Topological oscillations of the magnetoconductance in disordered GaAs layers. S. S. Murzin1,4, A. G. M. Jansen2,4, and I. Claus3,4 1Institute of Solid State Physics RAS, 142432, Chernogolovka, Moscow District., Russia 2Service de Physique Statistique, Magn´etisme, et Supraconductivit´e, D´epartement de Recherche Fondamentale sur la Mati`ere Condens´ee, CEA-Grenoble, 38054 Grenoble Cedex 9, France 3Center for Nonlinear Phenomena and Complex Systems, Facult´e des Sciences, Universit´e 4 Libre de Bruxelles, Campus Plaine, Code Postal 231, B-1050 Brussels, Belgium 0 4Grenoble High Magnetic Field Laboratory, Max-Planck-Institut fu¨r Festk¨orperforschung and 0 Centre National de la Recherche Scientifique, BP 166, F-38042, Grenoble Cedex 9, France 2 n Oscillatoryvariationsofthediagonal(Gxx)andHall(Gxy)magnetoconductancesarediscussedin a viewoftopologicalscalingeffectsgivingrisetothequantumHalleffect. Theyoccurinafieldrange J without oscillations of thedensity of states due to Landau quantization, and are, therefore, totally 9 different from the Shubnikov-de Haas oscillations. Such oscillations are experimentally observed in disordered GaAs layers in the extreme quantum limit of applied magnetic field with a good ] description bythe unifiedscaling theory of the integer and fractional quantumHall effect. l l a PACSnumbers: 73.50.Jt; 73.61.Ey;73.40.Hm h - s e The integer quantum Hall effect (QHE) is usually ob- states[7,8,9]. Thisso-calledunifiedscaling(US)theory m served at high magnetic fields, ω τ ≫ 1 (ω = eB/m describes well the shape of the scaling flow diagram de- c c . is the cyclotron frequency, τ is the transport relaxation pictingthecoupledevolutionofG andG fordecreas- t xx xy a time), and its appearance develops from the Shubnikov- ingtemperaturesinheavilySi-dopedn-typeGaAslayers m deHaasoscillationsbasedontheLandauquantizationof with different thickness for a wide range of G values xx - the two-dimensional(2D) electron system. However,the [10]. Thesecondtheoryhasbeendevelopedinthe”dilute d scalingtreatmentoftheintegerQHE[1]predictstheexis- instanton gas” approximation (DIGA), firstly for non- n tenceoftheQHEwithouttheLandauquantizationofthe interacting [11] and then for interacting electrons [12]. o c electron spectrum. It could exist even at low magnetic Both theories are developed for a totally spin-polarized [ fieldω τ ≪1[2]intheabsenceofmagnetoquantumoscil- electron system. For 2πG ≫ 1 they predict an os- c xx lationsofthedensityofstates. TheQHEatlowmagnetic cillating topological term in the scaling β-function with 3 fields ω τ ≪ 1 has not been observed so far, probably the same periodicity. However, they differ in predictions v c 1 because extremely low temperatures are required [3]. In on the oscillation amplitude. The oscillating topological 0 addition, the QHE couldexistin a layerwhose thickness term in the β-function should lead to oscillations in the 6 d is much larger than the electron transport mean free magnetic-fielddependenceofG andG whicharenot xx xy 5 path l, i.e. d ≫ l, in the extreme quantum limit (EQL) related to oscillations in the density of states like, e.g., 0 of applied magnetic field, where only the lowest Landau for the case of the Shubnikov-de Haas oscillations. 