Effective $g$ Factor of Subgap States in Hybrid Nanowires

We use the effective $g$ factor of Andreev subgap states in an axial magnetic field to investigate how the superconducting density of states is distributed between the semiconductor core and the superconducting shell in hybrid nanowires. We find a steplike reduction of the Andreev $g$ factor and an improved hard gap with reduced carrier density in the nanowire, controlled by gate voltage. These observations are relevant for Majorana devices, which require tunable carrier density and a $g$ factor exceeding that of the parent superconductor.

Figure 1

(a) False-color electron micrograph of device 1, showing InAs nanowire (green), three-facet Al shell (blue), $\mathrm{Ti}/\mathrm{Au}$ contacts (yellow), and bottom gates (grey). (b) Schematic device cross section showing the orientation of the applied magnetic field, $B$, and Al shell relative to the bottom gate. (c) Magnitude of the effective $g$ factor, $|g^{*}|$, of the lowest subgap state showing a steplike dependence on the bottom-gate voltage, $V_G$. Error bars are root-mean-square difference between upper (electron) and lower (hole) branches. (d) Differential conductance, $dI/dV$, as a function of source-drain bias, $V_{\mathit{SD}}$, at gate voltage $V_G=0.0\mathrm{V}$. Dashed lines correspond to $|g^{*}|=34$. (e)–(g) Similar to (d), but taken at gate voltage (e) $V_G=-2.0\mathrm{V}$, (f) $V_G=-4.0\mathrm{V}$, and (g) $V_G=-7.5\mathrm{V}$, giving (e) $|g^{*}|=27$, (f) $|g^{*}|=6.6$, and (g) $|g^{*}|=4.3$.

Figure 2

(a) False-color electron micrograph of device 2, consisting of InAs nanowire (green) with two-facet Al shell (blue), $\mathrm{Ti}/\mathrm{Au}$ contact and side gates (yellow), and $\mathrm{Ti}/\mathrm{Al}/\mathrm{V}$ contact (purple). (b) Schematic device cross section showing the direction of the applied magnetic field, $B$, and the orientation of the Al shell relative to the back gate. (c) Effective $g$ factor, $|g^{*}|$, as a function of the applied back-gate voltage, $V_{\mathrm{BG}}$. (d) Subgap state evolution in $B$, measured at $V_{\mathrm{BG}}=4.9\mathrm{V}$. The white, dashed lines correspond to $|g^{*}|=19$. (e) Same as (d) but taken at back-gate voltage $V_{\mathrm{BG}}=-2.4\mathrm{V}$, giving $|g^{*}|=4.1$.

Figure 3

(a) Conductance as a function of $V_{\mathit{SD}}$ and $B$ from device 3 taken at $V_G=-5.0\mathrm{V}$. A quasicontinuous band of ABSs have $|g^{*}|=10$. The gap closing-reopening feature around $B=0.9\mathrm{T}$ coincides with the formation of a zero-bias peak persisting up to $B\sim 1.7\mathrm{T}$. The white arrow indicates a Zeeman splitting of $\sim 225\mu \mathrm{eV}$. (b) Linecuts taken from (a) display a hard superconducting gap evolving into a zero-bias peak in high field at a subgap-state-rich regime. (c) Similar to (a) but taken at $V_G=-9.0\mathrm{V}$. A discrete ABS with $|g^{*}|=5.7$ coalesces at zero energy around $B=1.0\mathrm{T}$. The white arrow at $B=1.8\mathrm{T}$ corresponds to a Zeeman splitting of $\sim 125\mu \mathrm{eV}$. (d) Linecuts taken from (c) show the emergence of a symmetric zero-bias peak with low base conductance.

Figure 4

(a) Conductance as a function of $V_{\mathit{SD}}$ and $B$ for device 4 at $V_{SG}=-5.1\mathrm{V}$ shows subgap states with $|g^{*}|=10$ coalescing at $B\sim 1.1\mathrm{T}$, while an excited state increases in energy. (b) Linecuts taken from (a) illustrate the formation of a zero-bias peak at high field. A pair of low-conductance excited states resembling gap closing and reopening are visible around $B=1.0\mathrm{T}$.