Energy Relaxation of Two-Dimensional Electrons in the Quantum Hall Effect Regime K

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JETP Letters, Vol. 71, No. 1, 2000, pp. 31Ð34. Translated from Pis’ma v Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 71, No. 1, 2000, pp. 47Ð52. Original Russian Text Copyright © 2000 by Smirnov, Ptitsina, Vakhtomin, Verevkin, Gol’tsman, Gershenzon.

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Energy Relaxation of Two-Dimensional Electrons in the Quantum Hall Effect Regime K. V. Smirnov, N. G. Ptitsina, Yu. B. Vakhtomin,1 A. A. Verevkin, G. N. Gol’tsman, and E. M. Gershenzon Moscow State Pedagogical University, Moscow, 113567 Russia Received November 19, 1999

Abstract—The mm-wave spectroscopy with high temporal resolution is used to measure the energy relaxation τ times e of 2D electrons in GaAs/AlGaAs heterostructures in magnetic fields B = 0Ð4 T under quasi-equilibrium τ conditions at T = 4.2 K. With increasing B, a considerable increase in e from 0.9 to 25 ns is observed. For high ν τÐ1 B and low values of the filling factor , the energy relaxation rate e oscillates. The depth of these oscillations ν ν τÐ1 and the positions of maxima depend on the filling factor . For > 5, the relaxation rate e is maximum when the Fermi level lies in the region of the localized states between the Landau levels. For lower values of ν, the τÐ1 ν relaxation rate e is maximum at half-integer values of when the Fermi level is coincident with the Landau τÐ1 level. The characteristic features of the dependence e (B) are explained by different contributions of the intralevel and interlevel electronÐphonon transitions to the process of the energy relaxation of 2D electrons. © 2000 MAIK “Nauka/Interperiodica”. PACS numbers: 73.50.Jt, 73.40.Hm

1 1. In the last decade, the electronÐphonon interac- The energy relaxation may occur at the expense of tion and the mechanisms of the energy relaxation of hot the transitions both between the Landau levels and carriers were one of the most important problems of the within them. The conditions of the experiment deter- physics of 2D structures. For the case of the scattering mine which of these two processes prevails. The object by acoustic and optical phonons in a wide temperature of most experimental studies is the spectrum of range in the absence of magnetic field, this problem has phonons involved in the interaction with electrons in been solved by many authors, both theoretically (e.g., magnetic field, as well as their angular distribution [6Ð [1, 2]) and experimentally (e.g., [3, 4]). For a magnetic 9]. The measurements of the Landau level population field perpendicular to the 2D layer, the electronÐ by magnetic tunneling spectroscopy [10] provided the τ phonon interaction becomes essentially different estimates of the energy relaxation time e for the inter- because of the quantization of the carrier energy in the level electron transitions. In magnetic fields B ≥ 4 T, 2D plane. this quantity was found to be equal to ≈100 ns, which τ far exceeds the values of e corresponding to zero mag- A number of theoretical publications present the netic field. As for the direct measurements of the energy calculation of the energy relaxation rate of electrons in relaxation times in magnetic field, no such experiments a real situation with allowance for the broadening of the had been described in the literature. Landau levels, as well as the appearance of a region of delocalized states near the center of the Landau level 2. We studied the inelastic relaxation of two-dimen- and localized states between the levels in the electron sional electrons in GaAs/AlGaAs geterostructures at energy spectrum [5Ð8]. It was shown that, in a quantiz- the temperature T = 4.2 K in magnetic field 0–4 T per- ing magnetic field, the spectrum of phonons involved in pendicular to the 2D plane. The measurements were the interaction with electrons is essentially altered, performed under the weak heating conditions when the which should result in a change in the energy relaxation free carriers could be considered as quasi-equilibrium ones. We measured the relaxation time of the mm-wave rate. photoconductivity caused by the nonresonance absorp- Because of the dependence of the density of elec- tion of electromagnetic radiation in the regime of Shub- tron states on magnetic field, the energy relaxation rate nikovÐde Haas oscillations of the resistance (the energy បω should experience oscillations similar to the resistance of the radiation quantum = 0.6 meV was much less បω ω oscillations in the ShubnikovÐde Haas effect. than C for B > 1 T, where C is the cyclotron frequency). For the measurements, we used 1 ≈ e-mail: [email protected] GaAs/AlGaAs structures with the concentration ns

32 SMIRNOV et al.

800 relaxation time of the mm-wave photoconductivity sig- 600 nal is equal to the energy relaxation time of the carriers