3 level is occupied. Such a layer has a three-dimensional 0 In the presented work we derive explicit expressions (3D) ”bare”(non-renormalized)electronspectrum with- / at outoscillationsofthedensityofstatesintheEQL.Inthis for the topological oscillations of the Hall conductiv- ity G for both theories, and compare them with ex- m situationtheQHEhasbeenobservedinheavilySi-doped xy periment for thick (d ≫ l) disordered heavily Si-doped n-type GaAs layers [4, 5]. - GaAs layers with rather large G and G compared d xx xy n Here,weaddresstheproblemofthearisingoftheQHE to unity. The layers studied before in Ref.[4, 5] have o in the absence of magnetoquantum oscillations of the a 3D ”bare” electron spectrum. However, below 4 K c density of states. In this case the variation with tem- thecharacteristicdiffusionlengths,L =(D τ )1/2 and ϕ zz ϕ v: perature of the diagonal conductance per square (Gxx) LT =(Dzz~/kBT)1/2,forcoherentdiffusivetransportin- and Hall conductance (G ) is due to diffusive inter- i xy crease to values larger than d, and the system becomes X ference effects (below G and G are taken in units xx xy 2Dforcoherentdiffusivephenomena(D isthediffusion zz r e2/h), which in a scaling approach can be described by coefficientofelectrons alongthe magnetic field, τ is the a ϕ the renormalization-group equations. For comparison, phase breaking time). accordingtotheconventionaltheory,thetemperaturede- pendenceoftheShubnikov-deHaasoscillationspreceding The US theory describes the renormalization group the QHE is due to thermal broadening of the Fermi dis- flow of the conductances by the equation [7], tribution [6]. At the moment, two theories give explicit expressionsfortherenormalization-groupequations. The first theory has been derived for both integer and frac- s−s0 =−ln(f/f0), (1) tional QHE and for any value of G [7]. It is based xx ontheassumptionthatacertainsymmetrygroupunifies the structure of the integer and fractional quantum Hall for a real parameter s monotonically depending on tem- 2 perature, where G≡G +iG , f =f(s ) and In the ”dilute instanton gas” approximation for the xy xx 0 0 case of interacting electrons [12] ∞ qn2 4 ∞ (−1)nqn2 4 f =−(cid:16)Pn=−∞ (cid:17) (cid:16)Pn=−∞ (cid:17) , (2) dGxx = −λ −D G2 e−2πGxxcos(2πG ) (9) 2 ∞ q(n+1/2)2 8 dlnL π 1 xx xy (cid:0) Pn=0 (cid:1) dG xy = −D G2 e−2πGxxsin(2πG ). (10) with q = exp(iπG). For |q|2 = exp(−2πGxx) ≪ 1, the dlnL 1 xx xy function f = −1/(256q2)+3/32+O(q2) and Eq.(1) is reduced to Here L≈(~Dxx/kBT)1/2 and D1 =64π/e≈74.0. Solv- ing the quotient of these equations by ignoring terms of s−s ≈i2π G−G0 +24 ei2πG−ei2πG0 (3) order exp(−4πG ) one obtains 0 (cid:0) (cid:1) (cid:16) (cid:17) xx πD In the first-order approximation by ignoring the last os- G =G0 − 1 F(GT )−F(G0 ) sin(2πG0 ) cillating term in Eq.(3), this equation has the solution xy xy λ (cid:2) xx xx (cid:3) xy (11) where F(x)=1/4π3 2π2x2+2πx+1 exp(−2πx). (cid:0) (cid:1) G1 =G0 −(s−s )/2π, G1 =G0 . (4) Both theories have been developed for a totally spin- xx xx 0 xy xy polarized electron system. However, in a real system Inthesecond-orderapproximation,thesolutionlookslike electrons can have two different spin projections. For thecaseofnon-interactingelectrons,theelectronscanbe G = G1 + 12 e−2πG1xx −e−2πG0xx cos(2πG0 ) (5) describedintermsoftwoindependent,totallyspinpolar- xx xx π h i xy ized systems in the absence of spin-flip scattering. This G = G0 − 12 e−2πG1xx −e−2πG0xx sin(2πG0 ). (6) approach remains valid for interacting electrons as well, xy xy π h i xy if the triplet part of the constant of interaction is much smaller than the singlet one [13, 14], because only the This is a solution of Eq.