Spectroscopy of Spin-Orbit Quantum Bits in Indium Antimonide Nanowires

A double quantum dot in the few-electron regime is achieved using local gating in an InSb nanowire. The spectrum of two-electron eigenstates is investigated using electric dipole spin resonance. Singlet-triplet level repulsion caused by spin-orbit interaction is observed. The size and the anisotropy of singlet-triplet repulsion are used to determine the magnitude and the orientation of the spin-orbit effective field in an InSb nanowire double dot. The obtained results are confirmed using spin blockade leakage current anisotropy and transport spectroscopy of individual quantum dots.

Figure 1

(a) An electron moving with momentum $\overrightarrow{k}$ through the wire experiences a spin-orbit field ${\overrightarrow{B}}_{\mathrm{SO}}$ which rotates the spin by $\pi$ after a distance $l_{\mathrm{SO}}$. The vector $\overrightarrow{E}$ indicates the likely direction of the electric field. In the case of spin-orbit coupling due to structural inversion asymmetry, ${\overrightarrow{B}}_{\mathrm{SO}}\propto \overrightarrow{E}\times \overrightarrow{k}$ [32]. (b) Schematic of a double quantum dot in an InSb nanowire. Regions close to the source electrode, drain electrode, and the two dots (dot L and dot R) remain undepleted by the gates. The gates LB, CB, and RB define the left, central, and right barriers. The gates LP and RP are the left and right plungers used to control the electron number on each dot. (c) Scanning electron microscopy of a nanowire device similar to the one used in the measurements. (d) Charge stability diagram of the double dot for source-drain voltage $V_{\mathrm{sd}}=1\mathrm{mV}$. Typical charging energy is 10 meV. Numbers in brackets correspond to the charge occupation on the left and the right dots. The inset shows the charge stability diagram near the $(1,1)\to (0,2)$ charge transition for $V_{\mathrm{sd}}=5\mathrm{mV}$. The detuning axis $\epsilon$ is indicated by the dashed arrow.

Figure 2

(a) Left: blocked parallel configuration. $\phi$ is the angle between the nanowire axis ${\overrightarrow{n}}_W$ and $\overrightarrow{B}$. Right: microwaves applied to the LP gate induce EDSR. Tunneling to the right dot is allowed when the left qubit is rotated to antiparallel configuration. (b) $V$-shaped EDSR resonances with slopes providing $g_L$ and $g_R$ for $\phi =130°$ and $V_{\mathrm{sd}}=8\mathrm{mV}$. A larger $g$ factor was assigned to the larger dot, i.e., to the dot with smaller orbital energy (the orbital energy is 5 meV for the left dot and 7.5 meV for the right dot). $V$-shaped lines with half the slope are two-photon transitions. The enhanced current around $B=0$ is due to spin mixing in the absence of microwaves (see 29, Section S2). Resonances at constant $f$ are due to photon-assisted tunneling enhanced by cavity modes. (At each frequency, the maximum current is normalized to 1 pA and a constant offset is subtracted for clarity.) The inset shows the energy spectrum of weakly coupled double dots with arrows illustrating the observed transitions. (c) Current versus $f$ and $\phi$ for $B=35\mathrm{mT}$. The vertical axis on the left is rescaled to $g=hf/{\mu }_{B}B$. (At each field, a constant current offset is subtracted for clarity.) The $B$-field orientation with respect to nanowire in (b),(c) is indicated by cartoonlike drawings of arrows over the cylinders.

Figure 3

(a) Current, in color, versus $f$ and $B$ for the detuning $\epsilon \approx 0.5\mathrm{meV}$ ($V_{\mathrm{sd}}=-5\mathrm{mV}$). The dashed lines are fits to a model described in Ref. 29, Section S4. The line colors match transitions indicated in (c). (At each frequency, a current offset is subtracted for clarity.) (b) Diagram illustrating a strongly coupled double quantum dot realized by applying a more positive voltage to the central gate. (c) Energy diagram deduced from (a) and used to extract the $S\mathrm{\text{−}}T$ spin-orbit gap ${\Delta }_{\mathrm{SO}}^{\mathrm{DD}}$. Arrows indicate transitions observed in (a). In the absence of coupling, the triplet and the singlet state would simply cross, as indicated by dashed lines.

