Effective Mass Theory of Carbon Nanotubes with Vacancies in Magnetic Fields

Transcription

1 Journal of the Physical Society of Japan Vol. 7 No. 2 February 21 pp Effective Mass Theory of Carbon Nanotubes Vacancies in Magnetic Fields Masatsura Igami Takeshi Nakanishi 1 and Tsuneya Ando 2 Institute of Materials Science University of Tsukuba Tennodai Tsukuba Ibaraki Department of Applied Physics and DIMES Delft University of Technology Lorentzweg CJ Delft The Netherlands 2 Institute for Solid State Physics University of Tokyo Kashiwanoha Kashiwa Chiba Received June 29 2 An effective-mass theory is developed on transport of non-doped carbon nanotubes local and short-range impurities in the presence of a magnetic field. The conductance is shown to be scaled completely by the field component in the direction of impurities. In a weak-field regime the conductance strongly depends on strength of potential and the difference in the number of impurities at A and B sublattices N AB. In a strong-field limit the conductance is reduced to e 2 /π h if impurities exist only on A or B sublattices and vanishes in all other cases. These results are intuitively understood by localized charge distribution of the wave function in magnetic fields. KEYWORDS: carbon nanotube conductance effective-mass theory magnetic field impurity potential 1. Introduction Carbon nanotubes CN s are quasi one-dimensional materials which are built by wrapping a graphene sheet into tubes on nanometer scale. 1 Electronic structure of a single CN has been studied theoretically which predicted that CN becomes either metallic or semiconducting depending on its chiral vector i.e. boundary conditions in the circumference direction These predictions have been confirmed by Raman experiments 12 and direct measurements of local density of states by scanning tunneling spectroscopy STS In view of mesoscopic physics CN s are classified into a new kind of quantum wires which have unique geometric and electronic structures different from conventional ones fabricated using semiconductor heterostructures. The purpose of this work is to study effects of short-range impurities on transport of CN s in magnetic fields based on an effective-mass theory. It is well known that defects play important roles in electronic states of CN s and graphene sheets. Effects of topological defects i.e. five- or seven-member rings in nanotube junctions were calculated and a universal power-law dependence on the conductance was shown Electronic structure of CN s various caps was also studied. 19 In a graphene sheet a finite width localized edge states are formed near the Fermi level when the boundary is in a certain specific direction In an effective-mass scheme these defects can often be obtained by local boundary conditions imposed on envelope functions. For example such edge states are easily obtained by imposing rigid boundary conditions along a line in a certain direction. A point defect also leads to peculiar electronic states. STM images of a graphene sheet simulated in the pres- ence of a local point defect showed that a 3 3 interference pattern is formed in the wave function near the impurity. 23 This is intuitively understood by a mixing of wave functions at K and K points. Effects of scattering on impurity potential in CN s were studied theoretically and it was proved that a Born series for back-scattering vanishes identically for scatterers having a potential a range larger than the lattice constant This intriguing fact was related to Berry s phase acquired by a rotation in the wave-vector space in the system described by a k p Hamiltonian. 26 The k p scheme was extended to the study of effects of strong and short-range impurity potentials. 27 The scattering matrix obtained analytically shows a rule of the conductance quantization in the limit of strong scatterers that the conductance vanishes for N AB 2 and is quantized into one and two times e 2 /π h for N AB =1 and respectively where N AB is difference in the number of impurities at A and B sublattices. This result is in agreement that for the lattice-vacancy obtained numerically in a tight-binding model See ref. 33 for a review on the effective-mass description. In the presence of a magnetic field perpendicular to the axis the band structure is known to be strongly modified due to a formation of two dimensional Landau states This leads to various peculiar electronic properties including transport Therefore the magnetic field is expected to have a strong influence on the conductance in the presence of short-range impurities. In fact the conductance in the presence of a simple latticevacancy has been shown to exhibit strong dependence on N AB in a magnetic field A peculiar feature is the existence of the universal dependence on the field component in the direction of the vacancy which will be called conductance scaling in the following. 481

