Oscillatory nonlinear differential magnetoresistance of highly mobile 2D electrons in high Landau levels
aa r X i v : . [ c ond - m a t . m t r l - s c i ] J u l Oscillatory nonlinear differential magnetoresistance of highly mobile 2D electronsin high Landau levels
X. L. Lei
Department of Physics, Shanghai Jiaotong University, 1954 Huashan Road, Shanghai 200030, China
We examine the current-induced magnetoresistance oscillations in high-mobility two-dimensionalelectron systems using the balance-equation scheme for nonlinear magnetotransort. The reportedanalytical expressions for differential magnetoresistivity at high filling factors in the overlappingLandau-level regime, which show good agreement with the experimental observation and the nu-merical calculation, may be helpful in extracting physical information from experiments.
In addition to the universally existing Shubnikov-deHaas oscillations (SdHO), many different kinds of mag-netoresistance oscillations were discovered in the pastfew years in high-mobility two-dimensional (2D) elec-tron systems (ES) subject to a weak perpendicular mag-netic field and have become a field of great interest.These resistance oscillations always occur at low tem-peratures and are related to magnetotransport of 2Delectrons occupying high Landau levels (LLs). Amongthem the microwave-induced magnetoresistance oscilla-tions and the related zero-resistance states were thecentral focus of most experimental andtheoretical studies. Recently,the oscillatory behavior in the nonlinear magnetotrans-port has attracted much attention: in a 2D system evenwithout irradiation, a relatively weak current can inducedrastic suppression and strong oscillations of the differ-ential magnetoresistance, and may result in a state ofzero-differential resistance.
Several theoretical models have been proposed inan attempt to explain this interesting nonlinearphenomenon.
Numerical examinations based onthe current-control transport scheme were shown in goodagreement with the experimental observation for differ-ential magnetoresistivity as a function of the ratio ofthe current density to the magnetic field in both themagnetic-field sweeping and the current-sweeping config-urations covering both separated and overlapping Lan-dau level regimes. It is reported recently that by a systematic analysisof current-induced magnetoresistance oscillations, impor-tant physical information about electron-electron inter-action on the single particle life time can be extracted. From the point of view of experiment, an analytical ex-pression for differential magnetoresistivity, even appliesonly within limited ranges, is highly desirable because itcan be of great help to extract important physical infor-mation from experimental data. So far, a reliable ana-lytical expression derived from experimentally confirmedtheoretical models is still lacking.We examine this issue based on the balance-equationscheme of nonlinear magnetotransport, which dealswith a 2D system consisting of N s electrons in a unitarea of the x - y plane and subjected to a uniform mag-netic field B = (0 , , B ) in the z direction. These elec-trons, scattered by randomly distributed impurities and by phonons in the lattice, perform an integrative driftmotion under the influence of a uniform electric field E in the x - y plane. For high mobility and high carrier-density systems in which effects of electron-impurity andelectron-phonon scatterings are weak in comparison withtheir internal thermalization, the steady transport stateat lattice temperature T is described by the electron aver-age drift velocity v and an electron temperature T e . Theysatisfy the following force- and energy-balance equations: N s e E + N s e ( v × B ) + f ( v ) = 0 , (1) v · f ( v ) + w ( v ) = 0 . (2)Here f ( v ) = f i ( v ) + f p ( v ) is the damping force againstthe electron drift motion due to impurity and phononscatterings respectively, f i = X q k | U ( q k ) | q k Π (cid:0) q k , q k · v (cid:1) , (3) f p = 2 X q ,λ | M ( q , λ ) | q k Π (cid:0) q k , Ω q λ + q k · v (cid:1) × (cid:20) n (cid:18) Ω q λ T (cid:19) − n (cid:18) Ω q λ + q k · v T e (cid:19)(cid:21) , (4)and w ( v ) is the electron energy-loss rate to the lattice dueto electron-phonon interactions having an expression ob-tained from the right-hand side of Eq. (4) by replacing the q k factor with Ω q λ , the energy of a wavevector- q phononin branch λ . In these equations, n ( x ) = 1 / (e x −
1) is theBose function, U ( q k ) is the effective impurity potential, M ( q , λ ) is the effective electron-phonon coupling matrixelement, and Π ( q k , ω ) is the imaginary part of the elec-tron density correlation function at electron temperature T e in the presence of the magnetic field.The density correlation function Π ( q k , ω ) of a 2D elec-tron gas in the magnetic field can be written in the Lan-dau representation asΠ ( q k , ω ) = 12 πl B X n,n ′ C n,n ′ ( l B q k /
