltonian for other filling fators. The amiltonians for ν /5 and ν /7 are given, resetively by: ( ) ( ) ( ) ( ) 5 for,,,, ν g i z i σ µ σ σ σ η σ σ σ σ η σ σ σ σ σ ξ (9.) ( ) ( ) ( ) ( ) ( ) 7 for,,,, ν σ µ σ σ σ η σ σ σ σ η σ σ σ σ η σ σ σ σ σ ξ i z i g (9.) We have obtained the most effetive amiltonians in Eqs.(9.), (9.) and (9.) for ν /, ν /5 and ν /7, resetively. They rodue the quantum transitions among the degenerate ground states of D. These amiltonians an be exatly diagonalized by using the method of referene [85]. 9. Isomorhi maing from FQ states to one-dimensional fermion states We have obtained the most effetive amiltonian namely Eqs.(9.), (9.) and (9.) for the three filling fators. The amiltonians yield the quantum transitions among the ground states of D. In this setion we find the isomorhi maing from the degenerate ground states to many-fermion states in a one dimensional system. Thereby

196 the eigen-energy roblems an be solved exatly. We first examine the following maing from a single sin state to a fermion state. The down-sin state is maed to the vauum state, and the u-sin state is maed to a state with one fermion where is the reation oerator., (9.5) Then the sin oerators σ, σ, and z σ are maed to the oerators of the fermion system as follows: σ ( ) z, σ, σ (9.6) When the three sin-oerators σ, σ, and z σ are oerated to the two sin states, the following six results are obtained: σ, σ σ, σ z σ, z σ When the three oerators,, ( ) are multilied to the two fermion states and the following results are obtained:,, ( ), ( ) where we have used the roerty of the fermion oerator. The obtained six equations are orresonding to the six relations in sin system orretly as: σ, σ 96

197 97, σ σ ( ) ( ), z z σ σ Thus we have obtained the isomorhi maing from the single sin states to the fermion states. Next we onsider the isomorhi maing from many-sin states to many-fermion states. The u-sin at the site is maed to the reation oerator of the site. The multilying order of the reation oerators is the same as the order of the u-sins. Then all the many-sin states are maed to all the many-fermion states by one to one orresondene. We write two examles of the maings as follows:,,,,,,,,,,,,,,,,,, (9.7) The oerators i and i satisfy the anti-ommutation relations as follows: i i i i, }, { δ (9.8a) }, { i i i, }, { i i i (9.8b) The maing mentioned above yields the following orresondene from the roduts of the sin oerators σ σ, σ σ, σ σ and σ σ to the roduts of the fermion oerators as:, σ σ σ σ (9.9a), σ σ σ σ (9.9b) It will be roven below that the maing of (9.9a,b) is the isomorhi maing from many-sin states to many-fermion states. The roof is done via the following three stes: namely Proerty, Proerty and Proerty. (Proerty ) The oerator is ommutable with i and i for i and i. Also the oerator is ommutable with i and i for i and

198 98 i. oth oerators and are ommutable with i and i for, i i. Examle : ( ) ( ) (9.a) Examle : ( ) ( ) (9.b) Thus the roduts of the reation and annihilation oerators in (9.9a) an be moved in front of the site. Also the roduts in (9.9b) an be moved in front of the site. Aordingly it is suffiient to rove the following equations: (Proerty ) ( ) ( ) σ σ σ σ (9.a) ( ) ( ) σ σ σ σ (9.b) ( ) ( ) σ σ σ σ (9.) ( ) ( ) σ σ σ σ (9.d) ( ) ( ) σ σ σ σ (9.e) ( ) ( ) σ σ σ σ (9.f) ( ) ( ) σ σ σ σ (9.g) ( ) ( ) σ σ σ σ (9.h) Thus it is verified that the maing of Eqs.(9.9a) and (9.9b) is isomorhi due to Eqs.(9.a, b,, d, e, f, g and h). Next the sin oerator z i σ of the z-diretion is maed as follows:

199 99 i i z i σ (9.) This maing is orret as follows: (Proerty ) ( ) i i i i i i z i σ (9.a) ( ) i i i i z i σ (9.b) The roerties Eqs.(9.-) larify that the maings Eqs.(9.9a, b) and (9.) are isomorhi. Aordingly amiltonian Eq.(9.) is equivalent to the following form: ( ) ( ) [ ] ( )( ),,,, i i i g µ η ξ (9.) We will exatly solve the eigen-value roblem of this amiltonian in the next setion. 9. Diagonalization of the most effetive amiltonian at filling fator of / It has been roven that the amiltonian Eq.(9.) is equivalent to Eq.(9.). We will solve exatly the eigen-value roblem of the amiltonian Eq.(9.). As disussed in Chater, there are two kinds of sites in the unit ell at ν. So we introdue new oerators a and b defined as follows: a, b, a, and b (9.5) where is the ell number (unit-onfiguration number). Then, the amiltonian Eq.(9.) beomes ( ) ( ) [ ] ( )( ) J J b b a a g a b a b b a b a µ η ξ, (9.6) where J is the total number of ells given by N J ( N is the total number of eletrons). We aly a Fourier transformation for the oerators a, a, b, and b, and obtain ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) i i b b a a g b a e a b e a b b a µ η ξ, (9.7)

200 where a n J ( ) e in a, b n J e in b( ) (9.8) π and integer, -π < π. J amiltonian (9.7) is the sum of the terms with the argument. We onsider the single term with a given. The term is exressed by the following matrix: µ g i ξ ηe i ξ ηe µ g. (9.9) This matrix has two eigen-values: λ ( ) and ( ) ( ) µ g ξ η os ξη λ as λ (9.a) ( ) µ g ξ η os λ (9.b) ξη We introdue new annihilation oerators A ( ) and ( ) A given by A A ( ) a( ) ( ) a( ) ξ ηe i ( ) b( ) ξ η ξη os ξ ηe i ( ) b( ) ξ η ξη os, (9.a). (9.b) Of ourse the new oerators satisfy the anti-ommutation relations as follows; i ( ), A ( q) } δ i q { A δ (9.),, i ( ), A ( q) }, { A ( ), A ( q) } { A i (9.d) Then the amiltonian Eq.(9.7) is rewritten to a diagonal form as ( ( ) A ( ) A ( ) λ ( ) A ( ) A ( ) g ) λ. (9.) µ Thus we have sueeded in diagonalizing of the amiltonian (9.). 9. Magneti field deendene of the Sin-Polarization at filling fator of / The diagonal form, Eq.(9.), means that all the eigen-states are exressed by the diret rodut of the reation oerators A s ( ) as follows:

201 Eigen-state ( ) ( ) ( ) ( ) A A A A s s s sn N Thus the eigen-state is seified by the set { n ( ) n ( ) ; π < π} n s ( ) A ( ) A ( ) s s, where. The eletron-sin olarization γ e is obtained by alulating the thermo-dynami mean value numerially: γ e N J N σ z i i N N i ( i i ) N ( a ( )a( ) b ( )b( ) ) J ( a a b b ) J ( A s ( )A s ( )) s, (9.) where is the thermal average and the minus sign omes from the negative harge of eletron. Also we have emloyed Eqs.(9.), (9.5), (9.8) and (9.). Equation (9.) means that the eigen-energy for n s ( ) is ( ) λ and the eigen-energy for n s ( ) is zero. Then the oltzmann fator is ( ( ) k T ) for n s ( ) and ex( k T ) for ( ) s ex λ s n s where k is the oltzmann onstant and T is the temerature. Aordingly, the robability for n s ( ) is given by ex ex ex ( λs( ) kt ) ( λ ( ) k T) s ( λ ( ) k T) s and the robability for n s ( ) is given by. These robabilities yield the thermal average of A s ( )A s ( ) as ( λs( ) kt ) ( λ ( ) k T) ex As ( ) As ( ) ex s, (9.) whih gives ex ( ) ( ) ( λs ( ) kt ) As As tanh( λs ( ) ( kt )). ex( λs ( ) kt ) (9.5) Substitution of Eq.(9.5) into Eq.(9.) yields γ e tanh( λs( ) k T). J s (9.6) Sine the total number of eletrons is a marosoi value, we an relae the

202 summation by an integration: γ e π π d tanh( λs( ) kt ). (9.7) π s Thus the eletron-sin olarization at the filling fator of / has been exressed by Eq. (9.7). First, we study the low field behavior of the sin olarization. Equation (9.) means λ exists in the following region: that ( ) ( ) µ g ξ η µ g ξ η λ Also ( ) λ is in the region: ( ) µ g ξ η µ g ξ η λ When the magneti field takes a value between and η ( µ g ) and ( ) λ is ositive for any value of : λ ( ), ( ) < > λ for < < ξ η ( µ g ) ξ, ( ) Therefore, tanh( λ ( ) k T ) is nearly equal to and ( ( ) k T ) λ is negative tanh λ is nearly equal to at a low temeratures ( T ). Then, the sin-olarization is almost zero beause the sum in the right hand side of Eq.(9.7) is nearly equal to zero. When the magneti field inreases beyond the value ξ η ( µ g ), the sin-olarization inreases ontinuously until it reahes the maximum value of. This theoretial roerty at ν / is in agreement with the exerimental data shown in Fig.9.. When the quality of a quantum all devie is bad, the imurities and lattie defets rodue many random-otentials. Then the lateau on the olarization urve is rounded by the random otentials. This effet resembles the roerty due to thermal vibrations. So we introdue the effetive temerature T whih reresents the sum of the random otential effet and the thermal effet. The arameter values η ξ. and ( k T ξ ). are alied to the right hand side of Eq.(9.7). Thereby the integration is numerially alulated by using a omuter rogram. The theoretial result is shown with blue urve in Fig.9.. Exerimental data [77] are also lotted by red dots on the figure. The theoretial urve rerodues exerimental data reasonably well. In the next setion, the arameter deendene of the olarization is disussed.