Ω in the absence of the bolometric effect, and it is deter- ', 400 mined by the frequency dependence of the quantity ∆U: R 200 ∆U(∆f = 0) 0 ∆U(∆f ) = ------. 200 ( π∆ τ )2 1 + 2 f e 100 , mV

U 0 The frequency stability of the backward-wave tube ∆ oscillators allows one to perform such measurements at –100 ∆ 7 τ Ð8 frequencies f > 10 Hz ( e < 10 s). To measure large 0 1 2 3 4 relaxation times, the electromagnetic radiation of a B, T backward-wave tube oscillator is modulated by supply- ing a modulating voltage at the frequency ∆f to the Fig. 1. (᭹) Photoconductivity signal in the mm-wave range anode circuit of the tube. ∆U and (ᮀ) the oscillatory contribution to the resistance of ∆ τ the sample R' versus the magnetic field B. R' = R Ð R(B), The measurements of e in quasi-equilibrium condi- where ∆R(B) is the contribution of the magnetoresistance to tions impose stringent requirements on the sensitivity the resistance of the sample; ∆R(B) linearly increases with increasing B. of the measuring equipment because of the weak dependence of the resistance of the structure R on the electron temperature. The sensitivity of the equipment used in our experiments allowed us to perform the mea- ∆ rel. units U, surements with the minimum electromagnetic radiation 100 ≈ power incident on the sample and the dc power Pe min 5 × 10Ð17 watt per electron, which corresponds to an increase in the temperature of free two-dimensional ∆ ≈ τ –1 –1 carriers by Te 0.1Ð0.3 K; this value is quite suitable B = 2.75 í, e = 0.08 s τ –1 –1 for the measurements at the temperature T = 4.2 K. B = 2.71 í, e = 0.12 s 3. The measurements of the nonresonant mm-wave 10–1 photoconductivity ∆U showed that, in a magnetic field, the quantity ∆U exhibits oscillations similar to the ShubnikovÐde Haas oscillations of the resistance (Fig. 1). The measured values of ∆U shown in the fig- 10–3 10–2 10–1 100 101 102 ∆ ure correspond to the low-frequency modulation of the f, MHz mm-wave radiation (∆f = 2 kHz). The signal is a bipolar one: in the vicinity of the resistance minimum in the Fig. 2. Dependence of the signal magnitude on the modula- ShubnikovÐde Haas oscillations, ∆U corresponds to the tion frequency for two close values of magnetic field: B = 2.75 and 2.71 T. For low frequencies ∆f, the values of ∆U are growth of resistance with the absorption of electromag- normalized to the photoconductivity signal at a frequency of netic radiation, and in the vicinity of the maximum in 1 kHz. the resistance R it corresponds to a decrease in the resistance. The first maximum observed in the signal at B ≈ 0.4 T corresponds to the cyclotron resonance at the fre- 5 × 1011 cmÐ2 and the mobility µ = 2 × 105 cm2/Vs at T = quency of the mm-wave radiation. The signal ∆U is 4.2 K. As in our previous experiments [4], for the direct asymmetric about zero: it is considerably greater at the measurements of the energy relaxation time, we used minimum in the resistance R. The bipolarity of the sig- the mm-wave spectroscopy with high temporal resolu- nal is related to different mechanisms of photoconduc- tion. tivity (the electron gas heating near the resistance max- The specific feature of the case under study is that in imum and the hopping mechanism at the resistance magnetic field the energy relaxation time varies over minimum), and it was also observed in other experi- wide limits, from 10Ð9 to 10Ð7 s. To measure small relax- ments (e.g., [11]). τ Ð8 ∆ ation times e < 10 s, we used a mm-wave spectrome- The photoconductivity signal U was measured as a ter in which the electromagnetic radiation was incident function of the frequency ∆f in magnetic fields from 0 on the sample from two backward-wave tube oscilla- to 3.6 T. As an illustration, in Fig. 2 we present the tors shifted in frequency by ∆f. The absorption of elec- dependences of the photoconductivity signal ∆U on ∆f tromagnetic radiation by free carriers or bonded carri- for two close values of B (B1 = 2.75 T and B2 = 2.71 T) ers in the region of localized states leads to a variation corresponding to the vicinity of the minimum in R. One in the sample resistance ∆R and the appearance of a can see that the frequency dependences noticeably dif- photoconductivity signal ∆U at the frequency ∆f. The fer from each other. From the measurements of ∆U(∆f),

