Magnetoconductivity oscillations induced by intersubband excitation in a degenerate 2D electron gas
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J u l Magnetoconductivity oscillations induced by intersubband excitation in a degenerate2D electron gas
Yu.P. Monarkha ∗ B. Verkin Institute for Low Temperature Physics and Engineering, 47 Nauky Ave., 61103, Kharkiv, Ukraine
Magnetoconductivity oscillations and absolute negative conductivity induced by nonequilibriumpopulations of excited subbands in a degenerate multisubband two-dimensional electron system arestudied theoretically. The displacement from equilibrium, which can be caused by resonant mi-crowave excitation or by any other reason, is assumed to be such that electron distributions can nolonger be described by a single Fermi level. In this case, in addition to the well-known conductiv-ity peaks occurring at the Shubnikov-de Haas conditions and small peaks of normal intersubbandscattering, sign-changing oscillations with a different shape are shown to be possible. We foundalso that even a small fraction of electrons transferred to the excited subband can lead to negativeconductivity effects.
PACS numbers: 73.40.-c, 75.47.-m, 73.50.-h, 73.50.Pz, 73.63.Hs
I. INTRODUCTION
The transport properties of a 2D electron gas in a per-pendicular magnetic field have attracted much interest because of unexpected discoveries and new physics. Inaddition to the amazing quantum Hall effects observedin a degenerate 2D electron gas under equilibrium con-ditions , new experiments revealed resistivity oscilla-tions and zero-resistance states , if a 2D electron gasformed is GaAs/AlGaAs heterostructures is exposed tomicrowave (MW) radiation. These oscillations are con-trolled by the ratio of the radiation frequency, ω , to thecyclotron frequency, ω c . The zero-resistance states (ZRS)are assumed to be caused by instability of an electronsystem with absolute negative conductivity, σ xx <
0, re-gardless of the actual mechanism of MW-induced resis-tance oscillations (MIRO) which is still under debate (fora review, see Ref. 10).Among different theoretical mechanisms proposed forthe explanation of MIRO, here we would like to high-light the displacement and inelastic models. Thedisplacement mechanism is based on a peculiarity of or-bit center migration ( X → X ′ ) when an electron ab-sorbs a photon and simultaneously is scattered off impu-rities. The authors of the inelastic mechanism noticedthat photon-assisted scattering affects the distributionfunction of electrons f ( ε ) in such a way that it acquiresa nonequilibrium oscillating correction (a sort of popula-tion inversion) whose derivative leads to a sign-changingcontribution to σ xx .MW-induced magnetoconductivity oscillations similarto MIRO and even ZRS were observed in a nondegener-ate 2D electron gas formed on the free surface of liquidhelium . The important distinction of these new os-cillations is that they are observed only if the excitationenergy of the second surface subband ∆ , ≡ ∆ − ∆ istuned to the resonance with the MW field (∆ , = ~ ω )by varying the pressing electric field (a sort of Stark ef-fect in the 1D potential well formed at the surface). Itshould be noted also that the shape of these oscillations strikingly differs from the usual shape of magnetoint-ersubband oscillations described theoretically and ob-served for semiconductor heterostructures under condi-tions that two subbands are occupied. Instead of simplepeaks of σ xx expected at the conditions of alignment ofLandau levels belonging to different subbands, the shapeof MIRO observed in a 2D electron gas on liquid heliumrepresents rather a derivative of peaks.The oscillations reported for electrons on liquid heliumwere explained by a nonequilibrium population ofthe excited subband which triggers quasi-elastic intersub-band scattering of electrons with the same peculiarity oforbit center migration as that noticed in the displacementmodel. Thus, the intersubband mechanism of MIRO andZRS has something in common with the both displace-ment and inelastic mechanisms though it does not usethe concept of photon-assisted scattering which is impor-tant for these two models. Extensive studies of MIRO ina nondegenerate 2D electron gas on liquid helium haverevealed a number of remarkable effects associated withthe ZRS regime: in-plane redistribution of electrons ,self-generated audio-frequency oscillations , and incom-pressible states . An explanation of these novel obser-vations is based on the concept of electron density do-mains : regions of different densities appear to eliminatethe regime of negative conductivity.It should be noted also that even the delicate theoreti-cal predictions reported for the intersubband mechanismof MIRO which concern the effect of Coulomb inter-action on conductivity extrema were clearly observed inthe experiment . Still, this mechanism of MIRO wasdescribed only for a nondegenerate multisubband elec-tron system using an important simplification: f ( ε ) ∝ exp ( − ε/T e ), where ε is the in-plane energy, and T e isthe electron temperature. It is not clear how the Pauliexclusion principle affects this mechanism; and the the-ory does not indicate in what respect the results obtainedfor electrons on liquid helium can be applied to a degen-erate 2D electron system similar to those investigated