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

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Energy Relaxation of Two-Dimensional Electrons in the Quantum Hall Effect Regime K JETP Letters, Vol. 71, No. 1, 2000, pp. 31–34. Translated from Pis’ma v Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 71, No. 1, 2000, pp. 47–52. Original Russian Text Copyright © 2000 by Smirnov, Ptitsina, Vakhtomin, Verevkin, Gol’tsman, Gershenzon. CONDENSED MATTER 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 = 0–4 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 τ–1 B and low values of the filling factor , the energy relaxation rate e oscillates. The depth of these oscillations ν ν τ–1 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 τ–1 ν relaxation rate e is maximum at half-integer values of when the Fermi level is coincident with the Landau τ–1 level. The characteristic features of the dependence e (B) are explained by different contributions of the intra- level and interlevel electron–phonon 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 electron–phonon 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 [6– [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 electron– 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 [5–8]. 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 nikov–de Haas oscillations of the resistance (the energy "ω should experience oscillations similar to the resistance of the radiation quantum = 0.6 meV was much less "ω ω oscillations in the Shubnikov–de 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 0021-3640/00/7101-0031 $20.00 © 2000 MAIK “Nauka/Interperiodica” 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 –8 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. (d) Photoconductivity signal in the mm-wave range anode circuit of the tube. ∆U and (h) the oscillatory contribution to the resistance of ∆ τ the sample R' versus the magnetic field B. R' = R – 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 × 10–17 watt per electron, which corresponds to an increase in the temperature of free two-dimensional ∆ ≈ τ –1 –1 carriers by Te 0.1–0.3 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 Shubnikov–de 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- Shubnikov–de 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 resis- tance. The first maximum observed in the signal at B ≈ 0.4 T corresponds to the cyclotron resonance at the fre- 5 × 1011 cm–2 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 10–9 to 10–7 s. To measure small relax- ments (e.g., [11]). τ –8 ∆ 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.
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