Weak antilocalization in quantum wells in tilted magnetic fields

Weak antilocalization is studied in an ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ quantum well. Anomalous magnetoresistance is measured and described theoretically in fields perpendicular, tilted, and parallel to the quantum well plane. Spin and phase-relaxation times are found as functions of temperature and parallel field. It is demonstrated that spin dephasing is due to the Dresselhaus spin-orbit interaction. The values of electron spin-splittings and spin-relaxation times are found in the wide range of two-dimensional density. Application of in-plane field is shown to destroy weak antilocalization due to competition of Zeeman and microroughness effects. Their relative contributions are separated, and the values of the in-plane electron $g$ factor and characteristic size of interface imperfections are found.

Figure 1

Calculated energy diagram and squared wave function for the investigated structure.

Figure 2

The conductivity as a function of perpendicular magnetic field measured at $B_{\Vert }=0.3\mathrm{T}$ and $T=1.4\mathrm{K}$ for two neighbor pairs of potential probes as shown in inset. Arrows in inset marked as $B_{\perp }^{\Vert }$ show perpendicular component of in-plane magnetic field, which varies along the sample due to imperfection of magnet system.

Figure 3

The low-field magnetoconductivity at different temperatures for $B_{\Vert }=0$ (symbols). Solid curve is the best fit by Eq. (1) with the parameters ${\tau }_{\varphi }=72\mathrm{ps}$ and ${\tau }_s=8.8\mathrm{ps}$.

Figure 4

The temperature dependences of the phase and spin-relaxation times for $V_g=0$ and $B_{\Vert }=0$.

Figure 5

The values of $\hslash {\Omega }_R$, $\hslash {\Omega }_D$, $\hslash {\Omega }_3$ (circles), and ${\tau }_s$ (squares) as functions of electron density for the investigated structures. Open and solid symbols are the experimental data for $T=0.45\mathrm{K}$ and $1.5\mathrm{K}$, respectively. The curves are the result of self-consistent calculations (Ref. 26) in which we used $\gamma =18\mathrm{eV}{\mathrm{\AA }}^3$. The zero value of $\hslash {\Omega }_R$ at $n\simeq 4.5\times {10}^{15}{\mathrm{m}}^{-2}$, $V_g=-1.8\mathrm{V}$, shows that at this gate voltage the quantum well becomes effectively symmetric.

Figure 6

The conductivity as a function of $B_{\perp }$, measured at $T=1.4\mathrm{K}$ for different in-plane magnetic fields $B_{\Vert }$: 0, 0.075, 0.125, 0.225, 0.325, 0.425, 0.525, and $0.625\mathrm{T}$. The dashed curves are the best fit with only Zeeman contribution for some values of $B_{\Vert }$. Dotted curve illustrates the result of the fit with only Zeeman contribution carried out in the range of $B_{\perp }∕B_{tr}$ from 0 to 0.3. The solid curves are the best fit when both roughness and Zeeman splitting are taken into account carried out with $B_{\varphi }=0.0625\mathrm{mT}$ and $B_s=0.28\mathrm{mT}$ found at $B_{\Vert }=0$ and ${\Delta }_s$ and ${\Delta }_r$ as fitting parameters. For clarity, the plots are separated in vertical direction by the value of $0.1G_0$.

Figure 7

The value of the fitting parameters ${\Delta }_s$ and ${\Delta }_r$ corresponding to solid lines in Fig. 6 as a function of $B_{\Vert }^2$ (symbols). Upper line is Eq. (12) with ${\tau }_s=8.8\mathrm{ps}$, $D=0.09{\mathrm{m}}^2∕\mathrm{s}$, and $g=1.7$, lower one is Eq. (15) with $l=370\mathrm{nm}$ and $d^{2}L=75{\mathrm{nm}}^3$.

Figure 8

The longitudinal magnetoconductivity for $T=1.4\mathrm{K}$ (symbols). Solid curve is Eq. (17) with $B_{\varphi }=0.0625\mathrm{mT}$, $B_s=0.28\mathrm{mT}$, and ${\Delta }_r\left(B_{\Vert }\right)$ and ${\Delta }_s\left(B_{\Vert }\right)$ shown in Fig. 7 by solid curves. Dotted and dashed curves are calculation results when only roughness $\left({\Delta }_s=0\right)$ or Zeeman effect $\left({\Delta }_r=0\right)$ is taken into account, respectively.