Weak localization in low-symmetry quantum wells

Theory of weak localization is developed for electrons in semiconductor quantum wells grown along [110] and [111] crystallographic axes. Anomalous conductivity correction caused by weak localization is calculated for symmetrically doped quantum wells. The theory is valid for both ballistic and diffusion regimes of weak localization in the whole range of classically weak magnetic fields. We demonstrate that in the presence of bulk inversion asymmetry the magnetoresistance is negative: The linear in the electron momentum spin-orbit interaction has no effect on the conductivity while the cubic in momentum coupling suppresses weak localization without a change of the correction sign. Random positions of impurities in the doping layers in symmetrically doped quantum wells produce electric fields, which result in a position-dependent Rashba coupling. This random spin-orbit interaction leads to spin relaxation, which changes the sign of the anomalous magnetoconductivity. The obtained expressions allow determination of electron-spin relaxation times in (110) and (111) quantum wells from transport measurements.

Figure 1

Symmetrically doped (110) quantum well structure. Electrons propagating in the QW plane interfere due to the presence of self-intersecting paths where they acquire the same phase passing the loop clockwise and anticlockwise. Nonmirror symmetric positions of impurities in the doping layers result in a random electric field, which leads to spin relaxation via the Rashba effect.

Figure 2

Magnetoconductivity with account for $\mathit{k}$-cubic BIA spin-orbit interaction with ${\Omega }_3\tau =0,0.05,0.1,0.2,0.3,0.4,0.5$ (from bottom to top) at the dephasing rate $\tau /{\tau }_{\varphi }=0.01$. Solid lines represent the exact result, dashed lines are obtained in the diffusion approximation. Inset: Fermi contours in two spin subsystems at $\hslash {\Omega }_3/E_{\mathrm{F}}$ = 0.01, 0.2, and 0.5.

Figure 3

Zero-field conductivity correction in the presence of $\mathit{k}$-cubic term ${\Omega }_3$. The dephasing rate $\tau /{\tau }_{\varphi }=0.001,0.005,0.01,0.02,0.03,0.05,0.07,0.1$ (from bottom to top).

Figure 4

Magnetoconductivity at different values of the spin-flip rate $\tau /{\tau }_s=0,0.05,0.1,0.2,0.3$, and 0.4 (from bottom to top). The dephasing rate $\tau /{\tau }_{\varphi }=0.01$. Dashed curves are calculated in the diffusion approximation, Eq. (51).

Figure 5

Zero-field conductivity correction as a function of the dephasing rate at the spin-flip scattering rate $\tau /{\tau }_s=0,0.005,0.01,0.02,0.03,0.05,0.1,0.2,0.3$, and 0.4 (from bottom to top).