insemiconductor structures.In this work we develop a theory of magnetoconduc-tivity oscillations in a degenerate 2D electron gas whichare induced by nonequilibrium population of excited sub-bands. We introduce a new definition of the extended dy-namic structure factor of a multisubband 2D electron sys-tem which incorporates the concept of quasi-Fermi levels( imref ) and describes the contribution of elastic inter-subband scattering to the momentum relaxation rate un-der conditions that electron distribution is strongly dis-placed from equilibrium and cannot be attributed to sim-ple heating of electrons. We demonstrate that nonequi-librium populations of excited subbands can lead to mag-netointersubband oscillations whose shape differs fromthe shape of usual oscillations caused by the equilibriumpopulation of the second subband and the alignment ofstaircases of Landau levels . This induces importantchanges in quantum magnetotransport of a degenerate2D electron system and can even lead to negative linearresponse conductivity. II. MAGNETOTRANSPORT INMULTISUBBAND 2D SYSTEMS
Electrons formed on the free surface of liquid heliumhave a rather low density n e . × cm − , therefore attemperatures which are comparable with the Fermi tem-perature they are already localized in sites of the Wignerlattice . Above the Wigner solid transition tempera-ture this system can be considered as a nondegenerateCoulomb liquid where the Pauli exclusion principle isunimportant. Electrons on a liquid helium film repre-sent a remarkable exception: for a special arrangementof various substrates they can form a 2D Fermion sys-tem even at T = 0.Electrons in semiconductor structures usually have theeffective mass which is much smaller than the free elec-tron mass. Therefore, at low temperatures these elec-trons can be described as a 2D Fermi gas. A 2D electronsystem formed in a semiconductor device can have morethan one subband . There is a number of exper-iments demonstrated importance of intersubband scat-tering for electron transport in a 2D system . Theseresults represent properties of an equilibrium system,when the gate potential and the Fermi level position ina GaAs/AlGaAs heterostructure provide the second sub-band occupancy. There is also a possibility of changingcarrier density by illuminating samples with light due toelectron-hole pair generation . In this work, we shall fo-cus on magnetotransport properties of a 2D electron sys-tem under conditions that electron populations of excitedsubbands deviate substantially from equilibrium and can-not be described by a single chemical potential.The energy spectrum of a multisubband 2D electronsystem in crossed magnetic ( B ) and electric ( E k ) fields isdescribed by three quantum numbers ( l , n , and X ; herewe shall ignore the spin variable): E l,n,X = ∆ l + ε n + eE k X , (1) n n n n n n Subband l = 2 F Subband l = 1n
FIG. 1: Schematic illustration of a two-subband 2D elec-tron system in a magnetic field. The energy spectrum of theground (blue) and the first excited (red) subbands representsa staircase of Landau levels. The position of the Fermi-levelat equilibrium is shown by the pink horizontal line. where ∆ l is the subband energy ( l = 1 , , ... ), X is thecoordinate of the center of the cyclotron motion, ε n isthe usual Landau spectrum ε n = ~ ω c ( n + 1 / , (2)( n = 0 , , ... ), and ω c = eB/m e c is the cyclotron fre-quency. In the center-of-mass reference frame movingwith regard to the laboratory frame with the drift veloc-ity u d , the electric field E ′k → S A / πℓ B , where ℓ B = p ~ c/eB is the radius of thecyclotron orbit at n = 0, and S A is the surface area.The schematic view of Landau levels of a two-subbandsystem is shown in Fig. 1. The Landau levels of the ex-cited subband are up-shifted by ∆ , ≡ ∆ − ∆ as com-pared to respective levels of the ground subband. In con-trast with the model considered previously , the equilib-rium Fermi energy ε F is assumed to be smaller than theintersubband excitation energy ~ ω , = ∆ , (here ω , is the excitation frequency). It is obvious that at certainmagnetic fields defined by the condition ω , /ω c = m (here m = 1 , , ... ) Landau levels of the excited subbandbecomes completely aligned with high enough Landaulevels of the ground subband which triggers elastic inter-subband scattering.At strong magnetic fields directed perpendicular to theelectron layer, magnetotransport of a 2D electron gas iswell described by the center-migration theory , ifthe collision broadening of Landau levels is taken intoaccount. For semiconductor electrons, there are twoscattering mechanisms important at low temperatures:Coulomb scattering from charged centers and surfaceroughness scattering . Both of them represent essentiallyelastic scattering process. Each experimental realizationof a 2D electron system has its own specific nature ofscatterers. The details of this nature are not importantfor the effect considering in this work, and they can be in-corporated in the theory by changing the matrix elementsof electron scattering. As we shall see, the important pa-rameters of the theory are the Landau level broadeningand the momentum collision rate at zero magnetic field.Therefore, here we shall model the scatterers by artificialheavy atoms interacting with electrons by an arbitrarypotential V int ( | R e − R a | ) (here R e and