(3) for fixed s. However, for our interaction between electrons with the same spin leads experimentweareinterestedinthesolutionforfixedtem- toa renormalizationofthe conductanceinthis case. For perature T. In the first-order approximation it should the small spin-splitting in strongly disorderedGaAs, the coincide with the result of the first-order perturbation conductances of the electron systems with different spin theory for the electron-electron interaction in coherent projection(G↑ andG↓)areapproximatelyequalto half diffusive transport leading to logarithmic temperature- ij ij dependent corrections in the diagonal conductance the measured conductance, i.e. G↑ij ≈ G↓ij ≈ Gij/2. It allowsus to comparequantitatively the experimentalre- GT =G0 +λ/2π ln(T/T ), (7) sults with the theories. For large spin-splitting this is xx xx 0 impossible, because G↑ and G↓ are different, and only ij ij withoutanytemperaturedependenceintheHallconduc- the sum G↑ +G↓ can be measured. tance [13]. Therefore, s = −λln(T) in this approxima- ij ij The investigated heavily Si-doped n-type GaAs lay- tion. For totally spin-polarized electron system λ = 1 erssandwichedbetweenundopedGaAswerepreparedby [14]. molecular-beamepitaxy. Thenominalthicknessdequals Insecondorder,s willoscillateasa functionofG0 at xy 100 nm for the layers 2, 3, 6, and 140 nm for layer 7. fixed temperature T and will give additional oscillating The Si-donor bulk concentration n equals 1.8, 2.5, 1.6, term in Eq.(5), but the relation between s and T is un- and 3×1017 cm−3 for samples 2, 3, 6, and 7 as derived knownandthe amplitude ofthe G oscillationscannot xx from the period of the Shubnikov-deHaas oscillationsat be found. In this respect we note, that the last term in B < 5 T. The mobilities of the samples at T = 4.2 K Eq.(5) shows maxima at integer G0 as opposed to the xy are 2400, 2500, 2600 and 2600 cm2/Vs, and the electron expectedminimafortheintegerQHE.Thedifferencebe- densities per square N as derived from the slope of the tween G1 and GT can be ignored in the exponents of s xx xx Hall resistance R in weak magnetic fields (0.5−3 T) Eq.(6). Thereforethe HallconductivityG oscillatesas xy xy at T = 4.2 K are 1.26, 2, 2.08 and 2.86×1012 cm−2 for afunctionofthe”bare”HallconductanceG0 andhence xy samples 2, 3, 6 and 7, respectively. For all samples the as a function of the magnetic field B, with amplitude electron transport mean free path l is around 30 nm at AUS = 12 e−2πGTxx −e−2πG0xx (8) zeromagneticfield. Thedetailedstructureofthesamples xy π h i is described in Ref.[4]. = 1π2e−2πG0xxh(T0/T)λ−1i, (RIxnx,Fpiegr.1sqtuhaerem)aagnndetHotarlaln(Rspxoyr)trdesaitsataonfceth(beodthiaggoivneanl in units of h/e2), and of the diagonal (G ) and Hall xx as found by substituting GT (Eq.(7)) for G1 in Eq.