On the Transport Diamonds and Zero Current Anomaly in InGaAs/InP and GaAs/AlGaAs
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un On the Transport Diamonds and Zero Current Anomaly in InGaAs/InP andGaAs/AlGaAs
S. Fujita ∗ Department of Physics, SUNY at Buffalo, Buffalo, New York 14260, USA
H. C. Ho † Sincere Learning Centre, Kowloon, Hong Kong SAR, China (Dated: December 2, 2018)In the quantum Hall effect (QHE) the differential resistivity r xx ≡ r vanishes within a rangewhere the Hall resistivity forms a plateau. A microscopic theory is developed, starting with acrystal lattice, setting up a BCS-like Hamiltonian in terms of composite bosons, and using statisticalmechanical method. The main advantage of our bosonic theory is its capability of explaning theplateau formation in the Hall resistivity, which is assumed in the composite fermion theories. Inthe QHE under radiation, the resistivity vanishes within a range with no plateau formation. Thisis shown in terms of two-channels model, one channel excited by radiation where the supercurrentsrun and the other (base) channel in which the normal currents run. The transport diamonds (TD)and the zero direct current anomaly (ZCA) occur when the resistivity r is measured as a functionof magnetic field and direct current (DC). The spiral motion of an electron under a magnetic fieldcan be decomposed into two, the cyclotron motion with the cyclotron mass m ∗ and the guidingcenter motion with the magnetotransport M ∗ . The quantization of the motion generates magneticoscillations in the density of states. The magnetoconductivity is calculated, using kinetic theory andquantum statistical mechanics. The TR and ZCA are shown to be a breakdown of QHE. The integerQHE minima are shown to become the Shubnikov-de Haas (SdH) maxima progressively as the DCincreases. The ZCA at low temperatures ( T = 0 . . PACS numbers: 73.43.-f, 73.43.Qt, 74.25.Ha
I. INTRODUCTION
Recently, Studenikin et al . [1] discovered transportdiamonds (TR) and zero-current anomaly (ZCA) inIn x Ga − x As/InP and GaAs/AlGaAs. The TR and ZCAoccur when the differential resistance r xx = dV xx /dI of a Hall bar sample after the red-light illumination isplotted in the plane of the magnetic field and the di-rect current (DC). See Fig. 4, which is reproduced fromRef. [1], Fig. 2. Diamond-shaped regions developing fromSdH minima are called transport diamonds and a sharpdip in r xx appearing at a narrow horizontal line at zeroDC is called a zero-current anomaly . The details of theexperiments and theoretical backgrounds can be foundin Ref. [1]. The original authors [1] suggested an in-terpretation: the breakdown of the quantum Hall effect(QHE). We shall show in the present work that this isindeed the case based on the composite (c-)boson model,the model originally introduced by Zhang, Hansson andKivelson [2] and later developed by Fujita’s group [3]. Inthe prevalent theories [4], the QHE is discussed in termsof the c-fermions. [5] The formation of the Hall resis-tivity plateau where the resistivity vanishes, is assumed,however. In our c-boson theory, the plateau formation is ∗ Electronic address: fujita@buffalo.edu † Electronic address: [email protected] explained from the first principles. In Sec. II, we reviewour microscopic theory of the QHE. The QHE for a sys-tem subjected to a radiation is developed and discussedin Sec. III. The density of Landau states and statisicalweight in two dimensions (2D) are calculated in Sec. IV.Shubnikov-de Haas (SdH) oscillations [6] in the magne-toconductivity and de Haas-van Alphen (dHvA) oscilla-tions [7] in the magnetic susceptibility, are jointly calledmagnetic oscillations. They originate in the oscillationsin statistical weight. Magnetic oscillations are often dis-cussed, using a so-called Dingle temperature [8]. But thisis a phenomenological treatment, which must be avoided.Following our previous work [9], we present a microscopictheory of the SdH oscillations for a 2D system in Sec. V.The TR and ZCA are discussed in Sec. VI.
II. INTEGER QUANTUM HALL EFFECT
If a magnetic field B is applied slowly, then the clas-sical electron can continuously change from the straightline motion at zero field to the curved motion at a finite B (magnitude). Quantum mechanically, the change fromthe momentum state to the Landau state requires a per-turbation. We choose for this perturbation the phononexchange attraction between the electron and the fluxon (elementary magnetic flux). Consider an electron witha few fluxons. If the magnetic field is applied slowly,the energy of the electron does not change but the cy-clotron motion always acts so as to reduce the magneticfields surrounding the electron. Hence the total energyof the composite of an electron and fluxon is less thanthe electron energy plus the unperturbed field energy. Inother words, the composite (c-)particle is stable againstthe break-up, and it is in a bound (negative energy)state. The c-particle is simply a dressed electron car-rying Q fluxons. Q = 1 , , . . . . Originally, the c-particlewas introduced as a composite of one electron attachedwith a number of Chern-Simons gauge objects. [4] Theseobjects are neither bosons nor fermions, and hence thestatistics of the composite is not clear. The basic parti-cle property (countability) of the fluxons is known as theflux quantization, see Eq. (II.15). We assume that thefluxon is an elementary fermion with zero mass and zerocharge, which is supported by the fact that the fluxon,the quantum of the magnetic field B , cannot disappearat a sink unlike the bosonic photon, the quantum of theelectric field E [3]. Fujita and Morabito [10] showed thatthe center-of-mass (CM) of a composite moves follow-ing the Ehrenfest-Oppenheimer-Bethe’s (EOB) rule [11]:the composite is fermionic (bosonic) if it contains an odd(even) number of elementary fermions. Hence the quan-tum statistics of the c-particle is established.At the Landau level (LL) occupation number, alsocalled the filling factor , ν = 1 /Q , Q odd, the c-bosonswith Q fluxons are generated and can condense below cer-tain critical temperature T c . The Hall resistivity plateauis caused by kind of the Meissner effect as explained later.We develop a theory for GaAs/AlGaAs heterojunc-tion, the theory which can also be applied to InGaAs/InPquantum