2 482 Masatsura Igami Takeshi Nakanishi and Tsuneya Ando Vol. 7 The outline of the paper is as follows: A formalism of a perturbation approach for calculating the T matrix in a magnetic field based on an effective-mass theory is developed and origin of the conductance scaling is described in the 2. In 3 we present analytical results of magnetoconductance and give general discussions in a weak and a strong-field limit. Results of numerical calculation using an effective-mass theory and a tight-binding model are given in 4. Section 5 is devoted to summary and discussion. 2. Effective Mass Approximation 2.1 Hamiltonian We show the lattice structure of a graphite sheet and its first Brillouin zone in Fig. 1 the definition of coordinates. A unit cell has two carbon atoms denoted by A and B and L and η are the chiral vector and angle respectively. Electronic structure of a graphite sheet near the Fermi level consists of π-π bands. These two bands have conical dispersion and touch at K and K point in the first Brillouin zone. The Shrödinger equation near the Fermi level in an effective-mass scheme is given by in the Landau gauge where L is the circumference and H is the strength of the field. We consider the impurity potential used in the former study 27 which is given by V A j = V = j HF r =εf r H = H + V 2.1 where H is a k p Hamiltonian for an ideal graphite sheet V is impurity potential and F r is an envelope function. Hamiltonian is given by the following 4 4 matrix KA K A KB K B ˆkx iˆk y ˆkx +iˆk y H = γ ˆk x +iˆk y ˆkx iˆk y 2.2 where γ is a band parameter and ˆk =ˆk x ˆk y is a wave vector operator defined by ˆk = i + e A. 2.3 c h Effect of a magnetic field is included through vector potential A. In this study we deal a magnetic field perpendicular to the axis. This leads to A =A x A y = LH 2π sin 2π L x 2.4 α=ab V α j δr r j 2.5 2u A j a j a + j Vj B = 2u B j b jb + j 2.6 Fig. 1. a Lattice structure of a graphite sheet. η is the chiral angle. The coordinates are chosen in such a way that x is along the circumference of a nanotube and y is along the axis. b The first Brllouin zone and K and K points. c The coordinates for a nanotube and definition of the angle θ H. where u A j and ub j are the strength of a δ function potential at an A site r j and a B site r j respectively and we have defined vectors a j and b j as a j = 1 ajk = 1 e iφ A j 2 /2 2 a jk b j = e iφa j /2 1 bjk = 1 e iφ B j / b jk e iφb j /2 φ A j =K K r j + η φ B j =K K r j η + π/ where K and K are the wave vectors at K and K points respectively. 2.2 Landau wave function at ε= The k p Hamiltonian H in the presence of a magnetic field is related to H in its absence through H = P r 1 H P r where P r is a diagonal matrix defined by e fαx e fαx P r = e fαx e fαx 2.1 fα x =α cos 2πx L 2.11

3 21 Effective Mass Theory of Carbon Nanotubes Vacancies in Magnetic Fields 483 L 2 α = πl and l being the magnetic length defined by l c h/eh. With the use of the above relation between H and H Landau wave functions at ε= are determined exactly They are given by F K± r = 1 f K± r F K ±r = 1 f K ±r AL AL where A is the length of CN and F r f K± r = ±if + r f K ±r = The effective strength of the field is characterized by the parameter α. We have two regimes a weakfield regime α 1 and a strong-field regime α 1. In the strong-field regime F + r and F r are localized around x= and x= ± L/2 i.e. top and bottom of CN s respectively. Because of this localized feature of the Landau wave function envelope functions for A and B sites do not overlap each other in the strong-field regime. The group velocity at ε= is given by v = h γi α In the following we shall mainly confine ourselves to the case ε= where analytic expressions can be obtained for the wave functions and Green s functions. The extension to the case ε is straightforward except that numerical calculations are required for wave functions and Green s functions. 