Photoexcited transport has been a focus area in the study of transport at large filling factors in the high-mobility twodimensional electron system (2DES) over the past decade . Of interest here are the microwave-induced zeroresistance [1] states, which arise from the associated microwave-induced magnetoresistance oscillations [1][2][3]. Such zero-resistance states are believed, in one interpretation, to represent a photoinduced "absence of backscattering" condition of the high-mobility 2DES [54]. Thus, under microwave photoexcitation, at low temperatures, the magnetoresistance in a high-quality 2DES shows large, periodic-in-B -1 , magnetoresistance oscillations [1][2][3], where the extrema are "1/4-cycle shifted" with respect to cyclotron resonance and cyclotron resonance harmonics [1,5]. At lower temperatures, moderate microwave intensity transforms the oscillatory minima into zero-resistance states. Interesting experimental features examined by experiment include the 1/4-cycle phase shift [1,5,8], the nonlinear increase in the amplitude of the radiationinduced oscillations with the microwave power [18,39], observed correlations between the magnetoresistance oscillations and microwave reflection [8,25] from the 2DES, polarization sensitivity [22,28,29,31], and magnetoresistive response under bichromatic excitation [10,42]. Observed oscillatory phenomena in the photoexcited two-dimensional electron systems have been considered by the displacement model [43,46,48,49], the * rmani@gsu.edu microwave-driven electron orbital model [44,55], the inelastic model [51], and a memory effect theory [69].
A subject of experimental interest is the study of possible electron heating under photoexcitation, as the theory has predicted the possibility of variable, magnetic-field-dependent, microwave-induced electron heating in the 2DES in the largefilling-factor, low-magnetic-field limit [50,52]. Theory has examined the electron heating by microwave photoexcitation in a balance-equation scheme that takes into account photonassisted electron transitions as well as radiation-induced change of the electron distribution for high-mobility twodimensional systems. The results suggest that the electron temperature is a function of the magnetic field, the microwave intensity, and frequency, and it is determined by the balance between the energy absorption from the radiation field and the energy dissipation to the lattice through electron-phonon scattering.
This work indicates that microwave photoexcitation produces a small discernible increase in the electron temperature both at null magnetic field and at finite magnetic fields, in the examined range, in the GaAs/AlGaAs 2DES. The heating effect appears greater at null field in comparison to the examined magnetic-field interval, in qualitative agreement with theory [52].
Lock-in-based electrical measurements were performed on a photolithographically fabricated Hall bar from molecular beam epitaxy grown high-mobility GaAs/AlGaAs heterojunctions. The Hall bar sample was mounted at the lower end of a long cylindrical waveguide sample holder which is inserted into a variable temperature insert (VTI), within the bore of the superconducting magnet. The sample temperature was controlled and varied by pumping on and reducing the vapor pressure of the liquid helium within the VTI insert over the range 1.45 6 T 6 4.2 K. The sample was immersed in liquid helium for all the reported measurements. The Hall bar sample inside the VTI was preilluminated with red light to obtain a high-mobility state. At 1.47 K, sample electron density (n e ) was 2.4 × 10 11 cm -2 and mobility (μ e ) was 1.2 × 10 7 cm 2 /V s.
The microwave radiation over the 30 6 f 6 50 GHz band was generated with a commercially available microwave synthesizer and the specimen was illuminated with linearly polarized microwaves for the photoexcited transport measurements. The polarization of incident microwave was parallel to the long direction of the Hall bar device. The diagonal resistance (R xx ) is reported here at temperatures where microwave-induced magnetoresistance oscillations and Shubnikov-de Haas (SdH) oscillations are relatively strong, for a number of microwave source powers 0 6 P 6 4 mW. Here, we present the data at f = 48.5 GHz, which are representative of the observations over the above-mentioned frequency band.
