![]() The relation between the symmetry classes (orthogonal, symplectic, and unitary) and the weak localization (WL) and antilocalization (WAL). These symmetry arguments have been well known since the studies on the conventional 2D electron gases. If a system has both time-reversal and spin-rotational symmetries, then it is in the orthogonal class, in which the weak localization is expected. If a system has time-reversal symmetry but no spin-rotational symmetry, then it is in the symplectic class, in which the weak antilocalization is expected. If a system has no time-reversal symmetry, then it is in the unitary class, in which there is no weak localization or antilocalization. According to the classification of the ensembles of random matrix, there are three symmetry classes. Whether a system has weak localization or weak antilocalization depends on the symmetry (see Table 1). In contrast, the quantum interference can be negative, leading to a weak localization effect and totally opposite temperate and magnetic dependencies of conductivity. (b) and (c) The signatures of the weak antilocalization (WAL) in (b) temperature ( T) dependence of the conductivity σ and (c) magnetoconductivity, where B is the magnetic field. The open circles represent impurities and the arrows mark the trajectories that electrons travel. (a) Schematic illustration of time-reversed scattering loops in the quantum diffusion regime in disordered metals. This negative magnetoconductivity or positive magnetoresistivity is the more familiar signature of the weak antilocalization (see Fig. One can then observe that the conductivity goes down with increasing magnetic field. Because this correction requires time reversal symmetry, it can be suppressed by applying a magnetic field. If the quantum interference correction is positive, then it gives a weak antilocalization correction to the conductivity and the conductivity goes up with decreasing temperature (see Fig. Therefore, the quantum diffusion usually takes place at extremely low temperatures e.g., below the liquid helium temperature. The inelastic scattering has to be suppressed significantly to make the phase coherence length much longer than the mean free path. The phase coherence length is determined by inelastic scattering from electron–phonon coupling and interaction with other electrons. The weak localization or weak antilocalization arises due to this correction in the conductivity in the quantum diffusive regime. 1(a)) can give rise to a correction to the conductivity. This is the quantum diffusive regime, in which the quantum interference between time-reversed scattering loops (see Fig. If the mean free path is much shorter than the system size and the phase coherence length, then the electrons suffer from scattering but can maintain their phase coherence. The mean free path measures the average distance that an electron travels before its momentum is changed by elastic scattering from static scattering centers, while the phase coherence length measures the average distance that an electron can maintain its phase coherence. The quantum diffusion in disordered metals can be defined by the mean free path ℓ and the phase coherence length ℓ ϕ. These results pave the way toward silicon-compatible spintronic devices.Weak antilocalization is a transport phenomenon in the quantum diffusion regime in disordered metals. We also report here the numerical methods and code developed for calculating the magneto-resistance in the ballistic regime, where the commonly used HLN and ILP models for analyzing weak localization and anti-localization are not valid. As the density becomes larger than ∼6 × 10 11 cm −2, the spin precession length becomes shorter than the mean free path, and the system enters the ballistic spin transport regime. The mobility and the mean free path increase with increasing hole density, while the spin precession length decreases due to increasingly stronger spin–orbit coupling. From the magneto-resistance, we extract the phase-coherence time, spin–orbit precession time, spin–orbit energy splitting, and cubic Rashba coefficient over a wide density range. In this letter, we report the observation of a gate-induced crossover from weak localization to weak anti-localization in the magneto-resistance of a high-mobility two-dimensional hole gas in a strained germanium quantum well. For practical spin field-effect transistors, another essential requirement is ballistic spin transport, where the spin precession length is shorter than the mean free path such that the gate-controlled spin precession is not randomized by disorder. Gate-controllable spin–orbit coupling is often one requisite for spintronic devices.
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