203 Fig.9. Calulated sin-olarization urve for., ( ). Red dots are exerimental data [77] in this figure. η ξ k T ξ at ν. 9.5 Parameter Deendene of Polarization We examine how the sin-olarization at ν deends uon the values of η ξ and k T ξ. The theoretial sin-olarization urve for the values η ξ.,.,., and. is shown in Figs. 9.5a 9.5d resetively where k T ξ is set to.. Fig.9.5a ν / η ξ. k T ξ. Fig.9.5b ν / η ξ. k T ξ. Fig.9.5 ν / η ξ. Fig.9.5d ν / η ξ. k T ξ. Fig.9.5 Deendene of sin-olarization urves on the value of η ξ for k T ξ.

204 Figures 9.5a-d show that the deendene of the olarization on η ξ is weak. When the theoretial urve is examined in details, the olarization value is almost zero in < ritial. The ritial magneti field ritial is about.9t,.5t,.t and.t in Figs.9.5a, b, and d, resetively. Thus the arameter deendene aears orretly in these figures. We next examine the deendene of the sin olarization uon the value of k T ξ, whih is shown in Figs.9.6a 9.6d for a fixed value of η ξ.. Fig.9.6a ν /,η ξ., k T ξ.5 Fig.9.6b ν /,η ξ., k T ξ. Fig.9.6 ν /,η ξ., k T ξ.7 Fig.9.6d ν /,η ξ., k T ξ.5 Fig.9.6 Deendene of sin-olarization urves on the value of k T ξ The shae of the orner in the blue urves beomes shar for a small value of k T ξ and loose for a large value of k T ξ. The arameters η ξ. and k T ξ. rerodue the exerimental data well [77]. The shae of the urve robably deends on size and quality of a devie used in the exeriment. We have already enountered similar henomena for the measurement of the all resistane.

205 9.6 Sin-Polarization for other filling fators We alulate the sin-olarization at the twelve filling fators of /, /5, /7, /5, /7, /9, 7/5, 8/5, /, /, / and / below. These twelve filling fators are lassified into the following four tyes: (Tye ) filling fators ν, 5, 7 where < ν < (Tye ) ν,, where <ν < (Tye ) ν 7, 5, 5 where ν > (Tye ) ν,, where the denominators are even integers We first alulate the sin-olarizations of Tye namely ν /, /5 and /7. The olarization at ν / has already been alulated in setion 9.. The most uniform eletron onfiguration at ν /5 and ν /7 are given in Chater. Figures 9.7a and 9.7b exress the nearest and seond nearest interations via the Coulomb transition. ξ η η ξ η η Fig.9.7a: Eletron onfiguration for ν /5 ξ η η η ξ η η η Fig.9.7b: Eletron onfiguration for ν /7 At ν /5 the most effetive amiltonian is given by Eq.(9.) as i,, [ ξ ( σ σ σ σ ) η( σ σ σ σ ) η( σ σ σ σ ) ],, µ g ( ) σ z i 5

206 6 (9.8) We an exatly solve the eigen-value roblem of the amiltonian Eq.(9.8) whih is omosed of the following matrix: g e g e g i i µ η η η µ ξ η ξ µ (9.9) The eigen-values of this matrix are desribed by the symbols ( ) λ, ( ) λ, and λ ( ). Thereby the eletron sin-olarization for ν /5 is given by γ e J tanh λ s ( ) k T ( ) s 6π d tanh λ s ( ) k T ( ) s π π. (9.5) The olarization is alulated by using a omuter rogram. We set the values to. ξ η and. ξ T k same as in the ase of ν /. The theoretial olarization urve at ν /5 is obtained as in Fig.9.8b whih agrees well with the exerimental data [77]. It is noteworthy that the olarization for ν /5 is almost roortional to the magneti field strength near zero fields. This roerty is different from that of ν /. The amiltonian for ν /7 has been already obtained by Eq.(9.) as ( ) ( ) ( ) ( ) ( ),,,, i z i g σ µ σ σ σ η σ σ σ σ η σ σ σ σ η σ σ σ σ σ ξ (9.5) The four eigenvalues ( ) λ, ( ) λ, λ ( ), and λ ( ) are derived from diagonalization of the following matrix g e g g e g i i µ η η η µ η η µ ξ η ξ µ (9.5) The value of η ξ. is also alied for ν /7, and then the sin-olarization urve is alulated as shown in Fig.9.8. ere we didn t use the best fitting arameters. In site of using the same value namely ξ η., the three theoretial urves of the sin-olarization show good agreement with the exerimental data as in Figs.9.8a.

207 Fig.9.8a ν /, η ξ., k T ξ. Fig.9.8b ν /5, η ξ., k T ξ. Fig.9.8 ν /7, η ξ., k T ξ.5 Fig.9.8d ν /5, ζ η.5, k T η. Fig.9.8e ν /7, ζ η., k T η.8 Fig.9.8f ν /9, ζ η., k T η.6 Fig.9.8g ν 7/5, τ ξ., k T ξ.5 Fig.9.8h ν 8/5, κ τ., k T τ.5 7

208 Fig.9.8i ν /, τ ξ., k T ξ.5 Fig.9.8 ν /, k T υ.7 Fig.9.8k ν /, k T η.5 Fig.9.8l ν /, k T τ Fig.9.8a-l: Calulated sin-olarization urves (Red dots are exerimental data [77]) We study the ases of Tye, suh as ν /5, /7, and /9. The eletron onfiguration with the minimum lassial Coulomb energy is given in Chater for ν /5, /7, and /9. Figures 9.9a exress the most uniform onfiguration. The nearest eletron air has the ouling onstants η and the seond nearest eletron air has the ouling onstant ζ. These ouling onstants have already examined as in Eqs.(9.6b), (9.6), (9.8) and (9.9). η ζ η η ζ ζ Fig.9.9a: Eletron onfiguration for ν /5 η η ζ η η ζ Fig.9.9b: Eletron onfiguration for ν /7 8

209 η η η ζ η η η ζ Fig.9.9: Eletron onfiguration for ν /9 We alulate the sin olarization of the filling fator ν /9. The most effetive amiltonian is given by the following matrix form: µ g η i ζe µ g η η η µ g η i ζe η µ g (9.5) This matrix has four eigen-values λ ( ), ( ) sin-olarization γ e is given by λ, λ ( ), and λ ( ). The γ e J ( λ ( ) k T ) d tanh( λ ( ) k T ) tanh s s π s 8π π s (9.5) Similarly, we get the sin-olarization for ν /5 and /7. The theoretial value of γ e is drawn by blue urves in Figs.9.8d, e and f for ν /5, /7, and /9, resetively. These theoretial results well rerodue the exerimental data. We examine the ase of Tye where the filling fators are larger than, suh as ν 7/5, 8/5, and /. Figures 9.a, 9.b, and 9. show the eletron-onfigurations with the minimum lassial Coulomb energy. Therein a double-line indiate a Landau orbital ouied by two eletrons, one of whih has u sin and the other down sin. Single bold line indiates a Landau orbital ouied by an eletron. The sin exhange interations at only between eletrons laed in singly ouied orbitals, beause sins in the doubly ouied orbitals annot fli owing to the Pauli rinile. As an examle, the ouling onstants of the sin exhange interations are illustrated by ξ and τ in Fig.9.a for the filling fator of 7/5. There are three singly ouied orbitals and two doubly ouied orbitals in every unit-onfiguration. Therefore three eletrons an fli er seven eletrons inside a unit-onfiguration. 9

210 ξ τ ξ τ τ τ Fig. 9.a: Eletron onfiguration for ν 7/5 τ κ τ τ κ κ Fig. 9.b: Eletron onfiguration for ν 8/5 τ τ ξ ξ ξ τ Fig. 9.: Eletron onfiguration for ν / The symbol τ reresents the ouling onstant between the eletrons in two orbitals searated by a doubly ouied orbital. Then, the amiltonian for ν 7/5 is ( ) ( ) ( ) [ ] ( ),,,, i z i g σ µ σ σ σ σ τ σ σ σ σ τ σ σ σ σ ξ (9.55) The eigenvalues of the amiltonian are obtained by diagnalizing the following matrix. g e g e g i i µ τ τ τ µ ξ τ ξ µ (9.56) The three eigen-values of the matrix Eq.(9.56) are desribed by ( ) λ, ( ) λ and λ ( ). Using these eigen-values, the sin-olarization is exressed as follows: γ e 7 J tanh λ s ( ) k T ( ) s 7 π d tanh λ s ( ) k T ( ) s π π. (9.57) ere seven eletrons exist in a unit-onfiguration. The oeffiient /7 in the right hand side of Eq.(9.57) means that the ontribution of one eletron is /7 times full

211 olarization for every unit-onfiguration. The ouling onstant τ is weakened by the sreening effet of the interosing eletron airs. Aordingly the ouling onstant τ beomes smaller than η. So we set τ ξ. whih is smaller than the value of η ξ.. The theoretial urve of the olarization is drawn in Fig.9.8g. Similarly we alulate the olarizations at ν 8/5 and /, the theoretial urves of whih are drawn in Figs.9.8h and 9.8i, resetively. These figures show that the theoretial sin-olarization agrees well with the exerimental data [77]. We next investigate the ase of Tye with ν /, /, and /. In this ase no lateau aears in the all resistane urve. Figures 9.a, 9.b, and 9. indiate the most uniform eletron-onfigurations for the filling fators of /, /, and /, resetively. These onfigurations show that the most effetive amiltonian is omosed of only one kind of the sin exhange interation as in Figs.9.a, 9.b, 9.. Then the sin-olarization is desribed by only one kind of eigen-energy. The theoretial results of the sin-olarization are shown in Figs l whih are also in agreement with the exerimental data. υ υ υ Fig.9.a: Eletron onfiguration for ν / η η η η η Fig.9.b: Eletron onfiguration for ν / τ τ τ τ Fig.9.: Eletron onfiguration for ν / Thus, the resent theory has satisfatorily rerodued the twelve urves of the exerimental sin olarization.

