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Microwave induced magnetoresistance oscillations and inelastic scattering time in double quantum wells
ARTICLE IN PRESS Physica E 42 (2010) 1075–1077
Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Microwave induced magnetoresistance oscillations and inelastic scattering time in double quantum wells S. Wiedmann a,b,, G.M. Gusev c, O.E. Raichev d, A.K. Bakarov c,1, J.C. Portal a,e a
LNCMI-CNRS, UPR 3228, BP 166, 38042 Grenoble Cedex 9, France INSA Toulouse, 31077 Toulouse Cedex 4, France c ~ Paulo, SP, Brazil Instituto de Fı´sica da Universidade de Sa~ o Paulo, CP 66318 CEP 05315-970, Sao d Institute of Semiconductor Physics, NAS of Ukraine, Prospekt Nauki 45, 03028 Kiev, Ukraine e Institut Universitaire de France, 75005 Paris, France b
a r t i c l e in fo
abstract
Article history: Received 26 August 2009 Accepted 23 November 2009 Available online 1 December 2009
The interference of microwave-induced resistance oscillations and magneto-intersubband oscillations in double quantum wells exposed to a continuous microwave irradiation is under study. By comparing experimental and theoretical magnetoresistance traces at different temperatures, we confirm that the inelastic mechanism of photoresistance explains our observations up to T C 4 K. For higher temperatures, our results suggest a deviation of the inelastic scattering time tin from the predicted T 2 dependence. & 2009 Elsevier B.V. All rights reserved.
Keywords: Double quantum wells Microwave induced resistance oscillations Inelastic scattering time
1. Introduction Resonant transitions of two-dimensional (2D) electrons between Landau levels lead to magnetoresistance oscillations which, unlike the Shubnikov–de Haas (SdH) oscillations, survive at high temperatures and can provide useful information about scattering mechanisms. Magnetotransport measurements in double quantum wells (DQWs) reveal the magneto-intersubband (MIS) oscillations owing to the possibility of electron transitions between Landau levels belonging to different subbands [1]. In 2D systems with one occupied subband, resonant transitions between Landau levels occur in the presence of a continuous microwave (MW) irradiation. This leads to the microwaveinduced resistance oscillations (MIROs) [2] and ‘‘zero-resistance states’’ [3] whose period is determined by the ratio of radiation frequency o to the cyclotron frequency oc ¼ eB=m , where m is the effective mass of electrons. For high microwave intensity, the resistance has resonant features associated with the fractional ratios e ¼ o=oc ¼ n=m, [4] which have been investigated up to denominator 4 [5] and recently up to denominator 8 in a 2D system with moderate mobility [6]. The DQWs exposed to MW irradiation show a peculiar magnetoresistance which can be explained by an interference between MIS oscillations and MIROs [7]. The microscopic mechanism describing the MIROs at low
Corresponding author at: LNCMI-CNRS, UPR 3228, BP 166, 38042 Grenoble Cedex 9, France. E-mail address: [email protected] (S. Wiedmann). 1 Present address: Institute of Semiconductor Physics, Novosibirsk 630090, Russia.
1386-9477/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2009.11.077
temperatures T is associated with a MW-generated non-equilibrium oscillatory component of the electron distribution function [8]. The corresponding contribution to magnetoresistance is proportional to the inelastic scattering time tin due to electron– electron scattering and decreases with increasing temperature according to tin pT 2 [8]. The theory based on this (so-called ‘‘inelastic’’) mechanism has been generalized to the two-subband case [6] and it satisfactory describes the magnetoresistance in DQWs exposed to microwave irradiation. The interference pattern of magnetoresistance oscillations appears because the photoinduced part of the electron distribution, which oscillates as a function of MW frequency, is modified owing to subband coupling and becomes also an oscillating function of the subband separation [7]. In this paper, we study electron transport in DQWs exposed to a continuous microwave irradiation with a focus on temperature dependence of magnetoresistance oscillations.
2. Experimental basics We have performed temperature-dependent measurements in balanced GaAs DQWs with a well width of 14 nm and a barrier width of db ¼ 1:4 nm in the presence of microwave irradiation (35–170 GHz). The samples have high mobility m C 106 cm2 =V s and high total sheet electron density ns C1012 cm2 . They are mounted in a VTI cryostat with a waveguide to deliver MW radiation down to the sample. The resistance R ¼ Rxx was measured by using a standard low-frequency lock-in technique (13 Hz) under continuous MW illumination. The subband separation D ¼ 3:05 meV is extracted from the MIS oscillation period.
