Tunable Crossed Andreev Reflection and Elastic Cotunneling in Hybrid Nanowires

A short superconducting segment can couple attached quantum dots via elastic cotunneling (ECT) and crossed Andreev reflection (CAR). Such coupled quantum dots can host Majorana bound states provided that the ratio between CAR and ECT can be controlled. Metallic superconductors have so far been shown to mediate such tunneling phenomena, albeit with limited tunability. Here, we show that Andreev bound states formed in semiconductor-superconductor heterostructures can mediate CAR and ECT over mesoscopic length scales. Andreev bound states possess both an electron and a hole component, giving rise to an intricate interference phenomenon that allows us to tune the ratio between CAR and ECT deterministically. We further show that the combination of intrinsic spin-orbit coupling in InSb nanowires and an applied magnetic field provides another efficient knob to tune the ratio between ECT and CAR and optimize the amount of coupling between neighboring quantum dots.

Popular Summary

Robust, practical quantum computing requires quantum bits, or qubits, that can tolerate environmental noise. One potential type of building block of such qubits is a “Majorana bound state,” a special case of a type of fermionlike quasiparticle that is its own antiparticle. The top-down approach for finding such states is very demanding from the material-science perspective, and it is unclear if it is achievable. An alternative bottom-up approach proposes to form Majorana states by coupling quantum dots via two processes that must be perfectly balanced. Here, we show that a system that consists of two quantum dots coupled by a semiconducting-superconducting hybrid material can achieve this fine-tuning of the interactions.

The two processes that must be balanced are elastic cotunneling, in which an electron hops between two sites via an intermediate state, and crossed Andreev reflection, wherein two electrons enter or exit a superconductor simultaneously and split into two separate leads. In this work, we show that both processes happen via discrete states appearing in a semiconducting-superconducting hybrid and that they can be controlled by either an electrostatic gate or an external magnetic field. We back these findings with theoretical calculations and put them to use to realize Majorana states in such systems.

Future work will focus on a deterministic approach to the creation of Majorana bound states using the understanding of the microscopic process governing the interactions between the quantum dots.

Figure 1

Correlation between ABS and CAR or ECT processes. (a),(b) Illustration of the ECT (a) and CAR (b) processes. (c) Scanning electron micrograph of device $A$. (d) Schematic illustration of our devices and experimental setup. An InSb nanowire (green) is coated by a thin Al shell (blue, $\mathrm{Al}+\mathrm{Pt}$ for device $A$), on top of seven finger gates (red). Two $\mathrm{Cr}/\mathrm{Au}$ leads (yellow) are attached to both sides of the wire. (e) Spectroscopy configuration: Yellow bars depict voltage bias in normal ($N$) contacts, while blue rectangles represent the superconductor ($S$). Blue curves sketch the desired voltage profile defined with the gates; voltage barriers are not to scale. (f) Configuration with QDs: Applying low voltages on $V_{\mathrm{LO}}$,$V_{\mathrm{LI}}$ and $V_{\mathrm{RI}}$,$V_{\mathrm{RO}}$ forms a QD on the left and right side of the superconducting segment. (g),(h) Measurement of the ECT-induced current (g) and the CAR-induced current (h), as in Ref. [15], around a charge degeneracy point. (i) $G_{\mathrm{LL}}$ as a function of $V_{\mathrm{L}}$ and $V_{\mathrm{PG}}$ when setting the gates in the tunneling spectroscopy configuration. (j) $G_{\mathrm{RL}}$ as a function of $V_{\mathrm{L}}$ and $V_{\mathrm{PG}}$ in the same settings as (b). $G_{\mathrm{LL}}$ and $G_{\mathrm{RL}}$ are calculated by taking the numerical derivative after applying a Savitzky-Golay filter of window length 11 and polynomial order 1 to the measured $I_{\mathrm{L}}$ and $I_{\mathrm{R}}$ currents, respectively. (k) CAR- and ECT-induced currents as a function of $V_{\mathrm{PG}}$ measured using the $N\leftrightarrow N+1$ transition in both QDs. The values of $V_{\mathrm{LI}}$ and $V_{\mathrm{RI}}$ are kept constant during measurements in (b), (c), and (e).

