The high mobility GaAs/AlGaAs 2D electron system exhibits lots of fascinating phenomena at low magnetic fields and liquid helium temperatures including, for example, the microwave induced effects and a giant magnetoresistance (GMR) effect [15,22,[31][32][33][34][35][36][37][38][39][40]. Studies have shown that a supplementary dc-current can be used to obtain in-situ tunability of the GMR effect at a fixed temperature, and these studies have suggested that, this effect could be responsive to the heating of carriers [22]. To examine the influence of the ac bias current on carrier temperature in this 2D electron systems, we used magnetotransport measurements in the high mobility GaAs/AlGaAs two-dimensional electron gas system (2DES) to follow the lineshape of (Shubnikov-de Haas) SdH oscillations as a function of the ac-bias-current.

Studies of the SdH oscillations in the 2DES plays an important role in the revelation of the quantum mechanical constitution of matter, and they serve to interrogate the properties of charge carriers, such as effective mass and carrier temperature [10,[41][42][43][44][45][46][47][48][49][50][51][52][53]. SdH oscillations are observable in the diagonal electrical resistance of electronic systems at weak magnetic fields and low temperatures, and they are responsive to the ratio between the Fermi energy, E F , and the cyclotron energy, ω c , as ω c = eB/m * is directly proportional to the magnetic field. Here, e (m * ) is the electron charge (effective mass). SdH oscillations are periodic in the inverse magnetic field and peaks in the diagonal magnetoresistance R xx occur when Fermi level sweeps through the centers of the Landau levels [54][55][56][57][58][59] Electron scattering influences the lineshape of SdH oscillations. Dingle showed that electron scattering sets the effective temperature that is manifested in the lineshape of SdH oscillations [55].

In this study, lock-in based four-terminal magnetoresistance measurements were carried out under a perpendicular magnetic field to investigate the magnetotransport properties of two Hall bar devices, see inset of the figure 1(a), fabricated from GaAs/AlGaAs heterostructures with mobility equal to 6.6 × 10 6 and 1.1 × 10 7 cm 2 V -1 s -1 . GaAs/ AlGaAs heterostructures were grown using molecular beam epitaxy (MBE), and the spacer and cap layer thicknesses are 700 Å and 100 Å, respectively, as an electron donor layer, 5 Å thick Si doped (δ-doping) AlGaAs layer with the concentration of 10 12 cm -2 was grown at the top of the spacer. The width of the Hall bar channel is 200 Å, and the distance between potential contacts is 400 Å. voltage (V xx ) measurements were taken at different ac current bias over the span (0 I ac 12 µA) at different bath temperatures (1.6 K T b 4.2 K), as these temperatures were realized by submerging the sample in pumped liquid helium. The measured voltages were converted into diagonal resistances R xx using Ohms law. The SdH line shape of background subtracted longitudinal magnetoresistance (∆R xx ) was quantitatively analyzed by fitting them with a simplified form of Lifshitz-Kosevich theory [41,54,[56][57][58][59] to determine the elevated temperature of the carriers due to the increment of the current bias; the dependency of the carrier temperature on current bias was obtained.

Figure 1(a) shows the diagonal magnetoresistance at I ac = 2 µA, as bath temperature of the sample varied from 1.67 K to 3.00 K. The figure shows that, while the R xx at zero magnetic field is increasing with the sample temperature, the amplitudes of the SdH oscillations are decreasing with the temper ature. Figure 1(b) shows the magnetoresistance at 1.67 K, as the ac-bias current I ac varied from 2 µA to 12 µA. Here, R xx corresponding to the magnetic fields below 0.03 T, is hardly influenced by the bias current while it is substantially changing with the magnetic field after 0.03 T. The R xx under 0.03 T is similar in shape to the weak localization effect. [60] In between 0.03-0.07 T, magnetoresistance is decreasing both with the magnetic field and I ac , and at 0.01 T, R xx reduces by 29% when current goes from 2 µA to 12 µA. Even though the magnetic field at which SdH oscillations appear in R xx is shifting with the temper ature (figure 1(a)), we do not see a significant change in the starting point of SdH oscillation regime with the current bias.

Both an increase in the temperature and the bias current caused a decrease the amplitude of SdH oscillations, which suggests a carrier heating effect produced by the ac bias. Therefore, to understand and quantify the ac-bias effect, we have examined the lineshape of the SdH oscillations. To examine the amplitudes of SdH oscillations, we have removed the background R xx from the data, and background subtracted diagonal resistances ∆R xx are shown in figure 2. Data were taken for two different samples with two different mobility values, and both of them show similar variation of oscillation amplitudes with the bias current. Upper abscissa axis of the curves in figure 2 indicates the inverse magnetic field divided by the period of the oscillations. It can be clearly see that all the minima are located along consecutive integer numbers, and it illustrates the periodicity of oscillations with the inverse magnetic field.