well. GaAs forms a zincblende lattice. We as-sume that the interface is in the plane (001). The Ga ions form a square lattice with the sides directed in [110]and [1¯10]. The “electron” (wave packet) with a negativecharge − e will then move isotropically with an effectivemass m . The As − ions also form a square lattice ata different height in [001]. The “holes”, each having apositive charge (+e), will move similarly with an effec-tive mass m . A longitudinal phonon moving in [110]or in [1¯10] can generate a charge (current) density varia-tion, establishing an interaction between the phonon andthe electron (fluxon). If one phonon exchange is consid-ered between the electron and the fluxon, a second-orderperturbation calculation establishes an effective electron-fluxon interaction (cid:12)(cid:12) V q V ′ q (cid:12)(cid:12) ¯ hω q (cid:0) ε | p+q | − ε p (cid:1) − (¯ hω q ) , (II.1)where q (¯ hω q ) is the phonon momentum (energy), V q ( V ′ q )the interaction strength between the electron (fluxon)and the phonon. If the energies (cid:0) ε | p+q | , ε p (cid:1) of the finaland initial electron states are equal, the effective interac-tion is negative (attractive) as seen from Eq. (II.1).Following Bardeen-Cooper-Schrieffer (BCS) [12], westart with a Hamiltonian H with the phonon variables eliminated: H = X k X s ε (1) k n (1)k s + X k X s ε (2) k n (2)k s + X k X s ε (3) k n (3)k s − v X ′ q X ′ k X ′ k ′ X s h B (1) † k ′ q s B (1) kq s + B (1) † k ′ q s B (2) † kq s + B (2) k ′ q s B (1) kq s + B (2) k ′ q s B (2) † kq s i , (II.2)where n ( j ) k s is the number operator for the “electron”(1)[“hole”(2), fluxon (3)] at momentum k and spin s withthe energy ε ( j ) ks . We represent the “electron” (“hole”)number n ( j ) k s by c ( j ) † k s c ( j ) k s , where c ( c † ) are annihilation (cre-ation) operators satisfying the Fermi anticommutationrules: n c ( i ) k s , c ( j ) † k ′ s ′ o ≡ c ( i ) k s c ( j ) † k ′ s ′ + c ( j ) † k ′ s ′ c ( i ) k s = δ k , k ′ δ s , s ′ δ i,j , n c ( i ) k s , c ( j ) k ′ s ′ o = 0 . (II.3)We represent the fluxon number n (3) k s by a † k s a k s , with a ( a † ), satisfying the anticommutation rules, (II.3). B (1) † kq s ≡ c (1) † k+q / s a † − k+q / − s ,B (2) kq s ≡ c (2) k+q / s a − k+q / − s . (II.4)The prime on the summation in Eq. (II.2) means therestriction: 0 < ε ( j ) ks < ¯ hω D , ω D = Debye frequency.If the fluxons are replaced by the conduction elec-trons (“electrons”, “holes”) our Hamiltonian H is re-duced to the original BCS Hamiltonian, Eq. (24) ofRef. [12]. The “electron” and “hole” are generated,depending on the energy contour curvature sign [13].For example, only “electrons” (“holes”), are generatedfor a circular Fermi surface with the negative (posi-tive) curvature whose inside (outside) is filled with elec-trons. Since the phonon has no charge, the phononexchange cannot change the net charge. The pair-ing interaction terms in Eq. (II.2) conserve the charge.The term − v B (1) † k ′ q s B (1) kq s , where v ≡ (cid:12)(cid:12) V q V ′ q (cid:12)(cid:12) (¯ hω A ) − , A = sample area, is the pairing strength, generatesthe transition in the “electron” states. Similarly, theexchange of a phonon generates a transition in the“hole” states, represented by − v B (2) k ′ q s B (2) † kq s . Thephonon exchange can also pair-create and pair-annihilate“electron” (“hole”)-fluxon composites, represented by − v B (1) † k ′ q s B (2) † kq s , − v B (2) k ′ q s B (1) kq s . At 0 K, the systemcan have equal numbers of − (+) c-bosons, “electrons”(“hole”) composites, generated by − v B (1) † k ′ q s B (2) † kq s .The c-bosons, each with one fluxon, will be called thefundamental (f) c-bosons. At a finite temperature, thereare moving (non-condensed) fc bosons. Their energies w ( j ) q are obtained from [14]: w ( j ) q Ψ( k , q ) = ε ( j ) | k+q | Ψ( k , q ) − (2 π ¯ h ) − v ∗ × Z ′ d k ′ Ψ( k ′ , q ) , (II.5)where Ψ( k , q ) is the reduced wavefunction for the fc-boson; we neglected the fluxon energy. The v ∗ denotesthe strength after the ladder diagram binding, see below.For small q , we obtain a solution of Eq. (II.5) as w ( j ) q = ω +(2 /π ) v ( j )F q, w = − ¯ hω D exp( v ∗ D ) − − , (II.6)where v ( j )F ≡ (2 ε F /m j ) / is the Fermi velocity and D ≡ D ( ε F ) the density of states per spin. The brief derivationof Eqs. (II.5) and (II.6) is given in Appendix A. Notethat the energy w ( j ) q depends linearly on the momentummagnitude q .The system of free fc-bosons undergoes a Bose-Einsteincondensation (BEC) in 2D at the critical temperature[15] k B T c = 1 .
24 ¯ hv F n / . (II.7)A brief derivation of Eq. (II.7) is given in Appendix B.The interboson distance R ≡ n / calculated from thisexpression is 1 . hv F ( k B T c ) − . The boson size r cal-culated from Eq. (II.6), using the uncertainty relation( q max r ∼ ¯ h ) and | w | ∼ k B T c is (2 /π )¯ hv F ( k B T c ) − ,which is a few times smaller than R . Hence, the fc-bosons do not overlap in space, and the model of freebosons is justified. For GaAs/AlGaAs, m ∗ = 0 . m e , m e = electron mass. For the 2D electron density10 cm − , we have v F = 1 . × cm s − . Not allelectrons are bound with fluxons since the simultaneousgenerations of ± fc-bosons is required. The minority car-rier (“hole”) density controlls the fc-boson density. For n = 10 cm − , T c = 1 .
29 K, which is reasonable.In the presence of Bose condensate below T c the un-fluxed electron carries the energy [15] E ( j ) k = q ε ( f )2 k + △ , (II.8)where the quasi-electron energy gap △ is the solution of1 = v D Z ¯ hω D dε ε + △ ) / × n h − β ( ε + △ ) / io − , β ≡ k B T . (II.9)Note that the gap △ depends on the temperature T . Atthe critical temperature T c , there is no Bose condensateand hence △ vanishes.Now the moving fc-boson below T c has the energy ˜ w q obtained from˜ w ( j ) q Ψ( k , q ) = E ( j ) | k+q | Ψ( k , q ) − (2 π ¯ h ) − v ∗ × Z ′ d k ′ Ψ( k ′ , q ) . (II.10) We obtain after solving Eq. (II.10):˜ w ( j ) q = ˜ w + (2 /π ) v ( j )F q ≡ w + ε g + (2 /π ) v ( j )F q, (II.11)where ˜ w ( T ) is determined from1 = D ν Z ¯ hω D dε h | ˜ w | + ( ε + △ ) / i − . (II.12)The energy difference:˜ w ( T ) − w ≡ ε g ( T ) (II.13)represents the T -dependent energy gap . The energy ˜ w q is negative. Otherwise, the fc-boson should break up.This limits ε g ( T ) to be | w | at 0 K. The fc-boson energygap ε g declines to zero as the temperature approaches T c from below.The fc-boson, having the linear dispersion (II.11) canmove in all directions in the plane with the constantspeed (2 /π ) v ( j )F as seen from Eq. (II.11). The su-percurrent is generated by the ± fc-bosons condensedmonochromatically at the momentum directed along thesample length. The supercurrent density (magnitude) J ,calculated by the rule: (charge e ∗ ) × (carrier density n ) × (drift velocity v d ), is J ≡ e ∗ n v d = e ∗ n (2 /π ) (cid:12)(cid:12)(cid:12) v (1)F − v (2)F (cid:12)(cid:12)(cid:12) . (II.14)The induced Hall field (magnitude) E H equals v d B . Themagnetic flux is quantized BA = n φ ( h/e ) , n φ = fluxon density . (II.15)Hence, we obtain ρ H ≡ E H J = v d Ben v d = 1 en n φ (cid:18) he (cid:19) . (II.16)If n φ = n valid at ν = 1, we obtain ρ H = h/e inagreement with the plateau value observed.The model can be extended to the integer QHE at ν = P , P = 1 , , . . . . The field magnitude is less. The LLdegeneracy eBA/h is linear in B , and hence the lowest P LL’s must be considered. The fc-boson density n per LLis the electron density n e over P and the fluxon density n φ is the boson density n over P : n = n e /P, n φ = n /P. (II.17)At ν = 1 / ± c-fermionshave effective masses. The Hall resistivity ρ H has a B -linear behavior while the resistivity ρ is finite. In ourtheory the integer P is the number of the LL’s occupiedby the c-fermions.Our Hamiltonian in Eq. (II.2) can generate and stabi-lize the c-particles with an arbitrary number of fluxons.For example, a c-fermion with two fluxons is generatedby two sets of the ladder diagram bindings, each betweenthe electron and the fluxon. The ladder diagram bindingarises as follows. Consider a hydrogen atom. The Hamil-tonian contains kinetic energies of the electron and theproton and the attractive Coulomb interaction. If we re-gard the Coulomb interaction as a perturbation and usea perturbation theory, we can represent the interactionprocess by an infinite set of ladder diagrams, each lad-der step connecting the electron line and the proton line.The energy eigenvalues of this system is not obtained byusing the perturbation theory but they are obtained bysolving the Schr¨odinger equation directly. This exampleindicates that a two-body bound state is represented byan infinite set of ladder diagrams and that the bindingenergy (the negative of the ground-state energy) is cal-culated by a non-perturbative method.Jain introduced the effective magnetic field [5] B ∗ ≡ B − B ν = B − (1 /ν ) n e ( h/e ) (II.18)relative to the standard field for the composite (c-)fermion. We extend this idea to the bosonic (odd-denominator) fraction. This means that the c-particlemoves field-free at the exact fraction. The movement ofthe guiding centers (the CM of the c-particle) can occuras if they are subjected to no magnetic field at the exactfraction. The excess (or deficit) of the magnetic field issimply the effective magnetic field B ∗ . The plateau in ρ H is formed due to kind of the Meissner effect. Con-sider the case of zero temperature near ν = 1. Only thesystem energy E matters. The fc-bosons are condensedwith the ground-state energy w , and hence the systemenergy E at ν = 1 is 2 N w , where N is the number of − fc-bosons (or + fc-bosons). The factor 2 arises sincethere are ± fc-bosons. Away from ν = 1, we must addthe magnetic field energy (2 µ ) − A ( B ∗ ) , so that E = 2 N w + (2 µ ) − A ( B ∗ ) . (II.19)When the field is reduced, the system tries to keepthe same number N by sucking in the flux lines.Thus the magnetic field becomes inhomogeneous out-side the sample, generating the extra magnetic field en-ergy (2 µ ) − A ( B ∗ ) . If the field is raised, the systemtries to keep the same number N by expeling out theflux lines. The inhomogeneous fields outside raise thefield energy by (2 µ ) − A ( B ∗ ) . There is a critical field B ∗ c = (4 µ | w | ) / . Beyond this value, the supercon-ducting state is destroyed, which generates a symmetricexponential rise in the resistance R . In our discussionof the Hall resistivity plateau we used the fact that theground-state energy w of the fc-boson is negative, thatis, the c-boson is bound. Only then the critical field B ∗ c = (4 µ | w | ) / can be defined. Here the phonon ex-change attraction played an important role. The repul-sive Coulomb interaction, which is the departure point ofthe prevalent fermionic theories [4, 5], cannot generate abound state.In the presence of the supercondensate, the non-condensed c-boson has an energy gap ε g . Hence, the non-condensed c-boson density has the activation energytype exponential temperature-dependence:exp[ − ε g / ( k B T )] . (II.20)Some authors argue that the energy gap ε g for the integerQHE is due to the LL separation = ¯ hω . But the separa-tion ¯ hω c is much greater than the observed ε g . Besides,from this view one cannot obtain the activation-type en-ergy dependence.The BEC occurs at each LL, and therefore the c-bosondensity n is smaller for high P , see Eq. (II.17), and thestrengths become weaker as P increases. The most signif-icant advantage of our bosonic theory is that we are ableto explain why the plateaus in the Hall resistivity is de-veloped when the resistivity is zero as the magnetic fieldis varied. This plateau formation is phenomenologicallyassumed in the fermionic theories. [4, 5] III. QUANTUM HALL EFFECT UNDERRADIATION