Figure 2

Detailed study of CAR and ECT through an ABS. (a) Two possible paths for ECT: An electron hops from the left QD to the center ABS, followed by an escape from the ABS to the right QD (solid gray arrow), and the processes in the opposite order (dashed green arrow). (b) Two possible paths for CAR: An electron from the left QD enters the ABS followed by another electron arriving from the right QD (solid gray arrow) and the same processes in reversed order (dashed green arrow). (c) $E_{\mathrm{ABS}}$, and $u$, $v$ as a function of $\mu$ calculated in the atomic limit, where $E_{\mathrm{ABS}}=\sqrt{{\mathrm{\Gamma }}^2+{\mu }^2}$ with $\mathrm{\Gamma }=160\mathrm{\mu }\mathrm{eV}$ [33]. $\mu$ and $V_{\mathrm{PG}}$ are related via $\mu =-e\alpha (V_{\mathrm{PG}}-V_0)$, where $\alpha$ is the gate lever arm and $V_0=35\mathrm{mV}$ is an offset. Comparing data to theory, we estimate $\alpha \sim 0.01$. (d) $G_{\mathrm{LL}}$ as a function of $V_{\mathrm{L}}$ and $V_{\mathrm{PG}}$ showing a single ABS. (e) A toy-model calculation of the transmission probability as a function of $\mu$. (f) A high-resolution measurement of CAR and ECT amplitudes while tuning $V_{\mathrm{PG}}$. The background noise level is approximately 30 pA (see Supplemental Material [29]).

Figure 3

CAR and ECT mediated by spin-polarized ABS. (a) ECT process mediated by a spin-polarized ABS between QDs in the $\uparrow \uparrow$ spin configuration. (b) CAR process mediated by spin-polarized ABS between QDs in the $\uparrow \downarrow$ spin configuration. (c) Calculation of the transmission probability of ECT and CAR via an atomic-limit ABS as a function $\mu$ at the four possible spin configurations of the QDs. Spin-orbit coupling is included in the calculation as a small spin-flipping factor ($\sigma =0.2$) to allow for opposite-spin ECT and same-spin CAR (see Supplemental Material [29] for model details). Other model parameters are $\mathrm{\Gamma }=160\mathrm{\mu }\mathrm{eV}$ and $E_Z=100\mathrm{\mu }\mathrm{eV}$. (d) A high-resolution CAR and ECT amplitudes while tuning $V_{\mathrm{PG}}$ with $\overrightarrow{B}=80\mathrm{mT}$ (applied along the nanowire direction) at the four possible spin configurations of the QDs.

Figure 4

Tuning CAR and ECT with magnetic field orientation. (a)–(d) Spherical plots: The center of every colored tile corresponds to a specific magnetic field orientation. Each panel is taken at fixed $V_{\mathrm{PG}}$ and $|\overrightarrow{B}|=80\mathrm{mT}$. $V_{\mathrm{PG}}=475\mathrm{mV}$ in (c), while $V_{\mathrm{PG}}=480\mathrm{mV}$ in (a), (b), and (d). The QD spin configuration is $\downarrow \uparrow$ for all panels. See Fig. S7 [29] for data corresponding to other spin configurations. (a) CAR-induced current as a function of the magnetic field direction, extracted with the same method detailed in Fig. S1 [29] and used in the rest of the paper. (b) ECT-induced current as a function of the magnetic field orientation. (c) Energy of the lowest-energy ABS extracted from local tunneling spectroscopy as a function of the magnetic field orientation. (d) Ratio of the ECT and CAR currents from (b) and (a). Continuous lines highlight the locus of points where $I_{\mathrm{CAR}}=I_{\mathrm{ECT}}$; among them, the points with maximum current are marked with crosses. (e) Interpolation of data shown in (a)–(c) along the $\phi =-50°$ meridian. (f) Interpolation of data shown in (a)–(c) along the $\phi =+40°$ meridian. (g) $I_{\mathrm{CAR}}$ along the $I_{\mathrm{CAR}}=I_{\mathrm{ECT}}$ curves shown in (d). Negative-$\phi$ points are parametrized and plotted on the left, positive-$\phi$ points on the right. (h) Schematic defining $\theta$ and $\phi$: $\theta =\pm 90°$ is the direction parallel to the nanowire. $\theta =\phi =0°$ is the direction perpendicular to the substrate.