212 There is an interesting relation between the shae of the olarization urve and the dimension of the matries in Eqs.(9.9), (9.9), (9.5), (9.5), (9.56) and so on. The struture of energy gaes deends on the dimension of the matries. Therefore the dimension determines the shae of the olarization urves. For ν < the dimension is equal to the numerator of the filling fator. So the dimension beomes four at ν /7 same as at ν /9. Thereby the shae of the olarization urve at ν /7 resembles that at ν /9. Also the shae of the olarization urve at ν /5 resembles that at ν /7 and so on. The harater derived from the theory is onsistent with the exerimental data of olarization. Thus the sin olarization is originates from eletron only, not hole. Eseially the following harater aears. When the numerator of ν is an odd integer, the sin olarization is almost roortional to the magneti field strength in the neighborhood of. On the other hand, when the numerator of ν is an even integer, the sin olarization is almost zero u to a ritial value of the magneti field. 9.7 Small shoulder in Polarization Curve If we arefully observe the exerimental sin olarization urve (Fig.9.), then we find small shoulders in it. The small shoulders don t aear in the results of the revious setions. So we need to find a new origin of the small shoulders. [8, 8] R. E. Peierls studied an eletron system in a one dimensional rystal and onsidered the lattie distortion with the eriod doubling the unit ell. The lattie distortion rodues new band gas. Thereby the total energy beomes lower than that without the distortion. This effet is alled sin Peierls effet [86]. The henomena were observed in organi omounds [87-89] and inorgani hain omounds suh as TTF-CuS C (CF ), (MEM)-(TCNQ) and CuGeO [9]. In the resent theory, the sin olarization of FQ states is derived from the amiltonians (9.), (9.), (9.) and so on. If we onsider a new distortion with the double eriod of the unit onfiguration, the sin hain amiltonian of FQ system resembles that of the sin-peierls effet. For an examle ν we hange the distane between nearest orbitals in the first unit-onfiguration longer, the one in the seond unit-onfiguration shorter and so on. Then we have the four kinds of the ouling onstants ξ, ξ, η and η as shown in Fig.9.. The value of ξ is larger than that of ξ beause the distane for the ξ interation ath is shorter than that for ξ. Also, η > η holds. This distortion with the double eriod of the unit-onfiguration rodues additional energies. We all the distortion interval modulation. [8, 8]

213 ξ η ξ' η' ξ η ξ' η' Fig.9. Couling onstants of interations aused by distortion with double eriod 9.7. Distortion-deendene of the total energy We exress the distane between nearest orbitals by the symbol r for non-distorted ase. We onsider the distortion (interval modulation) where the orbital interval beomes r d for an odd number of unit-onfiguration and r d for an even number. Thereby the lassial Coulomb energy W inreases. (Simle examle of the funtion form): Three eletrons A, and C are laed on the straight line. When the distane between A and is r d and the distane between and C is r d, then the lassial Coulomb energy is given by ( ) d [ πε ( r d) ] e /[ πε ( r d) ] e /[ πε r ] e /[ πε r ] e / r Although this examle is oversimlified, the inreasing of W in the real ase is also roortional to d. Aordingly the inreasing value er eletron is given by ( d ) W N f (9.58) C r where f C is the onstant arameter whih is deendent on a quantum all devie. Next we examine d -deendene of the ouling onstants ξ and ξ. When d >, the ouling onstant ξ is weaker than ξ beause the ξ interation ath is longer than that of ξ. When d <, ξ is stronger than ξ beause the ξ interation ath beomes shorter than that of ξ. So there is a linear term of d in ξ and ξ as follows: ( d r ), ξ ξ f ( d r ) ξ f (9.59) ξ ξ ξ

214 where ξ is the ouling onstant in the non-distortion ase and ξ f is the roortionality onstant. In order to simlify Eqs.(9.58) and (9.59), we define a new dimensionless quantity t as t ( ) ξ ξ f ( ) r d (9.6) Then the ouling onstants ξ and ξ is exressed as ( ) ( ) t t, ξ ξ ξ ξ (9.6) The inreasing value of the lassial Coulomb energy W is also exressed by this dimensionless quantity t as follows: t C N W ξ (9.6a) where C is the dimensionless oeffiient as C ξ ξ f f C (9.6b) Now we alulate the sin exhange energy. The ouling onstants for ν are illustrated in Fig.9. and the sin exhange amiltonian is obtained as ( ) ( ) [ ] ( ) ( ) [ ] ( )( ),,,,,, z z z z g σ σ σ σ µ σ σ σ σ η σ σ σ σ ξ σ σ σ η σ σ σ σ σ ξ, (9.6) This amiltonian Eq.(9.6) is rewritten from Eqs.(9.9a) and (9.9b) as, ( ) ( ) [ ] ( ) ( ) [ ] ( )( ),,,,,, i i i g µ η ξ η ξ (9.6) Using the ell number, we introdue new oerators a a a a,,,,,,, as follows: a,, a,, a,, a, (9.65) Fourier transformation yields a, J e i a ( ), a, J e i a ( ), a, J e i a ( ), a, J e i a ( ), (9.66)

215 5 where J is the total number of unit-ells (unit-onfigurations) and ( ) integer J π ( ) π π < -. Substitution of Eqs.(9.65) and (9.66) into (9.6) gives ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) i i a a a a a a a a g a a e a a e a a a a a a a a a a a a µ η ξ η ξ. (9.67) For a given value of, the term in Eq.(9.67) is exressed by the following matrix M : g e g g e g M i i µ ξ η ξ µ η η µ ξ η ξ µ. (9.68) The four eigen-values of M are denoted by the symbols ( ) λ, ( ) λ, λ ( ) and ( ) λ ( ) λ λ λ λ whih are obtained by ( ) ( )( ) ( ) ( ) ( ) ( ) g os ξ ξ α ξ ξ α ξ ξ α ξ ξ α µ λ ( ) ( )( ) ( ) ( ) ( ) ( ) g os ξ ξ α ξ ξ α ξ ξ α ξ ξ α µ λ ( ) ( )( ) ( ) ( ) ( ) ( ) g os ξ ξ α ξ ξ α ξ ξ α ξ ξ α µ λ ( ) ( )( ) ( ) ( ) ( ) ( ) g os ξ ξ α ξ ξ α ξ ξ α ξ ξ α µ λ (9.69) where ξ η ξ η α (9.7a) Therein we have assumed η η ξ ξ beause the interval modulation is exeted to give almost the same effet for the ouling onstants ξ and η. The ratio ξ ξ is equal to η η beause of Eq.(9.7a): η η ξ ξ β (9.7b) This ratio is exressed by the distortion arameter t by using of Eq.(9.6): ( ) ( ) t t η η ξ ξ β (9.7) We show the two eigen values ( ) λ and ( ) λ by red and blak urves for β,

216 blue urves for β. and green urves for β. in Fig.9.. Fig.9. Eigenvalues of sin interation via interval modulation The red and blak urves indiate the eigenvalues λ ( µ g ) and ( µ g ) The blue urves indiate the eigenvalues λ ( µ g ) and ( µ g ) The green urves indiate the eigenvalues λ ( µ ) and ( µ ) g λ for β.. λ for β.. λ for β.. g The differene λ ( ) λ ( ) is minimal at π equal to (namely non-distorted ase), the energy ga λ ( π ) λ ( π ) Eq.(9.69) gives the differene between λ ( ) and λ ( ) at π as ( π ) λ ( π ) α ξ ξ α t ξ as seen in Fig.9.. When β is disaears. λ (9.7) where we have used Eq.(9.6). Thus the energy ga is roortional to t. The eigen-energies Eq.(9.69) yield the diagonal form of the amiltonian as follows: λ ( ) A ( ) A ( ) λ( ) A ( ) A ( ) λ ( ) A ( ) A ( ) λ ( ) A ( ) A ( ) µ g. (9.7) Alying Eq.(9.) we get ex ( ) ( ) ( λs ( ) kt ) As As ( tanh( λs ( ) ( kt ))) (9.7) ex λ k T ( ( ) ) The thermal average of the amiltonian (9.7) is s s λ s s g µ (9.7) ( ( ) ( tanh( λ ( ) ( k T )))) Sine the total number of eletrons is a marosoi value, we an relae the 6

217 summation by integration as N π π ( λ ( ) ( ( ( ) ) s tanh λs kt )) s µ g d (9.75a) ( W N ) ( N ) E Total N (9.75b) We numerially alulate the integration in Eq.(9.75a). The alulated result of the sin exhange energy at.[ T ] is drawn by the dashed red urve in Fig.9.. The dashed blak urve exresses the lassial Coulomb energy for the arameter C. 5 (see Eq.(9.6a)). The sum of the two urves gives the total energy N shown E Total by the blue urve. The total energy takes a minimum value at a non-zero t as in Fig.9.. So the minimum oint is realized in a low temerature, and then the interval modulation ours atually. [8, 8] Fig.9. Deendene of total energy uon t Further alulations are arried out for various values of the magneti field in the ase of C. 5. The results are shown in Fig.9.5. The interval modulation ours in the region of.87[ T ] < <.5[ T ] as seen in Fig

218 8 Fig.9.5 Deendene of total energy uon t We alulate the total energy for another ase ν /5. The ouling onstants are illustrated in Fig.9.6. ξ η η ξ' η' η' ξ η η ξ' η' η' Fig.9.6 Couling onstants for ν /5. The interations in Fig.9.6 rodue the ν /5 amiltonian. The amiltonian is exressed by the following matrix for a given wave number : g e g g g g e g i i µ η η η µ ξ ξ µ η η µ η η µ ξ η ξ µ (for ν /5) (9.76) This matrix has the six eigen-values ( ) λ, ( ) λ, ( ) λ, ( ) λ, ( ) 5 λ and ( ) 6 λ, the -deendenes of whih are shown in Fig.9.7.

