Zero-bias conductance peak in Majorana wires made of semiconductor/superconductor hybrid structures

Motivated by a recent experimental report Mourik et al. [Science 336, 1003 (2012)] claiming the likely observation of the Majorana mode in a semiconductor-superconductor hybrid structure, we study theoretically the dependence of the zero-bias conductance peak associated with the zero-energy Majorana mode in the topological superconducting phase as a function of temperature, tunnel barrier potential, and a magnetic field tilted from the direction of the wire for realistic wires of finite lengths. We find that higher temperatures and tunnel barriers as well as a large magnetic field in the direction transverse to the wire length could very strongly suppress the zero-bias conductance peak as observed in recent experiments. We also show that a strong magnetic field along the wire could eventually lead to the splitting of the zero bias peak into a doublet with the doublet energy splitting oscillating as a function of increasing magnetic field. Our results based on the standard theory of topological superconductivity in a semiconductor hybrid structure in the presence of proximity-induced superconductivity, spin-orbit coupling, and Zeeman splitting show that the recently reported experimental data are generally consistent with the existing theory that led to the predictions for the existence of the Majorana modes in the semiconductor hybrid structures in spite of some apparent anomalies in the experimental observations at first sight. We also make a prediction for the future observation of Majorana splitting in finite wires used in the experiments.

Figure 1

(Left) The differential conductance for a fixed Zeeman potential $\mathbf{V}=(1,0)$ meV and different tunneling barrier $U=38$ and 42 meV. The green dotted, red dashed, and orange lines denote different temperature $0,60,120$ mK, respectively. (Right) Zero-bias conductance peak as a function of temperature for Zeeman potential $\mathbf{V}=(1,0)$ meV and different tunneling barrier $U=38$ and 42 meV, respectively. The peak decreases monotonously with temperature. ($L=4.5\mu \text{m},$ spin-orbit coupling $\alpha =0.2$ meVÅ, induced pairing potential $\Delta =0.5$ meV.)

Figure 2

The differential conductance for a fixed $V_y=0$ meV and different $V_x=0,0.5,1,2,3,4$ meV. The red solid and blue dashed lines denote temperatures $T=0$ and 60 mK, respectively. At $V_x=1$ meV, the system is in the topological phase with quantized ZBCP. For length $L=4.5\mu \text{m},$ the Zeeman splitting $V_x\gg 1$ meV reduces the gap, and leads to a stronger overlap between the MFs that results in a splitting of the ZBCP peaks. Additionally, the finite temperature will decrease ZBCP. (Parameters used in the plot are $U=38$ meV, $L=4.5\text{a}\mu \text{m},$ spin-orbit coupling $\alpha =0.2$ meVÅ, induced pairing potential $\Delta =0.5$ meV.)

Figure 3

The differential conductance for a fixed $V_x=1$ meV and different $V_y=0.1,0.4,0.6$, and $0.8$ meV. The red solid and blue dashed denote temperatures $T=0$ and 60 mK, respectively. At $V_x=1$ meV, the system is in the topological phase with quantized ZBCP. $V_y$ will reduce the quasiparticle gap and hence suppress the ZBCP. Finite temperature also reduces the ZBCP. ($U=38$ meV, $L=4.5\mu \text{m},$ spin-orbit coupling $\alpha =0.2$ meVÅ, induced pairing potential $\Delta =0.5$ meV.)

Figure 4

(Left panel) ZBCP as a function of $V_y$ for a fixed $V_x=1$ meV at zero temperature denoted by the blue solid line. ZBCP shows a plateau for small extent of $V_y$ and is then suppressed by larger $V_y.$ The red dashed line denotes the quasiparticle energy gap. The peaks appear after gap closure. (Right panel) Blue solid and red dashed lines denote different temperature $T=60$ and 120 mK, respectively. At finite temperature the ZBCP is suppressed. ($L=4.5$ $\mu \mathrm{m},$ spin-orbit coupling $\alpha =0.2$ meVÅ, induced pairing potential $\Delta =0.5$ meV.)

Figure 5

ZBCP as a function of angle $\theta ={\tan }^{-1}(V_y/V_x)$ for a fixed $\left|\mathit{V}\right|=1$ meV at $T=0,60,120$ mK denoted by the red dashed, blue dotted and orange solid lines, respectively. ($L=4.5\mu$m, spin-orbit coupling $\alpha =0.2$ meVÅ, induced pairing potential $\Delta =0.5$ meV.)

Figure 6

(a)–(c) The evolution of the charge density $u^2-v^2$ of the split MFs generated by overlap for $L=4.5\mu \text{m}$ wire with different values of applied Zeeman-field $V_x$. Increasing $V_x$ increases the overlap between MFs, which in turn leads to a finite charge ${\left|u\right|}^2-{\left|v\right|}^2\ne 0$ associated with the split Majorana fermion. (d) The splitting of the Majorana fermion energy resulting from the same overlap as a function of Zeeman-field $V_x$. The red line is calculated result, while the blue dashed line is a fit to the analytically expected overlap as discussed in the text.