Anomalous proximity effect under Andreev and Majorana bound states

TL;DR

This work analyzes the anomalous proximity effect in a Rashba nanowire-based CN/DN/S junction to differentiate trivial zero-energy Andreev bound states from topological Majorana bound states. Using a lattice Bogoliubov–de Gennes model and Green's-function calculations, it shows that both phases exhibit a zero-energy LDOS peak and odd-frequency spin-triplet pairing in the clean limit, but the peak’s robustness to scalar disorder depends on the superconducting segment length—robust only when $L_S$ is long enough to suppress Majorana hybridization. For short $L_S$, disorder splits the zero-energy features in both phases, making them hard to distinguish, whereas long $L_S$ in the topological phase preserves the ZEP and triplet correlations, offering a reliable Majorana signature. Spin-singlet correlations are consistently suppressed near zero energy, underscoring the unconventional nature of the proximity effect and supporting its use as a diagnostic tool for Majorana physics in disordered hybrid structures.

Abstract

We theoretically study the anomalous proximity effect in a ballistic normal metal/diffusive normal metal/superconductor junction based on Rashba semiconductor nanowire model. The system hosts two distinct phases: a trivial helical phase with zero-energy Andreev Bound States and a topological phase with Majorana Bound States. We analyze the local density of states and induced pair correlations at the edge of the normal metal region. We investigate their behavior under scalar onsite disorder and changing the Superconductor and diffusive regions lengths in the trivial helical and topological phases. We find that both phases exhibit a zero-energy peak in the local density of states and spin-triplet pair correlations in the clean limit, which we attribute primarily to odd-frequency spin-triplet pairs. Disorder rapidly splits the zero-energy peak in the trivial helical phase regardless of the lengths of the superconductor and diffusive normal regions. The zero-energy peak in the topological phase show similar fragility when the superconductor region is short. However, for long superconductor regions, the zero-energy peak in the topological phase remain robust against disorder. In contrast, spin-singlet correlations are suppressed near zero energy in both phases. Our results highlight that the robustness of the zero-energy peak against scalar disorder, contingent on the Superconductor region length, serves as a key indicator distinguishing trivial Andreev bound states from topological Majorana bound states.

Figures (9)

• Figure 1: Schematic of the CN/DN/S junction. The left part of the system is a normal region (N, green area) with total length $L_{\rm N}$. The normal region is divided into two parts: a clean normal region (CN) with length $L_{\rm CN}$ and a disordered normal region (DN) with length $L_{\rm DN}$, with scalar onsite disorder represented as white dots. The total length of the normal region is $L_{\rm N} = L_{\rm CN} + L_{\rm DN}$. The right part of the system is a superconducting region (S) of length $L_{\rm S}$. The system is subjected to an external Zeeman field $B$ along the $x$ direction. The chemical potential in the normal region $\mu_N$ is lower than that of the superconductor $\mu_S$. Panel (a) symbolizes the system in the trivial regime with ABSs (light blue) at the DN/S interface, whereas panel (b) symbolizes the system in the topological regime with MBSs (light red) at the ends of the S region.
• Figure 2: Low energy spectrum for CN/DN/S junction with short S region as a function of the Zeeman field $B$ at $W=\mu_S$ for $L_{\rm DN}=2a$ (a,b), $L_{\rm DN}=20a$ (d,e), and $L_{\rm DN}=40a$ (g,h). Panels (a,d,g) show the spectrum for one disorder realization, whereas panels (b,e,h) show the average spectrum over 200 disorder realizations. The light gray curves in (a,b,d,e,g,h) show the spectrum for the clean case ($W=0$). Panels (c,f,i) show the wavefunction probability density of the zero-energy states with one disorder realization (solid) and for ensemble average (dashed) with $L_{\rm DN}=2a$ (c), $L_{\rm DN}=20a$ (f), and $L_{\rm DN}=40a$ (i). The top panels in (c,f,i) show the wavefunction probability density for the trivial zero-energy ABSs at $B=0.825B_c$, whereas the bottom row shows the wavefunction probability density for the MBSs at $B=1.2B_c$. The orange and pink curves in (a,b,d,e,g,h) show the values of $B$ for which the wavefunction probability density is shown in (c,f,i). The parameters used are: $L_{\rm S}=20a$, $L_{\rm N} = 40a$, $a=50$ nm, $\mu_S=0.5$ meV, $\mu_N=0.1$ meV, $\Delta=0.25$ meV, $\alpha=20$ meV nm, $t=1$ meV, and $W=\mu_S$ unless otherwise stated.
• Figure 3: Lowest positive energy level $E$ as a function of scalar disorder strength $W$ in the trivial $B=0.825B_c$ (a,b,c,d,e) and topological $B=1.2B_c$ (f,g,h,i,j) phases for $L_{\rm S}=20a$ (a,f), $L_{\rm S}=40a$ (b,g), $L_{\rm S}=60a$ (c,h), $L_{\rm S}=80a$ (d,i), and $L_{\rm S}=100a$ (e,j). The parameters used are: $L_{\rm N} = 40a$, $a=50$ nm, $\mu_S=0.5$ meV, $\mu_N=0.1$ meV, $\Delta=0.25$ meV, $\alpha=20$ meV nm, $t=1$ meV, $B=0.825B_c$ for the trivial phase and $B=1.2B_c$ for the topological phase.
• Figure 4: Ensemble-averaged LDOS for short S region ($L_{\rm S}=20a$) junctions evaluated at the leftmost site of the N region ($i=1$) as a function of energy $E$ and scalar disorder strength $W$ with $L_{\rm DN}=2a$ (a,f), $L_{\rm DN}=20a$ (b,g), and $L_{\rm DN}=40a$ (c,h). Panels (d,i) show line cuts of the LDOS in (a,f,b,g) for $W/\mu_S=0,1,2$. Panels (e,j) show the spatial profile of the LDOS at $E \approx 0$ for $W/\mu_S=0,1,2$ with $L_{\rm DN}=20a$. Top row (a,b,c,d,e): trivial phase ($B=0.825B_c$). Bottom row (f,g,h,i,j): topological phase ($B=1.2B_c$). Other parameters used are: $L_{\rm S}=20a$, $L_{\rm N} = 40a$, $a=50$ nm, $\mu_S=0.5$ meV, $\mu_N=0.1$ meV, $\Delta=0.25$ meV, $\alpha=20$ meV nm, $t=1$ meV, and $\eta=0.001$ meV.
• Figure 5: Ensemble-averaged Absolute value of the spin-triplet pair correlations $|\Vec{d}|$ for short S region ($L_{\rm S}=20a$) junctions evaluated at the leftmost site of the N region ($i=1$) as a function of energy $E$ and scalar disorder strength $W$ with $L_{\rm DN}=2a$ (a,f), $L_{\rm DN}=20a$ (b,g), and $L_{\rm DN}=40a$ (c,h). Panels (d,i) show line cuts of the spin-triplet pair correlations in (a,f,b,g) for $W/\mu_S=0,1,2$. Panels (e,j) show the spatial profile of the spin-triplet pair correlations at $E\approx 0$ for $W/\mu_S=0,1,2$ with $L_{\rm DN}=20a$. Top row (a,b,c,d,e): trivial phase ($B=0.825B_c$). Bottom row (f,g,h,i,j): topological phase ($B=1.2B_c$). Other parameters used are: $L_{\rm S}=20a$, $L_{\rm N} = 40a$, $a=50$ nm, $\mu_S=0.5$ meV, $\mu_N=0.1$ meV, $\Delta=0.25$ meV, $\alpha=20$ meV nm, $t=1$ meV, and $\eta=0.001$ meV.