Magnetic penetration depth in topological superconductors: Effect of Majorana surface states and application for UTe$_2$

Abstract

In this study, we examine how orbital degrees of freedom and Majorana surface states influence the magnetic penetration depth in the superconductor UTe$_2$. Using a two-orbital model, we analyze pairing states belonging to the irreducible representations of the $D_{2h}$ crystal symmetry: $A_u$, $B_{1u}$, $B_{2u}$, and $B_{3u}$. For bulk nodal states such as $B_{2u}$, we find that the penetration depth for screening currents along the antinodal direction and the cylindrical axis scales as $T^2$, in strong contrast to the conventional $T^4$ law. This behavior originates from quasiparticles near the point nodes contributing to the interorbital paramagnetic current. We further show that Majorana surface states can dominate the low-temperature response. The fully gapped $A_u$ state hosts Majorana cones, which produce a $T^3$ dependence of the penetration depth when the ratio of penetration depth to coherence length ($κ$) is small. In contrast, the other pairing states exhibit Majorana Fermi arcs: the exponent is $n=2$ along the dispersive direction, while along the dispersionless direction it depends on whether the arcs terminate at endpoints. The exponent $n=2$ in the dispersive direction is robust, while it in the dispersionless direction relies on the presence or absence of the endpoints of the arcs and deviates from $n=2$ when endpoints are absent. Our results demonstrate that penetration-depth measurements provide a direct probe of Majorana surface states in low-$κ$ superconductors. For larger $κ$, the surface contribution becomes negligible and the temperature dependence is governed by bulk quasiparticles.

Figures (9)

• Figure 1: An external magnetic field is screened by the superconducting current, $J_b$, resulting in an exponential decay of the magnetic field over the characteristic length scale of the magnetic penetration depth, $\lambda_b$. The blue curve illustrates the probability density of the Majorana surface states, which are localized within the coherence length $\xi_a$ from the boundary.
• Figure 2: Cylindrical Fermi surfaces and gap structures for (a) $A_u$, (b) $B_{1u}$, (c) $B_{2u}$, and (d) $B_{3u}$ states, respectively, where $k_a{a},~k_b{b}\in[-\pi,\pi]$ and $k_c{c}\in [-2\pi,2\pi]$. The color represents the magnitude of the superconducting gap. The $A_u$ and $B_{1u}$ states are fully gapped, whereas the $B_{2u}$ and $B_{3u}$ states have point nodes.
• Figure 3: (a) Typical structure of energy spectra for Majorana cone systems. The result is a slab geometry of the $A_u$ state with open $(010)$ surfaces. (a) Typical structure of energy spectra for Majorana Fermi arc systems. The result is a slab geometry of the $B_{1u}$ state with open $(010)$ surfaces.
• Figure 4: (a) Slab geometry with open surfaces perpendicular to the $x$-axis. An external magnetic field applied in the $z$-direction is screened by a Meissner current flowing along the $y$-axis. (b) Schematic illustration of the method of electrical mirror imaging. A planar current flowing in a thin slab reproduces the external magnetic field.
• Figure 5: Temperature dependence of magnetic penetration depth estimated from the bulk Meissner kernel. The insets in (a) and (b) show that $\log(\Delta\lambda)$ is proportional to the temperature, suggesting that exponential temperature behavior of $\Delta\lambda$. $n$ denotes the exponent in the power-law fitting $\Delta\lambda \propto T^n$.