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Fractional microwave-induced resistance oscillations
ARTICLE IN PRESS
Physica E 40 (2008) 1332–1334 www.elsevier.com/locate/physe
Fractional microwave-induced resistance oscillations I.A. Dmitrieva,,1, A.D. Mirlina,b,2, D.G. Polyakova a
Institut fu¨r Nanotechnologie, Forschungszentrum Karlsruhe, 76021 Karlsruhe, Germany Institut fu¨r Theorie der kondensierten Materie, Universita¨t Karlsruhe, 76128 Karlsruhe, Germany
b
Available online 12 September 2007
Abstract We develop a systematic theory of microwave-induced oscillations in magnetoresistivity of a 2D electron gas in the vicinity of fractional harmonics of the cyclotron resonance, observed in recent experiments. We show that in the limit of well-separated Landau levels the effect is dominated by the multiphoton inelastic mechanism. At moderate magnetic field, two single-photon mechanisms become important. One of them is due to resonant series of multiple single-photon transitions, while the other originates from microwave-induced sidebands in the density of states of disorder-broadened Landau levels. r 2007 Elsevier B.V. All rights reserved. PACS: 73.40.c; 78.68.n; 73.43.f; 76.40.+b Keywords: Magnetooscillations; Photoconductivity; 2D electron gas
1. Introduction Recently, much attention has been attracted to the discovery of microwave-induced resistance oscillations (MIRO) [1], followed by the spectacular observation of zero-resistance states (ZRS) in the oscillation minima [2,3]. Two microscopic mechanisms of the MIRO have been proposed: the ‘‘displacement’’ mechanism related to the effect of microwaves on the impurity scattering [4–6], and the ‘‘inelastic’’ mechanism accounting for nonequilibrium oscillatory changes in the electron distribution [7,8]. Both mechanisms rely on the energy oscillations of the density of states (DOS) nðeÞ of disorder-broadened Landau levels (LLs) and reproduce the observed phase of the o=oc –oscillations (here, o and oc ¼ eB=mc are the microwave and the cyclotron frequencies). The inelastic mechanism yields temperature-dependent MIRO with the amplitude, proportional to the inelastic scattering time tin / T 2 , while the displacement contribution is Corresponding author.
E-mail address: [email protected] (I.A. Dmitriev). Also at: A.F. Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia. 2 Also at: Petersburg Nuclear Physics Institute, 188300 St. Petersburg, Russia. 1
1386-9477/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2007.09.003
T-independent, in disagreement with the experiments. At relevant T1 K, the inelastic effect dominates and the corresponding theory [7,8] reproduces the experimental observations [1–3]. Further experimental investigations at elevated microwave power led to the discovery of ‘‘fractional’’ MIRO and ZRS [9,10] located near the fractional harmonics of the cyclotron resonance, o=oc ¼ 12; 32; 52; 23; . . . ; to be contrasted with the integer MIRO [1–3] (less pronounced FMIRO features were also observed earlier [11]). These remarkable observations motivated the present study where we address multiphoton effects and effects of the microwave radiation on the electronic spectrum, which govern the fractional MIRO in the case of separated LLs. 2. Inelastic mechanism of the MIRO We start by including the multiphoton processes in the theory [7,8]. In a classically strong magnetic field, oc ttr b1, the diagonal resistivity rxx reads [7] Z ¼ de~n2 ðeÞqe f ðeÞ, (1) rxx =rD xx 1 2 2 is the Drude resistivity, vF the where rD xx ¼ ðe vF n0 ttr Þ Fermi velocity, ttr the transport scattering time, n0 ¼ m=2p,
ARTICLE IN PRESS I.A. Dmitriev et al. / Physica E 40 (2008) 1332–1334
and n~ ðeÞ ¼ nðeÞ=n0 the dimensionless density of states (DOS) of disorder-broadened LLs. The MIRO originate from the microvave-induced oscillations in the distribution function f ðeÞ, which obeys the kinetic equation tin X f ðeÞ f T ðeÞ ¼ An n~ ðe noÞ½f ðe noÞ f ðeÞ. (2) 4tq n Here An ¼ An describes the probability of n-photon absorption (emission) and f T ðeÞ is the thermal distribution. The leading contribution to the integer MIRO comes from the single-photon A1 ¼ A1 Po , where Po ¼ ðtq =ttr ÞðevF E o Þ2 =o2 ðo þ oc Þ2 .
