3. Magnetism. p e ca (3.3) H = B = (0, 0, B), p p e c A( r), (3.1) A = 1 2 ( B r) = 1 ( By, Bx, 0) = p 2 e (

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1 3 Magetism 31 Couplig of matter to a magetic field: Diamagetism ad paramagetism A exteral magetic field ca couple to matter ad electros i two differet ways we cosider the o-relativistic case: 1 through the miimal couplig, expressed as p p e c A r, 31 where p is the mometum of the electro ad A is the vector potetial of the electromagetic field, ad through the spi of the electro, as gµ B σ B = e h mc S B, 3 where g is the gyromagetic factor for the free electro, µ B = e h mc > 0 is the Bohr mageto ad σ are the Pauli matrices We will be usig throughout the course atomic uits Sice the distace betwee atoms i solids is of a few agstrom, the atural legthscale is the Bohr radius a 0 : a 0 = h me = 05 agstrom = cm The eergy scale is give i E 0 = me4 h = e a 0 = erg = Ry = 7 ev 57 The miimal couplig 31 is resposible for the diamagetism i the system The couplig to the spi, eq 3, is the Zeema couplig ad is resposible for the paramagatism i the system From equatios 31 ad 3, we ca write the Hamiltoia of a electro i a magetic field B as H = p e ca e h m mc σ B + e h 1 dv 4m c r dr l σ + V r, }{{} 33 spi-orbit couplig where V r is the lattice potetial This Hamiltoia ca be derived out of the Dirac equatio Let us cosider a static magetic field applied alog the c directio: B = 0, 0, B, A = 1 B r = 1 By, Bx, 0 symmetric gauge ad p e A c = p e p A + A p + e c c A = p + eb c p xy + yp x p y x xp y + e B 4c x + y The, the Hamiltoia 33 becomes H = p m + µ Bl z + σ z B + e B 8mc x + y + e h We ote that L z = xp y yp x = h i l z = 1 x i y y, x S z = h σ z = h x y y x 58 = hl z, 1 4m c r dv dr l σ + V r 34

2 Here, L = h l is the orbital agular mometum ad S = h σ is the spi From eq 34, we have that the exteral magetic field couples liearly to the orbital agular mometum ad the spi How does the system react to the applicatio of exteral magetic field? It is to be expected that the system will magetize, therefore we should aalyze the thermal expectatio value of the magetizatio: M = µ = Trρ µ = B F S, where F S is the free eergy desity of the system that ca get magetized Remider: df S = SdT Md B The static isothermal magetic susceptibility is the give by χ αβ = M α B β = F S, B β B α where χ αβ is a tesor with compoets α ad β If we cosider the equilibrium state, to which the system relaxes after applicatio of a magetic field, we ca hadle the system withi equilibrium thermodyamics: F S = k B T l e βe Here, E are the eigevalues of the electro system i the presece of the exteral magetic field The, M = F S B = 1 E B e βe, which correspods to e βe M = µ = µ B l + σ = e h mc L + S The magetizatio is obtaied out of the thermal average of the magetic momet µ Sice the electros have egative charge e < 0, the total magetic momet has the opposite sig to orbital L ad spi S momets 59 If B = 0, 0, B, the χ zz = B M z = 1 1 Z k B T 1 Z k B T E B E B E B e βe e βe with Z = e βe beig the partitio fuctio i the caoical esemble I order to obtai the eigevalues of the Hamiltoia 34, we apply perturbatio theory up to the d order i the magetic field see QM II course The result reads as E = E 0 + µ B l z + σ z B + e B 8mc x + y + µ l z + σ z m B B E 0 E 0, m m where E 0 are the eigevalues of the uperturbed Hamiltoia I the limit B 0, E = µ B l z + σ z, B E B = e 4mc y + x + µ B m l z + σ z m E 0 E 0 35 m I the absece of collective magetism due to iteractio amog the electros, we have for B 0 that M z = 1 E Z B = 1 µ B l z + σ z = µ z = 0, Z i e, the magetizatio disappears, sice the magetic momets thermally compesate From eq 35 we ca divide the static susceptibility ito three terms: with χ zz = χ C + χ vv + χ dia, χ C = µ B l z + σ z e βe = µ z k B T e βe k B T >