The experiments by Mani et al . [16] and Zodov et al .[17] indicate that the applied radiation excites a largenumber of “holes” in the system. Using these “holes”and the preexisting “electrons” the phonon exchange canpair-create ± c-bosons, that condense below T c in theexcited (upper) channel. The c-bosons condensed withthe momentum along the sample length are responsiblefor the supercurrent. In the presence of the condensed c-bosons, the non-condensed c-bosons have an energy gap ε g , and therefore they are absent below T c . The fermioniccurrents in the base channel cannot be suppressed bythe supercurrents since the energy levels are different be-tween the excited and base channels. These c-fermionscontribute a small normal current. They are subject tothe Lorentz force: F = q ( E + v × B ), and hence theygenerate a Hall field E H proportional to the field B . Thisis the main feature difference from the usual QHE (underno radiation).In the neighborhood of the QHE at ν = 1, the currentcarriers in the base and excited channels are, respectively,c-fermions and condensed c-bosons. The currents are ad-ditive. We write down the total current density j as thesum of the fermionic current density j f and the bosoniccurrent density j b : j = j f + j b = en f v f + en b v b , (III.1)where v f and v b are the drift velocities of the c-fermionsand c-bosons. The Hall fields E H are additive, too. Hencewe have E H = E H ,f + E H ,b = v f B + v b B. (III.2)The Hall effect condition ( E H = v d B ) applies separatelyfor the c-fermions and c-bosons. We therefore obtain R H = E H j = v f B + v b Bn f v f + n b v b e . (III.3)Far away from the midpoint of the zero-resistance stretch,the c-bosons are absent and hence the Hall resistivity R H becomes B/ ( en f ): R H = B/ ( en f ) (far away) , (III.4)after the cancellation of v f . At the midpoint the c-bosonsare dominant. Then, the Hall resistivity R H is approxi-mately equal to h/e since E H j ∼ = v b Ben b v b ∼ = he n φ n b = he (midpoint) , (III.5)where we used the flux quantization [ B = ( h/e ) n φ ], andthe fact that the flux density n φ equals the c-boson den-sity n b at ν = 1. The Hall resistivity R H = E H /j is notexactly equal to h/e since the c-fermion current den-sity en f v f is much smaller than the supercurrent density en b v b , but it does not vanish. In the horizontal stretchthe system is superconducting, and hence the supercur-rent dominates the normal current: en b v b ≫ en f v f . Thedeviation ∆ R H is, using Eq. (III.3),∆ R H = v f B + v b Be ( n f v f + n b v b ) − v f Ben f v f ≃ n b Ben b v b ≃ he . (III.6)If the field B is raised (or lowered) a little from the mid-point, ∆ R H is a constant ( h/e ) due to the Meissnereffect. If the field is raised high enough, the supercon-ducting state is destroyed and the normal current setsin, generating a finite resistance and a vanishing ∆ R H .Hence the deviation ∆ R H and the diagonal resistance R xx are closely correlated as observed by Mani [16].In Fig. 2 of Ref. [16] (not shown here), we can see thatin the range, where the SdH oscillations are observedfor the resistance without radiation, the signature of os-cillations also appear for the resistance R xx with radi-ation. The SdH oscillations arise only for the fermioncarriers. This SdH signature in R xx should remain. Ourtwo-channel model is supported here.Mani et al ., Fig. 2 of Ref. [16], shows that the strengthof the superconducting state does not change much. The2D density of states for the conduction electrons asso-ciated with the circular Fermi surface is independent ofthe electron energy, and hence the number of the excitedelectrons is roughly independent of the radiaton energy(frequency). The “hole”-like excitations are absent withno radiation. We suspect that the “hole”-band edge isa distance ε away from the system’s Fermi level. Thismeans that if the radiation energy ¯ hω is less than ε ,the radiation can generate no superconducting state. Anexperimental confirmation is highly desirable here.If a bias voltage is applied to the system, then a normalcurrent runs in the base channel. In the upper channelthe supercurrent still runs with no potential drop. Theboth currents run in the same sample space. The ap-parent discrepancy in the electric potential here may be resolved by considering a static charge Q developed in thesystem upon the field application. That is, the systemwill be charged and the static potential V c = 12 CQ , (III.7)where C is the system’s capacitance, can balance the to-tal electric potential while the charging does not affectthe supercurrents. This effect may be checked by experi-ments, which is a critical test for our two-channel model.In summary, the QHE under radiation is the QHE atthe upper channel. The condensed c-bosons generate asuperconducting state with a gap ε g in the c-boson energyspectrum. The supercondensate suppresses the c-particlecurrents in the upper channel, but cannot suppress thenormal currents in the base channel. Thus, there is a fi-nite resistive current accompanied by the Hall field. Thisexplains the B -linear Hall resistivity. IV. THE DENSITY OF STATES ANDSTATISTICAL WEIGHT
We calculate magnetic oscillations in the statisticalweight for a 2D electron system. Let us take a dilute sys-tem of electrons moving in the plane. Applying a mag-netic field B perpendicular to the plane, each electronwill be in the Landau states with the energy given by E = ( N L + 1 / hω c , N L = 0 , , , . . . . (IV.1)The degeneracy of the Landau level (LL) is eBA/ π ¯ h, A = sample area . (IV.2)The weaker the field the more LL’s, separated by ¯ hω c , areoccupied by the electrons. The electron in the Landaustate can be viewed as circulating around the guidingcenter.We introduce kinetic momentaΠ x = p x + eA x , Π y = p y + eA y , (IV.3)in terms of which the Hamiltonian H for the electron is H = 12 m ∗ (cid:0) Π x + Π y (cid:1) ≡ m ∗ Π . (IV.4)The vector potential A = (1 / B × r can be writtenas A x = − (1 / By , A y = (1 / Bx , A z = 0. Usingthe quantum condition [ x, p x ] = [ y, p y ] = i ¯ h , [ x, y ] =[ p x , p y ] = 0, we obtain[Π x , Π y ] = − ( e ¯ h/i ) B. (IV.5)If we introduce( m ∗ ) / Π x ≡ P, ( eB ) − ( m ∗ ) / Π y ≡ Q, (IV.6)we obtain H = (1 / (cid:2) P + ω c Q (cid:3) , (IV.7) P x P y pPDPDP FIG. 1: A 2D Landau state is represented by the ring (shadedarea) of the phase-space volume 2 π Π∆Π. and [