(3)
Here E o is the amplitude of the circularly polarized microwave field [12], and tq 5tin ; ttr is the total (quantum) disorder-induced scattering time. If one assumes, in accord with the experimental conditions, that the temperature is high, 2p2 T=oc b1, the contribution to rxx of first order in Po takes the form [7,8] rxx =rD n2 ðeÞie þ ðtin =4tq ÞPo F ðoÞ, xx ¼ h~
(4)
where h. . . ie denotes the averaging over the period oc of the DOS, and the function F ðOÞ oscillates with O=oc , F ðOÞ ¼ Oh~n2 ðeÞqe ½~nðe þ OÞ þ n~ ðe OÞie . In the limit of separated LLs, oc tq b1, the DOS n~ ðeÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tq Re G2 ðdeÞ2 is a sequence of semicircles of width 2G ¼ 2ð2oc =ptq Þ1=2 , where de is the detuning from the center ðn þ 12Þoc of the nearest LL. In that case F ðOÞ ¼ ð16Oo2c =3p2 G3 ÞF½ðO N O oc Þ=G,
(5)
where N O is the integer number closest to O=oc , and the odd function FðxÞ is nonzero at jxjo2 (Fig. 1) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi FðxÞ ¼ xð1 þ jxjÞ jxjð2 jxjÞ 3x arccosðjxj 1Þ. (6)
3. Inelastic multiphoton mechanism of the fractional MIRO Provided G5oc , the oscillatory part of rxx (4) is finite only in narrow intervals ðo Noc Þo4G around the Φ (x) 2
-1
ωc 2ω
ω 2Γ
Fig. 2. Illustration of the processes leading to the fractional MIRO, for o=oc ¼ 1=2 þ G=2 and oc =G ¼ 7. Single-photon transitions within and between LLs (solid lines) are forbidden, while two-photon processes are allowed. The microwave-induced sidebands (8) (dashed line) make singlephoton processes possible.
integer values o=oc ¼ N [13]. Outside these intervals nðeÞnðe þ oÞ 0. Therefore, single-photon absorption is forbidden, so that f ðeÞ ¼ f T ðeÞ as long as multiphoton processes, given by An with jnj41, are not taken into account. Inclusion of two-photon processes leads to the appearance of the fractional MIRO in the frequency intervals jo ðN þ 1=2Þoc joG, N ¼ 0; 1; 2; . . . ; where nðeÞnðe þ 2oÞa0 (see Fig. 2). In these intervals, rxx =rD n2 ðeÞie þ ð3tin =32tq ÞP2o F ð2oÞ, xx ¼ h~
(7)
where we used A2 ¼ A2 ¼ 3P2o =8 [6,14]. The doubling of the argument of the function F in Eq. (7) [as compared to the integer MIRO, Eq. (4)] reflects the two-photon nature of the effect and leads to the emergence of the fractional MIRO at half-integer o=oc . The form and phase of the fractional oscillations (7) reproduce those for the integer MIRO, Eq. (4). Similarly to the integer case, there exists [15] a multiphoton contribution to the fractional MIRO governed by the displacement mechanism [4–6], which has a similar form but is a factor oc tin =tq Gb1 smaller [16] than the inelastic one (7). With increasing microwave power Po , the resistivity (7) in the oscillation minima becomes negative, which indicates a transition to the ZRS [17]. Remarkably, like in the integer case [8], the leading-order approximation (7) for the multiphoton inelastic effect is sufficient to describe the fractional photoresponse even at such high power, since the second order contribution / ðP2o tin =tq Þ2 remains small in the parameter G=oc . 4. Sideband mechanism of the fractional MIRO
1
-2
1333
1 -1 -2
Fig. 1. Function FðxÞ, Eq. (6).
