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

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 $|g^{*}|=8\pm 1$, 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 ${\Delta }_{\mathrm{SO}}=0.25\pm 0.05\mathrm{meV}$. This allows us to calculate the spin-orbit length ${\lambda }_{\mathrm{SO}}\approx 127\mathrm{nm}$ in such systems using a simple model.

Figure 1

(a) Sample schematic showing the Au grid covered in SiN (black) and the nanowire (red) on top. The source and drain contacts and the five individually contacted gates are indicated. (b) Differential conductance measured as a function of $V_{\mathrm{ds}}$ and plunger gate voltages. The other gates are set to $V_{g1}=-2.56\mathrm{V}$, $V_{g4}=-1.65\mathrm{V}$, and $V_{g5}=2.5\mathrm{V}$, respectively. Electron numbers are indicated in yellow, and the inset shows a scanning electron microscopy image of the wire on the electrode grid, with a $1\mu \mathrm{m}$ scale bar.

Figure 2

Evolution of the differential conductance peaks as a function of $B$ along the two cuts ($a$) and ($b$) in Fig. 1b.

Figure 3

Orbital splitting $\Delta \epsilon$ and singlet-triplet splitting ${\Delta }_{\mathrm{ST}}$ as a function of $V_G$ and $B$. In (a) and (b), $\Delta \epsilon$ was derived from the corrected difference between $\uparrow$ and $\Uparrow$ [see Figs. 1b, 2a]. In (c), ${\Delta }_{\mathrm{ST}}$ was calculated from the data in Fig. 1b, and in (d) both splittings ${\Delta }_{{\mathrm{ST}}^{+}}$ (□) and ${\Delta }_{{\mathrm{ST}}^0}$ (■) were extracted from Fig. 2b. For comparison, the small dots in (a) and (c) indicate the bare bias splittings for negative (blue) and positive (red) $V_{\mathrm{ds}}$.

Figure 4

(a) Inset: Singlet-triplet transition from cut ($b$) in Fig. 1 after gate voltages have been optimized. (a) Main panel: Enlargement of the transition between ${\mu }_{T^{+}}$ and ${\mu }_S$ exhibiting a clear anticrossing. (b) Energy levels extracted from the corrected peak splittings in (a) together with a fit to a simple model indicated by the red lines.