R a are radiusvectors of an electron and an atom respectively).In the model considering here, the interaction Hamil-tonian can be represented in terms of creation ( a † K ) anddestruction ( a K ) operators of atoms as H int = 1Ω v X e X K , K ′ exp [ − i ( K ′ − K ) R e ] ×× V | K ′ − K | a † K ′ a K , (3)where Ω v ≡ S A L z is the volume containing these atoms, K represents a 3D wave vector of an atom, and V | K ′ − K | isa Fourier-transform of the potential V int ( R ). For the ef-fective potential V a δ ( R e − R a ), conventionally describ-ing interaction with short-range scatterers, V Q = V a .Static defects resulting in elastic electron scattering aredescribed by the limiting case M a → ∞ (here M a is themass of an artificial atom). Surface defects can be mod-eled by a 2D layer of artificial atoms. Similar modelingcan be considered for a description of remote scatterers.In the case of a nondegenerate 2D electron gas, theproblem of finding the nonequilibrium magnetoconduc-tivity σ xx can be equally well solved by consideringthe momentum exchange at a collision in the labora-tory or in the center-of-mass reference frames.For nondegenerate electrons, a great simplification ap-pears because [1 − f ( ε n ′ ,X ′ )] ≃
1, and the quantity tobe averaged in the laboratory frame is independent of X . This allows one to restrict the averaging procedureto the Landau level index n only, assuming the distri-bution function f ( ε n ) ∝ exp ( − ε n /T e ) with an effectivetemperature T e .Magnetoconductivity σ xx of a degenerate 2D electronsystem can be found from the average friction force F fr acting on electrons due to interaction with scatterers (themomentum balance method ) or using a direct ex-pression for the current j x and calculating probabilitiesof electron scattering from X to X ′ (a version of the Tite-ica’s method ). In order to avoid complications withthe field term eE k X in the energy spectrum of degener-ate electrons, it is convenient to consider scattering pro-cesses in the center-of-mass reference frame moving with the drift velocity u d with regard to the laboratory ref-erence frame. In this moving frame, the driving electricfield E ′k is zero , and the electron spectrum coincideswith the Landau spectrum ε n . It is important that themomentum exchange at a collision Q ≡ K ′ − K in thecenter-of-mass frame is the same as in the laboratoryframe because of the linear relationship between a mo-mentum and the respective velocity. At the same time,one have to keep in mind that in the center-of-mass ref-erence frame the energy exchange at an elastic collisionacquires a Doppler shift correction, E ( a ) K ′ − E ( a ) K = − ~ Q · u d ≡ − ~ q · u d , (4)due to the quadratic dependence of the energy of an atomon its velocity. Here E ( a ) K = ~ K / M a and we used thenotation Q = { q , κ } with q and κ standing for the in-plane and vertical components respectively. It is quiteobvious that scattering probabilities should not dependon a choice of an inertial reference frame. Physically, thecorrection of Eq. (4) is equivalent to the energy exchangefor the electron spectrum considered in the laboratoryframe eE k ( X ′ − X ) = ~ q y V H , here we have taken intoaccount that X ′ − X = q y ℓ B due to the momentum con-servation and used the notation V H = cE k /B for the Hallvelocity ( u ( y )d ≃ − V H ).The momentum balance approach allows obtain-ing the effective collision frequency of electrons ν eff fromthe kinetic friction acting on the whole electron sys-tem F fr . In the linear transport regime, F fr is propor-tional to u d , and conventionally it can be written as F fr = − N e m e ν eff u d , where the proportionality factor ν eff defines electron magnetoconductivity σ xx ≃ e n e ν eff m e ω c , (5)and n e = N e /S A is electron density.The simplest way of obtaining ν eff is to considerthe momentum balance along the y -axis, F ( y )fr = − N e m e ν eff u ( y )d . Assuming u ( y )d ≃ − V H and using theBorn approximation for scattering probabilities in thecenter-of-mass frame, one can find F ( y )fr ( V H ) = − N e X q ~ q y ¯ W q ( V H ) , (6)where¯ W q ( V H ) = 2 πn (3D) a η ~ S A X l,l ′ ,n,n ′ f l ( ε n ) [1 − f l ′ ( ε n ′ )] ×× I n,n ′ ( x q ) U l ′ ,l ( q ) δ ( ε n ′ − ε n + ∆ l ′ ,l + ~ q y V H ) (7)is the probability of electron scattering with the in-planemomentum exchange equal ~ q , and ∆ l ′ ,l = ∆ l ′ − ∆ l .Here we have used the following notations: n (3D) a is thedensity of scatterers, η = 2 πℓ B n e is the filling factor, f l ( ε n ) is the electron distribution function, the functions U l ′ ,l ( q ) and I n,n ′ ( x q ) are defined by matrix elements ofthe interaction Hamiltonian U l ′ ,l ( q ) = 1 L z X κ V √ q + κ (cid:12)(cid:12)(cid:12)(cid:0) e − iκz (cid:1) l ′ ,l (cid:12)(cid:12)(cid:12) , (8) (cid:12)(cid:12)(cid:12)(cid:0) e − i q · r e (cid:1) n ′ ,X ′ ; n,X (cid:12)(cid:12)(cid:12) = δ X,X ′ − ℓ B q y I n,n ′ ( x q ) , (9) I n,n ′ ( x ) = [min( n, n ′ )]![max( n, n ′ )]! x | n − n ′ | e − x h L | n − n ′ | min( n,n ′ ) ( x ) i ,x q = q ℓ B /
