Direct Measurement of the Spin-Orbit Interaction in a Two-Electron InAs Nanowire Quantum Dot

Abstract

We demonstrate control of the electron number down to the last electron in tunable few-electron quantum dots defined in catalytically grown InAs nanowires. Using low temperature transport spectroscopy in the Coulomb blockade regime, we propose a method to directly determine the magnitude of the spin-orbit interaction in a two-electron artificial atom with strong spin-orbit coupling. Because of a large effective $g$ factor $|{g}^{*}|=8\ifmmode\pm\else\textpm\fi{}1$, the transition from a singlet $S$ to a triplet ${T}^{+}$ ground state with increasing magnetic field is dominated by the Zeeman energy rather than by orbital effects. We find that the spin-orbit coupling mixes the ${T}^{+}$ and $S$ states and thus induces an avoided crossing with magnitude ${\ensuremath{\Delta}}_{\mathrm{SO}}=0.25\ifmmode\pm\else\textpm\fi{}0.05\text{ }\text{ }\mathrm{meV}$. This allows us to calculate the spin-orbit length ${\ensuremath{\lambda}}_{\mathrm{SO}}\ensuremath{\approx}127\text{ }\text{ }\mathrm{nm}$ in such systems using a simple model.