2
x
Using the formalism developed in Ref. [6], it can be shown that the microwave illumination results in the appearance of ‘‘sidebands’’ in the DOS, located at distance o on both sides of every LL (see Fig. 2). To first order in Po and assuming again jo ðN þ 1=2Þoc joG, we obtain the following expression for the microwave-induced sidebands [16]: n~ ðsbÞ ðeÞ ¼ ðpPo =8oc tq Þ½~nðe þ oÞ þ n~ ðe oÞ,
(8)
ARTICLE IN PRESS 1334
I.A. Dmitriev et al. / Physica E 40 (2008) 1332–1334
where n~ ðeÞ is the unperturbed DOS. In the presence of the sidebands, single-photon transitions become possible (Fig. 2), n~ ðeÞ~nðsbÞ ðe oÞa0, resulting in the ‘‘sideband’’ contribution to the fractional MIRO,
for Functional Nanostructures of the DFG, by INTAS Grant No. 05-1000008-8044, and by the RFBR.
D 2 2 rðsbÞ xx =rxx ¼ ðptin =64oc tq ÞPo F ð2oÞ,
References
(9)
which has the same form as the leading two-photon inelastic contribution (7), but is a factor oc tq smaller. One more contribution originates from the sidebands oscillating in time with frequency 2o. ‘‘Oscillating sideband’’ contribution [16] is symmetric with respect to the detuning from the fractional resonances, in contrast to the antisymmetric F ð2oÞ, and is a factor ðoc tq Þ1=2 smaller than the two-photon contribution (7). 5. Conclusion In the limit of well-separated LLs, oc bG, the fractional MIRO are governed by the multiphoton inelastic mechanism, Eq. (7). At oc G, the sideband contribution (9) becomes relevant. Close to oc at which LLs start to overlap; specifically, at oc o4G, the effect is dominated by the resonant series of multiple single-photon transitions [10,18]. This effect appears at order ðtin Po =tq Þ2 . In the limit of strongly overlapping LLs, the fractional features get exponentially suppressed with respect to the integer MIRO [18,14]. Acknowledgments We thank S.I. Dorozhkin, K. von Klitzing, J.H. Smet, and M.A. Zudov for information about the experiments, and I.V. Gornyi for stimulating discussions. This work was supported by the SPP ‘‘Quanten-Hall-Systeme’’ and Center
[1] M.A. Zudov, R.R. Du, J.A. Simmons, J.R. Reno, Phys. Rev. B 64 (2001) 201311 (R). [2] R.G. Mani, J.H. Smet, K. von Klitzing, V. Narayanamurti, W.B. Johnson, V. Umansky, Nature 420 (2002) 646. [3] M.A. Zudov, R.R. Du, L.N. Pfeiffer, K.W. West, Phys. Rev. Lett. 90 (2003) 046807. [4] V.I. Ryzhii, Sov. Phys. Solid State 11 (1970) 2078. [5] A.C. Durst, S. Sachdev, N. Read, S.M. Girvin, Phys. Rev. Lett. 91 (2003) 086803. [6] M.G. Vavilov, I.L. Aleiner, Phys. Rev. B 69 (2004) 035303. [7] I.A. Dmitriev, A.D. Mirlin, D.G. Polyakov, Phys. Rev. Lett. 91 (2003) 226802. [8] I.A. Dmitriev, M.G. Vavilov, I.L. Aleiner, A.D. Mirlin, D.G. Polyakov, Phys. Rev. B 71 (2005) 115316. [9] M.A. Zudov, R.R. Du, L.N. Pfeiffer, K.W. West, Phys. Rev. B 73 (2006) 041303 (R). [10] S.I. Dorozhkin, J.H. Smet, K. von Klitzing, L.N. Pfeiffer, K.W. West, cond-mat/0608633. [11] M.A. Zudov, Phys. Rev. B 69 (2004) 041304 (R). [12] Here we consider passive circular polarization of the microwave field. Qualitatively similar results for arbitrary polarization will be presented elsewhere [16]. [13] S.I. Dorozhkin, J.H. Smet, V. Umansky, K. von Klitzing, Phys. Rev. B 71 (2005) 201306 (R). [14] I.A. Dmitriev, A.D. Mirlin, D.G. Polyakov, Phys. Rev. B 75 (2007) 245320. [15] X.L. Lei, S.Y. Liu, Appl. Phys. Lett. 88 (2006) 212109. [16] I.A. Dmitriev, A.D. Mirlin, D.G. Polyakov, accepted to Phys. Rev. Lett. [17] A.V. Andreev, I.L. Aleiner, A.J. Millis, Phys. Rev. Lett. 91 (2003) 056803. [18] I.V. Pechenezhskii, S.I. Dorozhkin, I.A. Dmitriev, J. Exp. Theoret. Phys. Lett. 85 (2007) 86.