2, and L mn ( x ) are the associated Laguerrepolynomials. When obtaining Eq. (7), we used the ad-vantages of describing scattering probabilities in the mov-ing frame - the summations over indexes X , X ′ and K are trivial leading to the factors n B = 1 / πℓ B and n (3D) a .Comparing the right side of Eq. (6) with the resultexpected for the linear regime N e m e ν eff V H , one can findthat ν eff = − m e V H X q ~ q y ¯ W q ( V H ) . (10)When expanding ¯ W q ( V H ) in V H , we can consider onlythe linear term ¯ W ′ q (0) V H [here the ’prime’ denotes thedifferentiation] because ¯ W q (0) depends only on the ab-solute value of q and, therefore, gives zero contributioninto ν eff .It is instructive to note that the same result for ν eff and σ xx can be found from the direct expression forthe electron current along x -direction (this method wasalso used for describing a nondegenerate electron sys-tem): j x = − en e X q ( X ′ − X ) q ¯ W q ( V H ) , (11)where we have to use the relationship ( X ′ − X ) q = ℓ B q y which follows from matrix elements of Eq. (9). TheEq. (11) and the definition of σ xx obviously yield theexpression for ν eff given in Eq. (10).To obtain a finite magnetoconductivity in the treat-ment presented above, one have to include higher ap-proximations by incorporating the collision broadeningof Landau levels Γ l,n (the broadening of electron den-sity of states). Following the ideas of the center migra-tion theory and the self-consistent Born approximation(SCBA) , in the right side of Eq. (7) we shall insert R dε R dε ′ δ l ( ε − ε n ) δ l ′ ( ε ′ − ε n ′ ); the subscripts l and l ′ in the respective delta-functions just mark the subbandwhere the level density belongs. Then, assuming the re-placement δ l ( ε − ε n ) → − π ~ Im G l,n ( ε ) [here G l,n ( ε ) isthe single-electron Green’s function], the average proba-bility of scattering with the momentum exchange ~ q can be represented in the following form:¯ W q ( V H ) = n (3D) a S A ~ X l,l ′ U l ′ ,l ( q ) D l,l ′ ( q, ω l,l ′ − q y V H ) , (12)where ω l,l ′ = ∆ l,l ′ / ~ , and D l,l ′ ( q, Ω) = 2 π ~ η Z dεf l ( ε ) [1 − f l ′ ( ε + ~ Ω)] ×× X n,n ′ I n,n ′ ( x q ) Im G l,n ( ε ) Im G l ′ ,n ′ ( ε + ~ Ω) (13)is a new generalization of the dynamic structure factor(DSF) of a multisubband 2D electron system. Expanding¯ W q in q y V H yields ν eff = n (3D) a m e ~ S A X q X l,l ′ q y U l ′ ,l ( q ) D ′ l,l ′ ( q, ω l,l ′ ) . (14)Thus, the effective collision frequency of a multisubband2D electron system is proportional to the derivative of theextended DSF D ′ l,l ′ ( q, ω l,l ′ ) with respect to frequency.There are two important approximations for the Lan-dau level density of states. The SCBA theory of Andoand Uemura yields the semi-elliptical shape of the densityof states − Im G n ( ε ) = 2 ~ Γ n s − ( ε − ε n ) Γ n , (15)where Γ n is the broadening parameter. In the case ofshort-range scatterers, Γ n is independent of Landau num-ber Γ n = Γ with Γ = r π ~ ω c ν , (16)where ν is the electron relaxation rate obtained for B =0. The cumulant expansion method yields the Gaussianshape of Landau levels − Im G n ( ε ) = √ π ~ Γ n exp " − ε − ε n ) Γ n , (17)which does not have the sharp cutoff of the density ofstates. Generally, the level shape is a kind of mixtureof elliptical and Gaussian forms , and the shape of thelowest level is close to a Gaussian.In the case of equilibrium Fermi-distribution, D l,l ′ ( q, Ω) has very useful properties which simplifysignificantly evaluation of ν eff and σ xx . For example,consider only the contribution from intrasubband scat-tering processes ( l ′ = l ). Then, D l,l ( q, Ω) coincides withthe conventional DSF of a 2D electron system whichsatisfies the condition D l,l ( q, − Ω) = e − ~ Ω /T e D l,l ( q, Ω) (18)The derivative of this relationship gives D ′ l,l ( q,
0) = ~ T e D l,l ( q,
0) and the linear (in q y V H ) term of Eq. (12)can be rewritten as δ ¯ W q ≃ − q y V H ~ T e ¯ W q (0) , (19)which allows representing σ xx in terms of the equilibriumprobability ¯ W q (0): σ xx ≃ e n e T e X q ( X ′ − X ) q ¯ W q (0) . (20)This equation coincides with the well-known result ob-tained previously , and it is similar to the Einsteinrelation between the conductivity and the diffusion coef-ficient.For the ground subband and the semi-elliptic shape ofLandau levels [Eq. (15)] induced by short-range scatter-ers, Eq. (20) transforms into the result of Ando and Ue-mura which indicates that the conductivity peak value( σ xx ) max = e π ~ ( n + 1 /
2) depends only on the Landaulevel index n and the natural constants . These ”check-points” of equilibrium transport regime, encourage us touse Eq. (14) for describing magnetotransport in nonequi-librium multisubband 2D electron systems.For a nonequilibrium filling of 2D subbands, the ex-tended DSF D l,l ′ ( q, Ω) generally has no a relationshipsimilar to Eq. (18). Only describing nondegenerate elec-trons and assuming f l ( ε ) ∝ N l exp ( − ε/T e ) it was pos-sible to introduce a version of the DSF S l,l ′ ( q, Ω)which had an important property resembling Eq. (18), inspite of the fact that the occupation of subbands was notequilibrium. Unfortunately, this version of the extendedDSF appears to be useless for degenerate electrons. Thenew definition of the extended DSF D l,l ′ ( q, Ω) givenin Eq. (13) transforms into ¯ n l S l,l ′ ( q, Ω) if the electronsystem can be considered as a nondegenerate gas [here¯ n l = N l /N e is the fractional occupancy of a subband]. III. QUASI-FERMI LEVEL APPROXIMATION
