2DEG under microwaves: Oscillatory photoresistivity and zero-resistance states. Alexander D. Mirlin

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1 2DEG under microwaves: Oscillatory photoresistivity and zero-resistance states Alexander D. Mirlin Forschungszentrum Karlsruhe & Universität Karlsruhe, Germany I.A. Dmitriev, D.G. Polyakov (FZ Karlsruhe) M.G. Vavilov (Yale University, USA) I.L.Aleiner (Columbia University, N.Y., USA) PRL 91, (2003) PRB 70, (2004) Physica E 25, 205 (2004) PRB (Rapid Comm.) 70, (2004) PRB 71, (2005) mirlin

2 Outline: Introduction: 2DEG magnetotransport Recent experiments: Oscillatory PhotoConductivity and Zero-Resistance States Microscopic theory: OPC due to microwave-induced oscillations in the distribution function From OPC to ZRS: spontaneous domain formation Theory vs. experiment Implications for thermodynamics: local compressibility Summary & Outlook

3 2D electron gas in strong magnetic fields Standard realization: GaAs/AlGaAs heterostructures Disorder: charged donors donors n i d ~ 100 nm 2DEG n e Typical experimental parameters: n e n i (1 3) cm 2, d 100 nm k F d 10 1 weak long-range disorder high mobility τ (k F d) 2 1 τ q τ, τ q transport and single-particle relaxation times

4 Magnetotransport resistivity tensor: ρ xx = E x /j x ρ yx = E y /j x Hall resistivity B E x E y j x classically (Drude Boltzmann theory): ρ xx = m e 2 n e τ ρ yx = B n e ec independent of B ρ ρ xy xx B

5 Quantum transport in strong magnetic fields Shubnikov - de Haas Oscillations and Integer Quantum Hall Effect (IQHE) Fractional Quantum Hall Effect (FQHE) composite fermions: e + 2 flux quanta

6 Experiment: Oscillatory photoresistance Photoresistivity: DC response of a 2DEG subjected to microwave radiation Zudov, Du, Simmons, and Reno, cond-mat/ ; PRB (2001) High-mobility 2DEG µ = cm 2 /Vs Microwave radiation at ω/2π = GHz Oscillations governed by ω/

7 Experiment: Zero-resistance states Mani, Smet, von Klitzing, Narayanamurti, Johnson, Umansky, Nature (2002);... Zudov, Du, Pfeiffer, West, PRL (2003);... ultra-high-mobility 2DEG µ = cm 2 /Vs Experiment ω/ ρ sin 2πω exp( π/ τ ph )

8 Experiment: Dependence on microwave power Mani et al. Zudov et al. 100 µw 30 µw 10 µw 3 µw 1 µw Maxima: δρ P Minima: ρ 0 at sufficiently strong P

9 Experiment: Temperature dependence Mani et al. Oscillations strongly enhanced with decreasing T Minima: ρ 0 Zero-resistance regions T seemingly activated behavior e 0/T with large T 0 10 K Zudov et al.

10 Oscillatory ac conductivity of a 2DEG sufficiently strong disorder constant DOS ν 0, no LLs Drude theory: σ xx (ω) = σ + (ω) + σ (ω) σ (D) ± (ω) = e 2 ν 0 v 2 F τ tr,0 4[1 + ( ± ω) 2 τ 2 tr,0 ] only one cyclotron peak at = ω no disorder, sharp LLs Kohn theorem absorption at ω = only. weak disorder oscillatory DOS LLs mixed by disorder cyclotron resonance harmonics at = ω/n.

