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Steady States of a Microwave Irradiated Quantum Hall Gas Assa Auerbach1, Ilya Finkler2, Bertrand I. Halperin2 and Amir Yacoby3, 1Physics Department, Technion, Haifa 32000, Israel. 2Physics Laboratories, Harvard University, Cambridge, MA 02138,USA. 3Department of Physics, Weizmann Institute of Science, Rehovot 76100, Israel. (Dated: February 2, 2008) 5 0 Weconsidereffectsofalong-wavelengthdisorderpotentialontheZeroConductanceState(ZCS) 0 ofthemicrowave-irradiated 2Delectron gas. Assumingauniform Hallconductivity,weconstructa 2 Lyapunovfunctionalandderivestabilityconditionsonthedomainstructureofthephoto-generated n fields. We solve the resulting equations for a general one-dimensional and certain two-dimensional a disorderpotentials,andfindnon-zeroconductances,photo-voltages,andcirculatingdissipativecur- J rents. In contrast, weak white noise disorder does not destroy the ZCS, but induces mesoscopic 9 current fluctuations. ] PACSnumbers: 73.40.-c,05.65.+b,73.43.-f,78.67.-n l l a h The observation of giant magnetoresistance oscilla- walls are pinned to the potential extrema, which results - tions inamicrowave-irradiatedtwo-dimensionalelectron in a non zero conductance and photo-voltage. (ii) The s e gas (2DEG) [1], has spurred intensive theoretical activ- simple “egg-carton” potential solved variationally, and m ity. Two distinct microscopic mechanisms for conduc- (iii) a generic non-separable potential depicted in Fig. 1, . tivity corrections have been proposed: (i) The displace- which is solved numerically. (ii) and (iii) exhibit two- t a ment photocurrent (DP)[2], which is caused by photo- dimensional domain wall pinning and frustration effects, m excitationofelectronsinto displacedguidingcentersand which result in circulating dissipative currents. (ii) the distribution function (DF) mechanism, whichin- - d volvesredistributionofintra-Landaulevelpopulationfor n large inelastic lifetimes[3]. o c Andreev, Aleiner and Millis[4] have noted that ir- [ respective of microscopic details, once the radiation is strong enough to render the local conductivity negative, 1 v the system as a whole will break into domains of photo- 0 generated fields and spontaneous Hall currents. In their 7 proposed domain phase, motion of domain walls can ac- 1 commodatetheexternalvoltage,resultinginaZeroCon- 1 ductanceState(ZCS)intheCorbinogeometry,oraZero 0 Resistance State for the Hall bar geometry, in apparent 5 agreement with experimental reports[1]. However, one 0 / may ask, what should be the effects of long-wavelength t a (relative to the cyclotron radius) disorder, which is ei- FIG. 1: Domain phase in the presence of disorder: the pho- m ther naturally present or deliberately introduced? What togeneratedpotentialφ(x,y)correspondingtoagenericlong- - isthenatureofthecouplingbetweenadisorderpotential wavelength disorder potential φd. The steady state is deter- d andthe photogeneratedfields [5], and couldthe disorder mined by numerically minimizing the Lyapunov functional n pin domain walls? Such pinning would affect the macro- (seetext). The domain walls (potential edges) are pinnedby o scopic transport and could destroy the ZCS. φd, yielding a finiteconductance. c : In this Letter we incorporate a long-wavelength dis- v i order potential φd(x) into the non-linear magneto- Non-Linear Magneto-Transport. In the presence of an X transportequations. Weexploreitseffectsonthedomain externalmicrowavefield, we use a localrelationbetween r structureandmacroscopictransportcoefficients. Forthe the dc current j(r) and the local electrostatic field E(r): a caseofaconstantHallconductivity, weconstructa Lya- punov functional[6]whichgreatlysimplifiesthedetermi- nationofthe stablesteady statesandtheir conductance. j=jd(E,r)+σHzˆ E . (1) × We use it to derive general stability conditions on domain walls in the strong radiationregime. We also show Here we assume at the outset that the Hall conductivity thatweak’white-noise’disorderisanirrelevantperturba- σH is aconstant,independentofrandE,whichleadsto tion, which does not destroy the ZCS. It does introduce, considerable simplifications. The dissipative conductiv- however,mesoscopic non linear current fluctuations. We ity, σαdβ(E)≡∂jαd/∂Eβ, satisfies find solutions for the following disorder potentials: (i) The general one-dimensional potential, where domain