219 Fig.9.7 -deendene of the six eigen-values for ν /5 Figure 9.7 indiates the energy gas between λ ( ) and λ ( ) and between λ ( ) λ at π. These gas are aused by the interval modulation. The total and ( ) 6 energy er eletron is equal to the sum as Eq.(9.77): 5 W N ξ (9.77a) C t N 6 π π ( λ ( ) ( ( ( ) ) s tanh λs kt )) 6 s µ g d (9.77b) ( W N ) ( N ) E Total N (9.77) The sum of W N and N is numerially alulated by a omuter. The result is shown in Fig

220 Fig.9.8 Deendene of the total energy uon t at ν /5 Therein we have used the arameter C. 5 whih is the same as in the ase of ν /. The total energy takes the minimum value at the dots in Fig.9.8. The t -value at the minimum energy deends uon the magneti field. The non-zero t -value aears in the region.5[ T ] < <.[ T ] for the exeriment [77] where the interval modulation ours atually Sin olarization in the ase with the interval modulation We alulate the sin-olarization γ e for the ase with the interval modulation. At ν, γ e is obtained by the integration as γ e π π d tanh( λs ( ) kt ) (9.78) π s where the four eigen-values λ ( ), λ ( ), λ ( ) and ( ) λ are given in Eq.(9.69). The sin-olarization γ e is numerially alulated via the following two methods namely easy method A and reise method. (Method A) Method A is the rough alulation using the fixed value of the distortion arameter t. That is to say the ratio ξ ξ η η is treated to be a onstant value for all the strength of magneti field. We use the fixed value ξ ξ η η.. The other arameters are adoted to be η ξ η ξ. and ( k T ξ ).5. Then a small energy ga aears between λ

221 and λ. We numerially alulate the integration in Eq.(9.78) and draw the grah of sin-olarization versus magneti field. A small shoulder aears in the theoretial urve of the eletron sin-olarization as seen in Fig.9.9. Fig.9.9 Method A: Theoretial urve of the sin-olarization for ν (Red dots are exerimental data [77]) This urve is slightly different from the exerimental data near the shar orners P and Q. So we hoose the different value as ( k T ) ξ. in order to make the urvature to be small in the orners P and Q. Then the small shoulder disaears. In order to maintain the existene of the small shoulder, we take a larger value ξ ξ η η.8. The alulated urve is exressed in Fig.9. whih is also slightly different from the exerimental data. Aordingly method A has some diffiulty to exlain the exerimental data. This inadequay is imroved by using the reise method. Fig.9. Method A: Calulated urve of the sin-olarization for ν (The arameters are hosen as ( T ) k. and ξ ξ η η.8) ξ

222 (Method ) We arry out more reise method where we alulate the t -deendene of the total energy er eletron. Some examles have been already shown in Figs.9.5 and 9.8. Therein we an obtain the t -value with the energy minimum as shown by dots. The t -value at the minimum oint gives the value of ξ ξ η η by Eq.(9.7). Using the magneti field deendene of sin-olarization. t -value, we an numerially alulate the Fig.9. Method : Calulated urve of the sin-olarization for ν η, ( k T ) Therein we have used the arameter-values ξ η ξ. ξ. and C.5. Figure 9. exresses that the alulated result is in good agreement with the exerimental data. The reason is simly disussed below: The magneti field is suffiiently strong near the orner P in Fig.9.9. In this region, almost all the sins have a down diretion. Then the number of u and down sin-airs dereases and so the sin exhange energy beomes small. Aordingly the total energy Eq.(9.75b) is nearly equal to the lassial Coulomb energy whih yields the energy minimum at t, namely non-distortion (non-interval-modulation). Thus the distortion aears only near the small shoulder. So the theoretial urve emloying method is in good agreement with the exerimental data. The reader may want to know how the shae of the olarization urve deends on the arameter C. We alulate the olarization urve for the following two ases C. and.65 in Fig.9.. Then the shae of the urve varies in only the neighborhood of the

223 small shoulder when hanging the arameter C. The alulated shoulder is too large for C. and too small for C.65. The alulated result for C the exerimental data as in Fig agrees with Fig.9. Sin-olarization via method for two ases with C. and.65 We next study the ase of ν /5. The sin-olarization γ e at ν /5 is given by γ e 6 π π 6 d tanh( λs ( ) kt ) (9.79) π s where the six eigen-energies λ for s,,,,5, 6 are numerially obtained from the s matrix (9.76). The two alulated urves via Method A or are shown in Fig.9., resetively. In the method we have alied C.5 same as in ν. The theoretial result via method is in good agreement with the exerimental data as seen in the right anel of Fig.9.. (Method A) (Method ) Fig.9. Sin-olarization for ν 5 (Red dots are exerimental data [77])

224 We next examine the ase of ν /7. The most uniform eletron onfiguration is illustrated in Fig.9.. Therein η ξ η ξ,,, are the four kinds of the ouling onstants. ξ η η ξ' η' η' ξ η η ξ' η' η' η η η' η' Fig.9. Couling onstants for ν /7. The eletron-onfiguration yields the ν /7 amiltonian desribed by the following matrix (9.8): g e g g g g g g e g i i µ η η η µ η η µ ξ ξ µ η η µ η η µ η η µ ξ η ξ µ (forν /7) (9.8) The sin-olarization γ e is given by ( ) ( ) π π λ π γ 8 tanh d 8 s s e T k (9.8) where s λ for 8,,,,5,6,7, s indiate the eigen-energies of the matrix (9.8).

225 (Method A) (Method ) Fig.9.5 Sin-olarization for ν 7 (Red dots are exerimental data [77]) The sin olarization an be evaluated from the eigen-energies. The results are shown in Figs.9.5. Method has used the same value.5 for the arameter C. 9.8 Phenomena similar to the sin-peierls effet at various filling fators We examine the ases of ν /5, ν /7, and ν /9 whih are lassified into Tye in setion 9.6. When the interval modulation ours, the most uniform eletron-onfigurations at ν /5, ν /7, and ν /9 beome like Figs.9.6a, 9.6b and 9.6, resetively. η ζ η' η ζ ' ζ η' ζ ' Fig.9.6a Couling onstants for ν /5 η η ζ η' η' ζ ' η η ζ η' η' ζ ' Fig.9.6b Couling onstants for ν /7 5

226 6 η ζ ' η η' ζ η η' η' η ζ ' η η' ζ η η' η' Fig.9.6 Couling onstants for ν /9 The ouling onstants yield the sin-exhange amiltonians for ν /5, ν /7, and ν /9 whih are reresented by the following matries, resetively. g e g g e g i i µ η ς η µ ς ς µ η ς η µ (for ν /5) (9.8a) g e g g g g e g i i µ η ς η µ η η µ ς ς µ η η µ η ς η µ (for ν /7) (9.8b) g e g g g g g g e g i i µ η ς η µ η η µ η η µ ς ς µ η η µ η η µ η ς η µ (forν /9) (9.8) The average value of η and η is exressed by η. Then Eq.(9.7) gives ( ) ( ) t t, η η η η (9.8) The ratios between the ouling onstants satisfy the following relations similar to Eq.(9.7). η ς η ς (9.8a) ( ) ( ) t t ς ς η η (9.8b) We re-exress the t-deendene of the lassial Coulomb energy by using the ouling

227 onstant η W N η (9.85) D t where D is a new dimensionless oeffiient. Using the eigen-values of the matries (9.8a, b, ), the sin olarizations is given by γ e γ e 6 γ e 8 π π π π d tanh( λs ( ) kt ) (for ν /5) (9.86a) π s π 6 d tanh( λs ( ) kt ) (for ν /7) (9.86b) π s π 8 d tanh( λs ( ) kt ) (for ν /9) (9.86) π s We numerially alulate the sin-olarization urves via method A and, the results of whih are shown in Figs. 9.7, 9.8 and 9.9. Therein the small shoulders originate from the interval modulation (distortion with double eriod). The theoretial result by Method is in better agreement with the exerimental data than that by Method A. (Method A) (Method ) Fig.9.7 Sin-olarization for ν 5 (Red dots are exerimental data [77]) 7

228 (Method A) (Method ) Fig.9.8 Sin-olarization for ν 7 (Red dots are exerimental data [77]) (Method A) (Method ) Fig.9.9 Sin-olarization for ν 9 (Red dots are exerimental data [77]) ere we shortly disuss the arameter values C and D. The inrease of the lassial Coulomb energy are exressed in Eqs.(9.6a) and (9.85) as W N ξ and C t W N η D t, resetively. So the arameter D may be almost equal to D C ξ η (9.87a) The ratio η ξ η ξ η ξ is.5 as in Method of Figs.9. and 9.5. Substitution of the value.5 and C. 5 into Eq.(9.87a) gives the value of D as D.5.5. (9.87b) The fitting values of D are.5 and. for ν /7 and /9, resetively as in Figs.9.8 8

