Weak Antilocalization Effect in an AlAs/AlGaAs Quantum Well

Weak antilocalization in a narrow AlAs quantum well containing a two-dimensional electron system with a large effective mass at low temperatures has been studied. Such quantum corrections are due to a strong spin–orbit coupling in it. The spin–orbit interaction constant has been determined from the approximation of experimental data by a theoretical model in the diffusion approximation. Additionally, this constant has also been independently measured from the modification of the single-particle g-factor in the quantum Hall effect regime in the same sample using electron paramagnetic resonance. Electron paramagnetic resonance spectroscopy and analysis of the weak antilocalization effect yield close values of the interaction constants β = 7.6 and 10.1 meV Å, respectively. Agreement between β values thus obtained becomes full if effects of the strong electron–electron interaction are taken into account in the weak antilocalization model by renormalizing the effective mass of the electron.

The conductivity of a two-dimensional electron system at low temperatures and in weak magnetic fields is modified because of the quantum mechanical effects of self-interference of electrons moving on closed diffusion trajectories. In the case of constructive interference, the resistance of the two-dimensional channel increases, and this effect is called weak localization [1,2]. Strong spin-orbit coupling allows spin flop at a scattering event along a trajectory and, as a result, mixing of the motion of the electron along two trajectories in the same direction and destructive interference of electron waves [3][4][5][6][7][8][9][10][11]. In this case, the resistance of the two-dimensional channel decreases, and this effect is called weak antilocalization. An external magnetic field destroys both weak localization and weak antilocalization.
It is of great importance to study the described quantum corrections to the conductivity. In particular, the shape of the magnetic field dependence of the magnetoresistance in the presence of weak localization effects can provide information on the coherence of electron waves. The analysis of the magnetic field destroying weak antilocalization allows one to extract the parameters of spin-orbit coupling representing one of the key interactions for both fundamental spin condensed matter physics and development of spintronic devices.
In this work, we detect for the first time the weak antilocalization effect in a narrow AlAs quantum well containing a two-dimensional electron system. The effective mass of the electron in this structure is m b = 0.25m 0 [12]. The large effective mass ensures a low kinetic energy compared to the characteristic Coulomb energy and is responsible for a significant role of the strong electron-electron interaction. In two-dimensional electron systems with a comparable effective mass, a number of striking multiparticle phenomena are observed, including Wigner crystallization [13], Stoner ferromagnetic transition in the quantum Hall effect regime [14,15], and condensation of spin waves [16]. Furthermore, as shown in [17,18], narrow AlAs quantum wells are also characterized by a quite strong spin-orbit coupling. The combination of these properties makes such material systems very promising for the study of fundamental collective phenomena associated with the spin degree of freedom of the electron. We emphasize that most theories used to describe the discussed quantum corrections are single-particle. Thus, the study of the weak antilocalization effect in strongly correlated electron system hosted in the AlAs quantum well is exceptionally important and, in particular, can result in refining existing theories and creation of new ones.
The analysis of quantum corrections to the conductivity of the two-dimensional channel allows one to determine the characteristic phase coherence time and the spin-orbit coupling constant β. The feature of this study is the possibility to independently determine the spin-orbit coupling constant from the modification of single-particle spin splitting in the quantum Hall effect regime [17].