Generally, it is very difficult to find f l ( ε ) if a systemis displaced from equilibrium. Therefore, in solid statephysics it is quite common to use the concept of a quasi-Fermi level or imref . In the following, we assume thatdisplacement from equilibrium is such that electron pop-ulations can no longer be described by a single chemicalpotential (or a Fermi level), nevertheless it is possibleto describe it introducing separate chemical potentials(quasi-Fermi levels) for each subband: f l ( ε ) = 1e ( ε +∆ l, − µ l ) /T e + 1 ≡ f F ( ε + ∆ l, − δµ l ) , (21)where δµ l = µ l − µ . The chemical potentials µ l are mea-sured from the bottom of the ground subband, while thezero of Landau energy ε is taken at the bottom of each subband. In most cases, it is sufficient to consider onlytwo subbands (the ground subband and the first excitedsubband), when electron populations of higher subbandscan be neglected. The form of Eq. (21) is quite accu-rate if electron-electron collisions are more important forintrasubband redistribution than for intersubband decayrates. Anyway, this form of f l ( ε ) is very useful becauseit allows obtaining σ xx in an analytical form for nonequi-librium populations of electron subbands.One can also introduce different electron tempera-tures for each subband ( T l,e ), still we shall assume that T l,e = T l ′ ,e = T e because in-plane energy relaxation be-tween different subbands is governed by electron-electroncollisions (electron spacing is usually much larger thanthe average distance between nearest subbands), whoserate is quite high for 2D electron systems . Regardingpossible heating of electrons ( T e > T ), we assume that T e is still much lower than the quasi-Fermi energies. Theopposite limiting case (nondegenerate electrons) was de-scribed in Refs. 19,20. It should be noted also that MIROobserved in a 2D electron gas on liquid helium are quitewell described even by the approximation T e = T in spiteof a substantial heating .Using the distribution function of Eq. (21) and thewell-known identity f F ( ε ) [1 − f F ( ε ′ )] = [ f F ( ε ) − f F ( ε ′ )] 11 − e ( ε − ε ′ ) /T e ,(22)it is possible to establish the following relationship forthe extended DSF D l ′ ,l ( q, − Ω) = e − ( ~ Ω+∆ l ′ ,l − µ l ′ ,l ) /T e D l,l ′ ( q, Ω) , (23)where µ l ′ ,l = µ l ′ − µ l . For a single subband ( l ′ = l ), thisproperty coincides with the property of the usual DSF ofa 2D electron gas given in Eq. (18).When considering the contribution from intersubbandscattering ν inter in Eq. (14), the property of Eq. (23) al-lows us to transform derivatives of the DSF whose fre-quency argument is negative into functions with a posi-tive argument D ′ l ′ ,l ( q, − Ω) = − e − ( ~ Ω+∆ l ′ ,l − µ l ′ ,l ) /T e D ′ l,l ′ ( q, Ω) ++ ~ T e e − ( ~ Ω+∆ l ′ ,l − µ l ′ ,l ) /T e D l,l ′ ( q, Ω) (24)Thus, a substantial part of D ′ l,l ′ ( q, ω l,l ′ ) entering Eq. (14)can be eliminated by reverse scattering processes due tothe first term in the right side of Eq. (24). Therefore,it is convenient to represent the contribution of inter-subband scattering to ν eff in the form containing onlypositive frequency arguments ( l > l ′ ). In this way,one can obtain two kinds of contributions: a normalcontribution proportional to D l,l ′ ( q, ω l,l ′ ), and an ab-normal (sign-changing) contribution proportional to thederivative D ′ l,l ′ ( q, ω l,l ′ ). To make a distinction between - , =0, n =0, all n =0.3, all n =1, all n FIG. 2: The quasi-chemical potential of the first excitedsubband µ − ∆ , (in units of ~ ω c ) versus the filling factor η = 2 πℓ B N /S A calculated for different conditions which areindicated in the figure legend. The level broadening Γ ,n isalso shown in units of ~ ω c . these contributions, we shall use the following notations: ν inter = ν N + ν A , where ν N = n (3D) a m e ~ S A X l>l ′ X q ~ T e q y U l ′ ,l ( q ) e − µ l,l ′ /T e D l,l ′ ( q, ω l,l ′ ) , (25) ν A = n (3D) a m e ~ S A X l>l ′ (cid:16) − e − µ l,l ′ /T e (cid:17) ×× X q q y U l ′ ,l ( q ) D ′ l,l ′ ( q, ω l,l ′ ) . (26)The normal contribution ν N exists even under the equilib-rium condition ( µ l,l ′ = 0), though at µ < ∆ l,l ′ it is verysmall due to f l ( ε ) present in D l,l ′ ( q, ω l,l ′ ). The abnor-mal terms ν A differ from zero only if electron distributionis somehow displaced from equilibrium ( µ l,l ′ > l = 2) has an extraelectron population δN , one expects that the all theseelectrons will occupy the lowest Landau level ( n = 0), iflow temperatures ( T e ≪ ~ ω c ) are considered and the fill-ing factor of the excited subband η = 2 πℓ B N /S A < µ = ∆ , + ε − T e ln (cid:18) − η η (cid:19) . (27) nn l - l , ( K ) l T=2K (N)
T=2K (A)
T=1K (N)
T=1K (A)
T=0.5K (N)
T=0.5K (A)
B = 0.45T n FIG. 3: The analytical (A) extension of the quasi-chemicalpotential of the l -subband (dashed, dash-dotted and dash-dot-doted lines) is compared with the results of numerical (N)calculations for narrow Landau levels (solid lines). The wavyshape of the solid lines increases with lowering temperaturetogether with the accuracy of the analytical approximation. In this equations, the last two terms represent the well-known high-field approximation for the chemical poten-tial .The influence of higher Landau levels and a finitebroadening Γ , on µ ( η ) is illustrated in Fig. 2 for ~ ω c /T e = 5 (in this figure µ − ∆ , and Γ ,n are givenin units of ~ ω c ). These results indicate that the simpleform of Eq. (27) describes the dependence µ ( η ) − ∆ , quite well if Γ , / ~ ω c ≤ .