JETP LETTERS Vol. 71 No. 1 2000 ENERGY RELAXATION OF TWO-DIMENSIONAL ELECTRONS 33

τ Ð1 τ –1, ns–1 we obtained the values of e (B) (Fig. 3). The absence e τ Ð1 0 of the experimental values of e for B = 2.9Ð3.3 and 10 2.3Ð2.5 T is related to the insufficient sensitivity of the experimental setup at high frequencies ∆f > 106 Hz (in these intervals of magnetic fields B, the photoconductivity signal is too weak; see Fig. 1). 10–1 The experiment shows that, with increasing B, the electronÐphonon interaction becomes less efficient τ Ð1 ≈ 7 6 5 4 3 ( e decreases), and, at B 1.2 T, the energy relaxation 10–2 rate decreases by an order of magnitude relative to its 0 1 2 3 4 value at B = 0. In the quantum Hall effect regime (B > B, í ν ⑀ បω < 1 T, the filling factor = F/ c 8), the dependence τ Ð1 (B) exhibits oscillations. The depth of these oscilla- Fig. 3. Dependence of the inverse relaxation time of the e τÐ1 tions increases with increasing B. In Fig. 3, the arrows photoconductivity signal e on the magnetic field B. The arrows indicate the values of B at which R takes its mini- indicate the values of B corresponding to the maximum mum values. The numbers near the arrows show the values photoconductivity signal (the minimum R). One can of the filling factor ν corresponding to the given magnetic τ Ð1 fields. see that, for B > 2.5 T, the quantity e is minimum in magnetic fields B corresponding to the resistance min- τ Ð1 In a major part of the magnetic field range, the val- imum, while, for B < 2 T, the minimum e corre- τ Ð1 sponds to the resistance maximum. ues of e obtained from our experiments describe the inelastic relaxation related to the electron transitions 4. We begin the analysis of our experimental results within the Landau levels. In fact, the energy of the mm- with some estimates. The measurements are performed wave radiation quantum is បω = 0.6 meV. For the max- at T = 4.2 K. The prevailing component of the electronÐ បω ≈ imum value of B, we obtain the estimate C 6 meV; phonon interaction is the scattering due to the deforma- then, for magnetic fields B > 1 T, we have បω ӷ បω . tion potential; the wave vector of a thermal phonon q = C ប In the experiment, the temperature was T = 4.2 K, and kT/ S is of the order of the wave vector of a two-dimen- ≈ Ӷ បω sional electron at the Fermi surface k (kT/បS ≈ k ). In kT 0.4 meV. Since kT C, in this magnetic field F F range the electrons occupy only the Landau levels with the absence of magnetic field, the wave vectors of the ⑀ ≤ ⑀ the energy F. Therefore, the absorption of a radia- phonons that take part in the electronÐphonon interac- បω Ӷ បω tion are limited in the direction perpendicular to the 2D tion quantum with the energy C can be accom- layer by the transverse dimension of the layer a : q⊥ < panied only by the electron transitions within the last 0 occupied Landau level. These transitions lead either to 1/a0, and, in the layer plane, according to the conserva- ⑀ < the carrier heating when F is coincident with the tion laws, we obtain the limitation q|| 2kF. Thus, all ≈ 6 Ð1 energy of the Landau level in the region of delocalized phonon states fill a cylinder of height 1/a0 10 cm states, or to nonresonant hopping between localized for typical GaAs/AlGaAs 2D structures and of radius q|| states when ⑀ falls between the Landau levels in the ≅ F (for the concentration of two-dimensional carriers nS region of localized states. In this case, the energy relax- 11 Ð2 6 Ð1 5 × 10 cm , we obtain q|| ≈ 4 × 10 cm ≈ 1/a). In a ation of excited carriers may occur only at the expense magnetic field, only the radius of the cylinder is of the electron transitions within the Landau level. changed; namely, q|| is limited by the magnetic length According to Kent et al. [9], the energy relaxation rate បր < × 5 Ð0.5 Ð1 for the intralevel relaxation strongly depends on the lB = eB : q|| 1/lB (1/lB = 3.9 10 T cm ). For Ӷ density of states; i.e., the energy relaxation rate is max- magnetic fields B < 4 T, we have q||B ≠ 0 q||B = 0. These imum when the Fermi level coincides with the Landau estimates show that, in the absence of magnetic field, level, and it is minimum when ⑀ falls in the region of the energy relaxation rate is determined by the emission F ≈ localized states. Only in low magnetic fields, in the of phonons with the energy kT, while, in magnetic transient region of B where kT < បω , interlevel transi- field, such phonons can be emitted only at low angles to C tions are possible owing to the presence of several par- the magnetic field direction, which considerably tially filled Landau levels. As a result of the absorption reduces the electron energy relaxation rate. Hence, the of electromagnetic radiation, the heated carriers may τ Ð1 ⑀ Ӷ បω substantial decrease in e observed in the experiment emit phonons of both low energies ph C and high ⑀ ≈ បω in magnetic field can be attributed to the change in the energies ph C (the interlevel ones). Although the spectrum of phonons involved in the electronÐphonon probability of the emission of high-energy phonons is interaction. much less than that of the low-energy phonons, the

