Separation of the Rashba and Dresselhaus terms using the Square and Rectangular Loop Arrays in InGaAs/InAlAs Quantum Wells

The spin interference effect [1,2] was investigated for square and rectangular loop arrays that were nanolithographically defined in InGaAs/InAlAs quantum wells both theoretically and experimentally. In theory, interference between the following spin wave functions were considered : $\Psi ^{CW}$=\textbf{R}$_{-x}$(\textit{$\theta $}$_{v})$\textbf{R}$_{y}$(\textit{$\theta $}$_{h})$\textbf{R}$_{x}$(\textit{$\theta $}$_{v})$\textbf{R}$_{-y}$(\textit{$\theta $}$_{h})\Psi _{i}$ (wave function after the clockwise path in a rectangular loop) and $\Psi ^{CCW}$=\textbf{R}$_{y}$(\textit{$\theta $}$_{h})$\textbf{R}$_{-x}$(\textit{$\theta $}$_{v})$\textbf{R}$_{-y}$(\textit{$\theta $}$_{h})$\textbf{R}$_{x}$(\textit{$\theta $}$_{v})\Psi _{i}$ (wave function after the counter-clockwise path), where the spin rotation operators \textbf{R}$_{\xi }$(\textit{$\theta $}$_{v,h})$ were obtained from solving the Poisson and Schrdinger equations self-consistently including the Rashba and Dresselhaus Hamiltonians. Then, the gate-dependence of the norm $\vert \Psi ^{CW}+\Psi ^{CCW}\vert ^{2}$, averaged over all directions for the initial spin ($\Psi _{i})$, were compared to the gate-dependence of the AAS oscillation amplitude in the experiment. We propose that the measurement of the spin interference effect is a reliable method for the simultaneous determination of the Rashba and Dresselhaus terms quantitatively. [1] Koga \textit{et al.}, PRB \textbf{70}, 161302(R) (2004);\textit{ ibid.} \textbf{74}, 041302(R) (2006). [2] Koga \textit{et al.}, phys. stat. sol. (C) \textbf{3}, 4220 (2006).