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S. Wiedmann et al. / Physica E 42 (2010) 1075–1077
First of all, we present in Fig. 1 the effect of MW irradiation (f ¼ 170 GHz) on a quantum well (QW) and a DQW with comparable density and mobility at a lattice temperature of 1.4 K. Fig. 1(a) shows SdH oscillations starting at 0.4 T. If the QW is exposed to microwave irradiation, here 170 GHz, MIROs and cyclotron resonance (CR at B ¼ 0:4 T) are visible. In DQWs, we observe MIS oscillations superimposed by low-field SdH oscillations in the absence of microwaves [see Fig. 1(b)]. With MW irradiation, also 170 GHz, a strongly modified MIS oscillation picture occurs. MIS peaks at 0.25, 0.5 and 0.65 T are enhanced for f ¼ 170 GHz, whereas MIS peaks at 0.4 T are damped. For B ¼ 0:3 T, the MIS peaks inverse sign. In Fig. 2 we present the dependence on MW frequency. We observe a strongly frequency-dependent oscillation picture where some MIS peaks or even groups of MIS peaks are enhanced, damped, or inverted (peak flip). For 35 GHz, all MIS peaks are inverted, except the features around 0.15 T.
3. Analysis
Fig. 1. (a) QW with and without a continuous MW irradiation. MIROs occur if the sample is exposed to MW irradiation. (b) MIS oscillations superimposed by lowfield SdH oscillations and interference of MIS and MIROs if the DQW is exposed to MW irradiation.
For temperature dependent measurements we have chosen the frequency range between 55 and 140 GHz and we focus here on measurements with a fixed frequency (85 GHz) at a constant MW electric field Eo . Power dependent measurements show that the enhanced MIS amplitudes are saturated at a microwave power Ps , in accordance with the theory [8]. We use low MW power P oPs to ensure that we are far away from the saturated regime. The estimated electric field is Eo C 2:0 V=cm. The theoretical model (see Ref. [7]) gives the dc resistivity X 2pðeF ej Þ rd ð2p2 Te =‘ oc Þ C 12D cos 2 r0 ‘ oc sinhð2p Te =‘ oc Þ j ¼ 1;2 2pD ; þD2 ð1Ao Þ 1 þcos
ð1Þ
‘ oc
where r0 is the zero-field resistivity, Te the electron temperature, 2 D ¼ expðp=oc tq Þ the Dingle factor, eF ¼ ‘ pns =2m the Fermi energy, and D ¼ e2 e1 the subband separation. The second term, proportional to the temperature damping factor, describes the SdH oscillations. The third term, quadratic in the Dingle factor, describes the MIS oscillations. Their modification by the radiation is given by the dimensionless factor Ao : Ao ¼
Fig. 2. Frequency dependence of magnetoresistance in DQWs under a continuous MW irradiation with a constant electric field of E ¼ 2:5 V=cm. The MIS peaks are enhanced, damped, or inverted. (The curves are shifted up for clarity, except the one for 35 GHz.)
Po ð2po=oc Þsinð2po=oc Þ 1 þPo sin2 ðpo=oc Þ
;
ð2Þ
where Po is proportional to the MW power and to the ratio of the inelastic scattering time tin to the transport time ttr . For linear MW polarization, and away from the cyclotron resonance, t eEo vF 2 o2c þ o2 Po ¼ in ; ð3Þ ttr ‘ o ðo2 o2c Þ2 pffiffiffiffiffiffiffiffi where vF ¼ ‘ pns =m is the Fermi velocity. The approximation of subband-independent densities and scattering times, which leads to the simple description given above, is reasonable under the condition D 5 eF . Fig. 3 presents the analysis of experimental data for different temperatures up to 8 K. The temperature-dependent quantum lifetime tq and transport time ttr are extracted from dark magnetoresistance measurements if Te CT. The comparison of experimental magnetoresistance to Eq. (1) at constant o and Eo shows that the theoretical temperature dependence of inelastic scattering time tin pTe2 fits well up to 3.5 K. Note that the experimental traces are superimposed by MIRO features n ¼ 1 and 2. Assuming that tin ¼ ‘ eF =lTe2 [8], where l is a numerical constant of order unity, we can find l C 1. Note that the
