Tunnel spectroscopy of Majorana bound states in topological superconductor/quantum dot Josephson junctions

We theoretically investigate electronic transport through a junction where a quantum dot (QD) is tunnel coupled on both sides to semiconductor nanowires with strong spin-orbit interaction and proximity-induced superconductivity. The results are presented as stability diagrams, i.e., the differential conductance as a function of the bias voltage applied across the junction and the gate voltage used to control the electrostatic potential on the QD. A small applied magnetic field splits and modifies the resonances due to the Zeeman splitting of the QD level. Above a critical field strength, Majorana bound states (MBS) appear at the interfaces between the two superconducting nanowires and the QD, resulting in a qualitative change of the entire stability diagram, suggesting this setup as a promising platform to identify MBS. Our calculations are based on a nonequilibrium Green's function description and is exact when Coulomb interactions on the QD can be neglected. In addition, we develop a simple pictorial view of the involved transport processes, which is equivalent to a description in terms of multiple Andreev reflections, but provides an alternative way to understand the role of the QD level in enhancing transport for certain gate and bias voltages. We believe that this description will be useful in future studies of interacting QDs coupled to superconducting leads (with or without MBS), where it can be used to develop a perturbation expansion in the tunnel coupling.

Figure 1

(a) Sketch of the transport setup with a voltage-biased nanowire S/QD/S configuration. The bias voltage $V_b$ is applied between the superconducting contacts and the electrostatic potential on the QD is controlled by the gate voltage $V_g$. (b) The model abstracted from the setup in (a). The leads are semi-infinite tight-binding chains with the end sites coupling to the QD with tunnel couplings ${\Gamma }_L$ and ${\Gamma }_R$.

Figure 2

DOS at the end site of an isolated lead for $t=10\Delta ,\mu =0,{\alpha }_0=2\Delta$, and $\delta ={10}^{-5}\Delta$ in the retarded Green's function. The different curves are the results for Zeeman energy $V_z=0$ (blue line), $V_z=0.5\Delta$ (black dotted-dashed line), $V_z=\Delta$ (green dotted line), and $V_z=2\Delta$ (red dashed line).

Figure 3

$dI/d{V}_b$ as a function of $e{V}_b$ and $E_{\downarrow }^{\left(0\right)}$ for (a) Zeeman energy $V_z=0$, (c) $V_z=0.1\Delta$, and (d) $V_z=0.2\Delta$. The color scale is limited to the range $[-0.001,0.005]e^2/h$. In (a), (c), and (d), the dotted black lines, added as guides to the eye, are given by $e{V}_b=\frac{2}{3}[E_g\pm (E_{\downarrow }^{\left(0\right)}-{\delta }_{\Gamma })]$ with a mark $n=3$ and $e{V}_b=\frac{2}{5}[E_g\pm (E_{\downarrow }^{\left(0\right)}-{\delta }_{\Gamma })]$ with a mark $n=5$ next to the peaks, and the black arrows point at the resonances given by Eq. (23). Note that $E_g$ is the energy gap which is also changed when $V_z$ varies. We do not show results for $e{V}_b<0.1\Delta$ because the number of harmonics which have to be taken into account when evaluating the Green's functions grow with decreasing $V_b$. We have chosen $\Gamma =0.9\Delta$ and $k_{B}T=0.01\Delta$. ${\delta }_{\Gamma }=0.13\Delta$ was found to give the best fit to the peaks. All other parameters are the same as in Fig. 2. (b) Shows $dI/d{V}_b$ as a function of $E_{\downarrow }^{\left(0\right)}$ for $V_z=0$, i.e., horizontal cuts in (a), for $e{V}_b=0.45\Delta$ (blue line, multiplied by 6), $0.6\Delta$ (red dotted line, multiplied by 4), and $0.8\Delta$ (green dashed line). (e) Schematic diagram of an $O\left({\Gamma }^3\right)$ tunnel process at $e{V}_b=0.8\Delta$ and $E_{\downarrow }^{\left(0\right)}=-0.2\Delta$ which empties an initially full QD. (f) Same as (e), but showing a process filling an initially empty QD with one electron. (g) Same as (e), but showing an $O\left({\Gamma }^5\right)$ tunnel process at $E_{\downarrow }^{\left(0\right)}=\Delta$.

Figure 4

$dI/d{V}_b$ as a function of $e{V}_b$ and $E_{\downarrow }^{\left(0\right)}$ with a large Zeeman energy, $V_z=1.7\Delta$ in (a), $V_z=1.8\Delta$ in (b), and $V_z=2.0\Delta$ in (c). The color scale is limited to the range $[-0.1,0.5]e^2/h$. The dashed black lines are guides to the eye given by $e{V}_b=\pm \frac{2}{3}(E_{\downarrow }^{\left(0\right)}-{\delta }_{\Gamma })$ marked with $n=3$ and $e{V}_b=\pm \frac{2}{5}(E_{\downarrow }^{\left(0\right)}-{\delta }_{\Gamma })$ marked with $n=5$, where ${\delta }_{\Gamma }=0.16\Delta$. (d) Shows $dI/d{V}_b$ as a function of $E_{\downarrow }^{\left(0\right)}$ for $V_z=2\Delta$, i.e., horizontal cuts in (c), for $e{V}_b=0.2\Delta$ (blue line), $0.3\Delta$ (red dotted line, multiplied by 10), and $0.4\Delta$ (green dashed line). We have used $\delta ={10}^{-3}$; the other parameters are the same as in Fig. 3. (e)–(g) Schematic diagrams of tunnel processes in the presence of MBS corresponding to (c) at $e{V}_b=0.3\Delta$. In (e) and (f), $E_{\downarrow }^{\left(0\right)}=-0.45\Delta$, while $E_{\downarrow }^{\left(0\right)}=0.75\Delta$ in (g). We have for illustrative purposes drawn electrons inside the MBS peaks, although in reality the charge will disappear into the superconductor and not be localized at the edge.

Figure 5

$dI/d{V}_b$ as a function of $e{V}_b$ and $E_{\downarrow }^{\left(0\right)}$ for the MBS-QD-MBS configuration with large coupling $\Gamma =3.5\Delta$, where $V_z=2\Delta$ and $\mu =0$. The horizontal dashed lines are guides to the eye indicating MAR peaks at $e{V}_b=\frac{E_g}{m}$, where $E_g\approx 0.71\Delta$, and $m=1,2,...,6$. The color scale is limited to the range $[-0.1,0.5]e^2/h$. The other parameters are the same as in Fig. 4.

Figure 6

A convergence check of the current for some example points along the line $E_{\downarrow }^{\left(0\right)}=0.5\Delta$ in Fig. 3 with an increasing number $K$ of Fourier components included. The relative error function is $(I_{\text{dc},K}-I_{\text{dc},K-1})/I_{\text{dc},K-1}\times 100(\%)$.