3 beig the paramagetic cotributio, χ vv = 1 Z m the va Vleck susceptibility ad e l z + σ z m µ B E 0 E 0 e βe > 0 37 m e χ dia = 8mc x + y = 6mc r < 0 38 the diamagetic cotributio The paramagetic cotributio, eq 36, is positive ad has a 1/T-temperature depedece: χ C = C T, with C = µ B k B 1 Z l z + σ z e βe beig the Curie costat Please ote that eve whe the magetizatio is zero, l z + σ z = 0, χ C is differet from zero ad the system is i the paramagetic state The paramagetic cotributio, eq 37, is costat ad positive sice the eigevalues of the excited states m are larger tha the groud state eergy: E 0 m > E 0 This term will be sigificat whe k B T E 0 m E 0 ie, at low temperature For k B T E 0 m E 0, the va Vleck term has a 1/T-behavior like the Curie term The diamagetic cotributio, eq 38, is egative, which implies that the magetizatio of the system for small fields has opposite sig to the magetic field This is a purely quatum effect It comes from the term p p e c A Usually, the paramagetic cotributio is larger tha the diamagetic cotributio, but if the total agular mometum disappears, there is o paramagetic cotributio ad we are left with oly the diamagetic cotributio see below This is called the Larmor diamagetism 61 I this sectio, we cosidered oe electro uder the ifluece of a magetic field We will geeralize these results to a system of N o-iteractig electros 3 Paramagetism of localized magetic momets We cosider a system of N atoms or ios with partially filled electro shells The total magetic momet per atom/io is give by J = L + S I this case, we are dealig with localized magetic momets, which correspod to the electros i the partially filled shells ad are localized o the atoms/ios, ie we are dealig with isulators Let us cosider the followig two cases 1 The electro shell has J = 0, which would be so i, for istace, shells with S = ad L = : J = L + S, J = L S, L S + 1,, L + S, J = L S = 0 I this case, the liear term L+ S disappears, ad oly the higher order terms va Vleck paramagetism ad Larmor diamagetism cotribute If the shell does ot have J = 0, the liear term does ot disappear ad will domiate the magetic behavior of the system Please ote that the groud state is S + 1-fold degeerate i zero field We ca defie the magetic momet as µ = gµ B J, where g is the Ladé factor g = 1 + JJ SS + 1 LL + 1 JJ + 1 6

4 We cosider ow the simplest Hamiltoia where we eglect the diamagetic cotributio: H = i µ B = i gµ B J z B, with B = 0, 0, B, ad calculate the susceptibility for the N atoms: Z = Tr e βh = Tr e β i µ BBJ z = = = N 1 e gµ BBJ+1β i=1 1 e gµ BBβ N N e gµ BBJ+ 1 β e gµ BBJ+ 1 β e gµ BBβ 1 e gµ BBβ 1 i=1 The, the free eergy is +J i=1 m J = J e βµ BBm J F = k B T l Z = Nk B T l egµ BBJ+ 1 β e gµ BBJ+ 1 β ad the magetizatio where M = B F, e gµ BBβ 1 e gµ BBβ 1 M = Ngµ B J + 1 coth βgµ B BJ + 1 = Ngµ B JB J gjµ B Bβ, B J x = J + 1 J J + 1 coth J x 1 x J coth J is the Brilloui fuctio This fuctio is show i Fig 31 For x 1 it is liear ad for x 1 B J x 1 ad M Ngµ B J, the saturatio magetizatio For small magetic fields, cothx 1 x + x so that B J x x J 3, ad Abbildug 31: The Brilloui fuctio M N J + 1 gµ B J gµ BJBβ J 3 χ = χ = C T M B For spi- 1 systems, The, ad = N gµ B JJ + 1 B 3k B T = N gµ B JJ + 1 B 0 3k B T JJ + 1 with C = N gµ B 3k B J = S = 1, g =, C = Nµ B k B B 1/ x = tahx M = Nµ B tah µb B k B T 64