Q, P ] = i ¯ h. (IV.8)Hence, the energy eigenvalues are given by ( N L +1 / hω c ,confirming Eq. (IV.1). After simple calculations, we ob-tain dxd Π x dyd Π y = dxdp x dydp y . (IV.9)We can then represent quantum states by small quasi-phase space cell of the volume dxd Π x dyd Π y . The Hamil-tonian H in Eq. (IV.4) does not depend on the position( x, y ). Assuming large normalization lengths ( L , L ), wecan represent the Landau states by the concentric shellsof the phase space having the statistical weight2 π Π∆Π · L L (2 π ¯ h ) − = eBA π ¯ h , (IV.10)where A = L L and ¯ hω c = ∆(Π / m ∗ ) = Π∆Π /m ∗ .Hence, the LL degeneracy is given by Eq. (IV.2). Figure 1represents a typical Landau state in the Π x -Π y space.As the field B is raised the separation ¯ hω c increases,and the quantum states are bunched together. As a resultof the bunching, the density of states N ( ε ) should changeperiodically.The electrons obey the Fermi-Dirac statistics. Consid-ering a system of free electrons, we define the Helmholtzfree energy F by F = N µ − k B T X ln h e ( µ − E i ) /k B T i , (IV.11)where µ is the chemical potential and the factor 2 arisesfrom the spin degeneracy. The chemical potential µ isdetermined from the condition ∂ F ∂µ = 0 . (IV.12)The total magnetic moment M for the system can befound from M = − ∂ F ∂B . (IV.13) Equation (IV.13) is equivalent to the usual condition thatthe total number of the electrons, N , can be obtained interms of the Fermi distribution function N = 2 X i f ( E i ) . (IV.14)The LL E i is characterized by the Landau oscillatorquantum number N L . Let us introduce the density ofstate dW/dE ≡ N ( E ) (IV.15)such that N ( E ) dE = the number of states having anenergy between E and E + dE . We write Eq. (IV.11) inthe form F = N µ − k B T Z ∞ dE dWdE ln h e ( µ − E ) /k B T i = N µ − k B T Z ∞ dEW ( E ) f ( E ) . (IV.16)The statistical weight (number) W is the total numberof states having energies less than E = ( N L + 1 / hω c . (IV.17)For a fixed pair ( E, N L ), the density of states is dW = L L (2 π ¯ h ) π Π∆ΠΘ[ E − ( N L + 1 / hω c ] , (IV.18)where Θ( x ) is the Heaviside step function:Θ( x ) = (cid:26) x >
00 if x < . (IV.19)We sum Eq. (IV.18) with respect to N L and obtain W ( E ) = C (¯ hω c )2 ∞ X N L =0 Θ[ ε − (2 N L + 1) π ] , (IV.20) C = 2 πm ∗ A (2 π ¯ h ) − , ε ≡ πE/ ¯ hω c . (IV.21)We assume a high Fermi-degeneracy such that µ ≃ ε F ≫ ¯ hω c . (IV.22)The sum in Eq. (IV.20) can be computed by using Pois-son’s summation formula [18] ∞ X n = −∞ f (2 πn ) = 12 π ∞ X m = −∞ Z ∞−∞ dτ f ( τ ) e − iωτ . (IV.23)We then obtain [19] W ( E ) = W + W osc , (IV.24) W = A ( m ∗ /π ¯ h ) E (IV.25) W osc = C ¯ hω c π ∞ X ν =1 ( − ν ν sin (cid:18) πνE ¯ hω c (cid:19) . (IV.26)The detailed calculations leading to Eqs. (IV.24)–(IV.26)are given in Appendix C. Only the first term ν = 1 inEq. (IV.26) will be important in practice for weak fields ε F ≫ ¯ hω c , (IV.27)which will be shown later.The term W , which is independent of B , gives theweight equal to that for a free electron system with nofield. Note that there are no Landau-like diamagneticterms proportional to the squared field B . V. SHUBNIKOV-DE HAAS OSCILLATIONS
Let us, first, consider the case with no magnetic field.We assume a uniform distribution of impurities with den-sity n I . We introduce a momentum distribution func-tion φ ( p , t ), defined such that φ ( p , t ) d p gives the rela-tive probability of finding an electron in the element d p at time t . This function will be normalized such that2(2 π ¯ h ) Z d p φ ( p , t ) = NA ≡ n, (V.1)where the factor 2 is due to the spin degeneracy.The electric current density j is given in terms of φ ( p , t )as j = − e (2 π ¯ h ) m ∗ Z d p φ ( p , t ) p . (V.2)The distribution function φ ( p , t ) can be obtained by solv-ing the Boltzmann equation for the stationary homoge-neous state, dropping t : e E · ∂∂ p φ ( p ) = n I m ∗ Z d Ω I ( p, θ )[ φ ( p ′ ) − φ ( p )] p, (V.3)where θ is the scattering angle, that is, the angle betweenthe initial momentum p and the final one p ′ , and I ( p, θ )is the differential cross section. Solving Eq. (V.3), weobtain the conductivity as [13] σ = 2(2 π ¯ h ) e m ∗ Z d p (cid:18) − dfdE (cid:19) E Γ( E ) , E ≡ p m ∗ , (V.4)where Γ is the energy ( E )-dependent relaxation rateΓ( E ) = n I Z d Ω I ( p, θ )(1 − cos θ ) pm ∗ . (V.5)The Fermi distribution function f ( E ) ≡ h e β ( E − µ ) + 1 i − (V.6)is normalized such that n = 2(2 π ¯ h ) Z d pf ( E )= Z ∞ dEν ( E ) f ( E ) , ν ( E ) ≡ N ( E ) A , (V.7) where ν ( E ) is the density of states per area. We canrewrite Eq. (V.4) as σ = e m ∗ Z ∞ dEν ( E ) (cid:18) − dfdE (cid:19) E Γ( E ) . (V.8)The Fermi distribution function f ( E ) drops steeplynear E = µ at low temperatures: k B T ≪ ε F (Fermi en-ergy). If the density of states varies slowly with energy E , then the delta-function replacement formula − dfdE = δ ( E − µ ) (V.9)can be used. Using Z ∞ dE N ( E ) (cid:18) − dfdE (cid:19) E = Z ∞ dE N ( E ) f ( E ) , (V.10)and comparing Eqs. (V.8) and the Drude formula σ = e m ∗ n γ , (V.11)we obtain nγ ( T ) = Z ∞ dEν ( E ) f ( E ) 1Γ ( E ) . (V.12)Note that the temperature dependence of the relaxationrate γ ( T ) is introduced through the Fermi distributionfunction f ( E ).Let us now consider a field-dressed electron (guidingcenter). We assume that the dressed electron is a fermionwith magnetotransport mass M ∗ and charge e . The ki-netic energy is represented by H ′ = 12 M ∗ (cid:0) Π x + Π y (cid:1) ≡ M ∗ Π . (V.13)We introduce a distribution function ϕ ( Π , t ) in theΠ x Π y -space normalized such that2(2 π ¯ h ) Z d Π ϕ (Π x , Π y , t ) = NA = n. (V.14)The Boltzmann equation for a homogeneous stationarystate of the system is e ( E + v × B ) · ∂ϕ∂ Π = Z d Ω Π M ∗ n I I (Π , θ )[ ϕ ( Π ′ ) − ϕ ( Π )] , (V.15)where θ is the scattering angle, that is, the angle betweenthe initial and final kinetic momenta ( Π , Π ′ ). In theactual experimental condition, the magnetic force termcan be neglected. Assuming this condition, we obtain thesame Boltzmann equation (V.3) for a field-free systemexcept the mass difference. Hence, we obtain σ = 2 e M ∗ (2 π ¯ h ) Z d p E Γ (cid:18) − dfdE (cid:19) . (V.16)As the field B is raised, the separation ¯ hω c becomesgreater and the quantum states are bunched together.The statistical weight W contains an oscillatory part, seeEq. (IV.26) W osc ∝ sin (cid:18) πε ′ ¯ hω c (cid:19) , ε ′ = Π ′ m ∗ . (V.17)Physically, the sinusoidal variations in Eq. (IV.26) ariseas follows. From the Heisenberg uncertainty principle(phase space consideration) and the Pauli exclusion prin-ciple, the Fermi energy ε F remains approximately un-changed as the field B varies. The density of states ishigh when ε F matches the N L -th level, while it is smallwhen ε F falls between neighboring LLs.If the density of states, N ( ε ), oscillates violently inthe drop of the Fermi distribution function f ( ε ) ≡ (cid:2) e β ( ε − µ ) + 1 (cid:3) − , one cannot use the delta-function re-placement formula (V.9). The width of df /dε is of theorder k B T . The critical temperature T c below