3. At Γ , / ~ ω c = 0 .
1, it is evendifficult to see the difference between results of numericalcalculations (not shown in Fig. 2) and the approximationΓ , = 0 illustrated in the figure by the red line. For thestrong broadening Γ , / ~ ω c = 1, the results of numeri-cal calculations (orange line) deviate substantially fromthe approximation given in Eq. (27), if η > .
2. Underthese conditions, the analytical form can be used only fora qualitative analysis or simple estimations. It is impor-tant that considering a 2D electron system with narrowLandau levels, the approximation of Eq. (27) can be usedeven for substantial values of the filling factor η ≤ . η >
1, one canfind a simple extension of the analytical form of Eq. (27)which can be used for the ground subband as well. There-fore, in the following equation, we shall use an arbitrarysubband index ( l ): µ l − ∆ l, = ∞ X n =0 (cid:20) ε n − T e ln (cid:18) n + 1 − η l η l − n (cid:19)(cid:21) ×× θ ( n + 1 − η l ) θ ( η l − n ) , (28)where θ ( x ) is the Heaviside step function, and η l =2 πℓ B N l /S A . This solution is found assuming that f l ( ε ) ≃ ε ≤ ε n − if n < η l < n + 1, thereforeit is a low temperature approximation. Fig. 3 illustratesthat at low temperatures ( T e ≤ η l = 1 , , ... . At high temperatures T e & . ~ ω c ,the deviations are strong because the numerical resultsshown by the red line approach the semi-classical formula µ l ( η l ) − ∆ l, ≃ π ~ N l /m e S A . In our numerical calcula-tions (here and below), the ratio of the effective electronmass to the free electron mass is fixed to the value 0.067which is typical for semiconductor heterostructures.In Fig. 3, the filling factor η l was varied by changingelectron density n l = N l /S A , while the magnetic fieldwas fixed. It is remarkable that the simple analytical ap-proximation given in Eq. (28) can be used also for thedescription of the well-known oscillations of the chem-ical potential µ ( B ) of a 2D electron system with a fixeddensity and narrow Landau levels (here we omit the sub-band index). This possibility is illustrated in Fig. 4 for n e = 1 . × cm − and T = 0 . η ( B ) is very close to an integer (1 , , ... ) asindicated in Fig. 4. IV. RESULTS AND DISCUSSION
According to Eqs. (25) and (26) the contribution fromintersubband scattering to the effective collision fre-quency as a function of the magnetic field is determinedby the extended DSF D l,l ′ ( q, Ω) and its derivative withrespect to frequency D ′ l,l ′ ( q, Ω) near the special pointsΩ = ω l,l ′ ≡ ∆ l,l ′ / ~ >
0. Considering the two sub-band model ( l = 2 and l ′ = 1), in Eq. (13) which de-fines D , ( q, Ω) the factor [1 − f ( ε + ~ Ω)] can be set tounity because the distribution function of electrons occu-pying the ground subband is very small at high energies: f ( ε + ∆ , ) ≪
1. The later inequality follows from thefact that the respective quasi Fermi level µ ≤ µ . Forthe regime of fixed density, µ < µ which is quite ob-vious according to Fig. 3. In the the regime of fixedchemical potential, µ = µ due to a reservoir of elec-trons . Therefore, the nonequilibrium DSF D , ( q, Ω) ( K ) B (T)
Analytical
Numerical =12
FIG. 4: Illustration of the efficiency of the analytical approx-imation given in Eq. (28) for the description of oscillationsof the chemical potential (Fermi energy) as a function of B under conditions that the collision broadening is small: theanalytical equation (solid red line), and numerical calculations(dashed blue line). The singular points, where the filling fac-tor η equals to an integer, are indicated. as a function of frequency is determined mostly by thedistribution of electrons occupying the excited subband f ( ε ) = (cid:26) − η η exp (cid:18) ε − ε T e (cid:19) + 1 (cid:27) − , (29)where we had used the approximation of Eq. (27) for µ assuming that η ≤ .