5 33 Pauli paramagetism of coductio electros I the previous sectio, we cosidered localized momets I the preset sectio, we shall deal with coductio electros i a metal The electros have a spi ad the applicatio of a magetic field iduces their magetizatio Therefore, we also expect a paramagetic cotributio from the coductio electros Let us cosider a miimal model where the spi of the coductio electros is coupled to a exteral magetic field: H =,σ ε k c σ c σ + µ BB c c c c 39 Abbildug 3: Eergy spectrum of the coductio electros i a exteral magetic field The z-compoet of all electro spis is ad S z = 1 c R c R c R c R = 1 c c c c R µ z = gµ B S z = µ B c c c c We calculate ow the magetizatio due to the applicatio of a exteral magetic field B = 0, 0, B: M = µ z = µ B c c c c 310 The Hamiltoia 39 ca be rewritte as H = ε k + µ BBc c + ε k µ BBc c so that we have free electros with slightly displaced oe-particle eergies: ε ± µ B B see Fig 3 Therefore, the expectatio value i eq 310 ca be calculated through the Fermi fuctio: M = µ B dε [fε + µ B Bρ 0 ε + µ B B fε µ B Bρ 0 ε µ B B], 65 where ρ 0 ε is the desity of states of the free electros For small B, we perform a Taylor expasio: M = µ BB dε ρ 0 ε df dε = µ BB dε ρ 0 ε df, df dε dε = δε ε F The, ad M = µ BBρ 0 ε F χ Pauli = µ Bρ 0 ε F is the Pauli susceptibility oly valid for small B Therefore, oe expects i metals at low temperature a costat cotributio to the susceptibility 34 Ladau diamagetism I the previous sectio, we dealt with the couplig of the spis of coductio electros to a exteral magetic field The coductio electros have also a well defied p, ad therefore we will also have a diamagetic cotributio due to the diamagetic couplig to the magetic field: p p e c A 66

6 I order to aalyze oly this couplig, we cosider spiless fermios i a magetic field I this case, the Hamilto operator is N p i e ca r i H = m i=1 = 1 p i ec m p ia i r i ec A i r i p i + e c A r i i For the magetic field give as B = 0, 0, B ad the Ladau gauge, A = 0, Bx, 0, div A = 0 p A = A p, the Hamiltoia reads H = p ix m + p iy m + p iz m eb mc p iyx i + e B mc x i i Sice we are cosiderig o-iteractig electros, we cocetrate o the H for oe electro: where H i = p x m + m ω o x ω o = eb mc p y mω o + p z m, is the cyclotro frequecy, which correspods to the classical frequecy of particles i a magetic field due to the Loretz force i the xy plae We cosider the followig oe-particle wavefuctio: Ψ r = Cϕxe ik yy e ik zz Here, C is a ormalizatio costat, ad the y- ad z-depedeces are give i plae waves sice the Hamiltoia does ot deped explicitly o either y or z The, HΨ r = p x m + m ω 0 = ECϕxe ik yy e ik zz x hk y mω h k z m Cϕxe ik yy e ik zz We are left with a displaced harmoic oscillator for the x coordiate: x x0 ϕx = ϕ, λ where x 0 = hk y = hck y mω 0 eb, h hc λ = = mω 0 eb ϕx are the Hermite polyomials The eigevalues of H are E,ky,k z = hω h k z m, with beig the Ladau quatum umber Istead of free electros with dispersio ε k = h k m, we ow have free electros with oe-particle eergies E,ky,k z, which are determied through three quatum umbers: the Ladau quatum umber, k y ad k z The eergy, however, has o explicit depedece o k y Therefore, we have degeeracy i the Ladau level This degeeracy is determied from the followig coditio for x 0 : x 0 = hk y mω 0 L x O the other had, we have periodic boudary coditios i y: The, k y = π L y l y with l y N π h l y L x l y mω 0L x L y, mω 0 L y π h 68

7 which implies that the umber of allowed l y, defiig the degeeracy of a Ladau level, is mω 0 L y L x π h = e B L x L y c π h Let us calculate the cotributio of the Ladau diamagetism to the susceptibility by cosiderig the thermodyamic relatios We ca work either i the caoical or i the grad caoical esemble: Φ = k B T l 1 + e βε α µ α = k B T L z e B L x L y dk z l 1 + e βε α µ π h c π h =0 = k BT V e B [ ] π h dk z l 1 + e β hω h k z m µ, c =0 which ca be writte as Φ = e k BTV B π h gµ hω c, =0 where gµ x = dk z l 1 + e βµ x h k z m I order to perform the sum over the Ladau quatum umber, we ca use the Euler-Maclauri formula: F + 1 = Fxdx + 1 =0 0 4 F 0 The, gµ hω = gµ hω 0 xdx + 1 d =0 0 4 dx gx x=0 1 µ = dy gy hω 0 d hω 0 4 dy gy, y=µ 69 with y = µ hω 0 x The grad caoical potetial the becomes Φ = k BTm π h 3 V It ca be writte as with µ dy gy }{{} B idepedet Φ = Φ 0 T, µ h e B 4m c Φ 0 T, µ = k BTm π h 3 V The, ad µ M = Φ B = e h 1m c B Φ 0 µ χ = M B = e h 1m c Φ 0 µ hω 0 4 µ Φ 0T, µ, Therefore, the Ladau susceptibility is χ Ladau = 1 3 µ B Φ 0 µ d dy gy µ dy gy = k µ BTm π h 3 V dy dk z l 1 + e βy h k z m For Φ 0, we directly take the grad caoical potetial for free electros without magetic field: Φ 0 = k B T l 1 + e βε µ, Φ 0 µ = e βε µ 1 + e = βε µ 1 e βε µ + 1 = fε k, Φ 0 µ = df dε k T 0 ρ 0 ε F 70