Microwave radiation-induced magnetoresistance oscillations are observable for B 6 0.2 T at T = 1.47 K in Fig. 1. It can be clearly seen in Fig. 1(a) that the amplitude of radiationinduced magnetoresistance oscillations grow nonlinearly with the microwave source power [18]. Strong SdH oscillations are also observable under both the dark and microwave-irradiated conditions for B > 0.2 T. In this study, we examine two observable characteristics in the R xx vs B traces with the parametric variation of P . These characteristics are (1) the R xx at zero magnetic field is upshifted to higher resistance values with the increment of the microwave source power as shown in the inset of Fig. 1 (the magnetic-field dependence of R xx or the line shape observed here will be a topic of study elsewhere), and (2) the amplitude of SdH oscillations decays with the increment of microwave source power at finite magnetic fields. We attribute these microwave-induced variations to a heating effect from the incident microwaves on the Hall bar device since the effect of increased microwave excitation on R xx is analogous to the effect of increasing the sample temperature, in the absence of photoexcitation. Our aim here is to convince one of electron heating for these two cases and extract the change of the electron temperature with photoexcitation.
We begin by considering the upshifting of the R xx traces with the incident microwave source power observable in the inset of Fig. 1(a). Since the null magnetic field R xx increment appeared as a result of microwave photoexcitation, which could also plausibly produce electron heating, we also examined the dependence of the null magnetic field R xx on the temperature. Thus, the resistance, R xx at B = 0, was measured as a function of the bath temperature, termed here the lattice temperature, T L , under dark conditions, i.e., without microwave photoexcitation. Although we examined the temperature dependence of R xx at B = 0 over a wide T interval, the temperature interval of interest for comparing with the upshift in R xx observed under microwave excitation turned out to be only 1. 47 the temperature T L , for this temperature interval. Figures 2(a
. This equation for the dark diagonal resistance can be inverted to obtain
Such inversion is carried out so that the zero-magnetic-field R xx can serve as a temperature gauge, even in the presence of microwave excitation. In the microwave irradiated condition, however, the diagonal resistance will serve as a gauge of the electron temperature, T e , not the lattice temperature, T L , since the electron system can potentially be decoupled from the lattice/bath in the presence of such drive.
Next, measurements as in Fig. 1 (the abscissa) at a set of five T L . The left side ordinates in Fig. 2(c) show the corresponding electron temperature scale obtained, as mentioned above, at each lattice temperature. The results of Fig. 2(c) suggest that 1T e /1P ≈ 0.1 K/mW of source microwave power at null magnetic field. In Fig. 2(c) the lines shown are simply guides to the eye. Since the ordinate scale in Fig. 1(b) is much expanded compared to the scale on the right ordinate of Fig. 2(c), the nonlinearity observable in Fig. 1(b) is not so evident in Fig. 2(c).
In the second part of our study, we examined the influence of microwave excitation on the SdH oscillations. For this purpose, we examined the SdH line shape over the span 2.3 < ω c /ω 6 5.2, which is indicated in Fig. 1(a). In order to facilitate line-shape fits, a monotonic background R xx term was subtracted from the raw magnetoresistance data to obtain the oscillatory 1R xx term. This term was then plotted vs the inverse magnetic field for different microwave source powers as shown in Figs. 3(a)-3(e) for T L = 1.47 K. Here, the red open circles represent data.
As mentioned previously, the amplitude of the SdH oscillations decays with the increment of microwave source power. To extract the amplitude of the SdH oscillations, a standard nonlinear least-squares fit was performed on 1R xx data with an exponentially damped sinusoidal function, i.e., 1R xx = -Ae -α/B cos(2πF /B), where A is the amplitude and F is the SdH frequency [15,[71][72][73]. Since, the parameter F is insensitive to the incident radiation at a constant lattice temperature, the F was fixed to a constant value. As an example, the fit of the 1R xx data at 1.47 K for various P spanning 0 6 P 6 4 mW are shown in Figs. 3(a)-3(e) as solid black lines. These panels propose good agreements between data and fit. For the sake of illustration, the temperature dependence of the SdH oscillation amplitude A 0 = Ae -α/B are exhibited vs the microwave power in Fig. 3(f). It can be clearly seen that A 0 decays exponentially with increasing lattice temperature as well as increasing microwave source power.
Next, we extracted the electron temperature, T e , at finite magnetic fields from the SdH oscillation amplitude. To extract this T e from the SdH amplitude in these data, the damping constant, α, in the fitting model,
m * /m e is effective electron mass ratio, T L is the lattice temperature, 1T e is the electron temperature increment with respect to the lattice, and T D is the Dingle temperature. In practice, it turned out that T D is very small compared to T L . Thus, T L + T D ≈ T L . And, the fit function became 1R xx = A 0 cos(2πF /B) = -Ae -λ(T L +1T e )(m * /m e )/B cos(2πF /B).