JETP LETTERS Vol. 71 No. 1 2000 34 SMIRNOV et al. former make a substantial contribution to the relaxation ShubnikovÐde Haas oscillations. Under weakly non- because of the large change in the electron energy in equilibrium conditions, the energy relaxation in high every event of emission. The energy relaxation rate for magnetic fields is determined by the electronÐphonon to these transitions is maximum when the Fermi level is transitions within the Landau level. The contribution of in the region of localized states between the Landau the electronÐphonon transitions between the Landau levels [9]. In this case, the probability of the emission levels manifests itself in intermediate magnetic fields. ⑀ Ӷ បω of phonons with the energy ph C is increased, This work was supported by the Russian Foundation < បω since, for kT C, the delocalized states at the Landau for Basic Research (grant no. 98-02-897), the Program ⑀ > ⑀ level with the energy F are filled to a greater extent, of the State Support of Leading Scientific Schools and the delocalized states at the Landau level with the (project no. 96-02-96426), the Program “Physics of ⑀ < ⑀ energy F are filled to a lesser extent, as compared Solid Nanostructures,” and the grant “Universities of to the case of the Fermi level lying in the region of delo- Russia.” calized states. The conditions corresponding to the energy relax- REFERENCES ation at the expense of the intralevel transitions are fully realized in our experiment for B > 2 T: the quantity 1. B. K. Ridley, Rep. Progr. Phys. 54, 169 (1991). τ Ð1 2. V. Karpus, Fiz. Tekh. Poluprovodn. 22, 439 (1988). e exhibits two deep minima at the magnetic fields ν 3. H. Sakaki, K. Hirakawa, J. Yoshino, et al., Surf. Sci. 142, corresponding to the minima in the resistance R ( = 3, 306 (1994); K. Hirakawa and H. Sakaki, Appl. Phys. τ Ð1 Lett. 49, 889 (1986). 4). At B = 3.6 T, the value of e is an order of magnitude less than that corresponding to the maximum in R. 4. A. A. Verevkin, N. G. Ptitsina, G. M. Chulcova, et al., Evidently, the depth of the oscillations should grow Phys. Rev. B 53, R7592 (1996); A. A. Verevkin, ν N. G. Ptitsina, G. M. Chulkova, et al., Pis’ma Zh. Éksp. with increasing B (at lower filling factors ) and with Teor. Fiz. 61, 579 (1995) [JETP Lett. 61, 591 (1995)]. τ Ð1 decreasing T. The oscillations of the quantity e are 5. H. A. J. M. Reinen, T. T. J. M. Berendschot, R. J. H. Kap- also observed for intermediate magnetic fields 1 T < pert, et al., Solid State Commun. 65, 1495 (1988). B < 2 T. However, at the values of B corresponding to 6. G. A. Toombs, F. W. Sheard, D. Neilson, et al., Solid Ð1 State Commun. 64, 577 (1987). the minimum resistance R, a maximum in τ is e 7. F. Dietzel, W. Dietsche, and K. Ploog, Phys. Rev. B 48, observed, which presumably testifies to the substantial 4713 (1993). contribution of the interlevel electronÐphonon transi- 8. K. Benedict, J. Phys. Condens. Matter 4, 4083 (1992). tions to the energy relaxation of two-dimensional electrons. In this interval of magnetic fields B, the depth of 9. A. J. Kent, R. E. Strickland, K. R. Stpickland, et al., the oscillations is much less than at low values of ν. Phys. Rev. B 54, 2019 (1996). 10. B. R. A. Neves, N. Mori, P. H. Beton, et al., Phys. Rev. Thus, we measured the energy relaxation times of B 52, 4666 (1995). 2D electrons in quasi-equilibrium conditions, in mag- 11. N. M. Grodnenskiœ, K. V. Starostin, and D. V. Galchen- netic fields corresponding to the quantum Hall effect kov, Pis’ma Zh. Éksp. Teor. Fiz. 43, 54 (1986) [JETP regime. We have shown that the quantization of the Lett. 43, 70 (1986)]. electron energy in magnetic field leads to a sharp decrease in the energy relaxation rate and to oscillations of the energy relaxation time that are similar to the Translated by E. Golyamina

JETP LETTERS Vol. 71 No. 1 2000