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the heating effect is negligible, Te C T. We have also performed temperature dependent measurements for lower MW power and the experimental data are again in perfect agreement with the inelastic model for 1:8 K oTe o2:8 K. A closer look to, e.g. Fig. 3(c) at T ¼ 6:0 K for 0:1 T o B o0:2 T, shows that the inverted MIS oscillations are not resolved in the theoretical plot and we have to introduce an enhanced inelastic scattering time tin with tin ¼ 3:5tin to fit the results. We have performed these temperature-dependent measurements for frequencies between 55 and 140 GHz (not shown here) and obtained the same results. Finally, Fig. 4 presents the extracted (enhanced) inelastic scattering time as a function of the electron temperature Te . The deviation starts at Tc ¼ 4 K, and for Tc 44 K we obtain a nearly constant tin C 190 ps. Similar saturation of tin with increasing temperature at T C 2 K has been recently observed [9] in onesubband system and attributed to a crossover to the displacement mechanism [10,11] of photoresistance. This explanation is not applicable to our sample, since the quantum lifetime, according to our estimates, is still much smaller than tin at Te 4 K. In summary, we have measured temperature dependence of magnetoresistance of double quantum wells at a constant frequency and weak MW intensity. By comparing the experimental data with the theory based on inelastic mechanism of MW photoresistance, we conclude that either this mechanism fails to describe the obtained oscillation picture at Te 44 K or temperature dependence of inelastic scattering time considerably deviates from the tin pTe2 law predicted theoretically. We believe that further theoretical studies of MW photoresistance, possibly involving new mechanisms, are necessary to explain our data.
Fig. 3. Temperature dependence at 85 GHz up to 8.0 K. Experimental traces (lower) are fitted to the theoretical model (upper). With increasing temperature, an enhanced inelastic scattering time tin (middle) is necessary to obtain experimental traces.
Acknowledgements This work was supported by LNCMI-CNRS, CNPq and FAPESP (Brasilian agencies), USP-COFECUB (Uc Ph 109/08) and microwave sources from the ANR project MICONANO. References
Fig. 4. Inelastic scattering time tin as a function of electron temperature Te . Deviation from inelastic mechanism occurs for Te 4 4 K, where T-dependence of tin is saturated.
electron temperature Te is at least 2.8 K (for this chosen MW electric field) because of heating of electrons due to microwave irradiation (see also Fig. 4). For the lattice temperatures T 4 3 K
[1] N.C. Mamani, G.M. Gusev, T.E. Lamas, A.K. Bakarov, O.E. Raichev, Phys. Rev. B 77 (2008) 205327. [2] M.A. Zudov, R.R. Du, J.A. Simmons, J.L. Reno, Phys. Rev. B 64 (2001) 201311(R). [3] R.G. Mani, J.H. Smet, K. von Klitzing, V. Narayanamurti, W.B. Johnson, V. Umansky, Nature 420 (2002) 646; M.A. Zudov, R.R. Du, L.N. Pfeiffer, K.W. West, Phys. Rev. Lett. 90 (2003) 046807. [4] M.A. Zudov, R.R. Du, L.N. Pfeiffer, K.W. West, Phys. Rev. B 73 (2006) 041301(R); R.G. Mani, J.H. Smet, K. von Klitzing, V. Narayanamurti, W.B. Johnson, V. Umansky, Phys. Rev. Lett. 92 (2004) 146801. [5] S.I. Dorozhkin, J.H. Smet, K. von Klitzing, L.N. Pfeiffer, K.W. West, JETP Lett. 86 (2007) 543; M.A. Zudov, R.R. Du, L.N. Pfeiffer, K.W. West, Phys. Rev. B 73 (2006) 041301(R). [6] S. Wiedmann, G.M. Gusev, O.E. Raichev, A.K. Bakarov, J.C. Portal, Phys. Rev. B 80 (2009) 035317. [7] S. Wiedmann, G.M. Gusev, O.E. Raichev, T.E. Lamas, A.K. Bakarov, J.C. Portal, Phys. Rev. B 78 (2008) 121301(R). [8] I.A. Dmitriev, M.G. Vavilov, I.L. Aleiner, A.D. Mirlin, D.G. Polyakov, Phys. Rev. B 71 (2005) 115316. [9] A.T. Hatke, M.A. Zudov, L.N. Pfeiffer, K.W. West, Phys. Rev. Lett. 102 (2009) 066804. [10] V.I. Ryzhii, Sov. Phys. Solid State 11 (1970) 2078; V.I. Ryzhii, R.A. Suris, B.S. Shchamkhalova, Sov. Phys. Semicond. 20 (1986) 1299. [11] A.C. Durst, S. Sachdev, N. Read, S.M. Girvin, Phys. Rev. Lett. 91 (2003) 086803.
Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Microwave induced magnetoresistance oscillations and inelastic scattering time in double quantum wells S. Wiedmann a,b,, G.M. Gusev c, O.E. Raichev d, A.K. Bakarov c,1, J.C. Portal a,e a
LNCMI-CNRS, UPR 3228, BP 166, 38042 Grenoble Cedex 9, France INSA Toulouse, 31077 Toulouse Cedex 4, France c ~ Paulo, SP, Brazil Instituto de Fı´sica da Universidade de Sa~ o Paulo, CP 66318 CEP 05315-970, Sao d Institute of Semiconductor Physics, NAS of Ukraine, Prospekt Nauki 45, 03028 Kiev, Ukraine e Institut Universitaire de France, 75005 Paris, France b
a r t i c l e in fo
abstract
Article history: Received 26 August 2009 Accepted 23 November 2009 Available online 1 December 2009
The interference of microwave-induced resistance oscillations and magneto-intersubband oscillations in double quantum wells exposed to a continuous microwave irradiation is under study. By comparing experimental and theoretical magnetoresistance traces at different temperatures, we confirm that the inelastic mechanism of photoresistance explains our observations up to T C 4 K. For higher temperatures, our results suggest a deviation of the inelastic scattering time tin from the predicted T 2 dependence. & 2009 Elsevier B.V. All rights reserved.
Keywords: Double quantum wells Microwave induced resistance oscillations Inelastic scattering time
1. Introduction Resonant transitions of two-dimensional (2D) electrons between Landau levels lead to magnetoresistance oscillations which, unlike the Shubnikov–de Haas (SdH) oscillations, survive at high temperatures and can provide useful information about scattering mechanisms. Magnetotransport measurements in double quantum wells (DQWs) reveal the magneto-intersubband (MIS) oscillations owing to the possibility of electron transitions between Landau levels belonging to different subbands [1]. In 2D systems with one occupied subband, resonant transitions between Landau levels occur in the presence of a continuous microwave (MW) irradiation. This leads to the microwaveinduced resistance oscillations (MIROs) [2] and ‘‘zero-resistance states’’ [3] whose period is determined by the ratio of radiation frequency o to the cyclotron frequency oc ¼ eB=m , where m is the effective mass of electrons. For high microwave intensity, the resistance has resonant features associated with the fractional ratios e ¼ o=oc ¼ n=m, [4] which have been investigated up to denominator 4 [5] and recently up to denominator 8 in a 2D system with moderate mobility [6]. The DQWs exposed to MW irradiation show a peculiar magnetoresistance which can be explained by an interference between MIS oscillations and MIROs [7]. The microscopic mechanism describing the MIROs at low
Corresponding author at: LNCMI-CNRS, UPR 3228, BP 166, 38042 Grenoble Cedex 9, France. E-mail address: [email protected] (S. Wiedmann). 1 Present address: Institute of Semiconductor Physics, Novosibirsk 630090, Russia.
1386-9477/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2009.11.077
temperatures T is associated with a MW-generated non-equilibrium oscillatory component of the electron distribution function [8]. The corresponding contribution to magnetoresistance is proportional to the inelastic scattering time tin due to electron– electron scattering and decreases with increasing temperature according to tin pT 2 [8]. The theory based on this (so-called ‘‘inelastic’’) mechanism has been generalized to the two-subband case [6] and it satisfactory describes the magnetoresistance in DQWs exposed to microwave irradiation. The interference pattern of magnetoresistance oscillations appears because the photoinduced part of the electron distribution, which oscillates as a function of MW frequency, is modified owing to subband coupling and becomes also an oscillating function of the subband separation [7]. In this paper, we study electron transport in DQWs exposed to a continuous microwave irradiation with a focus on temperature dependence of magnetoresistance oscillations.
2. Experimental basics We have performed temperature-dependent measurements in balanced GaAs DQWs with a well width of 14 nm and a barrier width of db ¼ 1:4 nm in the presence of microwave irradiation (35–170 GHz). The samples have high mobility m C 106 cm2 =V s and high total sheet electron density ns C1012 cm2 . They are mounted in a VTI cryostat with a waveguide to deliver MW radiation down to the sample. The resistance R ¼ Rxx was measured by using a standard low-frequency lock-in technique (13 Hz) under continuous MW illumination. The subband separation D ¼ 3:05 meV is extracted from the MIS oscillation period.