σd (E,r)=σd (E,r). (2) αβ βα 2 The vector function jd, in general, will depend explic- The Domain Phase. We now consider a homogeneous itly on the position r, due, e.g., to inhomogeneities in system, in the regime of strong microwave radiation at the2DEG,anditsdirectionmaynotbeperfectlyaligned frequencies slightly larger than the cyclotron harmonics with E. Eq. (1) is supplemented by the continuityequa- ω > mω ,m = 1,2,... i.e. positive detuning. Both DP c tion, j = ρ˙, where ρ is the charge density. We and DF mechanisms produce a regime of negative con- ∇ · − emphasize that equation (1) contains all microscopic in- ductivity j(E) E<0, which implies a minimum of g(E) · teractions at lengthscales shorter than the cyclotron ra- at a finite field E = E > 0, which was estimated[7] c dius l , which serves as an ultra-violet cut-off, of order to be of order ~ω| /|(el ). In order to satisfy equipoten- c c c 1µm. tialboundaryconditions,andtheconstraints dl E=0, · Writing E φ, we may relate changes in the elec- fielddiscontinuitiesandchargedensitysingularitiesmust ≡−∇ H trostatic potential φ to changes in ρ through the inverse form. capacitance matrix W: A second order expansion of g about E reads as c δφ(r)= d2r′W(r,r′)δρ(r′). (3) g (E)=g (E )+ 1(E E )2σ +λ E2 . (7) 0 0 c c c 2 − |∇· | Z If a time-independent steady state is reached, then we The clean system of Eq.(7), is governed by a ‘Mexican have simply j = 0, and the precise form of W is hat’ Lyapunovdensity, with a flat valley along E =E , unimportant.∇ · i.e. thesteadystatelocalconductivityis‘margin|al|ly’stac- In a Corbino geometry, one specifies the potential on bleeverywhereexceptinsidethedomainwalls. Thefield- the inner and outer boundaries of the sample, and one derivative coefficient λ σ l2 implements the ultra- looksforasolutionforφ(r)consistentwiththesebound- violet cut off, introducin≈g a dcodmwain-wall thickness scale ary conditions. Since we asssume σH to be a constant, l assumed here to be of the order of l . Domain walls dw c the Hallcurrentcannotcontribute to jinthe interior yield a positive contribution to G of order σ E2l per ∇· c c dw of the sample, so it does not appear in Kirchoff’s equa- unit length. In the absence of disorder, the system will tions. Consequently,the solutionforφ(r) is independent simplyminimizetotaldomainwallscontribution,subject of σH and we may, for simplicity set σH = 0. To re- to aforementioned constraints. cover the Hall current, one inserts the solution E into Achangeintheaveragefield E ,requiredifthereisa the second term in (1). change in the applied voltage Vh, cian be accommodated Condition (2) on jd allows us to define a scalar Lya- by a motion of domain walls, or a reorientation of the punov functional as local E. The relative corrections to zero conductance vanish as l over the sample length. This defines the E(r) dw G[φ]= d2rg(E), g = dE′ jd(E′). (4) clean ZCS phase described in Ref. [4]. Z Z0 · Long-Wavelength Disorder. In an inhomogeneous system, there will be a non-zero electrostatic field, E (r) A variation of (4) is given by d ≡ φ (r), present in the thermal equilibrium state, with d −∇ nomicrowaveradiationorbiasvoltage. Wemayaskhow δG= d2r jd δφ dsnˆ jdδφ. (5) this disorder field will alter the ZCS. ∇· − · Z Zbound Atweakdisorderfield, E <<E ,theLyapunovden- d c The second integral vanishes on equipotential bound- sity near E E is modifi|ed|to c aries,orintheabsenceofexternalcurrents. Theextrema ≈ of G are found to be steady states, with j=0. Using g(E,E )=g (E) σ (E) E E (r)+ (E2), (8) ∇· d 0 − 1 · d O d the positivity of the inverse capacitance matrix W, one may show that G[φ(t)] is indeed a Lyapunov functional, which yields a current density i.e. a non-increasingfunctionoftime, so thatits minima g′ σ′E E are stable steady states. In general, G may have multi- jd(r)= σ E + 0− 1 · dE, (9) 1 d ple minima. Any initial choice of φ(r) will relax to some − E local minimum of G, but not necessarily the “ground ′ where X ∂X/∂E, and the coefficient σ depends on 1 state” with lowest G. Nevertheless, we expect that in ≡ microscopic mechanisms. the presence of noise, the system might tend to escape We wish to elaborate on a physical issue regarding from high-lying minima and wind up in a state with G Eq.