Physica E 40 (2008) 1332–1334 www.elsevier.com/locate/physe
Fractional microwave-induced resistance oscillations I.A. Dmitrieva,,1, A.D. Mirlina,b,2, D.G. Polyakova a
Institut fu¨r Nanotechnologie, Forschungszentrum Karlsruhe, 76021 Karlsruhe, Germany Institut fu¨r Theorie der kondensierten Materie, Universita¨t Karlsruhe, 76128 Karlsruhe, Germany
b
Available online 12 September 2007
Abstract We develop a systematic theory of microwave-induced oscillations in magnetoresistivity of a 2D electron gas in the vicinity of fractional harmonics of the cyclotron resonance, observed in recent experiments. We show that in the limit of well-separated Landau levels the effect is dominated by the multiphoton inelastic mechanism. At moderate magnetic field, two single-photon mechanisms become important. One of them is due to resonant series of multiple single-photon transitions, while the other originates from microwave-induced sidebands in the density of states of disorder-broadened Landau levels. r 2007 Elsevier B.V. All rights reserved. PACS: 73.40.c; 78.68.n; 73.43.f; 76.40.+b Keywords: Magnetooscillations; Photoconductivity; 2D electron gas
1. Introduction Recently, much attention has been attracted to the discovery of microwave-induced resistance oscillations (MIRO) [1], followed by the spectacular observation of zero-resistance states (ZRS) in the oscillation minima [2,3]. Two microscopic mechanisms of the MIRO have been proposed: the ‘‘displacement’’ mechanism related to the effect of microwaves on the impurity scattering [4–6], and the ‘‘inelastic’’ mechanism accounting for nonequilibrium oscillatory changes in the electron distribution [7,8]. Both mechanisms rely on the energy oscillations of the density of states (DOS) nðeÞ of disorder-broadened Landau levels (LLs) and reproduce the observed phase of the o=oc –oscillations (here, o and oc ¼ eB=mc are the microwave and the cyclotron frequencies). The inelastic mechanism yields temperature-dependent MIRO with the amplitude, proportional to the inelastic scattering time tin / T 2 , while the displacement contribution is Corresponding author.
E-mail address: [email protected] (I.A. Dmitriev). Also at: A.F. Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia. 2 Also at: Petersburg Nuclear Physics Institute, 188300 St. Petersburg, Russia. 1
1386-9477/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2007.09.003
T-independent, in disagreement with the experiments. At relevant T1 K, the inelastic effect dominates and the corresponding theory [7,8] reproduces the experimental observations [1–3]. Further experimental investigations at elevated microwave power led to the discovery of ‘‘fractional’’ MIRO and ZRS [9,10] located near the fractional harmonics of the cyclotron resonance, o=oc ¼ 12; 32; 52; 23; . . . ; to be contrasted with the integer MIRO [1–3] (less pronounced FMIRO features were also observed earlier [11]). These remarkable observations motivated the present study where we address multiphoton effects and effects of the microwave radiation on the electronic spectrum, which govern the fractional MIRO in the case of separated LLs. 2. Inelastic mechanism of the MIRO We start by including the multiphoton processes in the theory [7,8]. In a classically strong magnetic field, oc ttr b1, the diagonal resistivity rxx reads [7] Z ¼ de~n2 ðeÞqe f ðeÞ, (1) rxx =rD xx 1 2 2 is the Drude resistivity, vF the where rD xx ¼ ðe vF n0 ttr Þ Fermi velocity, ttr the transport scattering time, n0 ¼ m=2p,
ARTICLE IN PRESS I.A. Dmitriev et al. / Physica E 40 (2008) 1332–1334
and n~ ðeÞ ¼ nðeÞ=n0 the dimensionless density of states (DOS) of disorder-broadened LLs. The MIRO originate from the microvave-induced oscillations in the distribution function f ðeÞ, which obeys the kinetic equation tin X f ðeÞ f T ðeÞ ¼ An n~ ðe noÞ½f ðe noÞ f ðeÞ. (2) 4tq n Here An ¼ An describes the probability of n-photon absorption (emission) and f T ðeÞ is the thermal distribution. The leading contribution to the integer MIRO comes from the single-photon A1 ¼ A1 Po , where Po ¼ ðtq =ttr ÞðevF E o Þ2 =o2 ðo þ oc Þ2 .