8. For larger η , we shall use theextension of Eq. (28).In the expression for the effective collision frequency ν eff , the DSF is affected by integration over q . For short-range scatterers, the respective integral can be easily cal-culated because R x q I n,n ′ ( x q ) dx q = ( n + n ′ + 1). There-fore, it is convenient to analyze the frequency dependenceof the dimensionless function J , ( ω/ω c ) = η Γ4 ~ ∞ Z D , ( q, ω ) x q dx q (30)instead of D , ( q, ω ). Here, for simplicity reasons, thecollision broadening of Landau levels Γ is assumed to beindependent of quantum numbers n and l . Employingthe Gaussian shape of Im G l,n ( ε ) given in Eq. (17) yields J , = η ∞ X n =1 ( n + 1) ∞ Z − ε / Γ exp (cid:0) − y (cid:1) (1 − η ) exp ( y Γ /T e ) + η ×× exp ( − (cid:20) y + ~ ω c Γ (cid:18) ωω c − n (cid:19)(cid:21) ) dy. (31) J , ( c ) , J ’ , ( c ) c FIG. 5: The frequency dependencies of the dimensionlessfunctions which define the shape of magnetooscillations of ν N and ν A calculated for two values of the magnetic field shownin the figure legend: J , ( ω/ω c ) (solid lines) and J ′ , ( ω/ω c )(dashed and dash-dotted lines). It is obvious that J , ( ω/ω c ) has prominent maxima nearthe conditions ω/ω c = 1 , , ... , if the 2D electron systemis pure enough and ~ ω c / Γ >
1. The results of numericalevaluations of the function J , ( ω/ω c ) and its derivative J ′ , ( ω/ω c ) are shown in Fig. 5 by the solid and dashed(dashed-dotted) lines respectively. The calculations wereperformed for N = 0 . N e and two values of the mag-netic field [ B = 0 . . B due to the factor (1 − η ) /η because the filling factor η (0 . ≃ .
165 while η (0 . ≃ . B affects notably also the positions of minima, max-ima and the zero-crossing (sign-changing) point of thederivative J ′ , ( ω/ω c ).Using the same approximations as those used for ob-taining Eq. (31), the abnormal contribution to the effec-tive collision frequency can be represented as ν A = ν p , πη (cid:18) ~ ω c Γ (cid:19) (cid:16) − e − µ , /T e (cid:17) Φ , ( B ) , (32)where we definedΦ , ( B ) = Γ ~ ω c J ′ , ( ω , /ω c ) (33)because the derivative J ′ , ( ω , /ω c ) contains the addi-tional factor ~ ω c / Γ according to Eq. (31). The dimen-sionless parameter p l,l ′ is determined by the followingmatrix elements p l,l ′ = B , B l,l ′ , B − l,l ′ = L − z X κ (cid:12)(cid:12)(cid:12)(cid:0) e − iκz (cid:1) l ′ ,l (cid:12)(cid:12)(cid:12) . (34) The accurate calculation of p , requires the knowledgeof the details of a particular 2D electron system such asthe wavefunctions of subband states which are not con-sidering in this work. For electrons on liquid helium , p , is a factor of two smaller than p , = 1. Therefore,in following numerical calculations we shall use a roughestimation: 2 p , ≃ l = 1) can be found as ν (1)intra ≃ ν p , πη (cid:18) ~ ω c Γ (cid:19) Φ , ( B ) , (35)whereΦ , ( B ) = ∞ X n =0 (2 n + 1) exp " − µ − ε n ) Γ . (36)At the same time, the contribution from electron scatter-ing within the first excited subband ν (2)intra has a very weakdependence on B because the distribution function f ( ε )given in Eq. (29) varies strongly near ε . Thus, ν (2)intra canbe considered as a small background value when the ratio N /N e ≪
1. The background value decreases also withnarrowing of the density of states. In the following, weshall neglect ν (2)intra and assume that ν intra ≃ ν (1)intra .Comparing ν A of Eq. (32) with ν (1)intra given in Eq. (35)indicates that the abnormal contribution contains the ad-ditional factor (cid:0) − e − µ , /T e (cid:1) which is zero under equi-librium conditions ( µ = µ = µ F ). If we have a nonequi-librium population of the second subband, then, accord-ing to Eq. (27) and Fig. 2, δµ becomes substantiallylarger than T e already at a small filling factor η . Forexample, Fig. 2 shows that µ > ∆ , if η > .