229 and 9.9. The fitting values of D are onsistent with the redited value namely Eq.(9.87b). The arameter D at ν /5 is.7 as in Fig.9.7. We annot understand why the arameter D is small at ν /5. Next we examine the ν /, 7/5, and 8/5 states whih have been lassified into Tye. The most uniform eletron-onfiguration and the ouling onstants are shown in Figs.9.a, 9.b and 9., resetively. ξ τ ξ τ ξ τ ξ τ ξ Fig.9.a Couling onstants for ν /. Double-line indiates a Landau orbital ouied by eletron air with u and down sins. ξ τ τ ξ τ τ ξ τ τ ξ τ τ ξ Fig.9.b Couling onstants for ν 7/5. Double-line indiates a Landau orbital ouied by eletron air with u and down sins. τ κ τ κ τ κ τ κ τ Fig.9. Couling onstants for ν 8/5. Double-line indiates a Landau orbital ouied by eletron air with u and down sins. There are doubly ouied orbitals in Figs.9.a, b and. As disussed in setion 9.6, the sin exhange fores at between eletrons in singly ouied orbitals. The eletron airs in doubly ouied orbitals have no olarization beause of anellation by u and down sin air. Therefore the eletron sin olarization is given by the following equations: 9

230 γ γ γ e e e π π π π d tanh( λs ( ) kt ) (for ν /) (9.88a) π s π 6 d tanh( λs ( ) kt ) (for ν 7/5) (9.88b) π s π d tanh( λs ( ) kt ) (for ν 8/5) (9.88) π s where the oeffiients /, /7 and /8 in Eqs.(9.88a), (9.88b) and (9.88) ome from the fat that the u and dawn sin airs anel the olarization. That is to say two eletrons er four eletrons at ν / have no olarization, four eletrons er seven eletrons at ν 7/5 have no olarization and also six eletrons er eight eletrons at ν 8/5 have no olarization. We numerially alulate the sin-olarization urves via method A and, the results of whih are shown in Figs. 9., 9. and 9.. (Method A) (Method ) Fig.9. Sin-olarization for ν (Red dots are exerimental data [77])

231 (Method A) (Method ) Fig.9. Sin-olarization for ν 7 5 (Red dots are exerimental data [77]) (Method A) (Method ) Fig.9. Sin-olarization for ν 8 5 (Red dots are exerimental data [77]) The ouling onstants at ν 8 5 are τ, κ, τ, κ as in Fig.9.. The ouling onstants and the lassial Coulomb energy are re-exressed by using τ as follows: ( t ), τ ( t) τ τ (9.89a) τ W N τ (9.89b) E t where E is a new oeffiient. The fitting value is E. 5 for ν 8 5. The exerimental data at ν has a very shar hange on the olarization urve from 8.5[ T] 6.5[ T] to 8[ T]. Also the shoulder at ν 8 5 aears from. It is very diffiult to fit the omliated behaviors of the exerimental data. The theoretial results in Figs.9. and 9. are onsistent with the to.5[ T] exerimental data at ν and 8/5, resetively. (This exerimental behavior annot rerodue without sin Peierls instability.)

232 The arameters η ξ ( k T ξ ), C, et al. may be deendent uon the gate voltage, material, shae of samle, et. We have used the same value.5 for C at ν /, /5 and /7. If we use the different values of C, we an find better fitting to the exerimental data than the resent results. (Short summary on the sin-olarizations of FQ states) U and down sins oexist at a low magneti field. In this ase, there are many degenerate ground states with different sin arrangements in the most uniform eletron-onfiguration. These many eletron states have the same eigen-energy of the amiltonian D. We have sueeded to diagonalize the artial amiltonian omosed of the Coulomb transitions among the degenerate ground states. Then the alulated results have rerodued the wide lateaus on the sin olarization urves of the exerimental data [77]. Furthermore we have studied the origin of the small shoulders. We take the interval modulation between Landau orbitals, the eriod of whih has the doubly eriod of the original unit onfiguration. Then the artial amiltonian with the modulation is diagonalized exatly. The total energy is deendent uon the interval modulation t as in Figs.9., 9.5 and 9.8. We have found the minimum total energy at the non-zero t -value. That is to say the interval modulation ours in some range of the magneti field. The sin-olarization is numerially alulated by emloying the eigen-state. Then the theoretial olarization urve rerodues the small shoulder and the wide lateau. The theoretial results are in good agreement with the exerimental data.

233 Chater FQE under a tilted magneti field We study the ase where the alied magneti field is tilted from the diretion erendiular to the surfae (or a thin layer of an eletron hannel) of the quantum all devie. In this hater, the diagonal resistane of an FQ state is examined how to deend on the tilted angle and the field strength. We also disuss the olarization behaviour of FQ states under a tilted magneti field.. Formulation in the tilted magneti field omogeneous magneti field has three omonents as (,, ) (.) x y z The vetor otential A should satisfy the following two onditions for a stati field: rot A,, and div A (.) ( ) x y z The omonent of the vetor otential is given by A y z, A z, A (.) x z y y x z We makes sure that the three omonents of A satisfy Eq.(.) as follows: A A z y Ax A A z y A x rot A,, ( x, y, z ) y z z x x y (.a) A A x y Az div A (.b) x y z The amiltonian of the single eletron is given by Eq.(.) as ( ea) m U ( y) V ( z) Thin (.5) Under a tilted magneti field, the eigen-equation of the amiltonian is i ez y ey z i exz i U ( y) V ( z) Thin ψ ( x, y, z) Eψ ( x, y, z) m x y z (.6) where ψ ( x, y, z) is the single eletron wave funtion. The amiltonian has no otential term deending uon x and therefore the wave funtion in the x-diretion is exressed by a lane wave as ψ ( x, y, z) ex( ikx) Ψ( y, z) (.7) In a quantum all devie, the width of the otential V Thin ( z) along the z-diretion is extremely narrow as in Fig... Therein the robability density in the z-diretion is large only

234 at and near z z where z is the osition of the otential minimum. Without loss of generality we take z to be the origin namely z (.8) When the onduting layer of the devie is ultra-thin, the terms e y z and e x z an be ignored beause of z z. Aordingly the eigen-equation (.6) beomes m i z Thin ψ, x y z e y i i U ( y) V ( z) ψ ( x, y, z) E ( x, y z) (.9) Consequently the x- and y-omonents of the magneti field don t ontribute to the FQ states. The theoretial result is as follows: When the magneti field is tilted from the z-diretion, the FQ state deends only on the z-omonent of the magneti field.. Comarison between resent theory and exeriments We omare the theoretial result with the exerimental data. The exerimental data of referene [, 5] are shown in Fig... Therein the diagonal resistane takes a loal minimum at.9 T for θ,.5 T for θ 8.5,.7 T for θ 6.5, 6. T for θ. and 7. T for θ 5.8 where θ indiates the tilted angle. θ θ θ θ θ Fig.. Diagonal resistane at ν for several angles [, 5] arxiv:ond-mat/9v [ond-mat.mes-hall] Mar The z-omonent of the magneti field is related to the total strength total : z total osθ (.) The z-omonent is numerially alulated for the five tilted angles as follows: z total osθ.9os.9 (.a) osθ.5os (.b) z total

235 z z z total osθ.7 os6.5.8 osθ 6.os..85 total total osθ 7. os (.) (.d) (.e) Eqs.(.a, b,, d and e) show that the z-omonent of the magneti field takes almost same value at the loal minima for the five tilted angles. Thus the exerimental results are in good agreement with the theoretial results. Figure. shows another exerimental data reorted in referene [9]. The diagonal resistane is lotted versus the z-omonent of the magneti field, where the blak urve indiates the diagonal resistane R xx for the tilted angle θ and the urle urve for θ.. The two urves are almost same as eah other. Thus these data indiate that only the z-omonent of the magneti field is effetive to the diagonal resistane for a devie with an ultra-thin eletron hannel. Fig.. all and diagonal resistanes versus z-omonent of magneti field for tilted two angles Ref.[9] arxiv:8.7v [ond-mat.mes-hall] Ot 8 On the other hand, the sin olarization behaves in a different deendene on the magneti field from the diagonal resistane. If the sin olarization exeriment similar to Kukushukin et al [77] is arried out under various tilted magneti fields, then the different behaviours may be observed. Therein the sin olarization for the diretion of the alied magneti field (not z-diretion) should be measured in various tilted magneti fields. The sin olarization urve versus magneti field strength (not z-omonent) may have the same shae as in Chater 9 under fixing the filling fator if the ouling onstants have the same values as in θ. 5

236 Chater Further exeriments We roose two exeriments in this hater where we suerose an osillated magneti field in addition to the stati strong magneti field. Thereby new henomena may be observed. The first roosal is a measurement of the energy ga and the seond one is an observation of tunnelling effet in the IQ and FQ states.. Diagonal resistane in the FQ state under a eriodially modulated magneti field or urrent We roose a measurement of the energy ga in this setion [9], [9]. The exeriment is as follows: () The strength of the stati magneti field is fixed to the value in order to maintain the FQ state with ν as an examle. (We may hoose another FQ state with another filling fator) () An a magneti field (or urrent) with a frequeny f is suerosed on the stati magneti field (or urrent). () The diagonal resistane R xx is measured with varying the frequeny f. Then R xx deends uon f and the ritial frequeny f aears as in Fig... R xx f Fig.. Diagonal resistane R xx versus f f As examined in the revious haters, there are two tyes of energy gas. For an examle at the filling fatorν : Tye : The valley deth is desribed by the differene between the energy at ν and one in its neighbours as in Table 5., ε Z/ at ± ν (.) Tye : The exitation-energy is the energy whih is neessary to destroy a nearest eletron-air. The value is given by Eq.(5.85) as 6