2.3 Green s function at ε= One particle Green s function at ε = is defined by [i H ]G = [i P 1 H P 1 ]G = In terms of Green s function G in the absence of a magnetic field given by [i H ] G =1 we have In r-representation this becomes G = PG P G jj = P r j G jj P r j. 2.2 With the use of Green s function obtained previously in the absence of a magnetic field 27 we have explicitly G jj = ia G jj GAB jj 2.21 G BA jj GBB jj F + r if r F ± 1 r = exp [±fα x] I α and I z being the modified Bessel function defined as I z = 1 dθ expz cos θ π g G jj = g + G AB jj = g1 ḡ 1 g G BB + jj = g G BA jj = ḡ1 g where g ± r j r j = exp [ ±f j + f j ] g r jj g 1 r j r j = exp [ f j f j ] g 1 r jj ḡ 1 r j r j = exp [ +f j f j ] g1r jj 2.23 and r jj =r j r j. The diagonal Green s function g ± represents propagation among same sublattice points and the off-diagonal terms g 1 and ḡ 1 represent propagation between different sublattices. In the above equation we have introduced f j fα x j 2.24 and the diagonal and off-diagonal Green s functions in the absence of a magnetic field g r jj = cos[πx +iy/l] g 1 r jj = sin[πx +iy/l] x=x j x j and y=y j y j. When impurities are localized in a distance of a few times of the lattice constant g 1 r jj becomes extremely large. This singular behavior is the origin of the conductance quantization of CN s the local lattice-vacancy which we have already discussed in the former study. 27 In the perturbation expansion of the T matrix diagonal and off-diagonal Green s functions always appear only between matrix elements of impurities and therefore in the form g jj g jj BB a + j G jj a j =a j a j g jj b + j GBB jj b j =b j b j g jj BB a + j GAB jj b j =a j b j g jj AB b + j GBA jj a j =b j a j g jj BA 2.27 = cosh[f j + f j ] + i sinh[f j + f j ] tan[φ jj ] = cosh[f j + f j ] i sinh[f j + f j ] tan[φ BB g jj AB = 1 [ g1 r jj e [f j f j ] e iφab jj 2 +g1r jj e [f j f j ] e ] iφab jj a i b j 1 g jj BA = 1 [ g 2 1 r jj e [f j f j ] e iφba jj jj ] +g 1 r jj e [f j f j ] e iφba jj ] b i a j where φ αβ jj =φα j φα j /2 α β = A or B. This means that we can always use g αβ jj for Green s function describing the propagation from a β site at r j to an α site at r j.

4 484 Masatsura Igami Takeshi Nakanishi and Tsuneya Ando Vol Scattering matrix The scattering matrix can be written formally as where the summation is taken over all impurities and T is the T matrix in the r-representation. The equation for T is given by T = 1 AL V iδ + 1 AL V i G ik T kj 2.35 Effect of the multiple scattering from a single impurity is exactly taken into account by introducing renormalized potential Ṽ by [ Ṽi = 1 1 ] 1 AL V ig ii Vi 2.36 Then the equation for T can be rewritten use of Ṽ as T = 1 ALṼiδ + 1 G ik T kj ALṼi In general we can write T = S = S + S [S ] αβ = δ αβ 2.3 [S 1 A ] αβ = i h T αβ 2.31 v α v β where v α and v β are velocity of channels α and β respectively and T is a T matrix defined by T = V + VGV + VGVGV + = V + VGT In the k-representation eq becomes T = T αβ = f + α r i T f β r j 2.33 [ 1 1 AL VG 1 1 AL V ] T T BA k k i T AB in terms of 2 2 matrices T etc. When ε = metallic CN s always have two open channels at K and K point. Therefore we can write the scattering matrix as r KK r KK t KK t KK r K K r K K S = t K K t K K t KK t KK r KK r KK 2.39 t K K t K K r K K r K K where t s and r s are transmission and reflection coefficients respectively. Using the Landau wave function at ε =S 1 can be written as S 1 αβ = ia γ I α f α r i + T f β r j 2.4 where summation is taken over all combination of impurities. Once transmission coefficients are obtained conductance is calculated by Landauer formula 4 given by G = e2 t µν π h µν where combinations of {µ ν} are given by {K K} {K K } {K K} and {K K }. The former two correspond to intra-valley scattering and the latter two correspond to inter-valley scattering between K and K points. 2.5 Scaling of conductance In the following we shall confine ourselves to the case that impurities are localized in a range smaller than circumference of CN i.e. r j r j L atε =. Then equation 2.23 can be rewritten as g ± r j r j = exp[±2f ] g 1 r j r j =g 1 r jj ḡ 1 r j r j =g 1r jj 2.42 f = fα x where r =x y is the center-ofmass of impurities. In eq effects of the magnetic field appear only in the diagonal Green s function which is determined by the local density of states at r. We can also replace eq. 2.4 by S 1 αβ = ia γ I α f α r + T S f β r 2.43 T S = T The factor I α appearing in eq is canceled by the normalization factor of the wave function F as in eq Therefore the magnetic field appears in the