Phase diagram and quantum transport in a semiconductor-superconductor hybrid nanowire with long-range pairing interactions

We investigate the effect of long-range pairing interactions on the phase diagram and the quantum transport properties in a semiconductor-superconductor hybrid nanowire. Both the power-law and exponential decay rates of long-range pairing are considered for comparison. It is found that a long-range topological phase hosting massive edge modes is induced when the exponent of the power-law decay is less than one. By diagonalizing the tight-binding model, we find that near-zero-energy Andreev bound states (ABSs) could be induced in the topologically trivial phase for slowly decayed long-range pairing interactions. For a one-dimensional long-range Kitaev chain, the shot noise is suggested to serve as a probe to discriminate Majorana bound states (MBSs) and topological massive modes. Our numerical results indicate that for realistic device parameters, the Fano factor of the shot noise is irrelevant to the topological property of the system, and it fails to distinguish the MBSs and ABSs. However, the noise Fano factor is generally consistent with the variation of the bound state energy, which can be used to detect the energy splitting of these bound states.

Figure 1

(a) The long-range superconducting nanowire is in the middle (blue lines and squares). It is in contact with two normal metallic leads (orange lines and disks) through tunnel couplings described by ${\mathrm{\Gamma }}_L$ and ${\mathrm{\Gamma }}_R$. Each site of the nanowire is coupled with every other site by a pairing amplitude $\mathrm{\Delta }\left(r\right)$. (b) Schematic diagram of the transport setup: A grounded spin-orbit-coupled semiconductor nanowire coated with an epitaxial superconducting shell in a parallel magnetic field $B$ is connected to two normal metallic leads $N_L$ and $N_R$. A pair of MBSs (${\gamma }_1,{\gamma }_2$) appear at both ends of the covered nanowire segment when the system is in the topological phase. A bias voltage $V$ is applied across the whole device.

Figure 2

Phase diagrams shown as a function of on-site energy $\epsilon$ and Zeeman splitting energy $V_Z$ with power-law decaying long-range pairing interactions. (a)–(e) $\beta =\infty$, 1.5, 1.0, 0.5, and 0 correspond to the short-range limit, weak short-range regime, critical line, strong long-range regime, and the long-range limit, respectively. The yellow, blue, and red colored regions highlight the trivial and different topological phases indicated by the integer winding numbers $W=0, W=+1$, and $W=-1$ in the short-range sector ($\beta >1$). The topologically trivial and nontrivial phases in the long-range sector ($\beta ⩽1$) labeled by noninteger winding numbers are colored green and orange, respectively. The topological phases in the absence of any long-range pairing interactions are shown as regions bounded by the cyan solid line ($W=+1$) and red solid line ($W=-1$). Here, we take $t=t_{\mathrm{SO}}={\mathrm{\Delta }}_0=1.0\mathrm{meV}$ for a clear view.

Figure 3

Phase diagram as a function of the decay rate $\beta$ and the Zeeman splitting energy $V_Z$ at the low-field regime with on-site energy fixed at $\epsilon =-2\mathrm{meV}$. The critical line formed by the boundary between the unconventional topological phase (orange region) and the trivial phase (green region) indicates a nonmonotonic critical Zeeman splitting energy $V_Z^C$ as $\beta$ decreases from 1 towards 0. Other parameters are the same as in Fig. 2.

Figure 4

Phase diagrams on the $\epsilon \text{−}{V}_Z$ plane of the nanowire with exponentially decaying long-range pairing interactions. (a)–(d) $\xi =0.1a, a, 5a$, and $\xi >L$ correspond to a short coherence length, a coherence length comparable with the lattice spacing, a coherence length larger than several lattice spacings, and the limit of long coherence length, respectively. Here, we set the Fermi momentum $k_F=\pi /2$ (half filling) and other parameters are the same as in Fig. 2.