237 Eexitation # ( )Z at ν (.) We numerially alulate these two values for the exeriment [, 5]. The value ε is obtained by the relation (7.a). Figure. shows the diagonal resistane versus magneti field strength where we an see the vanishing region of the diagonal resistane. The vanishing region aears between the magneti field strength and as [T], 6.[T] (.) 6 Fig.. Exerimental results of diagonal resistane R xx near ν in Ref. [, 5] Equation (7.a) gives the value ε by ε e( ) ( m ). (.) The exeriments [, 5] are arried out for the quasi-d eletron system in a GaAs/AlGaAs quantum-well. The effetive mass m is about.67 times eletron mass in GaAs. Substitution of Eq.(.) into Eq.(.) yields 9 ε.55.6 (.) ( ). (.5).5 [J] From Eqs.(.) and (.), the value of E exitation # is four times ε. Aordingly E exitation # 6.6 [J] (.6) for the exeriment [, 5]. It is noteworthy that the values of these energy gas deend uon the roerties of quantum all devie suh as, size, shae, thikness of the eletron hannel, quality and so on. There are two ossible mehanisms inreasing R xx as follows: Tye The filling fator ν deviates from / loally by the exitation via the osillating magneti field. This exitation ours when the energy due to the osillation is beyond the valley energy ε. So the ritial frequeny is given by f ε π (.7a) ( ) 7

238 The eletri urrent are sattered by the loal states with ν. Thereby the diagonal resistane beome large for f > f. Tye The suerosed osillating field breaks many nearest-eletron-airs. Then the many exited eletrons aear whih rodue the abrut inrement of R xx. In this ase the ritial frequeny f is given by f π (.7b) E exitation # ( ) Thus there are two auses (Tye and Tye) whih rodue the abrut inrement of the diagonal resistane. The resent author doesn t onlude now whih mehanism in the two tyes rodues the inrement of R xx mainly. We estimate the ritial frequenies for the two tyes: 9 Tye f ε ( π ) Gz 9 Tye ( ) 5 5Gz : for ν (.8a) : f for ν (.8b) Eexitation # π If an a magneti field is diffiult to be alied in the roosed exeriment, an eletromagneti irradiation instead of the a magneti field an be used [9, 9]. There is one more alternate method as follows: In an ordinal measurement of the diagonal resistane, a d urrent is alied between the soure and drain of the devie, and the d otential (diagonal) voltage are measured. In the resent roosal we additionally suerose an a urrent with a frequeny f between the soure and drain. Under the suerosition, the d omonent of the otential voltage versus d-urrent is deteted with hanging of f. Thereby we an get the frequeny deendene of the diagonal resistane (namely d voltage / d urrent).. Tunneling through a narrow barrier in a new tye of quantum all devie The value of the diagonal resistane is almost zero at an IQ state. This fat indiates that almost all the eletrons are not sattered by lattie vibrations and imurities in an IQ state. That is to say the oherent length of eletron beomes very long in an IQ state. For the frational quantum all states with seifi filling fators /, / et., the diagonal resistane is also almost zero [, 5]. This roerty makes it ossible to observe a tunnelling effet in a quantum all devie. In the revious artiles [9, 95] the tunneling devie was onsidered, but there are some diffiulties in it. So we remove the defets. The imroved devie is illustrated in Figs..,. and.5 for three tyes of the devie, resetively [96]. There is the otential barrier in the entral art of eah devie. (Tye ) A narrow insulating region is inserted in the middle of the onduting layer as shown in Fig... The length of the narrow barrier should be smaller than nm to realize a tunnelling effet. It would not be easy to fabriate this tye of devie. 8

239 Soure Side View Drain Magneti Field This area is enlarged below: Soure z y x all robes a b d Drain Gate A C D barrier Gate Potential robes Fig..: (Tye ) There is a narrow otential barrier near the enter of eletron hannel (Tye ) The seond tye of quantum all devie is illustrated in Fig.. where there are three gates. These gates are ut on to of the onduting layer. y adusting the voltage of the Gate, we an set an aroriate otential barrier. It would be easier to fabriate a devie with tye than tye. Side View Magneti Field Soure z y x all robes a b d Drain Soure Gate Gate Gate Drain A C D Gate Gate Gate Potential robes Fig..: (Tye ) Central-gate (Gate ) rodues the otential barrier. (Tye ) The devie is shematially drawn in Fig..5. There is an ultra-thin art in the entral art of the eletron hannel as shown in the enlarged side view. The ultra-thin art onnets the two quasi-d eletron systems to eah other. This tye of untion is a familiar one in the sueronduting quantum interferene devie (SQUID). When the eletri urrent exeeds some ritial value, this ultra-thin art lays a role of otential barrier. A devie with tye may be made more easily than tye. 9

240 Soure Side View Drain Magneti Field This area is enlarged below eletron hannel Soure z y x all robes a b d Drain ultra-thin art Gate A C D Gate Potential robes Fig..5: (Tye ) An ultra-thin art lays a role of otential barrier for slightly large urrent. We roose a new exeriment to find a tunnelling effet in an IQ or FQ state by emloying the devies shown in Fig This exeriment is arried out in the following roess: ) Magneti field is alied in the diretion erendiular to the quasi-d eletron system. The strength of the alied magneti field is adusted to yield an IQ or an FQ state. Also, by adusting two voltages of Gate and Gate resetively, the filling fators in both sides of the barrier are set to be the same value. ) Next, an osillating magneti field is suerosed on the d-magneti field. (We may aly a-urrent in addition to the onstant urrent instead of an osillating magneti field.) The frequeny of the osillation is exressed by the symbol f. ) Then we measure the voltage V between the otential robes and C versus eletri urrent I. The observation of ( I, V ) urve is exeted to give us interesting henomena of the tunnelling effet. When the eletri urrent flows beyond the narrow otential barrier, the quasi-artile tunnels from the higher energy osition to the lower one aomanying a stimulated emission of hoton. The stimulated emission is indued by the a magneti field with the frequeny f. This tunnelling henomenon is shematially drawn in Fig..6. Therein the energy differene of the quasi-artile from the high osition to the low osition is the value V Q where Q indiates the eletri harge of the quasi-artile. The emitted energy of a hoton is equal to πf. eause the total energy is onserved in this tunnelling henomenon, the voltage is determined by V πf Q (.9) When the eletri urrent beomes large, the stimulated emission yields multi-hotons [96]. Therefore the measured voltage V n is equal to n times V whih is given by n πf Q for n,,, (.) V n

241 Fig..6 Tunnelling of quasi-artile beyond a otential barrier I, urve. This henomenon is reminisent of the a-josehson effet. The aearane of the tunneling effet needs a long oherent-length of eletron. Therefore it is required that the diagonal resistanes in both sides of the otential barrier are nearly equal to zero. The vanishings are onfirmed by deteting the Then many voltage stes are observed in the ( V ) four voltages V A, V CD, V ab and V d between otential robes A and, between C and D, between all robes a and b, and between and d, resetively. All the voltages V A, V CD, V ab and V d should be nearly equal to zero. Also the voltage between robes and b is required to be equal to that between robes C and, beause the filling fator in the left side of the otential barrier should be equal to that in the right side. These onditions must be satisfied in whole measurements for the various values of the eletri urrent I. We next examine the ( V ) I, urve for the two ases namely IQE and FQE. Case A: Integer quantum all state The integer quantum all effet is aused by disrete Landau levels in a quasi-d eletron system. So the elementary harge e transfers aross the otential barrier in the new exeriment roosed here. Then the ( I, V ) urve behaves as the dashed urve in Fig..7 whih has a stair form derived from Eq.(.).

242 Fig..7 Dashed urve exresses the (I, V) urve in the Case A. (Case A) Integer quantum all effet Case : Frational quantum all state R.. Laughlin introdued a quasi-artile with frational eletri harge and roosed a trial wave funtion [9]. The quasi-artile and quasi-hole have frational harges and obey frational statistis [-] and [97]. Laughlin s theory suggests that the harge of the quasi-artiles aross the otential barrier is ν e at the frational number ν. We all it Case-. (Case -) The theory with frational harges (Laughlin s theory): Q ν e (.a) V π f ( ν e) ev ( hf ) ν (.b) Aordingly the value of the voltage ste V hanges with the different values of ν. Next we examine the ase of the omosite fermion theory whih is introdued by J. K. Jain [] and is develoed by many hysiists [98], [99]. The omosite fermion onsists of an eletron bound to an even number of magneti flux quanta. Therefore the omosite fermion has the elementary harge. That is to say the harge of the quasi-artiles aross the otential barrier is e at any filling fator ν. We all it Case-. (Case -) Comosite fermion theory : Q e (.a) V π f e ev hf (.b) ( ) In this book we have develoed a theory in haters - to exlain the frational quantum all effet. The Coulomb interation ats between two eletrons. The eletron air has a large binding energy at the seifi filling fators. The binding of the eletron air is so strong that the airing is ket in the tunnelling roess without ollase. So, the harge of the quasi-artiles aross the otential barrier may be equal to e. We all it Case-. (Case -) Present theory :

243 Q e (.a) V π f e ev hf (.b) Thus the ( V ) ( ) ( ) I, urves are different from eah others among the three theories. The theoretial urves are illustrated in Fig..8 for the three ases. Fig..8: ehaviours of voltage stes for three theories at the filling fator / The horizontal axis indiates the eletri urrent (arbitrary unit) If the exeriment roosed above is arried out and gets the urve of the voltage versus urrent, then the exerimental urve an distinguish the three ases as in Fig..8. Therein the following seven onditions should be satisfied in the exeriment in order to find the voltage stes. (Condition ) We should hoose the value of f muh smaller than the energy ga of the FQ state in order to rodue no exitations. fh << ε G namely f << ε G h (.) For examle, ε h is about 6.7 Gz at ν / for the exeriment [, 5] as estimated in Eq.(.8a). eause the value ε h varies from samle to samle, the frequeny f should be hosen to satisfy Eq.(.) for a devie used in the tunnelling exeriment. When we aly f Gz as an examle, the voltage ste is 6.7 µv for the Case -,.8 µv for the Case -, and.69 µv for the Case - resetively. (Condition ) The thermal exitation energy should be lower than the energy V Q πf not so as to disturb the tunnelling roess. k T << fh T << fh (.5) k where T is the temerature and k is the oltzmann onstant. In the ase of f Gz, T << fh k.8 K (.6)