which theoscillations can be observed is k B T c ∼ ¯ hω c . (V.18)Below T c , we may proceed as follows. Let us consider theintegral I = Z ∞ dEf ( E ) sin (cid:18) πE ¯ hω c (cid:19) , E ≡ Π M ∗ . (V.19)We introduce a new variable ζ ≡ β ( E − ν ), and extendthe lower limit to −∞ (low temperature limit): Z ∞ dE · · · e β ( E − µ ) + 1 = 1 β Z ∞− µβ dζ · · · e ζ + 1 → β Z ∞−∞ dζ · · · e ζ + 1 . (V.20)With the help of the integral formula Z ∞−∞ dζ e i αζ e ζ + 1 = π i sinh πα , (V.21)which is proved in Appendix D, we obtain fromEq. (V.19): I = − πk B T cos(2 πε F / ¯ hω c )sinh(2 π M ∗ k B T / ¯ heB ) . (V.22)Here, we used M ∗ µ ( T = 0) = m ∗ ε F , (V.23)since the Fermi momentum is the same for both dressedand undressed electrons. For very low fields, the oscilla-tion number in the range k B T becomes great, and hencethe sinusoidal contribution must cancel out. This effectis represented by the factor [sinh(2 π M ∗ k B T / ¯ heB )] − . We now calculate the conductivity, starting withEq. (V.8). For the field-free case, we may use Eqs. (V.9)and (V.10) to obtain nγ = ν ( ε F ) ε F Γ ( ε F ) . (V.24)For a finite B , the non-oscillatory part (background) con-tributes a similar amount: nγ = ν ( ε F ) ε F Γ( ε F ) , (V.25)calculated for the dressed electrons. The oscillatory partcan be calculated by using the integration formula I inEqs. (V.19) and (V.22). This part is much smaller than ν ( ε F ) ε F / Γ( ε F ) in Eq. (V.25), since the contribution islimited to the small energy range k B T . It is also smallby the sinusoidal cancellation. We, therefore, obtain nγ = ν ( ε F ) ε F Γ( ε F ) (1 + φ ) , (V.26) φ ≡ πk B Tε F cos(2 πε F / ¯ hω c )sinh(2 π M ∗ k B T / ¯ heB ) . (V.27)Strictly speaking, the contribution of the termswith ν = 2 , , · · · in the sum W osc in Eq. (IV.26)should be added. But this contribution, which carries[sinh(2 π νM ∗ k B T / ¯ heB )] − , is small sincesinh(2 π M ∗ k B T / ¯ heB ) ≫ . (V.28)In the present theory, the two masses m ∗ and M ∗ areintroduced naturally corresponding to the two physi-cal processes: the cyclotron motion of the electron andthe guiding center motion of the dressed electron. Thedressed electron is the same entity as the c-fermion withtwo fluxons in the QHE theory.In summary, the magnetoconductivity σ ( B ), given byEq. (V.16), may be written out as σ = e M ∗ nγ = e M ∗ ν ( ε F ) ε F Γ( ε F ) (1 + φ ) . (V.29)In contrast, the conductivity σ at zero field is σ = e m ∗ nγ = e m ∗ ν ( ε F ) ε F Γ ( ε F ) , (V.30)where we have assumed that the Fermi energy ε F remainsthe same for both cases. We note that the magnetocon-ductivity σ does not approach the conductivity σ in thelow field limit. In fact, we obtain in this limit ( φ = 0): σ − σ = e n (cid:18) M ∗ γ − m ∗ γ (cid:19) . (V.31)The difference arises from the carrier difference.If the “decay” rate δ = 2 π M ∗ k B T / ¯ he defined throughsinh( δ/B ) ≡ sinh(2 π M ∗ k B T / ¯ heB ) (V.32) -1 ( / )4 17 -1 ( / )4 9 -1 ( / )4 5 -1 w o microwaves/w microwaves/119 GHz0 7 K. R xx ( Ω -1 / δ FIG. 2: The resistance R xx versus the reduced inverse mag-netic field, B ′− . See Mani [19] for the actual reduction. Thenumber N in the abscissas is the intersection number betweenthe curves with (w/) and without (w/o) microwaves. is measured carefully, the magnetotransport mass M ∗ can be obtained directly through M ∗ = e ¯ hδ/ (2 π k B T ) . (V.33)Mani measured the SdH oscillations in GaAs/AlGaAs[19], Fig. 1, T = 0 . R xx linearly de-creasing with B − in the low field limit. For high puritysamples at very low temperature ( ∼ . γ is the natural linewidth arising from the LL separationdivided by ¯ h , that is, the cyclotron frequency ω c : γ = ω c = eB/m ∗ . (V.34)This generates the desired B − dependence for R xx .We fitted Mani’s data in Fig. 2 with R xx = A + Bx + [ E cos(2 πCx ) + F ] x sinh( Dx ) , (V.35)where A = 2 . B = − . C = 23 . D = 3 . E = 22 . F = 7 .
0. The fits agree with the data within theexperimental errors. Using Dx = δ/B , we obtain M ∗ = 0 . m e , (V.36)where m e is the gravitational electron mass. If m ∗ =0 . m e , then M ∗ /m ∗ = 4 .
5. These are reasonable num-bers.The relaxation rate γ = Γ( ε F ) can now be obtainedthrough Eq. (V.11) with the measured magnetoconduc-tivity. All electrons, not just those excited electrons nearthe Fermi surface, are subject to the electric field. Hence,the carrier density n appearing in Eq. (V.29) is the total density n of the dressed electrons. This n also appearsin the Hall resistivity expression ρ H ≡ E H j = v d Benv d = Ben , (V.37)where the Hall effect condition: E H = v d B, v d = drift velocity (V.38)was used.The dressed electrons are there whether the system isprobed in equilibrium or in nonequilibrium as long asthe system is subjected to a magnetic field. Hence theirpresence can be checked by measuring the susceptibilityor the heat capacity of the system. All (dressed) elec-trons are subject to the magnetic field, and hence themagnetic susceptibility χ is proportional to the carrierdensity n although the χ depends critically on the Fermisurface. We shall briefly discuss the magnetic momentand susceptibility.The magnetization M , that is, the total magnetic mo-ment per unit area, can be obtained from M = − ∂ F ∂B . (V.39)Using Eqs. (IV.12) and (IV.22), we obtain the magneti-zation M for the quasi-free electrons [13] M = 2 n µ ε F (cid:20) − (cid:18) ε F µ B B (cid:19) k B Tε F (cid:18) m ∗ M ∗ (cid:19) × cos(2 πε F / ¯ hω c )sinh(2 π M k B T / ¯ heB ) (cid:21) , (V.40)where µ B is the Bohr magneton. The magnetic suscepti-bility χ is defined by the ratio χ ≡ M B . (V.41)
VI. TRANSPORT DIAMOND AND ZEROCURRENT ANOMALY
We are now ready to discuss the TD and ZCA observedby Studenikin et al . [1]. Ref. [1], Fig. 1 is reproduced inFig. 3. The outstanding features are:(A) The differential resistance r ≡ dV /dI exhibit theSdH oscillations for higher DC, I DC = 50 µ A. Theenvelope of the SdH oscillations become smaller forweaker magnetic fields.(B) The background differential resistance for the SdHis zero.(C) The flat minima present at I DC = 0 indicate aQHE. The flat minimum means a zero resistance R ≡ V /I = 0 . (VI.1)0 FIG. 3: The differential resistance r of an InGaAs/InP Hallbar (width = 100 µ m) at different DC values, T = 270 mK.All curves except at I DC = 0 are shifted vertically by 0 .