S matrix only in the form f = fα x. Because fα x = L/2πl 2 cos θ H θ H = 2πx /L this means that the S matrix depends only on the field component H cos θ H in the direction perpendicular to the nanotube surface at the center-of-mass of impurities. This is the origin of the scaling law of conductance which we found in a previous numerical study. In this case the Green s functions are rewritten as g jj g jj BB g AB jj g BA jj = cosh[2f ] + i sinh[2f ] tanφ jj = cosh[2f ] i sinh[2f ] tanφ BB jj =Re[ AB i φ g 1 r jj e ] jj a j b j 1 =Re[ g1r BA i φ jj e ] jj b j a j T matrix In the following we shall exclusively consider the case that all impurities have a same strength i.e. u j =u. An extension to other cases is straightforward. By using eq the renormalized impurity potential can be written as u γl ζ 2.46 i g

5 21 Effective Mass Theory of Carbon Nanotubes Vacancies in Magnetic Fields ζ + ζ g 1 Γ BB t BB =1 ζ g 1 Γ BA t AB 1 ζ + ζ g 1 Γ t AB = ζ g 1 Γ AB t BB 1 ζ + ζ g 1 Γ BB t BA = ζ g 1 Γ BA t 2.51 where we define the following matrices: g = g jj ζ = 1 + 2iũ g Γ Γ AB = g jj BB = cosh[2fα x ] ũ = u L An examination of perturbation series respect to the impurity potential reveals that the T matrix can be written as T = 1 γl ζ2a i a + j AL i g t T BB = 1 γl ζ2b i b + j AL i g tbb T AB = 1 γl ζ2a i b + j AL i g tab T BA = 1 γl ζ2b i a + j AL i g tba. 2.5 As a result the equations for the T matrix can formally be written as 1 ζ + ζ g 1 Γ t =1 ζ g 1 Γ AB t BA t =t t BB =t BA t AB =t t BA =t BA Γ =Γ Γ BB =Γ BB Γ AB =Γ AB Γ BA =Γ BA 2.52 = g a + i a j Γ BB = g BB b + i b j = g AB a + i b j Γ BA = g BA b + i a j If there are N A and N B impurities at A and B site respectively t and Γ are an N A N A matrix t BB and Γ BB are an N B N B matrix t AB and Γ AB are an N A N B matrix and t BA and Γ BA are an N B N A matrix. In the case of a finite number of impurities we solve eq numerically and obtain T matrix using eq The above equations are essentially same as those in the absence of a magnetic field and the magnetic field appears mainly through the change in the diagonal Green s function from g to g = cosh[2fα x ]. We write the summation of T as T T AB T S = 2.54 T BA where T T AB T BA and are 2 2 matrices given by T T KK T KK = etc T K K T K K Then the reflection coefficients are given by r KK = ia = ia r K K r K K = ia r KK = ia e 2f T KK e 2f KK+iT AB KK+iT BA KK e 2f T K K e 2f K K itk AB BA K itk K T K K+ K K+ie 2f T AB K K ie 2f T BA K K T KK +TBB KK ie 2f TKK AB +ie2f TKK BA and the transmission coefficients are given by t KK =1+ ia t K K =1+ ia t K K = ia t KK 2.56 e 2f T KK+e 2f KK+iT AB KK it BA KK e 2f T K K +e 2f K K itk AB BA K +itk K = ia T K K K K+ie 2f T AB K K+ie 2f T BA K K T BB KK TKK ie 2f TKK AB ie2f TKK BA Similar expressions are obtained for r KK etc. and t KK etc. for waves incident from the right hand side. 3. Examples In the following the center-of-mass of impurities is always chosen at r = and θ H = i.e a top of CN s for simplicity. The conductance for arbitrarily values of θ H is obtained by replacing the strength of magnetic field H by H cos θ H because of the scaling shown above. 3.1 Impurities at same sublattices We first consider the case where N A impurities are introduced at A sites in the distance much smaller than L. In this case only the diagonal Green s function g ± r r appears in the perturbation series for the T matrix. Because g ± r r does not vary the distance so much it can be replaced by g ± r r where r is the center-of-mass of impurities. As a result T matrices are given by T AB =T BA = = and T = V A S V A S = j + V A S G T 3.1 2ũ j a j a + j 3.2 where G is diagonal Green s function at the center-ofmass of impurities. This result shows that the effective potential for localized impurities is given by the sum of their potentials. We write a position of impurity at an A site as r A j = n aj a + n bj b where a and b are primitive lattice translation vectors shown in Fig. 1 and n aj and n bj are integers. For this lattice point φ A j is given by