11 Beyond the Drude model: Return processes induce cyclotron resonance harmonics ε/ 7/2 5/2 3/2 1/2 ν(ε) Landau quantization quasiclassical memory effects interaction corrections

12 Disorder U(r): U(r)U(r ) = W ( r r ) B = 0: Quantum and transport relaxation times: τ 1 q,0 τ 1 tr,0 } = 2πν 0 dφ 2π W (2k F sin φ 2 ) { 1 (1 cos φ) White noise W (r) = 1 2πν 0 τ 0 δ(r) = τ q,0 = τ tr,0 = τ 0 Smooth disorder correlation length d k 1 F = Self-Consistent Born Approximation: τ tr,0 /τ q,0 (k F d) 2 1 (a) = + vx (b) ε + ω ε v x v x v x v x = + (c) validity conditions: d v F τ q,0, d λ B = ( c eb ) 1/2

13 σ(ω) = ± Oscillatory ac conductivity of a 2DEG e 2 v 2 F 4ω ν(ε) DOS, τ tr (ε) = τ tr,0 ν 0 ν(ε) Overlapping Landau levels: dε (f ε f ε+ω ) ν(ε) τtr 1 (ε + ω) (ω ± ) [ 2 τ 2 tr (ε) + τtr 2 (ε + ω) ] transport time, depends on ε and B due to ν(ε) LL width > DOS ν = ν 0 [ 1 2δ cos 2πε/ ] δ = exp( π/ τ q ) 1 [ ] Order δ 1 : σ(ω) = σ D (ω) 1 4δ cos(2πε F / ) sin(2πω/) 2πω/ ω = 0 SdH oscillations T > 2π 2 suppressed e 2π2 T / Order δ 2 : survives at high T! σ(ω) σ D (ω) = [ e 2π/τ q cos(2πω/ ) ] σ xx (ω) / σ xx (0) σ xx ( ) / σ xx ( =0) /3 1/2 / ω ω / τ q = π τ q = 10 1

14 Oscillatory photoconductivity: Two contributions n+3 Effect of microwaves on the collision integral Ryzhii 70 Ryzhii, Suris, Shchamkhalova 86 Durst, Sachdev, Read, Girvin 03 Vavilov, Aleiner 04 x + x ω n+2 n+1 n ε n = n + ee dc x Period and phase of oscillations in agreement with experiments but amplitude T -independent strong disagreement V Parametrically larger contribution: Effect of microwaves on the distribution function

15 Photoconductivity theory: distribution function oscillations microwave absorption = change of distribution function f ε σ(ω) ω 2 M(ω) 2 dε ω (f ε f ε+ω )ν(ε)ν(ε + ω) M(ω) 2 = σ(d) (ω) ω 2 ν 2 0 δf ε = 1 2 M(ω) 2 [ (f ε ω f ε )ν ε ω (f ε f ε+ω )ν ε+ω ] E 2 ω τ in τ in - inelastic relaxation time DOS ν(ε) = ν 0 [ 1 2δ cos 2πε/ ] δ = exp( π/ τ q ) 1 δf ε = δ τ in E 2 ω 2πω sin sin 2πε ν 0 ω M(ω) 2 ( ) ft ε Microwave-induced oscillatory δf ε

16 δf ε = Photoconductivity σ ph σ dc = σ D dc dε ( δf ) ε ν 2 (ε) ε ν0 2 = = 4σ D dc δ2 P ω πω sin 2πω P ω = 2σD (ω) ω 2 ν 0 E 2 ω τ in dimensionless microwave power In contrast to the SdHO, oscillations in photoconductivity are insensitive to temperature smearing and inhomogeneous broadening of LL! Up to now linear order in P ω. Need non-linear treatment Can the effect become large?

17 Non-linear photoconductivity: Kinetic equation Quantum kinetic equation for the distribution function f(ε): E 2 ω σ D (ω) 2ω 2 ν 2 0 ± ν(ε±ω)[f(ε±ω) f(ε)] + E2 dc σd dc ν 2 0 ν(ε) ε [ν 2 (ε) ε f(ε) ] = f(ε) f T (ε) τ in Dimensionless units for the strength of ac and dc fields: P ω = τ ( ) in eeω v 2 F ωc 2 + ω2 τ tr ω (ω 2 ωc 2)2, Q dc = 2 τ ( ) in eedc v 2 ( F π τ tr ) 2 Overlapping LLs δ = e π/τ q 1 f(ε) = f T (ε) + f osc (ε) + O(δ 2 ), look for a solution in the form ] f osc (ε) δ Re [f 1 (ε) e i2πε ωc. oscillations in non-equilibrium distribution function: f osc (ε) = δ 2π f T ε oscillatory photoconductivity 2πε sin 2πω P ω sin 2πω 1 + P ω sin 2 πω + 4Q dc + Q dc

