Tunneling characteristics of a chain of Majorana bound states

We consider theoretically tunneling characteristic of a junction between a normal metal and a chain of coupled Majorana bound states generated at crossings between topological and nontopological superconducting sections, as a result of, for example, disorder in nanowires. While an isolated Majorana state supports a resonant Andreev process, yielding a zero-bias differential conductance peak of height $2e^2/h$, the situation with more coupled Majorana states is distinctively different with both zeros and $2e^2/h$ peaks in the differential conductance. We derive a general expression for the current between a normal metal and a network of coupled Majorana bound states and describe the differential conductance spectra for a generic set of situations, including regular, disordered, and infinite chains of bound states.

Figure 1

(Color online) Illustration of a semiconductor with induced superconducting order parameter from an adjacent superconductor (s). Spatial variations in density or superconducting order parameter create crossings between TS and nontopological segment when the density crosses the critical density [vertical (red) lines]. MBS (circles) are located at each crossing point and the distances between them determine the coupling matrix elements $t_{ij}$ of the resulting Majorana network. A tunneling contact (N) probes the network by tunneling into the end Majorana mode, with tunneling density of states $\Gamma$.

Figure 2

(Color online) The conductance for tunneling into the end of a chain with one, two, five, and eight Majorana states coupled by $t=\Gamma$, illustrated with a small icon over each plot. The $n$ coupled Majoranas give rise to a $n$ states inside the gap, illustrated by the insets. The number of zeros and peaks with conductance $2e^2/h$ is also determined by $n$, see text.

Figure 3

(Color online) The conductance for tunneling into the end of an infinite Majorana chain with identical tunneling couplings. The parameters $\left|t\right|/\Gamma$ ranges from 5 to 1/8, from outside in.

Figure 4

(Color online) Both panels show the conductance for a chain with ten sites and one weak link with tunnel coupling $t_{\text{weak}}=0.25\Gamma$ while the other links have $t=\Gamma$. The weak links are the 1–2 and 5–6 connections for the left/right panel, respectively. The thick (blue) curve is the conductance for $kT=0.1\Gamma$, the thin gray curve for $kT=0$ while the crosses show the conductance for the chain truncate at the weak link also with $kT=0.1\Gamma$. It is clearly seen that the truncated approximation works well, because the chain after the truncations leads to structure barely resolvable, because $k_{B}T$ is not much smaller than the width $\left(\approx 0.125\right)$ of the additional resonances. It should be noted that the situation with only a single deviating link is not the generic situation but it is chosen here to illustrate the point, that the first weak link determines the effective length of the chain.