25 kΩfor clarity. (D) The SdH maxima and the QHE minima bothhave the right-left symmetry with varying magneticfields.(E) As the DC increases, the SdH maxima progressivelybecome the QHE minima.Our interpretation is as follows.(A) The SdH oscillations are described by formula(V.27). The oscillations are sinusoidal:cos(2 πε F / ¯ hω c ) = cos(2 πm ∗ ε F / ¯ heB ) , (VI.2)and the envelope is represented by πk B Tε F π M ∗ k B T / ¯ heB ) . (VI.3)The cyclotron mass m ∗ appears in Eq. (VI.2)and the magnetotransport mass M ∗ enters inEq. (VI.3). The two masses ( m ∗ , M ∗ ) correspondto the cyclotron motion and the guiding center mo-tion, respectively. We avoid the use of a Dingletemperature [5].(B) The background resistance h R i averaged over thefield B is zero: h R i ≡ (cid:28) VI (cid:29) = 0 . (VI.4)This behaviour is in agreement with formula(V.26). It arises from the fact that there is noLandau-like term proportional to the squared mag-netic field B in the statistical weight W in 2D, seeEq. (IV.24). (There is no Landau diamagnetism in2D in contrast to the 3D case.)(C) The flat minimum meaning zero resistance R = 0,indicates the existence of a superconducting state.The superconducting state is stable with an energygap. The supercurrents run with no scatterings by impurities and phonons. As is well known, themagnetic field is detrimental to the superconduct-ing state. If the excess magnetic field B ∗ relativeto the center field of the horizontal stretch exceedsa critical field, then the superconductivity is de-stroyed. The microscopic origin of this effect wasexplained in section 2. Briefly, the supercurrent iscomposed of the positively and negatively chargedpairon-currents. The excess magnetic field B ∗ gen-erates oppositely directed forces and breaks up ± pairons (Cooper pairs).(D) The magnetic field energy is quadratic in the ex-cess field B ∗ , see Eq. (II.19), which explains theright-left symmetry of the destroyment of the su-perconducting state.(E) The integer QHE occurring at the LL occupationnumbers ν = P = 1 , , . . . , have the quantized mag-netic fluxes: BA = 1 P Φ N φ = 1 P (cid:18) he (cid:19) N φ , (VI.5)where N φ is the fluxon number at ν = P = 1.Hence the QHE has maxima at B = 1 P (cid:18) he (cid:19) n φ , n φ ≡ N φ A . (VI.6)Equations (V.27) and (V.29) indicate that the re-sistivity r has minima whencos(2 πε F / ¯ hω c ) = 1 , (VI.7)whose solutions are2 πε F ¯ hω c = 2 πQ, Q = 1 , , . . . . (VI.8)We use ω c = eB/m ∗ , ε F = p / m ∗ , 2 πp = n e e ,and solve Eq. (VI.8) for B and obtain B = he Q n e , (VI.9)where n e is the electron density at ν = 1. FromEq. (II.17), we obtain n e = n φ . (VI.10)Both P and Q are positive integers. Hence we findfrom Eqs. (VI.6) and (VI.9) that the integer QHEmaxima and the SdH minima occur precisely atthe same magnetic fields B . Thus, the QHE min-ima progressively turn into the SdH maxima withincreasing DC.Fig. 2, Ref. [1] is reproduced in Fig. 4. The differentialresistance r ≡ dV /dI is plotted versus magnetic field (T)and direct current ( µ A). Diamond-shaped regions nearSdH minima are called transport diamonds (TD).1
FIG. 4: The differential resistance r of an InGaAs/InP Hallbar ( w = 100 µ m) is plotted versus magnetic field and DC.The ZCA position is indicated by arrow.FIG. 5: (a) The differential resistance of an InGaAs/InP Hallbar ( w = 100 µ m), T = 300 mK, is plotted versus magneticfield and DC; (b) the ZCA at two magnetic fields (1 .
58 T,1 .
70 T) indicated by vertical dashed lines in (a) for differenttemperatures.
Our interpretation of the TD is a break-down of thesuperconducting QH state due to the excess magneticfield and the direct current. The direct current by itselfgenerates a magnetic field, which is detrimental to thesuperconducting state.Fig. 5 is reproduced after Ref. [1], Fig. 3. In (a) trans-port diamonds are shown, which are similar to those inFig. 2. The temperature-dependence of the differentialresistance is shown in (b) in the range (0 . . r vs. DC near DC = 0 observed inFig. 3 and Fig. 5(a) is called the ZCA, the narrow hori-zontal line indicated by an arrow in Fig. 3. Its tempera-ture behavior at B = 1 .
58 T and B = 1 .
70 T is shown inFig. 5(b).The original authors [1] suspect that the origin of theZCA arises from the Coulomb gap in the one particledensity of states of interacting electrons. We propose adiffering interpretation:Let us consider the case of the SdH at B = 1 .
58 T,the top figure in Fig. 5(b). At the low temperatures T = (0 . .
2) K, the optical phonon population givenby the Plank distribution function can be approximatedby the Boltzmann distribution function: n ph = 1 e βε − ∼ = e − ε /k B T , (VI.11)where ε is the longitudinal optical phonon energy (as-sumed constant) and k B the Boltzmann constant. Thetemperature dependence is exponential. The phononpopulation n ph is rapidly changing with temperatureand dominates. The resistance R is proportional to theelectron-phonon scattering rate γ ph : R ≡ σ − ∝ γ ph = n ph v ph A , (VI.12)where v ph is the phonon speed, A the electron-phononscattering cross section, and n ph the phonon populationgiven in Eq. (VI.11).Studenikin et al . [1] observed that the temperature de-pendence of the ZCA follows the Arrhenius law: γ ∝ e − ε A /k B T (VI.13)with the activation energy ε A /k B = 1 . . (VI.14)This value may correspond to the optical phonon energy ε : ε A = ε . (VI.15)This finding supports our view that the temperature de-pendence of the ZCA arises from the electron-phononscattering.We next consider the ZCA for the QHE at B = 1 .
70 T.This ZCA is also temperature-dependent. As the tem-perature decreases from 1 . .
253 K, the negativepeak decreases in magnitude and its width becomes nar-rower. In the QHE under radiation, a supercurrent dueto moving pairons condensed run in the upper (excited)channel and a normal current due to electrons run in thebase channel. The resistance of the normal current isproportional to the electron-phonon scattering rate γ ph ,as shown in Eq. (VI.6). Then, the phonon populationapproximately decreases exponentially at low tempera-tures (below 1 . T is lowered, which explains the observedtemperature dependence.2The QHE at zero DC is destroyed either by increas-ing excess magnetic fields or by increasing DC-inducedmagnetic fields. But the ZCA indicates the destroymentis sharper for the case of increasing DC. This differenceshould arise from the direction of the magnetic field. TheDC running along the sample length is likely to be inho-mogeneous, stronger at the outer edge. Then the super-conductivity is destroyed at the edges first according toSilsbeeb rule. On the other hand, the applied magneticfield alone should keep the current homogeneous.Studenikin et al . [1] observed essentially same TD andZCA in heterojunction GaAs/AlGaAs. In particular, theQHE minima progressively turn to the SdH maxima asDC increases, and the ZCA is sharp near DC = 0. Thesame theory applies here. The authors thank Dr. S. Stu-denikin for enlightening discussions. Appendix A: DERIVATION OF EQS. (II.5) AND(II.6)