CONDENSED MATTER
Experiments were performed on a 4-nm AlAs/AlGaAs quantum well grown in the [001] direction using molecular beam epitaxy. The sample had the form of a standard 200-μm-wide Hall bar with drain/source and six potentiometric contacts (see Fig. 1a). Ohmic contacts to the two-dimensional channel were formed by the deposition and subsequent annealing of indium into the contact regions. The low-temperature values of the two-dimensional electron density and mobility were cm -2 and cm 2 /(V s). The sample was immersed in liquid He-3, whose vapor was evacuated so that the temperature of the sample was T = 0.5 K. The sample was placed in the center of a superconducting magnet, so that experiments could be carried out in magnetic fields up to 15 T. A typical longitudinal resistance of the sample at this temperature is shown in Fig. 1b; the positions of the first several minima of Shubnikov-de Haas oscillations are marked. In the electron paramagnetic resonance (EPR) spectroscopy experiments, an oversized waveguide was used to deliver microwave radiation to the sample. Backward wave oscillators, as well as a set of frequency multiplayers coupled to a microwave generator, were used as sources of radiation.
Electron paramagnetic resonance was detected by monitoring the magnetoresistance of the two-dimensional channel, which is extremely sensitive to microwave absorption, as shown in [19]. More precisely, EPR is detected as a peak in at a fixed microwave frequency under the smooth variation of the magnetic field. To improve the signal-to-noise ratio, we used the standard double lock-in technique, which was described in detail in [20][21][22]. Typical EPR peaks observed in the experiment near a filling factor of 3 are shown in Fig. 1c. The microwave frequency F used to excite spin excitons [23] is indicated next to each peak. It is clearly seen that the Q-factor of resonances is sufficiently high to accurately measure their magnetic field positions and, thereby, to determine the singleparticle Landé g-factor of the electron g* = hF/(μ B B), where h is the Planck constant and μ B is the Bohr magneton. The magnetic field dependence of the g-factor is shown in Fig. 1d. The shape and amplitude of the resonance peaks observed in the experiment were independent of the ramp rate of the magnetic field; therefore, the dynamic polarization of nuclei [24,25] was insignificant in the experiment.
We briefly describe the determination of the spinorbit coupling constant from the magnetic field dependence of the electron g-factor; it was presented in detail in our preceding work [17], where it was also demonstrated that the Dresselhaus spin-orbit coupling linear in the wave vector is the dominant type. The energy spectrum of the electron in-plane motion in strong quantizing fields is a set of spin-split Landau levels. The spin-orbit coupling in this basis is not × 11 = 7.7 10 n μ × 4 = 1.4 10 xx R diagonal and mixes Landau levels with different indices and different electron spin projections on the Oz axis. In this case, single-particle spin splitting and, hence, the g-factor of the electron are modified. In the second order of perturbation theory, this correction for strong magnetic fields can be represented in the form (1) where N is the index of the Landau level between spinsplit sublevels of which a spin transition occurs acompanied by the resonant absorption of a photon, A is a positive constant expressed in terms of known material parameters and world constants, and β is the Dresselhaus spin-orbit coupling constant. The approximation of the experimental magnetic field dependence of the g-factor by Eq. (1) is shown in Fig. 1d, where the indices of Landau levels are given next to the lines. We emphasize that β is the only parameter of the approximation. It is clearly seen that the data are reproduced very well by the theoretical curves obtained with the coupling constant β = 7.6 meV Å. Figure 1e shows the dependence of the measured g-factor at odd filling factors ν on ν 2 . According to Eq. (1), this dependence should be linear and the coupling constant β can be obtained from the slope of this straight line. The result is β = 7.6 meV Å. The intersection of the straight line with the y axis gives in the limit of high magnetic fields, where spin-orbit modification is insignificant. As shown in [18], the value is independent of the width of the quantum well and is in good agreement with the corresponding component of the tensor for the wide AlAs quantum well [26]. We emphasize that the nonparabolicity does not affect the coupling constant β thus determined [27,28].
We now analyze the weak antilocalization effect in the two-dimensional electron system under study. The experimental data for the dependence of the quantum corrections to the conductivity of the channel measured at a temperature of T = 0.5 K on the magnetic field applied perpendicularly to the plane of the system are shown by empty circles in Fig. 2a. For the sake of convenience, the conductivity curve is shifted along the vertical axis so that = 0 in the absence of magnetic field.
In agreement both with various theoretical models and with experimental data obtained by other authors, the measured magnetoconductivity of the two-dimensional channel first decreases with increasing magnetic field and then increases. The minimum is observed at a magnetic field determined by the spinorbit coupling constant, and the depth of the dip in the conductivity curve depends, in particular, on the phase coherence time of electron waves. Another key parameter in the theoretical description of quantum corrections is the so-called transport magnetic field at which the mean free path of the electron l is comparable with the magnetic length . The diffusion approximation is valid at magnetic fields B < B tr [3]. For the structure under study, B tr = 8 mT; consequently, the diffusion approximation is applicable in a relatively wide range. Within the model presented in [8,9], experimental data can be approximated with two independent parameters: the spin- When only one type of spin-orbit coupling, e.g., the Dresselhaus spin-orbit coupling linear in the wave vector, exists in the system, this theoretical approach allows an analytical solution. This is the case in narrow AlAs quantum wells [17]. The corresponding theoretical curve is shown in Fig. 2a. The experimental data at weak magnetic fields The spin-orbit coupling constant extracted from the analysis of the weak antilocalization effect is close to the value determined in the EPR experiments. The difference between these coupling constants can be attributed to the influence of the strong electronelectron interaction on the weak antilocalization φ τ effect. For example, a strong multiparticle interaction can significantly modify spin relaxation processes [29] and even enhance spin-orbit splitting [30]. Such an enhancement for the dimensionless parameter r s describing the strength of the electron-electron interaction in the structure under study is no more than 10% and cannot be responsible for the observed difference between the coupling constants β measured by different methods in this work. The development of an exact theoretical model to describe the weak antilocalization effect in the diffusion approximation in the presence of strong multiparticle correlations is beyond the scope of this work. Most of the transport processes in strongly correlated systems are accompanied by the modification of the spectrum of quasiparticles because of the multiparticle interaction, which can be taken into account in the simplest way by the Fermi liquid renormalization of the effective mass of quasiparticles [31][32][33][34]. This approach is used to describe various effects both at weak magnetic fields (e.g., the renormalization of the mass specifying the period of microwave-induced oscillations of the magnetoresistance [35]) and in quantizing magnetic fields (e.g., ferromagnetic phase transitions in the quantum Hall effect regime [14]). If the effective mass of particles involved in quantum interference along closed trajectories is m* = 1.33m b , experimental data are approximated well with the coupling constant β = 7.6 meV Å coinciding with the value determined from EPR spectroscopy. The corresponding curve at = 35 ps is shown in Fig. 2b. We emphasize that the quality of the approximation of experimental data with both the band and renormalized masses is the same. The chosen m* value is in satisfactory agreement with the masses obtained in [36,37] from the analysis of the behavior of Shubnikov-de Haas oscillations in similar structures at different temperatures.
To summarize, the weak antilocalization effect in a narrow AlAs quantum well containing a two-dimensional electron system with a large effective mass at low temperatures has been studied for the first time. The corresponding corrections to the conductivity have been described well within the diffusion model proposed in [8]. The approximation of the experimental data gives the Dresselhaus spin-orbit coupling constant β = 10.1 meV Å. A close spin-orbit coupling constant β = 7.6 meV Å has been independently determined from the modification of the single-particle g-factor measured using electron paramagnetic resonance in the quantum Hall effect regime in the same sample. The difference between these coupling constants is apparently due to the strong electron-electron interaction. Agreement between β values thus obtained becomes full if effects of the strong electronelectron interaction are taken into account in the weak antilocalization model within a simplified approach by renormalizing the effective mass of the electron.