Spin-Orbit Coupling and Anisotropy of Spin Splitting in Quantum Dots

In lateral quantum dots, the combined effect of both Dresselhaus and Bychkov-Rashba spin-orbit coupling is equivalent to an effective magnetic field $\pm B_{\mathrm{SO}}$ which has the opposite sign for $s_z=\pm 1/2$ spin electrons. When the external magnetic field is perpendicular to the planar structure, the field $B_{\mathrm{SO}}$ generates an additional splitting for electron states as compared to the spin splitting in the in-plane field orientation. The anisotropy of spin splitting has been measured and then analyzed in terms of spin-orbit coupling in several $\mathrm{AlGaAs}/\mathrm{GaAs}$ quantum dots by means of resonant tunneling spectroscopy. From the measured values and sign of the anisotropy we are able to determine the dominating spin-orbit coupling mechanism.

Figure 1

Schematic picture (a) and energy diagram (b) of the studied device under finite bias and finite magnetic field.

Figure 2

Differential conductance as a gray-scale plot for sample $A$ ($\hslash \omega =31.2\mathrm{meV}$) as a function of bias voltage and magnetic field for (a) $B_z$ being perpendicular to the quantum well plane, and (b) $B_{\parallel }$, $T=20\mathrm{mK}$.

Figure 3

Differential conductance as a gray-scale plot for sample $B$ ($\hslash \omega =13.8\mathrm{meV}$) as a function of bias voltage and magnetic field for (a) $B_z$ being perpendicular to the quantum well plane, and (b) $B_{\parallel }$, $T=20\mathrm{mK}$.

Figure 4

(a) Spin splitting of the state $A1$ of sample $A$ for both $B$-field configurations. (b) Spin splitting of the state $B1$ of sample $B$ for both $B$-field configurations. Solid lines represent the result of a linear fit to the high-field part of the data. Inset shows the spin splitting of the states $B2$ (triangles), $B3$ (circles), and $B4$ (stars) for both $B$-field configurations.