∆R xx data were fitted to a simplified formula based on Lifshitz-Kosevich theory to obtain the temperature increment induced by the I ac . Per the Lifshitz-Kosevich theory [56,57], oscillatory part of the diagonal conductivity in the SdH region can be expressed as a function of magnetic field, temperature, and chemical potential. Also, Ando has derived an expression for the SdH oscillations in diagonal conductivity of 2DES as a result of short-range scattering [58,59]. Further, he shows that in weak magnetic fields oscillations become nearly sinusoidal, with an exponentially damping with the inverse magnetic field and the effective temperature. Since the measured quantity in the experiment was resistance and measurements were taken at weak magnetic fields, we have used following semi-empirical expression to fit our data [41]. Where the damping factor α =(2π 2 k B m * / e) and ∆T = T e -T b . Here m * is the effective mass of the carriers, T b is the bath temperature, T e is the carrier temperature, and B is the magnetic field. Figure 3 shows the experimental data and the numerical fits of ∆R xx at a bath temperature T = 1.67 K with different I ac .

The fit extracted carrier temperatures with the bias current for different bath temperatures are shown in figure 4(a). All the curves converge towards the bath temperature when the ac-bias currents approach zero. According to this analysis, the carrier temperature is linearly increasing with the ac bias current. To clearly illustrate the linear relationship, we have plotted the increment of the carrier temperature with the bias current in the inset to figure 4(b). Curves at different sample temperatures are offset by 0.02 K for the sake of clarity. Figure 4(b) shows the calculated rate of carrier temperature increment with the ac current for both samples at different bath temperatures.

The electron temperature extracted from the amplitudes of SdH oscillations in the GaAs/AlGaAs 2D electron system in low magnetic fields appears to increase linearly with the applied ac bias current at the rate of 4.93 ± 0.04 mK µA -1 . The carrier temperature in a metallic system under Joule heating depends on both heating, and relaxation, which occurs due to both electron-electron scattering and electron-phonon scattering [36,[61][62][63][64][65][66][67][68][69]. The contribution to the energy relaxation from each mechanism depends also on experimental parameters such as the magn etic field and sample temperature. Since these experiments indicate that the electron temperature is higher than the lattice temperature, the electronic system is apparently in a non-equiibrium state with hot electrons in a lattice that is in thermal equilibrium with the liquid helium bath, i.e. T e > T b [36,51,65].

Below, we estimate the dependence of the electron temper ature on the current in order to compare with the results of figure 4. In the absence of an external magnetic field, an applied current bias, I, produces Joule heating as P = I 2 R, where R is the resistance of the specimen. This Joule heating boosts the electronic energy per unit area, u, as I 2 R = A∆u/τ e , where τ e is the energy relaxation time, and A = L × W , with L (W) as the length (width) of the Hall bar. From weak localization studies in such specimens, the inelastic length was determined, and from that, we obtained an energy relaxation time, τ e = 4.75 × 10 -9 s. The electronic energy is given by u = ED(E) f (E, T)dE, where D(E) is the 2D electronic density of states, E is the energy, and f (E, T) is the Fermi distribution function. Using the Sommerfeld expansion and neglecting higher order terms O([k B T] 4 ), u ≈ (nE F /2)+(nπ 2 (k B T) 2 /6E F ), where n is the electron density, and E F is the Fermi energy. Thus, the increase in the electron temperature, ∆T can be estimated from

The effective mass of the electron and fermi energy of the system are 0.064 and 9.2 meV respectively. Using standard Drude relations, the relationship between the temperature increment and bias current in the absence of a magnetic field becomes ∆T ≈ 6 2 τ e /πk 2 B e 2 W 2 nτ m I . Where τ m is the momentum relaxation time, and it is equal to 3.64 × 10 -10 s.

These relations suggest a linear variation in the electron temperature with the current, as and they indicate dT/dI ≈ 24 mK µA -1 in the absence of a magnetic field, which is compared here with the observed value dT/dI ac = 4.93 mK µA -1 obtained from this study of the low magnetic field SdH effect. The study of the faster than expected rise in the electron temperature with the current will constitute a topic for future investigation. The rate of increment of the carrier temperature with the bias current at different sample temperatures for two different AlGaAs samples with mobility equal to 6.6 × 10 6 cm 2 V -1 s -1 (blue squares) and 1.1 × 10 7 cm 2 V -1 s -1 (red circles). The inset shows the calculated Increment of the carrier temperatures with the bias current at different sample temperatures for sample with 6.6 × 10 6 cm 2 V -1 s -1 mobility. Data at different sample temperatures are offset by 0.02 K for the clarity.