ARTICLE IN PRESS 1076
S. Wiedmann et al. / Physica E 42 (2010) 1075–1077
First of all, we present in Fig. 1 the effect of MW irradiation (f ¼ 170 GHz) on a quantum well (QW) and a DQW with comparable density and mobility at a lattice temperature of 1.4 K. Fig. 1(a) shows SdH oscillations starting at 0.4 T. If the QW is exposed to microwave irradiation, here 170 GHz, MIROs and cyclotron resonance (CR at B ¼ 0:4 T) are visible. In DQWs, we observe MIS oscillations superimposed by low-field SdH oscillations in the absence of microwaves [see Fig. 1(b)]. With MW irradiation, also 170 GHz, a strongly modified MIS oscillation picture occurs. MIS peaks at 0.25, 0.5 and 0.65 T are enhanced for f ¼ 170 GHz, whereas MIS peaks at 0.4 T are damped. For B ¼ 0:3 T, the MIS peaks inverse sign. In Fig. 2 we present the dependence on MW frequency. We observe a strongly frequency-dependent oscillation picture where some MIS peaks or even groups of MIS peaks are enhanced, damped, or inverted (peak flip). For 35 GHz, all MIS peaks are inverted, except the features around 0.15 T.
3. Analysis
Fig. 1. (a) QW with and without a continuous MW irradiation. MIROs occur if the sample is exposed to MW irradiation. (b) MIS oscillations superimposed by lowfield SdH oscillations and interference of MIS and MIROs if the DQW is exposed to MW irradiation.
For temperature dependent measurements we have chosen the frequency range between 55 and 140 GHz and we focus here on measurements with a fixed frequency (85 GHz) at a constant MW electric field Eo . Power dependent measurements show that the enhanced MIS amplitudes are saturated at a microwave power Ps , in accordance with the theory [8]. We use low MW power P oPs to ensure that we are far away from the saturated regime. The estimated electric field is Eo C 2:0 V=cm. The theoretical model (see Ref. [7]) gives the dc resistivity X 2pðeF ej Þ rd ð2p2 Te =‘ oc Þ C 12D cos 2 r0 ‘ oc sinhð2p Te =‘ oc Þ j ¼ 1;2 2pD ; þD2 ð1Ao Þ 1 þcos
ð1Þ
‘ oc
where r0 is the zero-field resistivity, Te the electron temperature, 2 D ¼ expðp=oc tq Þ the Dingle factor, eF ¼ ‘ pns =2m the Fermi energy, and D ¼ e2 e1 the subband separation. The second term, proportional to the temperature damping factor, describes the SdH oscillations. The third term, quadratic in the Dingle factor, describes the MIS oscillations. Their modification by the radiation is given by the dimensionless factor Ao : Ao ¼
Fig. 2. Frequency dependence of magnetoresistance in DQWs under a continuous MW irradiation with a constant electric field of E ¼ 2:5 V=cm. The MIS peaks are enhanced, damped, or inverted. (The curves are shifted up for clarity, except the one for 35 GHz.)
Po ð2po=oc Þsinð2po=oc Þ 1 þPo sin2 ðpo=oc Þ
;
ð2Þ
where Po is proportional to the MW power and to the ratio of the inelastic scattering time tin to the transport time ttr . For linear MW polarization, and away from the cyclotron resonance, t eEo vF 2 o2c þ o2 Po ¼ in ; ð3Þ ttr ‘ o ðo2 o2c Þ2 pffiffiffiffiffiffiffiffi where vF ¼ ‘ pns =m is the Fermi velocity. The approximation of subband-independent densities and scattering times, which leads to the simple description given above, is reasonable under the condition D 5 eF . Fig. 3 presents the analysis of experimental data for different temperatures up to 8 K. The temperature-dependent quantum lifetime tq and transport time ttr are extracted from dark magnetoresistance measurements if Te CT. The comparison of experimental magnetoresistance to Eq. (1) at constant o and Eo shows that the theoretical temperature dependence of inelastic scattering time tin pTe2 fits well up to 3.5 K. Note that the experimental traces are superimposed by MIRO features n ¼ 1 and 2. Assuming that tin ¼ ‘ eF =lTe2 [8], where l is a numerical constant of order unity, we can find l C 1. Note that the
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the heating effect is negligible, Te C T. We have also performed temperature dependent measurements for lower MW power and the experimental data are again in perfect agreement with the inelastic model for 1:8 K oTe o2:8 K. A closer look to, e.g. Fig. 3(c) at T ¼ 6:0 K for 0:1 T o B o0:2 T, shows that the inverted MIS oscillations are not resolved in the theoretical plot and we have to introduce an enhanced inelastic scattering time tin with tin ¼ 3:5tin to fit the results. We have performed these temperature-dependent measurements for frequencies between 55 and 140 GHz (not shown here) and obtained the same results. Finally, Fig. 4 presents the extracted (enhanced) inelastic scattering time as a function of the electron temperature Te . The deviation starts at Tc ¼ 4 K, and for Tc 44 K we obtain a nearly constant tin C 190 ps. Similar saturation of tin with increasing temperature at T C 2 K has been recently observed [9] in onesubband system and attributed to a crossover to the displacement mechanism [10,11] of photoresistance. This explanation is not applicable to our sample, since the quantum lifetime, according to our estimates, is still much smaller than tin at Te 4 K. In summary, we have measured temperature dependence of magnetoresistance of double quantum wells at a constant frequency and weak MW intensity. By comparing the experimental data with the theory based on inelastic mechanism of MW photoresistance, we conclude that either this mechanism fails to describe the obtained oscillation picture at Te 44 K or temperature dependence of inelastic scattering time considerably deviates from the tin pTe2 law predicted theoretically. We believe that further theoretical studies of MW photoresistance, possibly involving new mechanisms, are necessary to explain our data.