8 The, χ Ladau = 1 3 χ Pauli The total cotributio of the coductio electros to the susceptibility is χ total = 3 χ Pauli, ad it is paramagetic 35 De Haas va Alphe effect The de Haas va Alphe effect is the periodic variatio of the magetic susceptibility as a fuctio of the iverse of the magetic field With this effect oe ca measure the Fermi surface of metals as well the effective mass of coductio electros The De Haas va Alphe effect is based o the eergy quatizatio of the electros due to the exteral magetic field Let us cosider the k x k y plae perpedicular to the magetic field Without applicatio of a magatic field, both k x ad k y are good quatum umbers, i e, the electroic states are characterized through lattice poits i the k x k y plae I a magetic field, the states are degeerate ad characterized by the Ladau quatum umber There are e BL x L y π hc degeerate states with eergy hω I order to uderstad quatum oscillatios i the presece of a magetic field, let us cosider the two-dimesioal free electro gas i the x y plae ad a magetic field applied alog the z-directio The Ladau states are quatized ad, sice there is o third dimesio, there is o cotiuum cotributio The eergy eigevalues are E = hω 0 + 1, ad the degeeracy for every level is mω 0 L x L y π h = e BL xl y π hc = pb, 71 with p = e BL xl y π hc If we have a system of N electros, pb 0 electros fill 0 Ladau levels, ad the remaiig N pb 0 electros fill the th level The total eergy is the give by E tot = 0 1 =0 pb hω hω 0 N pb 0 For larger B, the fillig of the th level decreases liearly The situatio is graphically expressed i Fig 33 Abbildug 33: Fillig of the Ladau levels i differet magetic fields If B = N p 0 1 B = p 0 N, the th level wo t be filled aymore, ad the Fermi eergy is the at the th 0 level Therefore, for the total eergy ad magetizatio we expect a 1 B-periodic behavior 36 Quatum Hall effect Sice i two dimesios oly discrete highly degeerate Ladau levels are preset, oe ca i such cases aalyze the Ladau quatizatio i detail 7

9 The desity of states of a two-dimesioal spiless o-iteractig electro gas i a strog magetic field is give by ρ d E = mω 0L x L y δ E hω π h This desity of states cosists of delta peaks at the Ladau eergies hω 0 + weighted by the degeeracy 1 With the help of semicoductor physics, it is possible to create a purely two-dimesioal electro gas This is doe o heterostructures of p-doped GaAs ad -doped Ga 1 x Al x As Usig the molecular beam epitaxy MBE techique, oe ca grow alterately the two semicoductors i order to form a heterostructure Each layer has a width of about several aometers Ga 1 x Al x As is -type doped, which geerates extra mobile electros i its coductio bad These electros migrate to fill the holes at the top of the GaAs valece bad ad partially ed up as states ear the bottom of the GaAs coductio bad There is of course a positive charge left o the door impurities which attracts these electros to the iterface ad beds the bads This is the source of electric field i the system Abbildug 34: The scheme of the eergy bads of the Ga 1 x Al x As/GaAs heterostructure The charge desity of the system is cm The charges bed i the y-directio util the created E y -field compesates the Loretz force: E y = v x c B = j xb ec = 1 I B = U y ec L y L x L y The trasfer of electros from Ga 1 x Al x As to GaAs cotiues util the dipole layer formed from the positive doors ad the egative iversio layer is sufficietly strog This dipole layer gives rise to a potetial discotiuity, which makes the Fermi level of GaAs equal to that of Ga 1 x Al x As The electroic states perpedicular to the separatio layer are localized so that i the layer betwee GaAs ad Ga 1 x Al x As there is a two-dimesioal electro gas A possible way to ivestigate this two-dimesioal electro gas is by cosiderig the Hall effect For that we shall itroduce this effect briefly Remider: Hall effect Uder applicatio of a electric field alog the x-directio ad a homogeeous magetic field alog the z-directio, the electros i a coductor of legth L x ad width L y, movig at velocity v x i the x-directio, feel a Loretz force i the y-directio: F L = e c v xb 73 Here, U y = U H is the Hall voltage, which is measured It is determied by U H = r H IB L x The proportioality coefficiet r H, r H = 1 ec, 74