2) Π ( n, n ′ , ω ) , (5)Π ( n, n ′ , ω ) = − π Z dε [ f ( ε ) − f ( ε + ω )] × Im G n ( ε + ω ) Im G n ′ ( ε ) , (6)where l B = p / | eB | is the magnetic length, C n,n + l ( Y ) ≡ n ![( n + l )!] − Y l e − Y [ L ln ( Y )] with L ln ( Y ) the associateLaguerre polynomial, f ( ε ) = { exp[( ε − ε F ) /T e ] + 1 } − is the Fermi function at electron temperature T e with ε F the Fermi level of the system, and Im G n ( ε ) is thedensity-of-states (DOS) function of the broadened LL n .The LL broadening depends on impurity, phonon andelectron-electron scatterings. In a high-mobility GaAs-based 2D system the dominant elastic scattering comesfrom impurities or defects in the background or closeproximity, and phonon and electron-electron scatter-ings are also not long-ranged because of the screening.The correlation lengths of these scattering potentials aremuch smaller than the cyclotron radii R c of electrons in-volving in the transport subject to a weak magnetic field,but much larger than R c /n for very high ( n ≫
1) LLs.In this case, the broadening of the LL is expected to bea Gaussian form Im G n ( ε ) = − (2 π ) Γ − exp[ − ε − ε n ) / Γ ] . (7)In this, ε n = ( n + ) ω c is the center of the n th LL, ω c = eB/m is the cyclotron frequency with m the effectivemass of the electron, and Γ, the half-width of the LL, is B / -dependent expressed as Γ = (2 ω c /πτ s ) / , (8)where τ s is the single-particle lifetime or quantum scat-tering time of the electron in the zero magnetic field. Thissingle-particle life time τ s , relating to impurity, phononand electron-electron scatterings, is generally tempera-ture dependent.The DOS function (7) for the n th LL is valid and canbe used in both separated and overlapping LL regimes.The total DOS (double spins) for a 2D system of unitarea in the presence of a magnetic field is given by g ( ε ) = − P n Im G n ( ε ) / ( π l B ), and the Fermi level ε F is determined by the electron sheet density N s from theequation R dεf ( ε ) g ( ε ) = N s . In the case of high LL fill-ing ε F is essentially the same as that of a 2D electrongas having the same carrier density N s without magneticfield, ε F = k F / m ( k F is the Fermi wavevector), and ν ≡ ε F /ω c is the the filling factor.When the LL width 2Γ is larger than the level spacing ω c (overlapping LLs), the Dingle factor δ = exp ( − π/ω c τ s ) (9)is smaller than 0.3. Keeping only terms of the lowestorder in δ or of the fundamental harmonic oscillation, wehave the following approximate expression for the DOS: g ( ε ) ≈ mπ (cid:2) − δ cos (2 πε/ω c ) (cid:3) . (10)For an isotropic system where the frictional force f ( v )is in the opposite direction to the drift velocity v , we canwrite f ( v ) = f ( v ) v /v . In the Hall configuration with thevelocity v in the x direction v = ( v, ,
0) or the currentdensity J x = J = N s ev and J y = 0, Eq. (1) yields, at agiven v , the transverse resistivity R yx = B/N s e , and the longitudinal resistivity and differential magnetoresistiv-ity as R xx = − f ( v ) / ( N s e v ) , (11) r xx = − ( ∂f ( v ) /∂v ) / ( N s e ) . (12)These formulas, expressing the nonlinear resistivity asa function of the drift velocity v without invoking electricfield, are convenient to direct relate to experiments wheretransport measurements are performed by controlling thecurrent. This is the basic feature of the balance equationapproach, in which the drift velocity v of the electronsystem, rather than the electric field, plays as the funda-mental physical quantity to affect electron transport. Inthis approach individual (relative) electrons, treated inthe reference frame moving at velocity v , do not directlyfeel a uniform electric field. The role of the drift velocity v is to provide the electron of wavevector q k with an en-ergy q k · v during its transition from a state to anotherstate induced by impurity or phonon scattering. Thisresults in an extra frequency q k · v in the density cor-relation function Π ( q k , ω + q k · v ) in Eqs.(3) and (4).Since in a magnetic field the density correlation functionis frequency periodic, Π ( q k , ω ) ∼ Π ( q k , ω + ω c ), due toperiodical LLs, change of the drift velocity v will leadto oscillation of related physical quantities. At low tem-perature and high LL filling, in view of Π ( q k , ω ) func-tion sharply peaking around q k ≃ k F , the effect of afinite velocity v , after the q k integration, is equivalentto shift the frequency in the Π function an amount of ω j ≡ k F v . Thus when ω j varies by a value of ω c , theresistivity experiences change of an oscillatory period. Itis thus convenient using a dimensionless parameter ǫ j ≡ ω j ω c = 2 mk F veB = r πN s me JB (13)to demonstrate the behavior of nonlinear magnetoresis-tivity oscillation.The numerical examination of impurity-related differ-ential resistivity shows that r i xx = − ( ∂f i ( v ) /∂v )( N s e )oscillates with changing ǫ j exhibiting an approximate pe-riod ∆ ǫ j ≈