(9): In the non-irradiated(dark) linear response the- close to the absolute minimum. ory, the current is driven by the electrochemical gradi- Using the boundary term in (5), the current across a ent ǫ = E E . Similarly, one may expect the pho- d Corbino sample is equal to the first derivative of G with tocurrent of−the DF mechanism to also depend on ǫ . In respect to the potential difference V between two edges, contrast, the ’upstream’ photocurrent, pumped by the andthe differentialconductanceis givenby the stiffness, DP mechanism, involves transitions between single par- or the second derivative: ticle states which feel the local electric field E(r). Thus, dI d2G due to both contributions, even if the DP mechanism = . (6) is relatively weak, E cannot be eliminated from Eq.(9) dV dV2 d 3 by a change of variables E ǫ . By our microscopic The disorder, however, will produce current fluctua- → estimate[8], for the pure DP mechanism, σ ,σ (E ) are tions across pre-existing domain walls, needed to satisfy c 1 c ′ close to the dark conductivity, and σ <<σ /E . boundary conditions on the sample. By Eq. (12), the 1 c c Localstabilityrequiresthatσd(r), ofEq.(2),hasnon- currentdensityintegratesacrossthedomainwalltoyield negative eigenvalues. The lower (transverse) eigenvalue a random number of order is given by δVφ¯ σ− = g0′ −σE1′E·Ed +O(Ed)2 ≥0 , (10) δI =±σ1Ecξd5/d2L1d/2 (14) so marginal stability occurs at E =E +σ′E E/σ . While the conductance will average out to zero at volt- c 1 d· c In a steady state, the normal current density (in di- ages V E ξ or if multiple domains are in series. The c d ≥ rection nˆ) is continuous across a domain wall. If E and randomcurrentsandconductancefluctuationsshouldbe 1 E2 are the fields on its two sides, we find by (10) and(8) observable in small samples, or as harmonic noise gener- that ation for oscillatory bias voltage. σ−(E1) E1·nˆ =σ−(E2) E2·nˆ +O(Ed2). (11) When E nˆ,i = 1,2 have opposite signs (as they do in i · the clean limit) (11) can only be satisfied for σ−(E1) = σ−(E2)= (Ed)2. This restrictsthefields at thedomain O wall to be at their respective marginally stable values. As a result, the current density (9) at the domain wall re- duces to j= σ (E )E + (E )2 (12) 1 c d d − O By Eq.(12), current conservationand Gauss’ theorem, FIG. 2: a) Photogenerated potential solutions φ, for one di- and we obtain a global condition on any closed domain mensional disorder φd(y), with zero current and zero voltage walls, boundary conditions. b) Map of ∇·E (red-positive, blue- negative) for solution φ(x,y) of Fig. 1, showing the domain 0= n j= 2πσQ2D+ (E2), (13) walls. Circulating dissipative currents are illustrated in one · − O d domain. I whereQ2D istheintegralofthe“2Ddisorder-chargeden- sity,” E /2π, over the area enclosed by the loop. d ∇· Finally, we note that generically, the differential con- One-DimensionalDisorder. Incontrasttoweakwhite- ductance of a sample in the Corbino geometry can be noise disorder, potentials with long-range correlations obtained by solving for the conductance of a linear sys- canberelevantperturbations. Considerthecaseofagen- tem with local conductivity given by σd(E(r)), in series eralone-dimensionaldisorder(seeFig.2a),whereφ (y)is d with resistive elements along the domain walls, which independent of the x-coordinate. At wavelengths larger arise from movement of the domain walls in response to than l , the Lyapunov functional is minimized if the dw a variation in the applied bias V. (There could also be system breaks up into parallel domains, so that E is ev- discontinuities in the current at discrete values of V, if erywhere aligned with E , and jd =0. These conditions d the system jumps discontinuously from one local mini- determine E(y) via (9), and the boundary conditions for mum of G to another.) We shall see that for weak long- arectangularCorbinogeometry(periodicboundarycon- wavelength disorder, the scale of the macroscopic conditions at x = 0 and x = L). The voltage difference V ductance is set by the domain-wall contribution. between the leads at y =0 and y =W , satisfies White-Noise Disorder. In the ZCS, we now show that aweak‘white-noise’disorderpotential,withacorrelation W V(I)= dy(E(y) E (y)) . (15) length ξ (of the order l l ), and root-mean-square d d dw c − valueφ¯ E ξ isanirrelev∼antperturbationwhichdoes Z0 d c d ≪ notintroducenewdomainwallsordestroytheZCS.This At strong radiation intensity and zero current, (12) and is shown by using an Imry-Ma comparison[9] of