(3)
Here E o is the amplitude of the circularly polarized microwave field [12], and tq 5tin ; ttr is the total (quantum) disorder-induced scattering time. If one assumes, in accord with the experimental conditions, that the temperature is high, 2p2 T=oc b1, the contribution to rxx of first order in Po takes the form [7,8] rxx =rD n2 ðeÞie þ ðtin =4tq ÞPo F ðoÞ, xx ¼ h~
(4)
where h. . . ie denotes the averaging over the period oc of the DOS, and the function F ðOÞ oscillates with O=oc , F ðOÞ ¼ Oh~n2 ðeÞqe ½~nðe þ OÞ þ n~ ðe OÞie . In the limit of separated LLs, oc tq b1, the DOS n~ ðeÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tq Re G2 ðdeÞ2 is a sequence of semicircles of width 2G ¼ 2ð2oc =ptq Þ1=2 , where de is the detuning from the center ðn þ 12Þoc of the nearest LL. In that case F ðOÞ ¼ ð16Oo2c =3p2 G3 ÞF½ðO N O oc Þ=G,
(5)
where N O is the integer number closest to O=oc , and the odd function FðxÞ is nonzero at jxjo2 (Fig. 1) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi FðxÞ ¼ xð1 þ jxjÞ jxjð2 jxjÞ 3x arccosðjxj 1Þ. (6)
3. Inelastic multiphoton mechanism of the fractional MIRO Provided G5oc , the oscillatory part of rxx (4) is finite only in narrow intervals ðo Noc Þo4G around the Φ (x) 2
-1
ωc 2ω
ω 2Γ
Fig. 2. Illustration of the processes leading to the fractional MIRO, for o=oc ¼ 1=2 þ G=2 and oc =G ¼ 7. Single-photon transitions within and between LLs (solid lines) are forbidden, while two-photon processes are allowed. The microwave-induced sidebands (8) (dashed line) make singlephoton processes possible.
integer values o=oc ¼ N [13]. Outside these intervals nðeÞnðe þ oÞ 0. Therefore, single-photon absorption is forbidden, so that f ðeÞ ¼ f T ðeÞ as long as multiphoton processes, given by An with jnj41, are not taken into account. Inclusion of two-photon processes leads to the appearance of the fractional MIRO in the frequency intervals jo ðN þ 1=2Þoc joG, N ¼ 0; 1; 2; . . . ; where nðeÞnðe þ 2oÞa0 (see Fig. 2). In these intervals, rxx =rD n2 ðeÞie þ ð3tin =32tq ÞP2o F ð2oÞ, xx ¼ h~
(7)
where we used A2 ¼ A2 ¼ 3P2o =8 [6,14]. The doubling of the argument of the function F in Eq. (7) [as compared to the integer MIRO, Eq. (4)] reflects the two-photon nature of the effect and leads to the emergence of the fractional MIRO at half-integer o=oc . The form and phase of the fractional oscillations (7) reproduce those for the integer MIRO, Eq. (4). Similarly to the integer case, there exists [15] a multiphoton contribution to the fractional MIRO governed by the displacement mechanism [4–6], which has a similar form but is a factor oc tin =tq Gb1 smaller [16] than the inelastic one (7). With increasing microwave power Po , the resistivity (7) in the oscillation minima becomes negative, which indicates a transition to the ZRS [17]. Remarkably, like in the integer case [8], the leading-order approximation (7) for the multiphoton inelastic effect is sufficient to describe the fractional photoresponse even at such high power, since the second order contribution / ðP2o tin =tq Þ2 remains small in the parameter G=oc . 4. Sideband mechanism of the fractional MIRO
1
-2
1333
1 -1 -2
Fig. 1. Function FðxÞ, Eq. (6).
2
x
Using the formalism developed in Ref. [6], it can be shown that the microwave illumination results in the appearance of ‘‘sidebands’’ in the DOS, located at distance o on both sides of every LL (see Fig. 2). To first order in Po and assuming again jo ðN þ 1=2Þoc joG, we obtain the following expression for the microwave-induced sidebands [16]: n~ ðsbÞ ðeÞ ¼ ðpPo =8oc tq Þ½~nðe þ oÞ þ n~ ðe oÞ,
(8)
ARTICLE IN PRESS 1334
I.A. Dmitriev et al. / Physica E 40 (2008) 1332–1334
where n~ ðeÞ is the unperturbed DOS. In the presence of the sidebands, single-photon transitions become possible (Fig. 2), n~ ðeÞ~nðsbÞ ðe oÞa0, resulting in the ‘‘sideband’’ contribution to the fractional MIRO,
for Functional Nanostructures of the DFG, by INTAS Grant No. 05-1000008-8044, and by the RFBR.