244 Aordingly the temerature should be ooled lower than a few mk. (Condition ) Figure.9 shows the to view of the devie. The urrent flows along the x-diretion and the all voltage yields along the y-diretion in Fig..9 (see also Figs..-5). Soure a all robes b d Drain y d x Gate A C D Potential robes Gate Fig..9 To view of the tunneling devie There is a otential barrier in the entral art of the devie. orizontal red lines indiate the entres of the y-diretion for many Landau orbitals with L. The intervals between the red lines are enlarged to bring them into view. The width of the devie is exressed by the symbol d. In the resent exeriment both the diagonal resistanes R, R in the left and right area left right should be almost zero. The vanishings of R left and R are observed by deteting the right voltages V A and V CD, resetively. So the following onditions are required: V A, V (.7) CD (Condition ) The all voltage of the left area is measured between the robes and b. Also that of the right area is measured between the robes C and. The osition-deendene of the eletri otential is illustrated in Fig... Therein V b ( y) has the argument y and indiates the eletri otential in the left side of the otential barrier at the osition y. Also V C ( y) indiates the eletri otential in the right side of the otential barrier at the osition y. The values of V b( ), V C( ), V b ( d ) and V C ( d ) in Fig.. indiate the otentials at the robes, C, b and, resetively: V ( b ) V, ( ) V C VC, V b ( d ) Vb, V C ( d ) V (.8) where d is the width of the devie as in Fig..9. The left anel of Fig.. shows the ase of Vb V V VC whih is named Case I. The right anel shows the Case II where Vb V V VC. In the Case I, VC( y) Vb( y) varies

245 with the hange of the oordinate y as in the left anel of Fig... In the Case II, the otential differene VC( y) Vb( y) is equal to VC V for any osition y as in the right anel of Fig... In this ase, the stimulated emission at any y-osition is rodued by the osillation with the same frequeny. The roerty may rodue the tunneling henomenon like a-josehson effet. V b V Voltage V b V Voltage V b (y) V b (y) V C (y) V V C (y) V Ste of Voltage V V Ste of Voltage C y C y y Case I yd y Case II yd Case I: Vb V V VC Case II: Vb V V VC Fig.. Position-deendene of eletri otential in both side of the otential barrier lue and sky-blue urve indiate the eletri otential in the left side of the otential barrier. Red urve indiates the eletri otential in the right side of the otential barrier. Consequently the following ondition is required in the resent exeriment. Vb V V VC (.9) That is to say the all voltage at the left area of the otential barrier, Vb V, is equal to that at the right area. Therefore the filling fator at the left area is the same as that at the right area. This ondition an be satisfied by adusting both voltages of the gates and. (Condition 5) If there are many imurities and lattie defets in the devie, the oherent length beomes very short. Then the tunnelling effet is disturbed by these imurities and lattie defets. Therefore it is neessary that the devie is of good quality to ensure a suffiiently long oherent length. (Condition 6) The diagonal resistane in the FQ states must be suffiiently small for finding the tunnelling effet. Figure. shematially indiates a omarison between the diagonal resistane and ( I, V ) urve. If the devie has a small diagonal resistane shown by the lower straight line, then we an observe the tunnelling effet. On the other hand if the devie has a large diagonal resistane shown by dashed line, the tunnelling effet will be masked by the large diagonal resistane. Aordingly the quantum all devie must be fabriated so as to have an ultra high mobility. 5

246 I, urve of a new exeriment Lower straight line indiates Ohm s law with the small diagonal resistane. Dashed line indiates Ohm s law with the large diagonal resistane. Fig.. ( V ) (Condition 7) As disussed in setion 8. of Chater 8, the onfinement of the quantum all resistane beomes weak for a small size of the devie at a frational (not integer) filling fator. Aordingly it is neessary to use an aroriate size of devie in order to have a large binding energy at the seifi filling fators. The seven onditions should be satisfied to find the tunneling effet. If the stair-like urve is observed in the exerimental ( I, V ) data, the voltage of the ste height determines the transfer harge roduing the tunneling effet. Thereby we an know whih urve among the three urves of Fig..8 is realized in the quasi-d eletron system. 6

247 Chater Disussion on Traditional theories At resent the dominating theories on the FQE are lassified into two tyes: the one was roosed by Laughlin and has been develoed by aldane and alerin [-, 7]. They have used the quasi-artile with frational harge. The other theory is investigated by Jain. is theory emloys a quasi-artile named omosite fermion [5-56] whih is an eletron aturing an even number of flax quanta. The wave funtions of these quasi-artiles are aroximately obtained in both theories, resetively. In setion 5. we have given the short omments omaring the resent theory with the traditional theories. Our goal is to understand the D eletron system whih is originally desribed by the total-amiltonian of the normal eletrons (not quasi-artiles). Therefore it is neessary to derive the wave funtion of eletrons from the wave funtion of the quasi-artiles. The derivation has not been done in the traditional theories. Furthermore we find other roblems whih are disussed in the setions.,.,. and.5. There is one more roblem: the all voltage is extremely large in omarison with the otential voltage under the onfinement of all resistane. (The all voltage is larger than 6 times of the otential voltage in the onfinement of the all resistane.) So it is neessary to take the eletri otential gradient into onsideration in order to larify the quantum all effet. Almost all the theories have negleted the eletri otential along the diretion of all voltage so as to deal with a simler amiltonian. Then the all voltage disaears. Thus the traditional theories have assumed nothing of the eletri otential gradient along the all voltage. In Se.. we investigate whether the eletri otential gradient exists in a entral art of a quantum all devie under a onfinement of the all resistane or not.. Eletri otential along all voltage in the quasi-d eletron system We onsider a thought exeriment (gedanken exeriment) emloying a following new devie: the devie is shown shematially in Fig.. whih has an additional robe at the entral art. The additional robe does not disturb the eletron flow from the soure to the drain beause of the small thikness of the additional robe in the y-diretion as in Fig... So the quantum all effet is observed in the new devie under a strong magneti field. Therein we examine the y-deendene of the eletri otential on the blue and ink dashed lines with the oordinates x x and a x x, resetively. X Z Potential robes Y Additional robe Magneti field x x a X x a x Y Additional robe all robe all robe Fig.. Quantum all devie with additional robe at entral art 7

248 U in Fig.. of Chater. We examine the eletri otential on the new devie as Fig... Sine the effet of the additional robe is small on the dashed blue line with x x, the eletri otential a U ( y) has the shae same as in Fig.. whih is drawn by the blue urve in Fig... Next we examine the eletri otential on the dashed ink line with x x in Fig... The otential is different from that on x x in only the neighborhood of the additional robe a with y d. For the other y-region the otential takes a similar behavior to that of x x. a So the otential shae an be exressed by the ink urve in Fig... Thus only the neighborhood of the additional robe has the different otential from that of x x. We have already onsidered the y-deendene of the eletri otential ( y) a Potential -ev U' (y) at xx V (V V ) / -ev y U(y) at xxa U" (y) at xx yd Fig.. y-deendene of eletri otential of the resent theory y On the other hand the traditional theories emloy the flat otential. That is to say, the y-deendene of the eletri otential is flat in the entral region of the devie. So the otential on the blue dashed line in Fig.. has the behavior as blue urve in Fig... Next the eletri otential on the ink dashed line in Fig.. is searated into two regions < y < d and d < y < d by the additional robe. The eletri otential urves in these two regions have the shae same as eah other beause of the following reason: The voltage of the additional robe is ( ) V beome ( V ) ( V ) V VC V V C V V V and so the two otential differenes of both sides (.a) (.b) Potential -ev V (V V ) / -ev y U' (y) at xx U" (y) at xx U(y) at xx a yd Fig.. y-deendene of eletri otential of the 8 y

249 The voltage V V is equal to C V C V. Thereby the eletri otential urves in the two regions should have the same shae. Aordingly the otential in the whole region < y < d beomes the ink urve in Fig... When the x-oordinate of the dashed line in Fig.. hanges from x x to x x, the a eletri otential also hanges. Then the otential-value in Fig.. ums from the ink to blue urve. Thus the traditional theories inlude the um struture in the eletri otential under the situation as in Fig... Probably this um struture is unrealisti. The diffiulty omes from the flatness assumtion of the eletri otential in a entral region of a quantum all devie. Our otential used in this book varies almost ontinuously with hanging the x-oordinate. If we omare the two otential behaviors in Figs.. and., the atual eletri otential may be nearly equal to the urve in Fig... That is to say the y-deendene of the eletri otential annot be ignored in the atual quantum all effet.. Laughlin theory and aldane-alerin hierarhy theory At the filling fator n, the quantum all resistane is deendent uon the elementary harge of eletron as R π e. The funtion form inludes the fator e in the denominator. If the quasi artiles has the frational harge ν e, then a new form of the all resistane is obtained by relaing e with ( ν e). That is to say, the quantum all resistane via the quasi-artiles is onfined to R ( ) π eν. owever the exerimental value of the all resistane is onfined to R π ( νe e) in the FQE. Thus the frational harge has some diffiulty. Next we omare the harge distributions between the resent theory and the Laughlin theory. The resent theory starts from the ground states of the amiltonian D as studied in haters -9. The ν state has the harge distribution of eletrons along the y-diretion as follows: y ex e () y s h e for ν, (.) ( ) ρ ( ) ( )( ( ) ( )) ( ) N s Fig.. Eletron distribution of the resent theory at ν 9