18 Non-linear photoconductivity: Results σ ph σ D dc Linear response: Q dc 0 = 1 + 2δ 2 [ 1 P ω 2πω sin 2πω 1 + P ω sin 2 πω not too strong P ω linear-in-p orrection σ ph strong P ω saturation: = 1 8δ 2 πω ω σ c dc despite δ 2 1, correction large near ω = k ] + 4Q dc + Q dc cot πω, P ω sin 2 πω 1 σ ph < 0 around minima for P ω > P ω = ( 4δ 2 πω sin 2πω 8 6 ) 1 sin 2 πω threshold power ρ ph /ρ D dc ρ ph /ρ (D) dc vs /ω ωτ q, 0 = 2π /5 1/4 1/3 1/2 / ω P (0) ω P ω ( = 0) = {0.24, 0.8, 2.4}

19 I-V characteristics and Zero-Resistance States Current-voltage characteristics at the minima: Linear response σ ph negative instability domains zero-resistance state spontaneous field E dc in the domains: determined from σ ph(e dc ) = 0 ee dc = πr c ( τtr 2τ in ) [ 1/2 (Eω ) ] 2 1/2 1 E dc 1 V/cm E ω can be measured by a local probe: Willett, Pfeiffer, West, PRL 2004

20 Stability conditions and Zero-Resistance States n t = j = 0 E = φ = Un j = ˆσ(E)E U Coulomb interaction continuity Poisson Evolution equation for fluctuations δn(r, t): [ t δn = ˆσ + E dˆσ ] E E Uδn de E 2 Stability conditions: Re (eigenvalues) > 0 Andreev, Aleiner, Millis, PRL 2003 σ xx = j x 0 and σ xx + E dσ xx E x de = j x E x 0

21 Cyclotron resonance broadening P ω [(ω ) 2 + τ 2 ] 1 n = 1 peak (CR) ( τ ) times higher and narrower? No! Reflection from 2DEG width Γ e2 c E F Also: heating; saturation of δf due to 0 f 1 ω = 50 GHz T = 0.8 K τ q = s ρ ph /ρ (D) dc 5 P ω (0) = / ω

22 τ q π Photoresistivity: Separated LLs ( 2ωc 1 separated LLs, width 2Γ = 2 πτ q ) 1/2 Linear response (Q dc 0) photoconductivity: σ ph 16ω [ c ω ( ) ] ω nωc 1 P 3π 2 ω Φ Γ Γ 2 Γ σ D dc Φ(x) = 3x [ 4π Re arccos( x 1) 1 x ] x (2 x ) 3 n ρ pc /ρ (D) dc /4 1/3 1/2 / ω Gaps in absorption spectrum shoulders method to measure τ q Linear-response σ ph < 0 around minima ZRS strong E dc : σ ph (E dc ) positive only if inter-ll elastic impurity scattering efficient ( ) 1/2 stronger spontaneous field Edc τtr ωc 2 τ q ev F

23 Shubnikov - de Haas oscillations Lower T and/or stronger B 2π 2 T / 1 SdHO superimposed on microwave-induced oscillations ρ ph /ρ (D) dc 5 ω = 120 GHz T = 0.8 K τ q = s P ω (0) = 0.05 Numerical solution of the kinetic equation for arbitrary T, P ω, and τ q / ω Suppression of the SdHO amplitude by microwaves Well separated LL gaps for microwave absorption regions where oscillations unaffected by microwaves. Experiment: Dorozhkin et al

