Majorana fermion induced nonlocal current correlations in spin-orbit coupled superconducting wires

Recent observation of zero bias conductance peaks in semiconductor wire/superconductor heterostructures has generated great interest, and there is a hot debate on whether the observation is associated with Majorana fermions (MFs). Here we study the local and crossed Andreev reflections of two normal leads attached to the two ends of a superconductor-semiconductor wire. We show that the MFs induced crossed Andreev reflections have significant effects on the shot noise of the device and strongly enhance the current-current correlations between the two normal leads. The measurements of shot noise and current-current correlations can be used to identify MFs.

Figure 1

(a) A schematic setup of experiment, two normal leads are coupled to the two ends of a semiconductor wire/superconductor heterostructure which supports MFs. (b) A local Andreev reflection process. An electron from one lead is reflected as a hole in the same lead. (c) A crossed Andreev reflection process. An electron from one lead is reflected as a hole in another lead and a Cooper pair is injected into the superconductor.

Figure 2

(a) The energy eigenvalues of a short quasi-1D wire versus chemical potential, the lowest energy states are indicated in red. The topological region is marked out. (b) Contour plot of differential conductance $G_1$ of the left lead as a function of chemical potential and electron incident energy $E$. The inset shows $G_1$ versus incident energy at a fixed chemical potential denoted by the vertical dashed line in (b). The height of the peak at $E=E_M$ is $\frac{2e^2}{h}\frac{t_L^2}{t_L^2+t_R^2}$. (c) The ground state wave function $|{\Psi }_R{|}^2$ in the topological regime, with chemical potential denoted by the dashed line in (b), projected onto the $x$-$y$ plane. The fermionic ground state is formed by two MF end states.

Figure 3

(a) Contour plot of Fano factor $P_{11}/e{G}_1$ for electrons with incident energy $E$ at chemical potential $\mu$. (b) Contour plot of $P_{12}/e\overline{G}$. (c) Contour plot of $C_{11}/e{\overline{I}}_1$ as a function of voltage bias $eV$ and chemical potential. (d) Contour plot of $C_{12}/e\overline{I}$. (e) The $P_{11}/e{G}_1$ (blue line) and $P_{12}/e\overline{G}$ (green line) as a function of incident energy $E$ at a fixed chemical potential denoted by the dashed lines in (a) and (b), respectively. (f) The $C_{11}/e{\overline{I}}_1$ (blue line) and $C_{12}/e\overline{I}$ (red line) as a function of voltage bias at fixed chemical potential denoted by the dashed lines in (c) and (d).

Figure 4

Random disorder with $\omega =16\Delta$ is present for all the figures. (a) and (b) The contour plots of the conductance $G_1$ and $G_2$, respectively. (c) and (d) The Fano factors $C_{22}/e{\overline{I}}_2$ and $C_{12}/e\overline{I}$, respectively. (e) The ground state wave function $|{\Psi }_{\mathbf{R}}{|}^2$ in the topological regime with $\mu =-129.4\Delta$ [indicated by dashed line A in (a)]. The dimensions of the wire are $N_x=50a$ and $N_y=5a$. (f) The ground state wave function in the trivial regime with $\mu =-120.8\Delta$ [indicated by dashed line B in (a)].

Figure 5

(a) Contour plot of $C_{11}/e{\overline{I}}_1$ as a function of voltage bias and magnetic field $V_x$. (b) Contour plot of $C_{12}/e\overline{I}$.

Figure 6

(a) The energy eigenvalues as a function of chemical potential. The topological regime with Majorana fermion end states are marked out. (b) Contour plot of $C_{11}/e{\overline{I}}_1$ as a function of voltage bias and chemical potential. (c) Contour plot of $C_{12}/e\overline{I}$.

Figure 7

$N_x=40$ for (a) and (b). $N_x=70$ for (c) and (d). (a) and (c) Contour plots of $C_{11}/e{\overline{I}}_1$ as a function of voltage bias $eV$ and chemical potential. (b) and (d) Contour plots of $C_{12}/e\overline{I}$.