Transport in superconductor–normal metal–superconductor tunneling structures: Spinful $p$-wave and spin-orbit-coupled topological wires

We theoretically study transport properties of voltage-biased one-dimensional superconductor–normal metal–superconductor tunnel junctions with arbitrary junction transparency where the superconductors can have trivial or nontrivial topology. Motivated by recent experimental efforts on Majorana properties of superconductor-semiconductor hybrid systems, we consider two explicit models for topological superconductors: (i) spinful $p$-wave, and (ii) spin-split spin-orbit-coupled $s$-wave. We provide a comprehensive analysis of the zero-temperature dc current $I$ and differential conductance $dI/dV$ of voltage-biased junctions with or without Majorana zero modes (MZMs). The presence of an MZM necessarily gives rise to two tunneling conductance peaks at voltages $eV=\pm {\mathrm{\Delta }}_{\mathrm{lead}}$, i.e., the voltage at which the superconducting gap edge of the lead aligns with the MZM. We find that the MZM conductance peak probed by a superconducting lead without a BCS singularity has a nonuniversal value, which decreases with decreasing junction transparency. This is in contrast to the MZM tunneling conductance measured by a superconducting lead with a BCS singularity, where the conductance peak in the tunneling limit takes the quantized value $G_M=(4-\pi )2e^2/h$ independent of the junction transparency. We also discuss the “subharmonic gap structure”, a consequence of multiple Andreev reflections, in the presence and absence of MZMs. Finally, we show that for finite-energy Andreev bound states (ABSs), the conductance peaks shift away from the gap bias voltage $eV=\pm {\mathrm{\Delta }}_{\mathrm{lead}}$ to a larger value set by the ABSs energy. Our work should have important implications for the extensive current experimental efforts toward creating topological superconductivity and MZMs in semiconductor nanowires proximity coupled to ordinary $s$-wave superconductors.

Figure 1

Schematic diagram of a superconductor–normal metal–superconductor (SNS) junction with a $\delta$-function potential barrier of strength $Z$.

Figure 2

Energy spectrum of a spinful $p\mathrm{SC}$ for different parameter regimes: (a) $|V_Z|<|{\mu }_p|$ and ${\mu }_p<0$ ($p_0$-SC with no topological channel), (b) $|V_Z|>{\mu }_p$ and ${\mu }_p>0$ ($p_1$-SC with one topological channel), (c) $|V_Z|<{\mu }_p$ and ${\mu }_p>0$ ($p_2$-SC with two topologi-cal channels).

Figure 3

Plots of (a) dc current $I$ and (b) normalized differential conductance $G/G_0$ versus bias voltage $V$ for an $sN{p}_0$ junction with various values of transparencies $G_N$. The parameters used for the $s\mathrm{SC}$ are ${\mu }_s=20\mathrm{K}$ and ${\mathrm{\Delta }}_s=0.01\mathrm{K}$. The parameters used for the $p_0$-SC are ${\mu }_p=-0.01\mathrm{K}, V_Z=0\mathrm{K}, {\mathrm{\Delta }}_p=0.2\mathrm{eV}\AA$, where the smallest gap is ${\mathrm{\Delta }}_{p_0}=0.01\mathrm{K}$. The smallest gap in the junction is ${\mathrm{\Delta }}_{\min }=0.01\mathrm{K}$.

Figure 4

Plots of (a) dc current $I$ and (b) normalized differential conductance $G/G_0$ versus bias voltage $V$ for an $sN{p}_1$ junction with various values of transparencies $G_N$ in the limit of large Zeeman field ($V_Z=2{\mu }_p$). The red dashed line at $G_M=(4-\pi )2e^2/h$ is the conductance value due to a single Andreev reflection from the MZM. The parameters used for the $s\mathrm{SC}$ are ${\mu }_s=200\mathrm{K}$ and ${\mathrm{\Delta }}_s=2.5\mathrm{K}$. The parameters used for the $p_1$-SC are ${\mu }_p=20\mathrm{K}, V_Z=40\mathrm{K}, {\mathrm{\Delta }}_p=0.0785\mathrm{eV}\AA$, where the smallest gap is ${\mathrm{\Delta }}_{p_1}=4\mathrm{K}$. The smallest gap in the junction is ${\mathrm{\Delta }}_{\min }=2.5\mathrm{K}$.