Dropping the “holes” from the Hamiltonian H inEq. (II.2), we obtain H c = X k X q (cid:16) ε | k+q / | + ε (3) | − k+q / | (cid:17) B † kq B kq − v X ′ q X ′ k X ′ k ′ B † k ′ q B kq , (A.1)where we suppressed the “electron” and spin indices. Us-ing the anticommutation rules (II.6), we obtain h H c , B † kq i = (cid:16) ε | k+q / | + ε (3) | − k+q / | (cid:17) B † kq − v X ′ k ′ B † k ′ q (cid:16) − n k+q / − n (3) − k+q / (cid:17) . (A.2)The Hamiltonian H c is bilinear in ( B, B † ), and can there-fore be diagonalized exactly: H c = X µ w µ φ † µ φ µ , (A.3)where w µ is the energy and φ µ the annihilation operator.We multiply Eq. (A.2) by φ µ from the right, take a grandcanonical ensemble average, denoted by angular brackets,and get w µ Ψ µ ( k , q ) = (cid:16) ε | k+q / | + ε (3) | − k+q / | (cid:17) Ψ µ ( k , q ) − v (2 π ¯ h ) Z ′ d k ′ Ψ µ ( k ′ , q ) × D − f F (cid:0) ε | k ′ +q / | (cid:1) − f F (cid:16) ε (3) | − k ′ +q / | (cid:17) E , (A.4)where h n p i = f F ( ε p ) is the Fermi distribution function.The reduced wavefunctionΨ µ ( k , q ) ≡ D B † kq φ µ E = h µ | ˆ n | k , q i (A.5) can be regarded as the mixed representation ofthe reduced density operator ˆ n defined through h k ′ , q ′ | ˆ n | k , q i ≡ D B † k , q B k ′ , q ′ E . The fc-boson energy w µ can be specified by ( N L , q ), and it will be denoted by w q since it is N L -independent. As T → f F ( ε p ) →
0. Drop-ping the fluxon energy and replacing q / q , we obtainEq. (II.5). We solve this equation, assuming ε F ≫ ¯ hω D .Using a Taylor series expansion, we obtain Eq. (II.6) tothe linear in q . Appendix B: DERIVATION OF EQ. (II.7)
The BEC occurs when the chemical potential µ van-ishes at a finite T . The critical temperature T c can bedetermined from n = (2 π ¯ h ) − Z d p (cid:2) e β c ε − (cid:3) − , β c ≡ ( k B T c ) − . (B.1)After expanding the integrand in powers of e − β c ε andusing ε = cp , we obtain n = 1 . π ) − ( k B T c / ¯ hc ) , (B.2)yielding Eq. (II.7). Appendix C: STATISTICAL WEIGHT FOR THELANDAU STATES
The statistical weight W for the Landau states in 2Dwill be calculated in this appendix. We write the sum inEq. (II.16) as2 ∞ X n =0 Θ( ǫ − (2 n + 1) π ) = Θ( ǫ − π ) + ψ ( ǫ ; 0) , (C.1) ψ ( ǫ ; x ) ≡ ∞ X n = −∞ Θ( ǫ − π − π | n + x | ) . (C.2)Note that ψ ( ǫ ; x ) is periodic in x and can therefore beexpanded in a Fourier series. After the Fourier expansion,we set x = 0 and obtain Eq. (C.1). By taking the realpart (Re) of Eq. (C.1) and using Eq. (IV.20), we obtainRe { Equation (C.1) } = 1 π Z ∞ dτ Θ( ǫ − τ )+ 2 π ∞ X ν =1 ( − ν Z ∞ dτ Θ( ǫ − τ ) cos ντ, (C.3)where we assumed ǫ ≡ πE/ ¯ hω c ≫ π against ǫ . The integral in the first term in Eq. (C.3)yields ǫ . The integral in the second term is Z ∞ dτ Θ( ǫ − τ ) cos ντ = 1 ν sin νǫ. (C.4)3We then obtainRe { Equation (C.1) } = 1 π ǫ + 2 π ∞ X ν =1 ( − ν ν sin νǫ. (C.5)Using Eqs. (IV.20) and (C.5), we obtain W ( E ) = W + W osc = C ¯ hω c (cid:16) ǫπ (cid:17) + C ¯ hω c π ∞ X ν =1 ( − ν ν sin (cid:18) πνE ¯ hω c (cid:19) . (C.6) Appendix D: DERIVATION OF EQ. (V.21)
Let us consider an integral on the real axis I ( y, α, R ) = Z R − R dx e i α ( x +i y ) e z + 1 , z = x + i y and α, R > . (D.1)We add an integral over a semicircle of radius R in theupper z -plane to form an integral over a closed contour. We then take the limit: R → ∞ . The integral over thesemicircle vanishes in this limit if α >
0. The integral onthe real axis, I ( y, α, ∞ ), becomes the desired integral inEq. (V.21). The integral over the closed contour can beevaluated by using the residue theorem. Note that ( e z +1) − has simple poles at z = π i , π i , . . . , (2 n − π i , . . . . We may use the following formula valid for a simple poleat z = z j : Res { p ( z ) /q ( z ) , z j } = p ( z j ) /q ′ ( z j ) , (D.2)where p ( z ) is analytic at z = z j , and the symbol Resmeans a residue. We then obtain I ( α, ∞ ) = 2 π i · ∞ X n =1 Res (cid:26) e i αz e z + 1 , z n = (2 n − π i (cid:27) = 2 π i · ∞ X n =1 e i α [(2 n − π i] e (2 n − π i = − π i e − απ − e − απ = π i 1sinh απ . (D.3) [1] S.A. Studenikin, G. Granger, A. Kam, A.S. Sachra-jda, Z.R. Wasilewski, and P.J. Poole, arXiv:1012.0043v1[cond-mat.mes-hall] (2010).[2] S.C. Zhang, T.H. Hansson, and S. Kivelson, Phys. Rev.Lett. , 82 (1989).[3] S. Fujita and Y. Okamura, Phys. Rev.
B69 , 155313(2004); S. Fujita, Y. Tamura, and A. Suzuki,
Mod. Phys.Lett. B , 817 (2001); S. Fujita, K. Ito, Y. Kumek, andY. Okamura, Phys. Rev. , 075304 (2004).[4] Z.F. Ezawa, Quantum Hall Effect (World Scientific, Sin-gapore, 2000); see also R.E. Prange and S.M. Girvin,eds.,
Quantum Hall Effect (Springer-Verlag, New York,1990); B.I. Halperin, P.A. Lee, and N. Read,
Phys. Rev.
B47 , 7312 (1993); M. Janssen, O. Viehweger, V. Fasten-rath, and J. Hajdu,
Introduction to the Theory of the In-teger Quantum Hall Effect (VCH, Weinbeim, Germany,1994).[5] J.K. Jain,
Phys. Rev. Lett. , 199 (1989), Phys. Rev.
B40 , 8079 (1989); ibid , , 7653 (1990); Surf. Sci. ,65 (1992).[6] L.W. Shubnikov and W.J. de Haas,
Proc. NetherlandsRoyal Acad. Sci. , , 130 and 163 (1932).[7] W.J. de Haas and P.M. van Alphen, Leiden Comm. , 208d,211a (1930);
Leiden Comm. , 220d (1932).[8] R.B. Dingle,
Proc. Roy. Soc.
A211 , 500 (1952). [9] S. Fujita, S. Horie, A. Suzuki, and D.L. Morabito,
Ind.J. PAP , , 850 (2006).[10] S. Fujita and D.L. Morabito, Mod. Phys. Lett. B , 753(1998).[11] P. Ehrenfest and J.R. Oppenheimer, Phys. Rev. , 311(1931); H.A. Bethe and R. Jackiw, Intermediate Quan-tum Mechanics , 2nd ed. (Benjamin, New York, 1968),p. 23.[12] J. Bardeen, L.N. Cooper, and J.R. Schrieffer,
Phys. Rev. , 1175 (1957).[13] S. Fujita and K. Ito,
Quantum Theory of ConductingMatter (Springer, New York, 2007), pp. 119–122, pp. 78–82, pp. 144–145.[14] S. Fujita, J.H. Kim, K. Ito, and M. De Llano,
Internat.J. Mod. Phys. B , 4129 (2009).[15] S. Fujita and S. Godoy, Quantum Statistical Theory ofSuperconductivity (Plenum, New York, 1996), pp. 184–186, pp. 202–204.[16] R.G. Mani et al ., Nature , , 646 (2002).[17] M.A. Zudov et al ., Phys. Rev. Lett. , 046807 (2003).[18] R. Courant and D. Hilbert, Methods of MathematicalPhysics , vol. 1 (Interscience-Wiley, New York, 1953),pp. 76–77.[19] R.G. Mani,
Physica E ,22