Current correlations in a Majorana beam splitter

We study current correlations in a $T$ junction composed of a grounded topological superconductor and of two normal-metal leads which are biased at a voltage $V$. We show that the existence of an isolated Majorana zero mode in the junction dictates a universal behavior for the cross correlation of the currents through the two normal-metal leads of the junction. The cross correlation is negative and approaches zero at high bias voltages as $-1/V$. This behavior is robust in the presence of disorder and multiple transverse channels, and persists at finite temperatures. In contrast, an accidental low-energy Andreev bound state gives rise to nonuniversal behavior of the cross correlation. We employ numerical transport simulations to corroborate our conclusions.

Figure 1

(a) The proposed experimental setup is a $T$ junction between a topological superconductor (TSC) and two metallic leads. Here the TSC is realized by a semiconductor nanowire, proximity coupled to a conventional $s$-wave superconductor under an applied magnetic field. (b) We model the TSC by a spinless $p$-wave superconductor. It is coupled to the leads through a normal-metal section N, whose length $d_{\mathrm{N}}$ is taken to zero. Scattering at the NP interface is described by the reflection matrix $r_{\mathrm{NP}}$ [see Eq. (3)], while scattering at the $T$ junction is described by the matrix $S_{\mathrm{J}}$ [see Eq. (4)].

Figure 2

(a) Zero-frequency cross correlations $P_{\mathrm{RL}}$ [defined in Eq. (1)] of the currents through the left and right leads as a function of bias voltage $V$ at various temperatures. $P_{\mathrm{RL}}$ is negative for all $V$ and approaches zero at voltages which are larger than both the resonance width and the temperature. (b) Total differential conductance, $dI/dV$, where $I=I_{\mathrm{R}}+I_{\mathrm{L}}$. At zero temperature $dI/dV$ exhibits a zero-bias conductance peak quantized to $2e^2/h$ [54]. A nonzero temperature widens the peak and reduces its height to a nonuniversal value. The inset shows the zero-temperature local density of states at zero energy in the wire in the absence of coupling to the leads in arbitrary units. The section of the wire not covered by the superconductor is $x\in [0,x_0]$, as depicted in Fig. 1.

Figure 3

Current cross correlation $P_{\mathrm{RL}}$ vs bias voltage $V$ at $\mu =0$ and $T=0$ for different realization of short-range Gaussian disorder. (a) $B=520\mathrm{mT}>B_{\mathrm{c}}$, the system is in the topological phase with a zero-energy Majorana bound state (MBS) at each end of the wire. The universal behavior of $P_{\mathrm{RL}}\left(V\right)$, (being negative and approaching zero at high voltage) is not affected by the presence of disorder. (b) For each realization of disorder the magnetic field is tuned to have an Andreev bound state (ABS) with zero energy at the end of the wire, while keeping the system in the topologically trivial phase, $B=170-200\mathrm{mT}<B_{\mathrm{c}}$ (see the text for more details). The behavior of $P_{\mathrm{RL}}\left(V\right)$ varies significantly for different realizations of disorder. In all cases $P_{\mathrm{RL}}>0$ for large $V$ in contrast to the topological case where it goes to zero.

Figure 4

(a) Cross correlation and (b) differential conductance at various chemical potentials $\mu$, corresponding to a different odd number of occupied transverse channels. The calculations are performed at $T=0$, $v_{\mathrm{dis}}=0$, and $B=520\mathrm{mT}$. The addition of occupied channels introduces extra subgap states which coexist with the Majorana bound state. These appear as peaks in the differential conductance spectra at finite $V$ [see (b) at $V\simeq 80\mu \mathrm{eV}]$. Above this voltage the behavior of $P_{\mathrm{RL}}$ is no longer universal.

Figure 5

Illustration of the tight-binding model corresponding to the system depicted in Fig. 1. Each lead is tunnel coupled to the sites adjacent to it. The purple sites are ones in which there is a nonvanishing induced pair potential [cf. Eq. (11)].