Figure 5

Plots of (a) dc current $I$ and (b) normalized differential conductance $G/G_0$ versus bias voltage $V$ for an $sN{p}_1$ junction with various values of transparencies $G_N$ in the limit of small Zeeman field ($V_Z=1.1{\mu }_p$). The red dashed line at $G_M=(4-\pi )2e^2/h$ is the conductance value due to a single Andreev reflection from the MZM. The parameters used for the $s\mathrm{SC}$ are ${\mu }_s=200\mathrm{K}$ and ${\mathrm{\Delta }}_s=2.5\mathrm{K}$. The parameters used for the $p_1$-SC are ${\mu }_p=20\mathrm{K}, V_Z=22\mathrm{K}, {\mathrm{\Delta }}_p=0.0785\mathrm{eV}\AA$, where the gaps are 2 K and 3.4 K with the smallest gap for the $p_1$-SC being ${\mathrm{\Delta }}_{p_1}=2\mathrm{K}$. The smallest gap in the junction is ${\mathrm{\Delta }}_{\min }=2\mathrm{K}$.

Figure 6

Plots of (a) dc current $I$ and (b) normalized differential conductance $G/G_0$ versus bias voltage $V$ for an $sN{p}_2$ junction with various values of transparencies $G_N$. The red dashed line at $2G_M=(4-\pi )4e^2/h$ is the conductance value due to a single Andreev reflection from two MZMs. The parameters used for the $s\mathrm{SC}$ are ${\mu }_s=20\mathrm{K}$ and ${\mathrm{\Delta }}_s=0.01\mathrm{K}$. The parameters used for the $p_2$-SC are ${\mu }_p=20\mathrm{K}, V_Z=0\mathrm{K}, {\mathrm{\Delta }}_p=2\times {10}^{-4}\mathrm{eV}\AA$, where the gap is ${\mathrm{\Delta }}_{p_2}=6.3\times {10}^{-3}\mathrm{K}$. The smallest gap in the junction is ${\mathrm{\Delta }}_{\min }={\mathrm{\Delta }}_{p_2}=6.3\times {10}^{-3}\mathrm{K}$.

Figure 7

Plots of (a) dc current $I$ and (b) normalized differential conductance $G/G_0$ versus bias voltage $V$ for a $p_{2}N{p}_2$ junction with various values of transparencies $G_N$. The parameters used for both $p_2$-SCs are ${\mu }_p=20\mathrm{K}, V_Z=0\mathrm{K}, {\mathrm{\Delta }}_p=2\times {10}^{-4}\mathrm{eV}\AA$, where the gap is ${\mathrm{\Delta }}_{p_2}=6.3\times {10}^{-3}\mathrm{K}$. The smallest gap in the junction is ${\mathrm{\Delta }}_{\min }=6.3\times {10}^{-3}\mathrm{K}$.

Figure 8

Plots of (a) dc current $I$ and (b) normalized differential conductance $G/G_0$ versus bias voltage $V$ for a $p_{2}N{p}_1$ junction with various values of transparencies $G_N$. The parameters used for the $p_1$-SC are ${\mu }_p=20\mathrm{K}, {\mathrm{\Delta }}_p=2\times {10}^{-4}\mathrm{eV}\AA$, and $V_Z=40\mathrm{K}$, where the gap is ${\mathrm{\Delta }}_{p_1}=0.011\mathrm{K}$. The parameters for the $p_2$-SC are ${\mu }_p=20\mathrm{K}, {\mathrm{\Delta }}_p=2\times {10}^{-4}\mathrm{eV}\AA , V_Z=0\mathrm{K}$, where the gap is ${\mathrm{\Delta }}_{p_2}=6.3\times {10}^{-3}\mathrm{K}$. The smallest gap in the junction is ${\mathrm{\Delta }}_{\min }={\mathrm{\Delta }}_{p_2}=6.3\times {10}^{-3}\mathrm{K}$.