Figure 4

(a) As the left electron tunnels to the right, it experiences a field ${\overrightarrow{B}}_{\mathrm{SO}}$. (b)–(d) The avoided crossing in the EDSR spectrum, as in Fig. 3a, for three directions of $\overrightarrow{B}$: $\phi =170°$, $\phi =110°$, and $\phi =90°$ ($V_{\mathrm{sd}}=-5\mathrm{mV}$). (At each magnetic field, an offset is subtracted for clarity.) (e) Transitions between (1, 1) states and $S(0,2)$ at finite $B$. The two singlet states are hybridized due to tunnel coupling. $T_{+}(1,1)$ and $T_{-}(1,1)$ are coupled to $S(0,2)$ due to ${\overrightarrow{B}}_{\mathrm{SO}}$. This SOI-induced coupling scales as $|{\overrightarrow{B}}_{\mathrm{SO}}\times \overrightarrow{B}|$ for small $\overrightarrow{B}$ [22]. (f)–(h) $I$ versus $\epsilon$ and $B$ for the same orientations of $\overrightarrow{B}$ as in (b)–(d) with microwaves off. (i) Extracted values of ${\Delta }_{\mathrm{SO}}^{\mathrm{DD}}$ (see 29, Section S6) and $I$ at $B=20\mathrm{mT}$ and $\epsilon =0.5\mathrm{meV}$ [indicated by dots in (f)–(h)] as a function of $\phi$. The solid line is a fit to ${\Delta }_{\mathrm{SO}}^{\mathrm{DD}}={\Delta }_{\mathrm{SO}}|\cos (\phi -{\phi }_0)|$, with ${\Delta }_{\mathrm{SO}}=5.2\pm 0.3\mu \mathrm{eV}$ and ${\phi }_0=1°\pm 5°$. The error bars are determined by the width of the EDSR resonance.

Figure 5

(a) Two electrons in the right quantum dot. The separation of the two electrons in the triplet state is of the order of the dot size. (b) Charge stability diagram close to the $(1,1)\to (0,2)$ transition at $B=1.4\mathrm{T}$, for $V_{\mathrm{sd}}=7\mathrm{mV}$ and $\overrightarrow{B}\parallel {\overrightarrow{n}}_W$. The transitions $T_{+}(1,1)\to S(0,2)$ and $T_{+}(1,1)\to T_{+}(0,2)$ are indicated by solid arrows. (c) Resonances corresponding to $T_{+}(1,1)\to S(0,2)$ and $T_{+}(1,1)\to T_{+}(0,2)$ as a function of $B$ for $\phi =180°$. The color maps in (c)–(e) indicate values of $dI/d\epsilon$ in arbitrary units. (d),(e) Avoided crossing for $\phi =180°$ and $\phi =90°$. The dashed lines are fits to the model from Ref. 9. (f) The gap ${\Delta }_{\mathrm{SO}}^{\mathrm{SD}}$ as a function of $\phi$. The solid line is a fit to ${\Delta }_{\mathrm{SO}}^{\mathrm{SD}}={\Delta }_{\mathrm{SO}}^{\prime }\sqrt{{cos}^2(\phi -{\phi }_0){cos}^2\theta +{sin}^2\theta }$, with ${\Delta }_{\mathrm{SO}}^{\prime }=230\pm 10\mu \mathrm{eV}$, ${\phi }_0=2°\pm 5°$, and $\theta =10°\pm 3°$. The error bars are determined by the average linewidth corresponding to $T_{+}(1,1)\to S(0,2)$ and $T_{+}(1,1)\to T_{+}(0,2)$ transitions. Note that the anisotropy of ${\Delta }_{\mathrm{SO}}^{\mathrm{SD}}$ depends on the relative positions of the two electrons in the right dot, which may be different from the nanowire axis. The out-of-plane ${\overrightarrow{B}}_{\mathrm{SO}}$ angle $\theta$ therefore may be nonzero due to confinement details of the right quantum dot. Measurements at the $(1,1)\to (2,0)$ transition yielded the same in-plane anisotropy for the left dot (data not shown).