10 is called the Hall coefficiet ad ca be either positive or egative depedig o whether the carriers are electros or holes I the classical Hall effect, we have that the Hall resistace U H I is proportioal to the magetic field I the two-dimesioal electro system that we itroduced, for large eough magetic fields ad low temperatures, this proportioality is ot aymore fulfilled I fact, what is observed is that h the Hall resistivity ρ xy shows steps ad plateaux at quatized values of ie, where i is a iteger ρ xy is ot liear i B, but remais costat over a large iterval of B ad jumps at a critical B to the ext quatized value h i 1e see Fig 35 As a fuctio of the Fermi eergy, the Hall coductivity of the two-dimesioal electro gas i a magetic field follows a step-like behavior The quatized values for the Hall coductivity are the values, where the Fermi eergy falls i the gap betwee two discrete Ladau levels There must be a mechaism that explais why the Fermi eergy either remais i a gap betwee two Ladau levels or i a regio of occupied states ad the curret-carryig state does ot chage as a fuctio of magetic field Abbildug 35: Schematic represetatio of the measured Hall resistace While the Hall resistivity shows quatized values, the logitudial resistivity ρ xx = 0 I the basis of the Ladau states, the matrix elemets of the curret operator are give by k y j x lk y = e m hω 0 + 1δ,l 1 δ,l+1 δky k y m, h k y j y lk y = eω 0 + 1δ,l 1 + δ,l+1 δky k mω y 0 Out of these matrix elemets, oe ca derive the Hall coductivity: σ xy = e e + 1 = + 1 π h h 75 Two-dimesioal systems are ot completely perfect, but have impurities ad impurity scatterig is possible Through this impurity scaterig ad, i geeral, a disorder potetial, the degeeracy of Ladau levels is lifted ad the Ladau levels exted to Ladau bads If the disorder is ot very strog, there are still gaps betwee the Ladau bads for various Ladau values I every Ladau bad, there are delocalized states i the middle of the bad, ie, where the origial eigevalue hω resided, ad localized states at the boudaries of the Ladau bads The localized states do ot cotribute to the curret trasport, which meas that whe the Fermi eergy is i the regio of localized states the diagoal coductivity σ xx disappears ad the o-diagoal Hall coductivity σ xy keeps the value that it takes i the gap betwee two Ladau levels Without disorder we caot accout for the plateau ature of the quatum Hall effect The observatio that i the quatum Hall effect the quatizatio of the coductivity is a iteger multiple of a uiversal costat meas that this 76

11 behavior ca oly be depedet o a very robust property of the twodimesioal systems: the geometry The quatizatio of the coductivity is obtaied i a oe-dimesioal chael I the quatum Hall effect curret arises from the states at the edge: EDGE STATES Up to ow, we have talked about the iteger quatum Hall effect Whe the quality of samples is high eough, at low T ad high B oe observes o-iteger steps: ρ xy = h fe, with f = p q p ad q are itegers, q is odd I order to uderstad the fractioal quatum Hall effect, oe has to iclude electroic correlatios The excitatios i such a system ca be described through quasiparticles with o-iteger charge Laughli wo the Nobel Prize i 1998 for suggestig a asatz wavefuctio to describe the fractioal quatum Hall effect 77

3. Magnetism. = e2. = erg = 2 Ry = 27.2 ev. E 0 = me4 H = p p e c A( r), (3.1) B = (0, 0, B), A = 1 2 ( B r) = 1 ( By, Bx, 0) = p 2 e (

3. Magnetism. = e2. = erg = 2 Ry = 27.2 ev. E 0 = me4 H = p p e c A( r), (3.1) B = (0, 0, B), A = 1 2 ( B r) = 1 ( By, Bx, 0) = p 2 e ( The eergy scale is give i 3 Magetism E 0 = me4 h = e a 0 = 043 10 10 erg = Ry = 7 ev 31 Couplig of matter to a magetic field: Diamagetism ad paramagetism A exteral magetic field ca couple to matter ad