Detrimental effects of disorder in two-dimensional time-reversal invariant topological superconductors

The robustness against local perturbations, as long as the symmetry of the system is preserved, is a distinctive feature of topological quantum states. Magnetic impurities and defects break time-reversal invariance and, consequently, time-reversal invariant (TRI) topological superconductors are fragile against this type of disorder. Nonmagnetic impurities, however, preserve time-reversal symmetry and one naively expects a TRI topological superconductor to persist in the presence of nonmagnetic impurities. In this work, we study the effect of nonmagnetic disorder on a TRI topological superconductor with extended $s$-wave pairing, which can be engineered at the interface of an Fe-based superconductor and a strongly spin-orbit coupled Rashba layer. We model two different types of nonmagnetic random disorder and analyze both the bulk density of states and edge state spectrum. Contrary to naive expectations, we find that the disorder strongly affects the topological phase by closing the energy gap, while trivial superconducting phases remain stable and fully gapped. The disorder phase diagram reveals a strong expansion of a nodal phase with increasing disorder. We further show the decay of the helical Majorana edge states in the topological phase and how they eventually disappear with increasing disorder. These results alter our understanding of effects of impurities and disorder on TRI topological phases and may help explain the difficulty of experimental observation of TRI topological superconductors.

Figure 1

Phase diagram of a clean hybrid structure consisting of an Fe-based $s_{\pm }$-wave superconductor and a 2D Rashba SOC layer, described by Eq. (1), as a function of $\mu -r$ and ${\lambda }_R$. Topologically trivial $s_{++}$-wave ($\nu =0$), nontrivial $s_{\pm }$-wave ($\nu =1$), and nodal phases are depicted by blue, red, and white, respectively. The color intensity coding corresponds to the energy gap size. Blue and red solid lines mark the phase boundaries.

Figure 2

Bulk DOS for different superconducting phases in the presence of disorder with topological $s_{\pm }$-wave (red), trivial $s_{++}$-wave (blue), and conventional $s$-wave (green) superconducting symmetries, all with similar energy gaps in the clean limit. Top row: Anderson disorder with $W=1.0,1.8$, and 2.4. Bottom row: Concentration disorder with $V=2$ for $c=1\%,4\%$, and $7\%$.

Figure 3

Energy of subgap states induced by $N$ randomly placed nonmagnetic impurities on a $51\times 51$ lattice as a function of impurity scattering strength $V$. Gray shade corresponds to the bulk bands in the clean limit. Cyan shade and dashed blue line mark the disorder strengths for the Anderson and concentration disorder used in Fig. 2. (a)–(d) Topological $s_{\pm }$-wave phase for (a) $N=1$, (b) $N=10$, (c) $N=20$, (d) single impurity vs dimers of impurities. (e), (f) Trivial $s_{++}$-wave phase for (e) $N=10$ and (f) $N=20$. (d) Single impurity and single-impurity dimer oriented either horizontally or diagonally.

Figure 4

(a) Critical disorder strength $W_c$ of Anderson disorder (red, left axis) and topological gap ${\mathrm{\Delta }}_{\mathrm{Topo}}$ in the clean limit (blue, right axis) as a function of ${\lambda }_R$. (b) Critical concentration $c^{*}$ for concentration disorder as a function of ${\lambda }_R$. Solid red line in (a) is a fit to $W_c\propto {\lambda }_R^{1/2}$; all other lines are only guides to the eye.

Figure 5

Phase diagram as a function of Rashba SOC ${\lambda }_R$ and chemical potential $\mu -r$ for Anderson disorder with $W=1.2$ (a), $W=2.4$ (b), and $W=3.6$ (c), representing disordered versions of the clean phase diagram Fig. 1. Blue and red dashed lines separate different phases in the clean system.

Figure 6

Phase diagram as a function of disorder strength $W$ and chemical potential $\mu -r$ for ${\lambda }_R=0$ (a), ${\lambda }_R=0.5$ (b), and ${\lambda }_R=1.0$ (c).

Figure 7

Local DOS for subgap energies $\left|E\right|<{\mathrm{\Delta }}_{\mathrm{Topo}}/4$ in the topological phase for a finite disk of radius $r=47$. 2D and 3D plots of the local DOS in the clean limit [(a), (d)] and for $W=1.8$ [(b), (e)] and $W=2.6$ [(c), (f)]. (g) Line cuts of the local DOS in same low-energy range through the disk for different disorder strengths $W$.