D 2 2 rðsbÞ xx =rxx ¼ ðptin =64oc tq ÞPo F ð2oÞ,
References
(9)
which has the same form as the leading two-photon inelastic contribution (7), but is a factor oc tq smaller. One more contribution originates from the sidebands oscillating in time with frequency 2o. ‘‘Oscillating sideband’’ contribution [16] is symmetric with respect to the detuning from the fractional resonances, in contrast to the antisymmetric F ð2oÞ, and is a factor ðoc tq Þ1=2 smaller than the two-photon contribution (7). 5. Conclusion In the limit of well-separated LLs, oc bG, the fractional MIRO are governed by the multiphoton inelastic mechanism, Eq. (7). At oc G, the sideband contribution (9) becomes relevant. Close to oc at which LLs start to overlap; specifically, at oc o4G, the effect is dominated by the resonant series of multiple single-photon transitions [10,18]. This effect appears at order ðtin Po =tq Þ2 . In the limit of strongly overlapping LLs, the fractional features get exponentially suppressed with respect to the integer MIRO [18,14]. Acknowledgments We thank S.I. Dorozhkin, K. von Klitzing, J.H. Smet, and M.A. Zudov for information about the experiments, and I.V. Gornyi for stimulating discussions. This work was supported by the SPP ‘‘Quanten-Hall-Systeme’’ and Center
[1] M.A. Zudov, R.R. Du, J.A. Simmons, J.R. Reno, Phys. Rev. B 64 (2001) 201311 (R). [2] R.G. Mani, J.H. Smet, K. von Klitzing, V. Narayanamurti, W.B. Johnson, V. Umansky, Nature 420 (2002) 646. [3] M.A. Zudov, R.R. Du, L.N. Pfeiffer, K.W. West, Phys. Rev. Lett. 90 (2003) 046807. [4] V.I. Ryzhii, Sov. Phys. Solid State 11 (1970) 2078. [5] A.C. Durst, S. Sachdev, N. Read, S.M. Girvin, Phys. Rev. Lett. 91 (2003) 086803. [6] M.G. Vavilov, I.L. Aleiner, Phys. Rev. B 69 (2004) 035303. [7] I.A. Dmitriev, A.D. Mirlin, D.G. Polyakov, Phys. Rev. Lett. 91 (2003) 226802. [8] I.A. Dmitriev, M.G. Vavilov, I.L. Aleiner, A.D. Mirlin, D.G. Polyakov, Phys. Rev. B 71 (2005) 115316. [9] M.A. Zudov, R.R. Du, L.N. Pfeiffer, K.W. West, Phys. Rev. B 73 (2006) 041303 (R). [10] S.I. Dorozhkin, J.H. Smet, K. von Klitzing, L.N. Pfeiffer, K.W. West, cond-mat/0608633. [11] M.A. Zudov, Phys. Rev. B 69 (2004) 041304 (R). [12] Here we consider passive circular polarization of the microwave field. Qualitatively similar results for arbitrary polarization will be presented elsewhere [16]. [13] S.I. Dorozhkin, J.H. Smet, V. Umansky, K. von Klitzing, Phys. Rev. B 71 (2005) 201306 (R). [14] I.A. Dmitriev, A.D. Mirlin, D.G. Polyakov, Phys. Rev. B 75 (2007) 245320. [15] X.L. Lei, S.Y. Liu, Appl. Phys. Lett. 88 (2006) 212109. [16] I.A. Dmitriev, A.D. Mirlin, D.G. Polyakov, accepted to Phys. Rev. Lett. [17] A.V. Andreev, I.L. Aleiner, A.J. Millis, Phys. Rev. Lett. 91 (2003) 056803. [18] I.V. Pechenezhskii, S.I. Dorozhkin, I.A. Dmitriev, J. Exp. Theoret. Phys. Lett. 85 (2007) 86.