Majorana zero modes in semiconductor-superconductor hybrid structures: Defining topology in short and disordered nanowires through Majorana splitting

TL;DR

This paper addresses the challenge of defining topology and robust Majorana zero modes in finite, disordered semiconductor-superconductor nanowires. Using a minimal 1D SM-SC model with Gaussian disorder, it computes the Majorana splitting $E_s$, the topological gap $\Delta_s$, and introduces the ratio $R=E_s/\Delta_s$ as an operational topology metric, analyzing how disorder and wire length affect exponential protection. The results show a disorder-induced crossover from exponential to algebraic decay of the splitting with length, with substantial mesoscopic fluctuations and a breakdown of exponential protection when the disorder is of order the pristine gap; this implies topology in finite disordered wires is not uniquely defined without context. The findings have direct implications for interpreting Majorana experiments in InAs/Al systems, highlighting the need for long, exceptionally clean wires and providing practical diagnostics (the $E_s/\Delta_s$ ratio and localization length) to assess topology in realistic devices.

Abstract

Majorana zero modes (MZMs) are bound midgap topological excitations at the ends of a 1D topological superconductor, which must come in pairs. If the two MZMs in the pair are sufficiently well-separated by a distance much larger than their individual localization lengths, then the MZMs behave as non-Abelian anyons which can be braided to carry out fault-tolerant topological quantum computation. In this `topological' regime of well-separated MZMs, their overlap is exponentially small, leading to exponentially small Majorana splitting, thus enabling the MZMs to be topologically protected by the superconducting gap. In real experimental samples, however, the existence of disorder and the finite length of the 1D wire considerably complicate the situation, leading to ambiguities in defining `topology' since the Majorana splitting between the two end modes may not necessarily be small in disordered wires of short length. We theoretically study this situation by calculating the splitting in experimentally relevant short disordered wires, and explicitly investigating the applicability of the `exponential protection' constraint as a function of disorder, wire length, and other system parameters in realistic models of nanowires currently being used experimentally. We find that the exponential regime is highly constrained, and is suppressed for disorder somewhat less than the topological superconducting gap. We provide detailed results and discuss the implications of our theory for the currently active experimental search for MZMs in superconductor-semiconductor hybrid platforms. A general consequence of our work is that `topology' in finite disordered wires may not be uniquely defined, necessitating a careful analysis which depends on the context.

Figures (27)

• Figure 1: Band structure as a function of Zeeman field $V_Z$ for pristine limit $\sigma=0$ (left column), $\sigma=0.1$ meV (middle column), and $\sigma=0.9$ meV (right column) for $L=1~\mu$m (top row) and $L=5~\mu$m (bottom row). The maximal MZM splitting energy $E_s$ (in the unit of meV) in the first lobe is indicated by the blue arrow, and the gap size $\Delta_s$ (in the unit of meV) is indicated by the red arrow, with their values at the top of each panel. The vertical dashed line indicates the topological phase transition point at $V_Z=1.02$ meV.
• Figure 2: The distribution of the maximal MZM splitting energy $E_s$ in the first lobe as a function of disorder strength $\sigma$ for (a) $L=1~\mu$m and (b) $L=5~\mu$m.
• Figure 3: The disorder-averaged maximal MZM splitting energy $\expval{E_s}$ in the first lobe as a function of disorder strength $\sigma$ for (a) short wires ($L=0.6,0.8,1,2~\mu$m from light to dark blue) and (b) long wires ($L=3,4,5,\dots,15~\mu$m from light to dark blue).
• Figure 4: The disorder-averaged maximal MZM splitting energy $\expval{E_s}$ in the first lobe as a function of the wire length $L$ for different disorder strengths $\sigma$ (from light to dark blue: $\sigma=0,0.1, 0.2,\dots,1$ meV). Lower-left inset: The exponential fit of $\expval{E_s}$ in the pristine limit $\sigma=0$ for $L\ge5~\mu$m with $\expval{E_s}= A_{s}\exp(-L/\xi_s)$, where the best fit $A_{s}=0.27(8)$ meV and $\xi_s=0.77(2)~\mu$m. Upper-right inset: The power-law fit of $\expval{E_s}$ for $L\ge5~\mu$m at the crossover point $\sigma=0.2$ meV in the log-log plot with $\expval{E_s} = \alpha_{s} L^{-\eta_{s}}$, where the best fit $\alpha_{s}=0.010(5)$ meV and $\eta_{s} = 1.7(2)$.
• Figure 5: The distribution of the gap size $\Delta_s$ for (a) short wires ($L=1~\mu$m) and (b) long wires ($L=5~\mu$m) as a function of disorder strength $\sigma$.