6 486 Masatsura Igami Takeshi Nakanishi and Tsuneya Ando Vol. 7 For a single impurity N 1 = 1 and N 2 = N 3 = in particular we have G = e π h 1 + 2ũ g Fig. 2. Schematic illustration of impurities in an armchair nanotube. The open and closed circles denote A and B lattice points respectively. a A b A 3 c AB and d A 3 B. φ A j =K K n aj a + n bj b+η = 2π 3 n aj + n bj +η mod 2π. 3.3 It is obvious from eq. 3.3 that φ A j takes only three independent values η η + 2π/3 orη + 4π/3. If there are N 1 impurities φ A 1 =η N 2 impurities φ A 2 =η+2π/3 and N 3 impurities φ A 3 =η+4π/3 T is given by T = iũ ia 1 3Mũ 2 +2iNũ g N+3iMũe 2f e iη P e iη P N+3iMũe 2f 3.4 N = N 1 + N 2 + N 3 M = N 1 N 2 + N 2 N 3 + N 3 N 1 P = N 1 + N 2 ω + N 3 ω where ω= exp2πi/3. When N 1 =N 2 =N 3 P vanishes and inter-valley transmission and reflection become absent. From eq. 3.4 we obtain an analytical expression for the conductance G = e2 1 3ũ 2 M 2 1+ π h 1 3ũ 2 M 2 +2Nũ g The conductance is independent of the parameter P. This shows that the conductance is always quantized into the conductance quantum e 2 /π h a half of the ideal conductance and independent of N and M in the strongfield limit g. Its origin is intuitively understood by the distribution of Landau wave functions along the circumference direction. In fact the Landau wave function at the K point is mainly localized in the A site at a top of CN s. When impurities exist in the A site at the top of CN s a scattering probability for an in-coming channel K increases due to the multiple scattering among the A sites and consequently the in-coming wave is reflected back into the same valley i.e. r K K = 1. At the K point the situation is completely opposite and the Landau wave function for the A site has no amplitude at the top of CN s. This leads to a complete transmission for the wave at the K point. Therefore the conductance is quantized into the conductance quantum in the strongfield limit. For impurities at B sites the same conductance is obtained except that the roles of the K and K points are interchanged. This shows that the conductance is given by G e 2 /π h independent of the field in the case of a strong scatterer ũ 1. In the case of A 3 impurities N 1 = N 2 = N 3 =1 [see Fig. 2 b] we have N=3 M=3 and P =. In the absence of a magnetic field a perfect reflection occurs and the conductance becomes zero in the limit of strong scatterers. In high magnetic fields the conductance is again quantized into e 2 /π h. 3.2 Pair of impurities: AB We next consider two neighboring impurities at an A site r A and at a B site r B. In this case the T matrix satisfies This leads to t =1 ζ g 1 Γ AB t BA t BB =1 ζ g 1 Γ BA t AB t AB = ζ g 1 Γ AB t BB t BA = ζ g 1 Γ BA t. 3.8 t = t BB = 1 1+Γ AB ζ g 1 2 t AB = t BA Γ AB ζ g 1 = 1+Γ AB ζ g where use has been made of Γ BA = Γ AB. In a weak-field regime f = α 1 we have g 1 and g AB L/a. In the case of strong scatterers therefore the conductance is approximately given by an ideal value 2e 2 /π h small corrections G = 2e2 π h [ sin 2 Γ AB 2 φ AB +4f 2 ] 3.1 where φ AB =φ A φ B /2. The deviation from the ideal conductance increases the the field but its magnitude is quite small because Γ AB 2 L/a 2 1. In the strong-field regime f 1 the diagonal Green s function becomes extremely large i.e. g = cosh[2f ] exp[2f ] 1. When the condition Γ AB ζ g 1 1 Γ AB g and t AB = t BA Γ 1 is satisfied we have t =t BB 1 1 AB g. This means that we can ignore effects of off-diagonal Green s function in the strongfield limit. The conductance is given by which is exponentially small. G 8e 4f Γ AB Impurities: A NA B NB Consider the general case N A impurities at the A site and N B impurities at the B site where N A > and N B >. In the absence of a magnetic field the conductance is quantized into zero one and two times of the conductance quantum e 2 /π h depending on the difference in the number of impurities at A and B sublattices N AB in

7 21 Effective Mass Theory of Carbon Nanotubes Vacancies in Magnetic Fields 487 the case of strong scatterers and L/a 1. This can be understood in terms of a pair-wise elimination by multiple scattering between A and B impurities as has been shown previously. 27 The situation does not change as long as the diagonal Green s function g is not so much different from unity. Corrections due to the magnetic field can be calculated and in the case N AB = for example the correction is given by G e2 a 2f π h C 2 L 3.12 where and C is a positive constant roughly proportional to ON A + N B. In the strong-field where the diagonal Green s function becomes much larger than the off-diagonal term we have t 1 ζ + ζ g 1 Γ 1 t AB t BB 1 ζ + ζ g 1 Γ BB 1 t BA Since t and t BB are same as those for impurities at same sites reflection and transmission coefficients are solved analytically and become r KK = r K K 1 r K K = r KK t KK = t K K = t K K = t KK These results indicate