(6). (G ) conductance are plotted for sample 2. At 4.2 K, xx xx xy Thisdependenceistotallydifferentfromtheexponential themagnetoresistanceshowsthetypicalbehaviorofbulk variationwithtemperatureoftheShubnikov-deHaasos- material with weak Shubnikov-de Haas oscillations for cillations. increasing field B and a strong monotonous upturn in 3 0.6 T = 4.2 K 0.46 K 1 0.0 2)e 0.4 00..2088 Rxy D Gxy sample 6 / h (x ) -0.2 D Gxx 00..697 Rx h / R , xy 0.2 2 (e 00..5466 B y-0.4 EQL R Gx xx D 0 , x x G -0.6 6 D ) h 2e/ D G 0.1 (x 4 -0.8 xx x G G G , y xx xy sample 7 Gx 2 2 4 6 8 G0 (e2/h) xy 0 0 5 10 15 20 FIG. 2: Residual variation for the diagonal ∆Gxx and for %(cid:3)(cid:11)7(cid:12) Hall conductance∆Gxy after subtraction of the4.2-K values atdifferenttemperatures,forsamples6and7. Numbersnear curvesindicate temperatures in K. FIG. 1: Magnetic field dependenceof the diagonal (Rxx, per square) and Hall (Rxy) resistance and of the diagonal (Gxx) and Hall (Gxy) conductance for sample 2 in a magnetic field perpendicular to the heavily doped GaAs layer (thickness pure layer with ballistic motion across the layer when 100 nm) at different temperatures. The arrow indicates the l/d ≫ 1. In our case, however, l/d ≈ 0.2÷0.3 in zero field BEQL of theextremequantum limit. magnetic field, which ratio even decreases in the EQL for the mean free path along the field. The 3D character of the ”bare” electron spectrum of the samples has been the extreme quantum limit (EQL) where only the low- confirmed in experiments in a tilted magnetic field [5]. est Landau level is occupied. At lower temperatures Note that the absence of oscillations at T = 4.2 K can R , R , G , and G start to oscillate. Minima of notbeexplainedbytemperaturebroadeningoftheoscil- xy xx xy xx G and of |∂G /∂B| arise at magnetic fields where latorystructures,becausedisorderbroadeningdominates xx xy Gxy at 4 K attains even-integer values, in accordance largelywith~/τ ≫kBT (foroursamples~/τkB >80K). with both theories mentioned above. These oscillatory In Fig.2 we plot the residual variation ∆G (T) = xx structures develop into the QHE at the lowest tem- G (T)−G0 as a function of G0 for sample 6 at dif- xx xx xy peratures where Rxy and Gxy reveal remarkable steps ferent temperatures, ∆Gxy =Gxy(T)−G0xy at T =0.46 near the values Rxy = 1/2 and 1/4, and Gxx = 2, K for sample 6, and ∆Gxx at T = 0.1 K for the thick- and 4. In the corresponding fields pronounced min- est sample 7. Here G0 is the conductance at T = 4.2 ij ima are observed in Rxx and Gxx. Note that, con- K taken as the ”bare” conductance (see below). Both trary to the QHE structures, the amplitude of the weak ∆G and ∆G oscillate with comparable amplitudes xx xy Shubnikov-de Haas oscillations below the EQL does not under the same conditions of applied field and tempera- depend on temperature because the thermal damping ture. The minima of ∆G are at even integer values of xx factor 2π2kBT/[~ωcsinh(2π2kBT/~ωc)] = 0.994 is close G0xy (slightly shifted in case of a superimposed smooth to 1 for B = 5 T at T = 1 K. Similar but less pro- variation of ∆G ) and the minima of ∆G are shifted xx xy nounced structures are observed for the other samples on+0.5unit inthe G0 scale,in accordancewith theory xy investigated. Moreover, for samples 3 and 7 additional [7]. minima of G and of |∂G /∂B| are observed, at fields xx xy The smoothly varying part of G , by ignoring the xx where G =6 at T =4 K. xy oscillatory part, decreases for decreasing temperature The size quantization could result in oscillatory struc- while that of G does not change. The temperature xy tures in the magnetotransport data in the EQL in a dependence of the smooth part Gsm of the diagonal xx 4 value for the ”bare” conductance G0 agrees with the US theory 7 saturation of Gsm around T =4.2 K.xx DIGA xx ) 0.1 h