1. As-suming this reasonable condition, we can neglect the ex-ponentially small term in the factor (cid:0) − e − µ , /T e (cid:1) andset this factor to unity even if µ is fixed (according toFig. 3, µ decreases with lowering N which also reducese − µ , /T e ).Another important distinction between ν A and ν intra iscaused by different behaviors of the dimensionless func-tions Φ , ( B ) and Φ , ( B ) illustrated in Fig. 6. Theboth functions oscillate with varying 1 /B , but the pe-riods of these oscillations are different. Assuming µ isfixed to µ F , the maxima of the positive function Φ , ( B )entering ν intra occur at ~ ω c = µ F / ( n + 1 /
2) due tothe Shubnikov–de Haas effect. In contrast to Φ , ( B ),the function Φ , ( B ), which determines ν A , is a sign-changing function having maxima and minima, accordingto the definition of Eq. (33) and Fig. 5; its zero-crossingpoints occur at magnetic fields which are close to thecondition ∆ , / ~ ω c = m (here m = 1 , , ... ).It is instructive to analyze ν N using the same ap-proximations and conditions. Direct transformation ofEq. (25) yields ν N = ν p , πη e − µ , /T e ~ ω c T e Γ J , ( ω , /ω c ) . (37) , ( B ) , , ( B ) B (T) , (B) , (B) N =0.1N e FIG. 6: Graphical illustration of the functions Φ , ( B ) andΦ , ( B ) which determine ν A and ν intra respectively. As compared to the contribution from intrasubband scat-tering of Eq. (35), here we have T e in the denomina-tor because for intersubband scattering one cannot usethe relationship f ( ε ) [1 − f ( ε )] → T e δ ( ε − ε F ). Theshape of oscillations caused by ν N is determined by thefunction J , ( ω , /ω c ) shown above in Fig. 5 by solidlines. This shape is in a qualitative accordance with re-sults obtained for magnetointersubband oscillations un-der equilibrium conditions . For nonequilibrium regimedescribed here, Eq. (37) contains also the exponentialfactor exp ( − µ , /T e ) which becomes very small even forrelatively weak excitations N = 0 . N e . It should benoted also that under conditions used here, the ampli-tude of Φ , is about 5 times larger than the respectiveamplitude of J , . Therefore, ν N can be neglected ascompared to ν A and ν intra .Typical dependencies of σ xx ( B ) are shown in Fig. 7.In the equilibrium case ( µ = µ ), ν eff = ν intra and σ xx ( B ) has maxima when ~ ω c = µ F / ( n + 1 /
2) accord-ing to the SCBA theory (blue dashed line). In thisfigure, the electron conductivity σ xx is normalized bythe first ( n = 0) peak value σ (0)max = e / π ~ found forthe Gaussian level density ( B ≃ .
827 T). Already asmall nonequilibrium electron population of the excitedsubband ( N /N e = 0 .
1) induces important changes into σ xx ( B ) shown in Fig. 7 by the red line. Besides addi-tional maxima and a substantial reduction of the SCBApeak at n = 3, there are sign-changing variations of σ xx ( B ) near B ≃ .
48 T, 0 .
24 T and 0 .
156 T, and quitedeep minima with regions where the linear response con-ductivity σ xx becomes negative. An increase in the elec-tron population of the excited subband ( N /N e = 0 . xx / ( ) m a x B (T) , intra N = 0.1N e , intra A N = 0.2N e , intra A FIG. 7: Magnetoconductivity normalized to σ (0)max = e / π ~ versus the magnetic field for different levels of the displace-ment from equilibrium: N ≃ N =0 . N e (red solid line), and N = 0 . N e (olive dash-dottedline). by the olive dash-dotted line. It should be noted that forsuch a population, η ( B ) becomes larger than unity inthe region of low B , and, therefore, the approximationof Eq. (27) defining µ fails. In this case, we had usedthe extension of the quasi-Fermi energy given in Eq. (28).Numerical calculations indicate also that reducing tem-perature from 1 K to 0 . ν A .Thus, the theoretical analysis given above indicatesthat the Pauli exclusion principle does not ruin the inter-subband mechanism of MIRO, if the electron distributionin the ground and excited subbands can be described bythe quasi-Fermi level approximation. Moreover, a sharpincrease of the imref of the excited subband as a func-tion of the filling factor shown in Fig. 2 reduces stronglythe compensational contribution from reverse intersub-band scattering [the exponential term in parenthesis ofEq. (32); under conditions of Fig. 7 this term does notexceed 0.04]. This means that magnetoconductivity os-cillations and ZRS induced by the resonant MW field,whose polarization direction is perpendicular to the elec-tron layer, can be realized in sufficiently clean semicon-ductor devices. The regions with negative linear responseconductivity attract a special interest, because they allowperforming complementary studies of ZRS in heterostruc-tures caused by a definite mechanism. These studies po-tentially can help also with the identification of the originof MIRO and ZRS in the conventional setup.0 V. CONCLUSION
We have presented a theory of quantum magnetotrans-port in a degenerate multisubband electron system underconditions that electron distributions over 2D subbandscannot be described by a single chemical potential. Usingthe concept of quasi-Fermi levels and the self-consistentBorn approximation, we expressed magnetoconductivityequations in terms of the extended dynamic structure fac-tor and its derivative with regard to frequency. We haveshown that a displacement from the equilibrium electrondistribution over excited subbands, which cannot be re- duced to trivial heating, leads to appearance of abnormalsign-changing contribution to the momentum collisionrate and magnetoconductivity. Calculations performedfor a simplified potential of scatterers indicate that evena small fraction of electrons (about 10%) transferred tothe first excited subband can drastically change the shapeof magnetointersubband oscillations an lead to negativelinear response conductivity. The theory can be appliedto electrons on helium films with a special arrangementsof substrates , and to multisubband 2D electron systemsof semiconductor devices. ∗ E-mail: [email protected] T. Ando, A.B. Fowler, and F. Stern,
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