that the conductance vanishes always in the strong-field limit independent of N AB. This can intuitively be understood again by the buildup of the wave function at the top side of the cylindrical nanotube in high magnetic fields. In fact the incident wave at the K is scattered strongly by the potential at A sites and that at the K points by the potential at B sites. There exists a magnetic field range where g 1 g g AB and g g BA because g AB g BA L/a 1. In such intermediate field range scattering is determined by N AB remaining impurities at A or B sublattices. Therefore the conductance is given by an expression similar to eq. 3.6 for impurities only at A or B sublattice points. This shows that the conductance is nearly quantized into e 2 /π h for sufficiently strong impurities ũ g 1. For N AB = 2 such as AB 3 impurities to be discussed in the next section for example the conductance nearly vanishes in the absence of a magnetic field increases and takes the value e 2 /π h in the intermediate field range and then decreases and vanishes in the limit of a strong magnetic field. For N AB =1 such as A 43 B 42 on the other hand the conductance stays at e 2 /π h for a wide range of the magnetic field and vanishes in the high-field limit. 4. Numerical Calculations The local site energy V at an impurity in a tightbinding model and the potential strength ũ in the effective-mass theory are connected to each other by following relations. 3 3 u = 2 a2 V γ = 2 γ a ũ = 1 a V L γ Fig. 3. Calculated conductance as a function of a magnetic field at ε= in the presence of a single impurity potential at an A site in armchair nanotubes for several ũ. L/ 3a=3 and θ H =. Solid and dashed lines are results obtained by the effective-mass theory and in a tight-binding model respectively. where γ is a nearest-neighbor hopping integral and a is the lattice constant. For examples we have V/γ = and.1 for ũ = and.1 respectively when L/a = 5. In this section we shall consider the case θ H = and ε = again. Figure 3 shows calculated conductance as a function of a magnetic field in the presence of a single impurity at an A site in armchair nanotubes for several ũ where we set L/ 3a=3. Solid and dashed lines show results obtained by an effective-mass theory and in a tight binding model respectively which show good agreement in all range of the magnetic field. In the case of a weak impurity potential ũ<1 the conductance decreases the increase of the field and approaches the conductance quantum e 2 /π h in the strong-field limit. In the case of a strong impurity potential ũ 1 the conductance is quantized into e 2 /π h independent of the field. These results are in agreement those obtained in CN a single lattice-vacancy. 28 Figure 4 shows calculated transmission and reflection probabilities in the presence of a single impurity potential at an A site in armchair nanotubes for ũ=1.. In the weak-field regime both intra- and inter-valley processes occur and this leads to scattering wave typical 3 3 periodicity around the impurity. In the high magnetic field inter-valley scattering is suppressed and intra-valley transmission for in-coming channel K and reflection for K have an equal amplitude i.e. t KK = r K K =1. Because of the peculiar wave functions in a high magnetic field we have t K K = r KK =1 if an impurity is situated at B site instead of A or when θ H =π. Figure 5 shows calculated conductance in the presence

8 488 Masatsura Igami Takeshi Nakanishi and Tsuneya Ando Vol. 7 Fig. 4. Calculated transmission and reflection probabilities dotted lines and conductance solid line as a function of a magnetic field in the presence of a single impurity potential at an A site in armchair nanotubes for ũ=1.. Fig. 5. Calculated conductance as a function of a magnetic field at ε= in the presence of A 3 impurities in armchair nanotubes for several ũ. L/ 3a=3 and θ H =. Solid and dashed lines are results obtained by the effective-mass theory and in a tightbinding model respectively. of A 3 impurities in armchair nanotubes. In this case the conductance is obtained by putting N=3 and M=3 in eq The conductance decreases impurity potential and vanishes in the limit ũ 1 in the absence of a magnetic field. The field dependence is opposite between the case ũ<1/3 and ũ>1/3 and disappears when ũ=1/3. In the strong-field limit the conductance is quantized into e 2 /π h independent of ũ. The vanishing conductance in the limit ũ corresponds to the case of an A 3 