TheamplitudesA oftheoscillationsofG ,andG , / ij xx xy 2 e conductancesareverysimilarasshowninFig.3wherethe ( U soufmtheAdijia+g∆onUaliscopnlodtutcetdanacseaGfusxmnxcftoiornalolfotuhressammopoltehswpaitrht ∆ = 24/πexp(−πG0 ). The values of ∆ = 0.044, D+ 3 U xx U A ij 0ti.v0e0l9y,,0a.r0e2s,manadlle0r.0th02anfotrhseacmoprrleessp2o,n3d,in6g, avnadlu7es, roefspAec-. ij 0.03 sample 2 The experimental data are rather well described by the 3 result of the US theory for A (Eq. 8) applied to the xy 6 totalconductanceoftwoindependentelectronsystemsof 7 opposite spin. Although showing a very similar depen- (cid:20)(cid:17)(cid:23) (cid:20)(cid:17)(cid:25) (cid:20)(cid:17)(cid:27) (cid:21)(cid:17)(cid:19) dence, A can not be deduced in frame of this theory. xx Gsm (e2/h) The DIGA theory predicts much larger amplitudes than xx experimentallyobserved,as shownby the dotted lines in Fig.3 for ADIGA+∆ according DIGA theory for sam- xy U FIG. 3: Amplitude Aij of the topological oscillation of the ples 3 and 7. Hall (solid symbols) and diagonal (open symbols) conduc- tance plus ∆U ≡ 24/πexp(−πG0xx) as a function of the In summary, due to topological scaling effects oscilla- smooth part Gsm of the diagonal conductance for four sam- tions of the diagonaland Hallmagnetoconductances can xx ples. The full line shows the dependence 24/πexp(−πGsxmx) existwhentherearenooscillationsofthedensityofstates following from the unified scaling theory. The dotted lines duetoLandauquantization. Theoscillationsobservedin show the result of the ”dilute instanton gas” approximation the extreme quantum limit of the applied magnetic field theory for samples 3 and 7. indisorderedGaAslayers,withthicknesslargerthanthe electrontransportmeanfreepath,fallintothiscategory. The oscillations of G are quantitatively well described conductance, taken as the midpoint value of the arrow xy by the unified scaling theory for the integer and frac- in Fig.2, is well described by the first-order electron- tional quantum Hall effect [7]. Their amplitude is much electron-interaction correction (Eq.(7)) with λ = 1.9 for smaller than the ”dilute instanton gas” approximation samples 2, 3, and 6, and λ = 2 for sample 7 in the tem- [12] predicts. peraturerangefrom0.15to 1 Kfollowedbya saturation around 4.2 K. These values are close to the theoretical We would like to thank I. S. Burmistrov for helpful upper limit λ = 2 for a system with two spins [13, 14], discussions,andN.T.Moshegov,A.I.Toropov,K.Eberl, corresponding to a negligibly small triplet part of the and B. Lemke for their help in the preparation of the electron-electron interaction. The choice of the 4.2 K samples. This work is supported by RFBR and INTAS. [1] H. Levine, S. B. Libby, and A. M. M. Pruisken, Phys. [7] B. P. Dolan, Nucl. Phys. B 554[FS], 487 (1999); Rev. Lett. 51, 1915 (1983); A. M. M. Pruisken in The cond-mat/9809294. Quantum Hall Effect, edited by R. E. Prange and S. M. [8] C.A. Lu¨tken and G.G. Ross, Phys. Rev. B 45, 11837 Girven,Springer-Verlag, 1990. (1992); 48, 2500 (1993). [2] D.E.Khmel’nitski˘ı, Pis’maZh.Eksp.Teor.Fiz.38,454 [9] C.P.BurgessandB.P.Dolan,Phys.Rev.B 63,155309 (1983) [JETP Lett. 38, 552 (1983)]; Phys. Lett. A 106, (2001). 182 (1984). [10] S. S. Murzin, M. Weiss, A. G. M. Jansen and K. Eberl, [3] Bodo Huckestein,Phys. Rev.Lett. 84, 3141 (2000). Phys. 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