1, with maxima at positions somewhat lowerthan integers ǫ j = n ( n = 1 , , , ... ) and minima some-what lower than half integers ǫ j = n +1 / n = 1 , , , ... ).The predicted results covering both separated and over-lapping Landau level regimes, are in good agreement withthe experimental observation of Ref. 25 in both magnetic-field sweeping and current-sweeping cases.Simple and accurate analytical expressions of theimpurity-induced resistivity r i xx can be derived for short-range potentials in the case of high filling factor ν withinthe overlapping LL regime.At temperature T ≪ ε F the impurity-induced linear( v →
0) resistivity r i xx (0) is given, to the lowest nonzeroorder in δ and the fundamental SdHO part, by r i xx (0) = R i0 (cid:2) δ − δD ( X ) cos(2 πν ) (cid:3) , (14)where X ≡ π T /ω c , D ( X ) ≡ X/ sinh( X ), and R i0 is thelow-temperature ( T ≪ ε F ) linear resistivity of the 2Delectron gas in the absence of magnetic field, which isdirectly related to the transport scattering time τ tr or thelinear mobility µ as R i0 = m/ ( N s e τ tr ) = 1 / ( N s eµ ).The expression (14) shows that r i xx (0), though oscillat-ing strongly with changing filling factor ν , is always pos-itive. Both its oscillatory (SdHO) and non-oscillatroyparts are τ s -dependent through the δ -involved terms inthe presence of a magnetic field and may change withtemperature. The linear resistivity R i0 of a 2D electronsystem without magnetic field, however, is not affectedby its single particle lifetime τ s and thus temperatureindependent.The impurity-related SdHO [the last term in Eq. (14)]exhibits minima at integer filling factor ν and quicklydiminishes with rising temperature T ≫ ω c / π due tothe temperature-dependent factor D ( X ). The tempera-ture dependence of the non-SdHO part of the impurity-induced linear resistivity of the 2D electron system in thepresence of a magnetic field can appear mainly throughthe temperature change of its single particle lifetime τ s entering the Dingle-factor coefficient 2 δ .In the case of overlapping LLs and high filling factor ν ,the nonlinear longitudinal differential resistivity r i xx canbe expressed, at temperature T e ≪ ε F , as r i xx = R i0 n δ G (2 πǫ j ) − δ D ( X e ) cos(2 πν ) S (2 πǫ j ) o , (15)in which X e ≡ π T e /ω c , S ( z ) = J ( z ) − J ( z ) , (16) G ( z ) = J ( z ) − J ( z ) − z (cid:2) J ( z ) − J ( z ) (cid:3) , (17) J k ( z ) being the Bessel function of order k . With the helpof power expansions and asymptotic expressions of Besselfunctions, we have S ( z ) ≃ − z , (18) G ( z ) ≃ − z (19)for z ≪
1; and S ( z ) ≃ r πz cos (cid:16) z − π (cid:17) , (20) G ( z ) ≃ r πz h cos (cid:16) z − π (cid:17) − z sin (cid:16) z − π (cid:17)i (21)for z ≫ ǫ j → T e → T ) Eq. (15) re-turns to Eq. (14). For nonlinear transport Eq. (15) showsthat the SdHO term, which diminishes when temperature T e ≫ ω c / π , is modulated with an oscillatory factor S (2 πǫ j ) by the finite current density through the dimen-sionless parameter ǫ j . Furthermore, the non-SdHO part Fun. (17) Fun. (19) Fun. (21) G ( p e j ) e j FIG. 1: (Color online) G (2 πǫ j ) function (17) and its approx-imate expression (19) and (21) for small and large argument z = 2 πǫ j . g = p / t s w j g = 0.75 g = 1 g = 0.5 r xx / R i e j = w j / w c FIG. 2: (Color online) The reduced differential magnetoresis-tivity r xx /R i0 calculated from Eq. (15) (excluding the SdHOpart) versus ǫ j = ω j /ω c for several fixed values of γ ≡ π/τ s ω j = 0 . , .
75 and 1. of the differential resistivity, which survives the temper-ature rising as long as T e ≪ ε F , exhibits strong oscil-lation with changing ǫ j , having an oscillation amplitude∆ r ≡ r i xx − R i0 as ∆ rR i0 = 2 δ G (2 πǫ j ) . (22)Since the Dingle-factor coefficient 2 δ varies smoothlywith changing ǫ j , it is apparent that the oscillatoryperiod and the maxima and minima positions of dif-ferential resistivity r i xx are determined mainly by the G (2 πǫ j ) function. For ǫ j > .
5, the r i xx oscillates with ǫ j approximately in a period ∆ ǫ j ≈
1, having maximaaround ǫ j ≈ n − / n = 1 , , , ... ) and minima around ǫ j ≈ n + 3 / n = 1 , , , ... ). Nevertheless, the accuratepositions of the maxima and minima vary somewhat with ǫ j because of the possible change of 2 δ with ǫ j and the ǫ / j -dependent coefficient at large ǫ j .The G (2 πǫ j ) function (17) is shown in Fig. 1, togetherwith its approximate expressions (19) and (21) for smalland large arguments z = 2 πǫ j . The reduced differentialmagnetoresistivity r xx /R i0 calculated from Eq. (15) (ex-cluding the SdHO part), is plotted in Fig. 2 as a functionof ǫ j for several fixed values of γ ≡ ( π/τ s ω j ) = 0 . , . ǫ j < .
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