Figure 9

Plots of (a) dc current $I$ and (b) normalized differential conductance $G/G_0$ versus bias voltage $V$ for a $p_{1}N{p}_1$ junction with various values of transparencies $G_N$. The parameters used for both $p_1$-SCs are ${\mu }_p=20\mathrm{K}, {\mathrm{\Delta }}_p=2\times {10}^{-4}\mathrm{eV}\AA$, and $V_Z=40\mathrm{K}$, where the gap is ${\mathrm{\Delta }}_{p_1}=0.011\mathrm{K}$. The smallest gap in the junction is ${\mathrm{\Delta }}_{\min }={\mathrm{\Delta }}_{p_1}=0.011\mathrm{K}$.

Figure 10

Plots of (a) dc current $I$ and (b) normalized differential conductance $G/G_0$ versus bias voltage $V$ for a $p_{0}N{p}_2$ junction with various values of transparencies $G_N$. The parameters used for the $p_0$-SC are ${\mu }_p=-0.01\mathrm{K}, V_Z=0\mathrm{K}, {\mathrm{\Delta }}_p=0.2\mathrm{eV}\AA$, where the gap is ${\mathrm{\Delta }}_{p_0}=0.01\mathrm{K}$. The parameters used for the $p_2$-SC are ${\mu }_p=20\mathrm{K}, V_Z=0\mathrm{K}, {\mathrm{\Delta }}_p=2\times {10}^{-4}\mathrm{eV}\AA$, where the gap is ${\mathrm{\Delta }}_{p_2}=6.3\times {10}^{-3}\mathrm{K}$. The smallest gap in the junction is ${\mathrm{\Delta }}_{\min }=6.3\times {10}^{-3}\mathrm{K}$.

Figure 11

Plots of (a) dc current $I$ and (b) normalized differential conductance $G/G_0$ versus bias voltage $V$ for a $p_{0}N{p}_1$ junction with various values of transparencies $G_N$. The parameters used for the $p_0$-SC are ${\mu }_p=-0.01\mathrm{K}, V_Z=0\mathrm{K}, {\mathrm{\Delta }}_p=0.2\mathrm{eV}\AA$, where the gap is ${\mathrm{\Delta }}_{p_0}=0.01\mathrm{K}$. The parameters used for the $p_1$-SC are ${\mu }_p=20\mathrm{K}, V_Z=40\mathrm{K}, {\mathrm{\Delta }}_p=2\times {10}^{-4}\mathrm{eV}\AA$, where the gap is ${\mathrm{\Delta }}_{p_1}=0.011\mathrm{K}$. The smallest gap in the junction is ${\mathrm{\Delta }}_{\min }=0.01\mathrm{K}$.

Figure 12

Plots of (a) dc current $I$ and (b) normalized differential conductance $G/G_0$ versus bias voltage $V$ for a $p_{0}N{p}_0$ junction with various values of transparencies $G_N$. The parameters used for both $p_0$-SCs are ${\mu }_p=-0.01\mathrm{K}, V_Z=0\mathrm{K}, {\mathrm{\Delta }}_p=0.2\mathrm{eV}\AA$, where the gap is ${\mathrm{\Delta }}_{p_0}=0.01\mathrm{K}$. The smallest gap in the junction is ${\mathrm{\Delta }}_{\min }=0.01\mathrm{K}$.

Figure 13

Energy spectrum of SOCSW for different parameter regimes: (a) $V_Z=0$ (nontopological), (b) $V_Z<\sqrt{{{\mu }_0^2+{\mathrm{\Delta }}_0^2}^{\phantom{0}}}$ (nontopological), (c) $V_Z=\sqrt{{{\mu }_0^2+{\mathrm{\Delta }}_0^2}^{\phantom{0}}}$ (transition), (d) $V_Z>\sqrt{{{\mu }_0^2+{\mathrm{\Delta }}_0^2}^{\phantom{0}}}$ (topological).