surface transversestability imposes that domain walls form pre- to bulk contributions to the Lyapunov functional. By cisely at maxima and minima of φ , given by the y ,i= (8), for a square domain of area L2, the negative contri- d i d 1,...,N To lowest order in φd, there will be a non-zero bution ofaligning E with the averageddisorderfield E , d photovoltage which only depends on these positions: scales as σ φ¯ E √L ξ . However the linear cost of its 1 d c d d − domainwallsgrowsas+σ E L l . Thereforeweakdis- c c d dw N ordercannotnecessarilybreakthesystemupintosmaller V(0)=E ( 1)NW 2 ( 1)iy + (E ), (16) c i d domains. − − − ! O i=1 X 4 where i = 1,3... are maxima. The differential conduc- find E > E , which corresponds to circulating dissipa- c tivity σ is given, to lowest order in φ , by tive currents, which match onto the tangential currents yy d at domain walls, given by Eq. 12. In Fig. 1, a generic 1 = 2Ec 1 2Ec , (17) two dimensional example is displayed. φd contains 20 σyy σ1W i |φ′d′(yi)| ≡ σ1E˜d Fourier components chosen from Gaussian distributions X with < φ (k2 > independent of k. The potential φ is d | | If P(E ) is the probability distribution for E at a ran- found by numerically minimizing G. Fig. 2b plots the d d dom point, it can be shown that E˜−1 = P(0). If 2Dchargedensitywheredomainwallsappearaslinesin- d E is taken from a Gaussian distribution, then E˜ = gularities. In both two dimensional examples, (18) and d d (2π)1/2Erms, where Erms is the root-mean-square value Fig.1, G is frustrated from perfect alignment of E and d d E (r) by the conditions E E and E = 0. This of Ed. If φd is a single sine wave, then E˜d =21/2πEdrms. frudstration underlies the|cir|c≥ulatcing dis∇sip×ative currents The transverse differential conductivity σ can also be xx which are illustrated in one of the domains in Fig. 2b. calculated, using (10), and is given, to first order in φ d by σ = σ E /E . For a Gaussian distribution, xx 1 d c h| |i one has σ = (2/π)σ , while for a single sine wave, xx yy In summary, we have introduced long wavelength dis- σ =(4/π2)σ . xx yy order into the transport theory of the microwave irradi- Two-Dimensional Potentials The simplest 2D choice atedquantumHallgas,usingtheLyapunovfunctionalas for φ is the separable ‘egg-carton’ potential, given by d an organizing principle for the stability of steady states. E0 We showed that weak white noise disorder is irrelevant φd = d (cos(x)+cos(y)) . (18) for the stability of the ZCS although it produces meso- √2 scopic current fluctuations. For a strong and long range potentials, the ZCS state breaks up into domains, which We construct a zeroth order trial solution for zero bias current jd = 0 by placing domain walls on the lines x = will generally result in a finite conductance and a pho- 0 tovoltaic effect. A microscopic theory necessary for the nπandy =mπ,forintegernandm. Thisyieldsconstant steadystatesdependenceonmicrowavepoweranddetun- electric fields in each square domain, of the form: ingfrequencyisdeferredtoaforthcomingpublication[8]. E Wehavenotconsideredeffectsofconductivityanisotropy E = c( xˆ yˆ) . (19) 0 √2 ± ± and variations in Hall coefficient. The latter will not affect the domain pattern or the longitudinal conductivity E is a gradient of a continuous potential, and satisfies forone-dimensionaldisorder,butmighthavelargeeffects 0 charge neutrality (13). Upon application of an external and be experimentally relevant in other geometries. voltageinanarbitrarydirection,domainwallswillmove, as in the one-dimensional case, and also tilt with E into a herringbone pattern. A variationalcalculation,assum- Acknowledgment. We thank E. Meron, A. Stern, R. ing that within eachdomain E is constant, finds, to first L. Willett and Y. Yacoby for helpful discussions. AA order in E , that σ = 0.83σ E0/E . Numerical cal- and AY are grateful for the hospitality of Harvard Uni- d xx 1 d c culations confirmthat the variationalsolutionis at least versity,Aspen Center for Physics and KavliInstitute for close to the exact answer. TheoreticalPhysics. Work was supported in part by the We have calculated analytically the first order (in E ) Harvard Center for Imaging and Mesoscale Structures, d correctionsto(19),forzeroexternalvoltage,byintegrat- NSF grant DMR02-33773, the US-Israel Binational Sci- ing Kirchoff’s laws[8]. 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