24 Temperature dependence: Inelastic relaxation time For not too strong microwave power: Dominant relaxation mechanisms: σ ph σ dc P ω τ in smooth part of the distriubution function f T (ε) phonons oscillatory part f osc (ε) e-e collisions 1 τ ee = πt 2 4ɛ F ln ɛ F max [ T, ( τ tr ) 1/2] overlapping LLs 1 τ ee Γ T 2 ɛ F ln ɛ F max [T, Γ ( τ tr ) 1/2] separated LLs = σ ph σ dc T 2

25 Two contributions to photoconductivity σ (1) ph : influence of microwave on impurity scattering, T independent, polarization-dependent σ (2) ph : related to the change of the distribution function = τ in, strongly T dependent, independent of direction of linear polarization σ (2) ph σ (1) ph τ in τ q 1 for relevant T Order-of-magnitude estimate of energy scales: E F 100 K ω 1 K τ 1 q τ 1 tr 1 K 10 mk T 1 K τ 1 in T 2 E F 10 mk

26 ω Theory vs experiment 2π GHz, τ q 10 ps ωτ q 2π (overlapping LLs) ρ ph /ρ D µw 30 µw 10 µw 3 µw 1 µw /51/4 1/3 1/2 / ω Zudov et al. Period, phase, shape OK T dependence at maximum τ in T 2 OK Mani et al Studenikin et al.

27 Low-B damping σ ph exp ( π/ τ ph ) Theory: τ ph /τ q = 0.5 Experiment: 1.6 (Mani et al.), 5.2 (Zudov et al.) experiment underestimates τ q? yes τ q disorder-induced damping of SdHO: 60 Uncertainty while extrapolating T 0; Both homogeneous and inhomogeneous broadening of LL are included. Γ M, Γ SdH (µev) Γ MIRO Γ SdH k B T (mev) Studenikin et al., cond-mat/ Photoconductivity oscillations: not sensitive to inhomogeneous broadening and thermal smearing good method to measure τ q

28 Implications for thermodynamics: Compressibility non-equilibrium oscillatory distribution function oscillatory contribution to local compressibility χ = n q / (eφ q ) = ν(ε) f(ε) dε wave vector q l 1 in ε δχ = δ P 2 χ 0 2πω sin 2πω 1 + P sin 2 πω + 4Q + Q δχ/ν δχ ν E dc /Edc 1 ZRS PSfrag replacements Eq. (8), Q = 0 Eq. (9) /ω Local compressibility measurements real space snapshot of the ZRS domain structure σ ph

29 Experiment in progress: Local potential probe Courtesy of A. Yacoby (Weizmann Inst., Rehovot) and J. Smet (MPI Stuttgart)

30 Recent extensions of the theory propagation of surface acoustic waves Robinson, Kennett, Cooper, Falko laterally modulated structures Dietel, Glazman, Hekking, von Oppen effect of long-range disorder on the ZRS Auerbach, Finkler, Halperin, Yacoby nature of phase transition into ZRS Alicea, Balents, M.P.A.Fisher, Paramekanti, Radzihovsky strong magnetic field: > ω: strong suppression of resistivity, ρ ph ρ dark experim. observed and explained by the theory (intra-ll transitions) Dorozhkin, Smet, Umansky, von Klitzing

31 Summary Microwaves magnetooscillations in the 2DEG conductivity negative linear-response conductivity instability domains zero-resistance states Two contributions to OPC: effect of radiation on (i) scattering off impurities and (ii) electron distribution function Parametrically larger contribution: Microwave-induced non-equilibrium distribution function. Magnitude of the effect proportional to τ in T 2 Implications for thermodynamics: oscillatory compressibility Future research (experimental and theoretical): detailed study of the domain physics effect of finite T on ZRS (experiment: activation?) noise in ZRS experiments at higher ω (ZRS in the regime of separated LL) spin-related effects

32 Acknowledgment Experimental data & discussions: J. Smet, K. von Klitzing (Stuttgart MPI), R. Mani (Harvard), M. Zudov, R. Du (Utah University), A. Yacoby (WIS, Rehovot), S. Studenikin (Ottava), S. Dorozhkin (Chernogolovka) Financial support: SPP Quanten-Hall-Systeme of the DFG