Figure 14

Plots of (a) dc current $I$ and (b) normalized differential conductance $G/G_0$ versus bias voltage $V$ for a nontopological–nontopological SOCSW junction with various values of transparencies $G_N$ and no Zeeman field. The parameters used for both SOCSWs are ${\mu }_0=0\mathrm{K}, V_Z=0\mathrm{K}, {\mathrm{\Delta }}_0=0.01\mathrm{K}, \alpha =0.5\mathrm{eV}\AA$, where the gap is ${\mathrm{\Delta }}_{\mathrm{SOCSW}}^{\mathrm{nontopo}}=0.01\mathrm{K}$. The smallest gap in the junction is ${\mathrm{\Delta }}_{\min }=0.01\mathrm{K}$.

Figure 15

Plots of (a) dc current $I$ and (b) normalized differential conductance $G/G_0$ versus bias voltage $V$ for a nontopological–nontopological SOCSW junction with various values of transparencies $G_N$ and finite Zeeman field. The parameters used for both SOCSWs are ${\mu }_0=0\mathrm{K}, V_Z=0.002\mathrm{K}, {\mathrm{\Delta }}_0=0.01\mathrm{K}, \alpha =0.5\mathrm{eV}\AA$, where the gap is ${\mathrm{\Delta }}_{\mathrm{SOCSW}}^{\mathrm{nontopo}}=0.008\mathrm{K}$. The smallest gap in the junction is ${\mathrm{\Delta }}_{\min }=0.008\mathrm{K}$.

Figure 16

Plots of (a) dc current $I$ and (b) normalized differential conductance $G/G_0$ versus bias voltage $V$ for a nontopological–topological SOCSW junction with various values of transparencies $G_N$. The red dashed line at $G_M=(4-\pi )2e^2/h$ is the conductance value due to a single Andreev reflection from the MZM. The nontopological SOCSW is not subjected to any Zeeman field and the topological superconductor has a small Zeeman field. The parameters used for the nontopological SOCSW are ${\mu }_0=0\mathrm{K}, V_Z=0\mathrm{K}, {\mathrm{\Delta }}_0=0.5\mathrm{K}, \alpha =0.5\mathrm{eV}\AA$, where ${\mathrm{\Delta }}_{\mathrm{SOCSW}}^{\mathrm{nontopo}}=0.5\mathrm{K}$. The parameters used for the topological SOCSW are ${\mu }_0=0\mathrm{K}, V_Z=15.0\mathrm{K}, {\mathrm{\Delta }}_0=10.0\mathrm{K}, \alpha =0.05\mathrm{eV}\AA$, where the gap is ${\mathrm{\Delta }}_{\mathrm{SOCSW}}^{\mathrm{topo}}=0.75\mathrm{K}$. The smallest gap in the junction is ${\mathrm{\Delta }}_{\min }=0.5\mathrm{K}$.

Figure 17

Plots of (a) dc current $I$ and (b) normalized differential conductance $G/G_0$ versus bias voltage $V$ for a nontopological–topological SOCSW junction with various values of transparencies $G_N$. The red dashed line at $G_M=(4-\pi )2e^2/h$ is the conductance value due to a single Andreev reflection from the MZM. The nontopological SOCSW is not subjected to any Zeeman field and the topological superconductor has a large Zeeman field. The parameters used for the nontopological SOCSW are ${\mu }_0=0\mathrm{K}, V_Z=0\mathrm{K}, {\mathrm{\Delta }}_0=0.5\mathrm{K}, \alpha =0.5\mathrm{eV}\AA$, where ${\mathrm{\Delta }}_{\mathrm{SOCSW}}^{\mathrm{nontopo}}=0.5\mathrm{K}$. The parameters used for the topological SOCSW are ${\mu }_0=0\mathrm{K}, V_Z=60.0\mathrm{K}, {\mathrm{\Delta }}_0=10.0\mathrm{K}, \alpha =0.05\mathrm{eV}\AA$, where the gap is ${\mathrm{\Delta }}_{\mathrm{SOCSW}}^{\mathrm{topo}}=0.42\mathrm{K}$. The smallest gap in the junction is ${\mathrm{\Delta }}_{\min }=0.42\mathrm{K}$.