lattice-vacancy. Figure 6 shows calculated conductance in the presence of an AB pair impurity in armchair nanotubes L/ 3a=3. In the effective-mass theory we have to introduce a cutoff n c which is determined by the condition that the corresponding wave length 2π/k c k c =2πn c /L should be comparable to the lattice constant a. Therefore we have roughly n c L/a. In actual calculations the cutoff is chosen as n c = 26. Because of the singularity of the off-diagonal Green s function the conductance is almost equal to G =2e 2 /π h in the absence of a magnetic field. It decreases monotonically the field and vanishes in the strong-field limit. Furthermore it does not depend on the strength of the impurity potential except near ũ= ± Γ AB corresponding to a resonance. 27 Calculated transmission and reflection probabilities in the presence of an AB pair impurity in armchair nanotubes for ũ=1. are shown in Fig. 7. In the absence of a magnetic field the back scattering in the same valley r KK and r K K and the transmission between different valleys t KK and t K K are absent because of a mirror symmetry of impurities about a plane containing the axis. 18 Inter-valley components in transmission and Fig. 6. Calculated conductance as a function of a magnetic field at ε = in the presence of an AB pair impurity locates along a circumference direction in armchair nanotubes for several ũ. L/ 3a = 3 θ H = and n c = 26. Solid and dashed lines are results obtained by the effective-mass theory and in a tightbinding model respectively. reflection coefficients gradually increase reach maximum around L/2πl and then decrease the field. In strong-field the conductance vanishes and the reflection occurs in the same valley.

9 21 Effective Mass Theory of Carbon Nanotubes Vacancies in Magnetic Fields 489 Fig. 7. Calculated transmission and reflection probabilities dotted lines and conductance solid line as a function of a magnetic field in the presence of an AB pair impurity locates along a circumference direction in armchair nanotubes for ũ=1.. Fig. 8. Calculated conductance as a function of a magnetic field at ε = in the presence of A 3 B impurities in armchair nanotubes for several ũ. L/ 3a = 3 θ H = and n c = 26. Solid and dashed lines are results obtained by the effective-mass theory and in a tight-binding model respectively. Figure 8 shows calculated conductance in the presence of AB 3 impurities in armchair nanotubes where L/ 3a=3 and n c = 26. In the case of a weak impurity potential ũ<1 the conductance decreases the increase of the field and approaches zero in the strong-field limit. In the case of a strong impurity potential ũ>1 we can neglect scattering from impurity at a B site because of the singularity of the off-diagonal Green s function in the weak-field limit. Consequently multiple scattering among A sites becomes dominant and the transmission probability of in-coming channel K increases the field until L/2πl 2 2. The conductance takes nearly a quantized values e 2 /π h in the intermediate field region. In the strong-field limit we cannot ignore an impurity at a B site. Therefore both in-coming channels K and K are strongly scattered and the conductance vanishes. Figure 9 shows calculated conductance for A 43 B 42 impurities localized in a circle a radius w/a 6. and a center at an atomic site. We have L/ 3a=3 and n c = 26. In the case of weak potential ũ=.1 the conductance decreases the increase of the field and vanishes in the strong-field limit as in the case of simpler examples. In the case of a strong impurity ũ 1 on the other hand the effective scattering from the A 43 B 42 impurities is reduced to that of a single impurity at an A site because N AB = 1. Further the conductance remains at e 2 /π h until L/2πl 2 =2 3 and vanishes in the strong-field limit. The conductance shows a sharp resonance around ũ.1 and the resonance position is quite sensitive to various parameters including n c. This is the origin of a large deviation between the results in a tight-binding model and an effective-mass theory for ũ=.1. Fig. 9. Calculated conductance as a function of a magnetic field at ε = in the presence of circular site-center A 43 B 42 impurities in armchair nanotubes for several ũ. L/ 3a = 3 θ H = and n c = 26. Solid and dashed lines are results obtained by the effective-mass theory and in a tight-binding model respectively. 5. Summary and Discussion In this paper we have studied effects of short-range impurities on transport of CN based on an effective-mass theory in the presence of a magnetic field perpendicular to the axis. The results are summarized as follows:

10 49 Masatsura Igami Takeshi Nakanishi and Tsuneya Ando Vol. 7 Green s functions describing propagation between same and different sublattices have different magneticfield and distance dependence. The former is safely replaced by the local density of states at the center-of-mass of impurities r and depends only on the field component in the direction of the r i.e H cos θ H. On the other hand the latter does not depend on the field and takes an extremely large value when impurities are localized in a distance of a few times of the lattice constant. As a result both T and S matrices are a function of H cos θ H leading to the conclusion that the conductance is also a function of H cos θ H. This is the origin of the conductance scaling we found in the previous numerical study. 28 The conductance exhibits characteristic field variation depending critically on the number of impurities. This is closely related to relative importance of diagonal and offdiagonal Green s functions at the center-of-mass of impurities. In the weak-field regime where the off-diagonal Green s function is much larger than the diagonal term the conductance strongly depends on the strength of potential and difference N AB in the number of impurities at A and B sublattices. In the limit of strong scatterers i.e. ũ 1 the conductance is almost equal to ideal 2e 2 /π h in the absence of the field and vanishes monotonically the field when N AB =. In the case N AB =1 the conductance is reduced to a half of the ideal value and does not depend on the field in intermediate fields. In the case N AB 2 the conductance vanishes in the absence of a field but increases up to a half of the ideal value in intermediate fields. In the strong-field limit Landau wave functions for A and B sites are localized in a spatial region different from each other i.e. around a bottom and a top of CN. Therefore an overlap integral of envelope functions for the A and B site becomes smaller and we can ignore contribution of the off-diagonal Green s functions and treat A and B sublattices separately. Consequently the conductance is quantized into e 2 /π h if impurities exist in only A or B sites. In other cases the conductance always becomes zero independent of N AB. There have been some experimental efforts to synthesize CN hetero atoms like boron and nitrogen. Electronic and geometric structures of B-doped multiwalled- CN s were studied STS measurements and the formation of a nanodomain of BC 3 islands was suggested. 41 In this system a doped boron will act as a local impurity in the host CN. Effects of a single B and N atom on transport in a 11-armchair nanotube were studied using an ab initio pseudopotential method in a Landauer formalism. 42 It was shown that doped B and N produce a quasi-bound impurity state at.7 ev below and.53 ev above the Fermi energy respectively and the conductance is given by e 2 /π h at the corresponding energy. The origin of this conductance is essentially same as that of a single impurity case which we have already discussed in ref. 27. In this case the strength of the impurity potential is estimated as V 1.5 and 2.2 ev for B and N respectively using eq of ref. 27 and eq This impurity strength is of the same order as that of the nearest-neighbor hopping integral γ.for a multi-wall nanotube L/a=3 for example we have ũ=1/6 in the case V =γ and the magnetic field corresponding to L/2πl 2 =1 and 5 is given by 4.9 and 24.7 T respectively. Another intriguing way to introduce local impurities is fluorination of the sp 2 carbon Electronic and geometric structures of a fluorinated graphene sheet are being studied using an ab initio pseudopotential method. 45 A doped fluorine makes a covalent bond an sp 2 carbon and the charge density distribution shows a typical 3 3 structure near the Fermi level. This means that the doped fluorine acts as a local defect in π-electron network of a graphene sheet. Acknowledgments Two of the authors M.I. and T.N. acknowledge the support of a research fellowship from the Japan Society for the Promotion of Science for Young Scientists. This work was supported in part by Grants-in-Aid for Scientific Research and for Priority Area Fullerene Network from the Ministry of Education Science Sports and Culture Japan. 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