In Fig. 4, we compare the microwave power variation of electron temperature, T e = T L + 1T e , at zero magnetic field extracted from the power variation of the zero-field R xx , with the power variation of the electron temperature at finite magnetic fields extracted from the SdH oscillations. Figures 4(a)-4(e) show that the electron temperatures increased with the incident microwave source powers, both in the absence of a magnetic field and also at small finite magnetic fields. However, the increase in the electron temperature at zero magnetic field under microwave excitation is approximately six times higher than the extracted electron temperature increase over the SdH oscillation region. Figures 4(a)-4(e) indicate that 1T e /1P ≈ 0.1 K/mW at null magnetic field and 1T e /1P ≈ 0.015 K/mW at finite magnetic fields.
Theory [52] suggests that stationary microwave photoexcitation can heat highly mobile electrons in the 2D system. The absorbed energy from the radiation field is transferred to the lattice by electron-phonon scattering through bulk LA, TA, and LO phonons. The electronic energy absorption rate is strongly magnetic field dependent and also shows oscillations with periodicity in the inverse magnetic field reflecting the periodicity of both microwave-induced magnetoresistance oscillations and Shubnikov-de Haas oscillations [52]. It turns out that the electron temperature, T e , reflects the features observed in the energy absorption rate [52]. Simulations for typical experimental parameters suggest that at lower magnetic fields, i.e., ω c /ω 6 1.4, energy absorption occurs via inter-Landaulevel transitions, leading to a significantly enhanced T e , with T e ≈ 10 K for electric fields at the specimen of the order of 3.5 V/cm at f = 50 GHz for a specimen mobility μ = 2.5 × 10 7 cm 2 /V s [52]. As the magnetic field increases, the inter-Landau-level transitions weaken and the absorbed energy decreases rapidly, leading to an electron temperature that is only slightly higher than the lattice temperature at ω c /ω ≈ 3 [52]. It is worth pointing out that analytic expressions for the absolute absorption coefficient as a function of the magnetic field, in this B-field range of interest where the absorption changes rapidly with the magnetic field, do not occur in the literature, to our knowledge.
The experimental results reported here are in qualitative agreement with these theoretical expectations. The observed increase of the T e is much greater at null magnetic field than in the regime of Shubnikov de Haas oscillations. Indeed, the rate of increase of the T e with P at null field is approximately six times greater than at finite magnetic fields. On the other hand, the maximum observed increase here in the electron temperature is ≈ 0.4 K at null magnetic field with a bath temperature of ≈ 1.47 K and a source power of 4 mW (see Fig. 4), while theory suggests that the increase can be as large as 10-20 K at T L ≈ 1 K. We attribute this difference to two factors: (a) our specimen mobility is smaller in comparison to the value used in the theoretical calculation. A larger mobility provides for a longer elastic mean free path and phase coherence length. It appears plausible that over these longer length scales/times, more energy can be absorbed from the radiation field, leading to enhanced heating. That is, a lower mobility in our specimens in comparison to the theoretical calculation might lead to a reduced heating effect in experiment, especially at null magnetic field. (b) In our setup, there is significant microwave attenuation between the source and the specimen. Thus, the power level at the specimen can be reduced by ≈ 10-16 dB or more compared to the power level at the source, which is the power level specified in experiment. This implies that the electric fields at the specimen are much lower, than the values utilized for the theoretical simulation. As a consequence, the presented curves in the theory [52] overestimate the heating effect. Thus, the results generally support the conclusion that there is electronic heating, although it is not as large as expected from theory.
This study indicates that microwave photoexcitation produces a small discernible increase in the electron temperature both at null magnetic field and at finite magnetic fields in the GaAs/AlGaAs 2D electron system. The heating effect appears greater at null field in comparison to the examined finite-field interval, in line with theoretical predictions. However, the increase in the electron temperature in the zero-field limit is smaller than theoretical predictions mostly because theory assumes no microwave attenuation between source and sample while, in our experiment, the attenuation appears substantial [52].