Fig. 3. Temperature dependence at 85 GHz up to 8.0 K. Experimental traces (lower) are fitted to the theoretical model (upper). With increasing temperature, an enhanced inelastic scattering time tin (middle) is necessary to obtain experimental traces.
Acknowledgements This work was supported by LNCMI-CNRS, CNPq and FAPESP (Brasilian agencies), USP-COFECUB (Uc Ph 109/08) and microwave sources from the ANR project MICONANO. References
Fig. 4. Inelastic scattering time tin as a function of electron temperature Te . Deviation from inelastic mechanism occurs for Te 4 4 K, where T-dependence of tin is saturated.
electron temperature Te is at least 2.8 K (for this chosen MW electric field) because of heating of electrons due to microwave irradiation (see also Fig. 4). For the lattice temperatures T 4 3 K
[1] N.C. Mamani, G.M. Gusev, T.E. Lamas, A.K. Bakarov, O.E. Raichev, Phys. Rev. B 77 (2008) 205327. [2] M.A. Zudov, R.R. Du, J.A. Simmons, J.L. Reno, Phys. Rev. B 64 (2001) 201311(R). [3] R.G. Mani, J.H. Smet, K. von Klitzing, V. Narayanamurti, W.B. Johnson, V. Umansky, Nature 420 (2002) 646; M.A. Zudov, R.R. Du, L.N. Pfeiffer, K.W. West, Phys. Rev. Lett. 90 (2003) 046807. [4] M.A. Zudov, R.R. Du, L.N. Pfeiffer, K.W. West, Phys. Rev. B 73 (2006) 041301(R); R.G. Mani, J.H. Smet, K. von Klitzing, V. Narayanamurti, W.B. Johnson, V. Umansky, Phys. Rev. Lett. 92 (2004) 146801. [5] S.I. Dorozhkin, J.H. Smet, K. von Klitzing, L.N. Pfeiffer, K.W. West, JETP Lett. 86 (2007) 543; M.A. Zudov, R.R. Du, L.N. Pfeiffer, K.W. West, Phys. Rev. B 73 (2006) 041301(R). [6] S. Wiedmann, G.M. Gusev, O.E. Raichev, A.K. Bakarov, J.C. Portal, Phys. Rev. B 80 (2009) 035317. [7] S. Wiedmann, G.M. Gusev, O.E. Raichev, T.E. Lamas, A.K. Bakarov, J.C. Portal, Phys. Rev. B 78 (2008) 121301(R). [8] I.A. Dmitriev, M.G. Vavilov, I.L. Aleiner, A.D. Mirlin, D.G. Polyakov, Phys. Rev. B 71 (2005) 115316. [9] A.T. Hatke, M.A. Zudov, L.N. Pfeiffer, K.W. West, Phys. Rev. Lett. 102 (2009) 066804. [10] V.I. Ryzhii, Sov. Phys. Solid State 11 (1970) 2078; V.I. Ryzhii, R.A. Suris, B.S. Shchamkhalova, Sov. Phys. Semicond. 20 (1986) 1299. [11] A.C. Durst, S. Sachdev, N. Read, S.M. Girvin, Phys. Rev. Lett. 91 (2003) 086803.