Figure 18

Plots of (a) dc current $I$ and (b) normalized differential conductance $G/G_0$ versus bias voltage $V$ for a nontopological–topological SOCSW junction with various values of transparencies $G_N$. The nontopological SOCSW has a finite Zeeman field and the topological superconductor has a small Zeeman field. Note that the MZM tunneling conductance is not quantized at $G_M=(4-\pi )2e^2/h$. The parameters used for the nontopological SOCSW are ${\mu }_0=0\mathrm{K}, V_Z=0.2\mathrm{K}, {\mathrm{\Delta }}_0=0.5\mathrm{K}, \alpha =0.5\mathrm{eV}\AA$, where ${\mathrm{\Delta }}_{\mathrm{SOCSW}}^{\mathrm{nontopo}}=0.3\mathrm{K}$. The parameters used for the topological SOCSW are ${\mu }_0=0\mathrm{K}, V_Z=15.0\mathrm{K}, {\mathrm{\Delta }}_0=10.0\mathrm{K}, \alpha =0.05\mathrm{eV}\AA$, where the gap is ${\mathrm{\Delta }}_{\mathrm{SOCSW}}^{\mathrm{topo}}=0.75\mathrm{K}$. The smallest gap in the junction is ${\mathrm{\Delta }}_{\min }=0.3\mathrm{K}$.

Figure 19

Plots of (a) dc current $I$ and (b) normalized differential conductance $G/G_0$ versus bias voltage $V$ for a topological–topological SOCSW junction with various values of transparencies $G_N$. The parameters used for both SOCSWs are ${\mu }_0=0\mathrm{K}, V_Z=15\mathrm{K}, {\mathrm{\Delta }}_0=1.17\mathrm{K}, \alpha =0.05\mathrm{eV}\AA$, where the gap is ${\mathrm{\Delta }}_{\mathrm{SOCSW}}^{\mathrm{topo}}=0.01\mathrm{K}$. The smallest gap in the junction is ${\mathrm{\Delta }}_{\min }=0.01\mathrm{K}$.

Figure 20

(a) Schematic diagram of an $s\mathrm{SC}$–SOCSW junction with a pair of ABS (one at each end of the topological region). The chemical potential of the topological and nontopological regions are $|{\mu }_{\mathrm{topo}}|<\sqrt{{V_Z^2-{\mathrm{\Delta }}_0^2}^{\phantom{0}}}$ and $|{\mu }_{\mathrm{nontopo}}|>\sqrt{{V_Z^2-{\mathrm{\Delta }}_0^2}^{\phantom{0}}}$, respectively. The parameters used for the $s\mathrm{SC}$ are ${\mu }_s=50\mathrm{K}$ and ${\mathrm{\Delta }}_s=0.67\mathrm{K}$. The SOCSW parameters are ${\mu }_{\mathrm{nontopo}}=211.18\mathrm{K}, V_Z=15\mathrm{K}, {\mathrm{\Delta }}_0=10\mathrm{K}, \alpha =0.05\mathrm{eV}\AA$, and length of the topological region, $L_{\mathrm{topo}}=0.6 \mu \mathrm{m}$. We use a dissipation term $i\mathrm{\Gamma }{\tau }_0\otimes {\sigma }_0$ in the BdG Hamiltonian of both the left and right superconductors with a dissipation strength $\mathrm{\Gamma }=0.05K$ to broaden the van-Hove singularity. (b) The energy of the Andreev bound state closest to zero energy versus the chemical potential ${\mu }_{\mathrm{topo}}$ in the topological region. The red, green, and purple dots indicate the value of the topological chemical potential used in (c), (d), and (e), respectively. Normalized differential conductance $G/G_0$ for the SOCSW for several chemical potential values in the topological region: (c) ${\mu }_{\mathrm{topo}}=0\mathrm{K}$, (d) ${\mu }_{\mathrm{topo}}=1.697\mathrm{K}$, and (e) ${\mu }_{\mathrm{topo}}=4.5\mathrm{K}$. Inset: the ABS conductance in the weak tunneling limit, which is the conductance for the smallest transparency in the main plot.