Quantum Dot Parity Effects in Trivial and Topological Josephson Junctions

An odd-occupied quantum dot in a Josephson junction can flip transmission phase, creating a $\pi$ junction. When the junction couples topological superconductors, no phase flip is expected. We investigate this and related effects in a full-shell hybrid interferometer, using gate voltage to control dot-junction parity and axial magnetic flux to control the transition from trivial to topological superconductivity. Enhanced zero-bias conductance and critical current for odd parity in the topological phase reflects hybridization of the confined spin with zero-energy modes in the leads.

Figure 1

(a) Schematic of a dot junction made from an InAs nanowire (green) containing a quantum dot (QD) with coupling and occupancy controlled by voltage $V_{\mathrm{dot}}$. A voltage bias, $V_b$, with a small ac component was applied across the single dot junction and the current, $I$, measured. Thin Al leads (purple) remain superconducting with applied axial magnetic field $B_{\parallel }$. The lobe structure in the destructive Little-Parks regime accesses trivial (S) or topological (T) superconductivity in the leads [10]. (b) The dot junction was embedded in an interferometer with a reference junction controlled by gate voltage $V_{\mathrm{Ref}}$. Current bias $I_b$ with small ac component was applied and voltage $V$ measured. Perpendicular field $B_{\perp }$ controlled interferometer phase. (c) False-color micrograph of a measured device showing a loop of thin Al deposited on ramps to contact the full-shell wire. Uncolored gates were set to $+2\mathrm{V}$.

Figure 2

Bias spectroscopy of the isolated dot junction with reference arm closed. (a) Differential conductance $dI/d{V}_b$ as a function of gate voltage $V_{\mathrm{dot}}$ and dc bias $V_b$ in the zero lobe ($B_{\parallel }=0$). Sweeping $V_{\mathrm{dot}}$ changes dot occupancy from even ($e$ state) to odd ($o$ state) to even. A uniform conductance (supercurrent) peak at $V_b=0$ is visible throughout the range of $V_{\mathrm{dot}}$. Negative differential conductance features (green) are visible in the $e$ state. Red (blue) marks indicate location of $e(o)$ state cuts in (c),(d). (b) Same as (a) except in the first lobe ($B_{\parallel }=120\mathrm{mT}$). A strong enhancement of the zero-bias conductance peak occurs in the odd-occupied state. (c) Lobe structure in bias spectroscopy as a function $B_{\parallel }$ for $e$ state. Green, gray, and black marks indicate cuts in (e). (d) Same as (c) for $o$ state, with green, gray, and black marks indicating cuts in (f). Enhanced zero-bias conductance persists through the first lobe and does not split with increasing magnetic field. We interpret the isolated zero-bias peak at $B_{\parallel }\sim 46\mathrm{mT}$ as a Kondo enhancement (see text). (e) Cuts from (c) in the $e$ state showing small supercurrent peaks and several subgap resonances in both the zeroth (green) and first (black) lobes, with a broad zero-bias dip in the destructive regime (gray). (f) Cuts from (d) in the $o$ state showing a large zero-bias conductance peak in the first lobe and a broad zero-bias peak in the destructive regime (gray).

Figure 3

Differential resistance $dV/d{I}_b$ of the interferometer as a function gate voltage $V_{\mathrm{dot}}$ controlling dot occupancy and $B_{\perp }$ controlling flux through the interferometer. Current bias $I_b$ was set to periodically exceed the total switching current of the interferometer. (a) The zeroth lobe ($B_{\parallel }=0$) with $I_b=2\mathrm{nA}$ showed a $\pi$ phase shift in the $o$ state relative to the $e$ state, indicating a $\pi$ junction. (b) Same as (a) except in the first lobe ($B_{\parallel }=120\mathrm{mT}$) with $I_b=1.7\mathrm{nA}$, showing no phase shift as a function of $V_{\mathrm{dot}}$.

Figure 4

Differential resistance $dV/d{I}_b$ of the interferometer as a function of bias current $I_b$ and perpendicular magnetic field $B_{\perp }$, (a),(c) in the zeroth lobe, along cuts through the $e(o)$ state [red (blue) marks in Fig. 3], showing relative $\pi$ phase shift, and (b),(d) in the first lobe, along cuts through the $e(o)$ state [red (blue) marks in Fig. 3], showing absence of phase shift. Note in (a),(c) that switching currents exceed retrapping currents. (b),(d) In the first lobe ($B_{\parallel }=120\mathrm{mT}$), switching and retrapping currents are comparable. (e) Relative phase shift of $\pi$ between $e$ state (red) and $o$ state (blue) in the zeroth lobe. (f) No phase shift between the $e$ state (red) and $o$ state (blue) in the first lobe ($B_{\parallel }=0$), where both phases align with the $e$ state (red) in the zeroth lobe. Critical current in the $o$ state (blue) exceeds the $e$ state (red) in the first lobe ($B_{\parallel }=120\mathrm{mT}$).