Figure 5

Phase diagrams on the $\beta \text{−}\mu$ plane with power-law decaying long-range pairing exhibit a dependence on the on-site pairing potential. (a) ${\mathrm{\Delta }}_0=0.25\mathrm{meV}$, (b) ${\mathrm{\Delta }}_0=0.025\mathrm{meV}$. With chemical potential fixed at $\mu =0$ ($\epsilon =2t$), the topological phase transition occurs at $\beta ={\beta }_C$. The other parameters are $t=t_{\mathrm{SO}}=V_Z=1.0\mathrm{meV}$.

Figure 6

The energy spectra for a finite Majorana nanowire with power-law decaying long-range pairing terms as a function of the chemical potential $\mu$ at various decay rates (a)–(f) $\beta =2$, 1.2, 0.5, 0.1, 0.05, and 0.02. The energies for the lowest-lying states are colored red. We choose $L=10\mu \mathrm{m}$ and other parameters are the same as in Fig. 5.

Figure 7

The lowest-energy spectrum on a $\beta \text{−}\mu$ plane with power-law decaying long-range pairing. (a) Zeeman field fixed at $V_Z=1.0\mathrm{meV}$ and wire length $L=4\mu \mathrm{m}$. (b) Details of the region bounded by the red box in (a) with a longer wire $L=10\mu \mathrm{m}$. The cyan dashed line indicates where the cut in (c) is taken. (c) The energy of the lowest-lying state $E_0$ (red solid line) and the first few excited states $E_n$ (gray solid line) vs the chemical potential $\mu$ at $\beta =0.01$. (d), (e) The spatial profiles of the lowest-energy states marked by the triangle and diamond symbols in (c). Other parameters are the same as in Fig. 5.

Figure 8

Differential conductance as a function of the bias voltage $V$ and Zeeman splitting energy $V_Z$ for a Majorana nanowire with power-law decaying long-range pairing at (a) $\beta =0.5$ and (b) $\beta =1.5$. (c), (d) $G_D$ and $G_A$ are the conductances associated with the direct tunneling process and the local Andreev reflection, respectively. The extent of $V_Z$ is indicated by the red box in (a). (e), (f) The same quantities plotted within the range of $V_Z$ specified by the red box of (b). $G_0=2e^2/h$ is the quantized conductance. The system parameters are taken as $L=2\mu \mathrm{m}, t=3\mathrm{meV}, {\mathrm{\Delta }}_0=0.25\mathrm{meV}, \alpha =0.1$ eV Å, $\mu =0$, and ${\mathrm{\Gamma }}_L={\mathrm{\Gamma }}_R=0.05\mathrm{meV}$.

Figure 9

Dependence of the shot noise Fano factors and low-energy spectrum on the Zeeman field in the presence of power-law decaying long-range pairing. (a) The phase diagram on the $\beta \text{−}{V}_Z$ plane. The colored dashed lines correspond to the cuts where (b)–(d) are taken. (b)–(d) The low-energy spectrum as a function of Zeeman splitting energy $V_Z$ for $\beta =1.5, \beta =1.0$, and $\beta =0.5$, respectively. (e)–(g) The noise Fano factor as a function of $V_Z$ and (h)–(j) the lowest energy vs $V_Z$ in the ranges denoted by I, II, and III in the upper panels (bounded in red boxes). The system parameters are taken as those in Fig. 8 and the wire length is taken as $L=4\mu \mathrm{m}$.

Figure 10

The phase diagram, the lowest energy, and shot noise Fano factor as a function of Zeeman splitting energy $V_Z$ and coherence length $\xi$ in the presence of exponentially decaying long-range pairing interactions. (a), (b) The phase diagram and the lowest-energy spectrum on the $\xi \text{−}{V}_Z$ plane. The dashed lines indicate where the cuts in the lower panels are taken. (c) and (d) illustrate the orange line cut of (b) and the corresponding shot noise Fano factor against $V_Z$ at $\xi =10a$. Similarly, (e), (f) and (g), (h) plot the pink and red line cuts and the corresponding Fano factors as functions of $V_Z$ for $\xi =5a$ and $\xi =0.5a$, respectively. Other parameters are the same as in Fig. 9.