250 Therein the orbitals with wave numbers of ( s ) π and ( s ) π are emty. The distribution along the x-diretion is uniform beause of the lain wave funtion as obtained in Eq.(.). When the devie length is nm and the strength of magneti field is T, we get the values ( e ()).59 ( nm) and h ( e).6 nm. In this ase, the alulated distribution is shown in Fig... That figure shows that the harge distribution is suffiiently uniform. Furthermore the erturbation state has more uniform distribution than the ground state of D beause the eletrons sread to all the Landau ground states via the quantum transitions due to the Coulomb interation. Fig..5 Quasi artile distribution of Laughlin wave funtion at ν Next we examine the harge distribution in the Laughlin theory. Figure.5 shows the distribution, the shae of whih is round. Thus Laughlin s distribution dose not sread to a whole region of D eletron system with a retangular shae in a usual ase. Aordingly it is neessary to suerose the wave funtions with different entre ositions in order to derease the lassial Coulomb energy. ut they doesn t make the orthogonal set.. Comosite Fermion theory J.K. Jain introdued a quasi-artile named omosite fermion whih is an eletron aturing even number of flux quanta. e omared his theory with the aldane-alerin hierarhy theory in Ref. [] and also summarized the omosite fermion theory. Therein he lassified the omosite fermions into sixteen tyes. We disuss here the states for the filling fators with the denominator smaller than 6: () ν n ( n ) ν, 5, : The FQ states are the IQ states of omosite fermion whih is an eletron aturing two flux quanta. () ν n ( n ) ν, 5, : The FQ states are the IQ states of omosite fermion whih is an eletron aturing two flux quanta. The effetive magneti field has the oosite diretion against the alied magneti field. () ν n ( n ) ν 5, : The FQ states are the IQ states of omosite fermion whih is an eletron aturing four flux quanta. 5

251 () ν n ( n ) ν 5, : The FQ states are ombined the ν IQ state with the IQ state of omosite fermion whih is a hole aturing four flux quanta. (5) ν n ( n ) ν 6 5, : The FQ states are ombined the ν IQ state with the IQ state of omosite fermion whih is an eletron aturing four flux quanta. (6) ν n ( n ) ν 5, 8 5, The FQ states are ombined the ν IQ state with the IQ state of omosite fermion whih is a hole aturing two flux quanta. (7) ν n ( n ) ν, 7 5, The FQ states are ombined the ν IQ state with the IQ state of omosite fermion whih is a hole aturing two flux quanta. The effetive magneti field has the oosite diretion against the alied magneti field. (8) ν n ( n ) ν 9 5, The FQ states are ombined the ν IQ state with the IQ state of omosite fermion whih is a hole aturing four flux quanta. We study the following five examles in more details. ) FQ state at ν In the artile [7] the ν FQ state is onstruted by ombining the ν IQ state with the omosite fermion state of hole for ν as in Fig..6. Therein the blak dots on the green sheet indiate the eletrons in the ν IQ state. The omosite fermions of hole are exressed by the white irles on the yellow sheet, eah of whih is bound with two flux quanta as in Fig..6. Alied magneti field Effetive magneti field Fig..6 Comosite fermion theory for ν The effetive magneti field is exressed by the red arrows, the diretion of whih is oosite against the alied magneti field. The total filling fator is the sum of ν and ν. Aordingly the filling fator of eletrons beomes ν. ) FQ state at ν 5 The ν 5 FQ state is exlained by the ombination of the ν IQ state and the omosite fermion state of hole with ν. Eah hole is bound with two flux quanta and the effetive magneti field is arallel to the alied magneti field as shown in Fig..7. 5

252 Alied magneti field Effetive magneti field Fig..7 Comosite fermion theory for ν 5 ) FQ state at ν 6 5 The ν 6 5 FQ state is rodued by ombining the ν IQ state with the omosite fermion state with ν 5. Therein the eletron of the ν IQ state is exressed by the blak irles on the sky-blue sheet in Fig..8. The residual eletrons are exressed by the blue dots on the ink sheet, eah of whih is bound by the four flux quanta as seen in Fig..8 shematially. That is to say, some eletrons are unbound with magneti flux quanta and the other eletrons are bound with flux quanta. owever all the eletrons exist in the same onduting thin layer and their wave funtions are overlaing with eah other. In the many-body roblem all the wave funtions of eletrons should satisfy the anti-symmetri relation. Also all the eletrons should be affeted by the same magneti field. Aordingly the ombination of the ν IQ state and the ν 5 omosite fermion state has some diffiulty. Alied magneti field Effetive magneti field Fig..8 Comosite fermion theory for ν 6 5 ) FQ state at ν 9 5 The ν 9 5 FQ state is reated by ombining the ν IQ state with the omosite fermion state of hole for ν 5 as illustrated in Fig..9. 5

253 Alied magneti field Effetive magneti field Fig..9 Comosite fermion theory for ν 9 5 Therein the blak dots on the green sheet indiate the eletrons in the ν IQ state. The omosite fermions of hole are exressed by the white irles on the yellow sheet, eah of whih is bound with four flux quanta as in Fig..9. 5) FQ state at ν 5 The ν 5 FQ state is rodued by ombining the ν IQ state with the omosite fermion state of hole with ν 5. Therein the eletron of the ν IQ state is exressed by the blak irles on the sky-blue sheet in Fig... The omosite fermions of hole are exressed by the white irles on the yellow sheet, eah of whih is bound by the four flux quanta. Alied magneti field Effetive magneti field Fig.. Comosite fermion theory for ν 5 Thus the omosite fermion theory uses many different kinds of quasi-artiles and different diretions of the effetive magneti field. When the filling fator is hanged by adusting the gate voltage (or the alied magneti field strength), the quasi-artile varies from hole to eletron, or the diretion of the effetive magneti field hanges from arallel to anti-arallel. Also the number of the bound flux-quanta hanges. These omliated assumtions are very artifiial. 5

254 . Effetive magneti field and flux quantization is alied to a quantum all devie. The strength of the field an be varied ontinuously and the strength is uniform at all ositions inside the quasi-d eletron system. In the omosite fermion theory some magneti flux quanta are atured by the omosite fermions and the other magneti flux yields the effetive magneti The external magneti field (,, ) field. Then the alied magneti field is equal to the sum of the effetive field (,, eff ) the bound field (,, bound ). (,, ) (,, ) (, ) (.) eff, bound and The omosite fermion theory ombines the IQ state of the normal eletrons with the IQ state of the omosite fermions for the ases () - (8) as exlained in Se... Therefore the theory emloys a uniform magneti field for the eletrons and a uniform effetive field are onstant everywhere on a for the omosite fermions. Thereby and eff eff quantum all devie. Aordingly the magneti field is also uniform beause of bound Eq.(.). In the omosite fermion theory the bound flux does not affet the other omosite fermions. Only the effetive field affets the omosite fermions. That is to say, the bound flux annot enetrate the wave funtion of the other omosite fermions. Equation (.) means that has a uniform value for every osition on a quantum all devie. Also the wave bound funtion of eah omosite fermion sreads on a quantum all devie. Therefore it is diffiult to assume that does not affet the omosite fermions. bound Furthermore the quasi D-eletron system has no seial boundary as in a sueronduting ring. Although there is the roerty that the number of Landau states with the lowest level is equal to the number of the flux quanta, this roerty does not mean the quantization of the magneti flux. In fat the roerty aears also in the three dimensional eletron-system under the uniform magneti field when we onsider a single momentum level along the magneti field diretion. There is no quantization of magneti flux in the -D eletron system. Consequently the equality of the two numbers does not mean the flux quantization..5 Sin-olarization of omosite fermion theory J. K. Jain examined the sin-olarization in the CF theory. e wrote in Ref []: For sinful omosite fermions, we write ν ν ν, where ν and ν are the filling fators of u and down sin omosite fermions. The ossible sin olarizations of the various FQE states are then redited by analogy to the IQE of sinful eletrons. For examle, the /7 state mas into ν, where we exet, from a model that neglets interation between omosite fermions, a sin singlet state at very low Zeeman energies (with ν ), a artially sin olarized state at intermediate Zeeman energies ( ν ), and a fully sin olarized state at large Zeeman energies ( ν ). The CF ylotron energy is roortional to and the Zeeman energy is roortional to the magneti field as exlained in the artile []. Then the CF energy ε is equal to CF 5

255 CF ( n ) α( n ) β ε ( n ) α( n ) β ε CF (.) where α and β are the oeffiients. Therein n ν L where L is the CF Landau level number ( L,,, ). When the filling fator ν of eletrons is smaller than /, ν is given by the CF level number ν and the number of attahed flux quanta as follows: ν ν ( ν ± ) (.5) When the sign in the denominator is lus the effetive magneti field is arallel to the alied magneti field. So the oeffiient β beomes ositive. On the other hand the effetive magneti field has oosite diretion against the alied field for the minus sign in Eq.(.5). Then β is negative. We obtain the sum of the CF ylotron energy and the Zeeman energy for various CF levels and CF sins. The results are shown in Fig.. where we find the rossing oints between the energies for sin-u and down states. The ratio of the field strengths at the rossing oints is ::9:6 and so on. Fig..a Energy of CF for ν ( ν ) Fig..b Energy of CF for ν ( ν ) lue dashed urves for u-sin CF, red for down sin CF. The axes are drawn with arbitrary sale. Fig..a CF state at ν 7 Fig..b Polarization of CF state at ν 7 For an examle ν 7 the energy setrum of the CF theory is given by Fig..b. So the omosite fermions ouy the lowest four levels whih are exressed by the bold urves as in Fig..a. Therein the ratio of the magneti field strengths, and is ::9. Aordingly the sin-olarization γ deends on the magneti field strength at zero temerature as in Fig..b. It is noteworthy that the effetive field is oosite against the alied magneti field at 55