Magnetization Signature of Topological Surface States in a Non-Symmorphic Superconductor

Abstract

Superconductors with nontrivial band structure topology represent a class of materials with unconventional and potentially useful properties. Recent years have seen much success in creating artificial hybrid structures exhibiting the main characteristics of 2D topological superconductors. Yet, bulk materials known to combine inherent superconductivity with nontrivial topology remain scarce, largely because distinguishing their central characteristic-the topological surface states-has proved challenging due to a dominant contribution from the superconducting bulk. In this work, a highly anomalous behavior of surface superconductivity in topologically nontrivial 3D superconductor In₂ Bi, where the surface states result from its nontrivial band structure, itself a consequence of the non-symmorphic crystal symmetry and strong spin-orbit coupling, is reported. In contrast to smoothly decreasing diamagnetic susceptibility above the bulk critical field, H_c2 , as seen in conventional superconductors, a near-perfect, Meissner-like screening of low-frequency magnetic fields well above H_c2 is observed. The enhanced diamagnetism disappears at a new phase transition close to the critical field of surface superconductivity, H_c3 . Using theoretical modeling, the anomalous screening is shown to be consistent with modification of surface superconductivity by the topological surface states. The possibility of detecting signatures of the surface states using macroscopic magnetization provides a new tool for the discovery and identification of topological superconductors.

Keywords: magnetization and magnetic susceptibility; non-symmorphic crystal symmetries; superconductivity; topological surface states.

Figure 1

Anomalous AC susceptibility of In₂Bi. a) Schematic crystal structure of In₂Bi. Bi atoms are shown in blue and In atoms in different shades of red, to distinguish between In atoms within the hexagonal planes (dark red) and those making up In chains (light red). The shaded areas denote the unit cell containing four In and two Bi atoms. Symmetry axes are indicated by arrows. b) ZFC and FC magnetization as a function of T at H = 10 Oe. Inset: photo of our typical cylindrical crystal; scale bar: 1 mm. c) AC susceptibility measured using h _ac = 0.1 Oe and frequency f = 8 Hz (red curves). Black curves: DC magnetization and its hysteresis for this sample. As a reference, the blue dashed curves show the standard response expected for surface superconductivity. The inset in the lower panel shows a zoom of χ′′ indicating the transition to the vortex state at H _c1. The vertical dashed lines indicate H _c2 and H _c3, and the arrows the sweep directions. d) Top: Lissajous loops for the representative DC fields indicated by the color‐coded dots in (c). Bottom: Corresponding waveforms m _ac(t) for the applied sinusoidal field h _ac(t) of amplitude 0.1 Oe.

Figure 2

Anomalous diamagnetic response at different temperatures and AC excitations. a) AC susceptibility as a function of the AC field amplitude (see legends). b) χ′(H) at T between 2 and 6 K measured with 0.5 K step; h _ac = 0.1 Oe. c) Hysteresis in M(H) between the increasing (black symbols) and decreasing (red) DC field H; T =  2 K, H _ts is indicated by an arrow. d) Phase diagram for all the critical fields (labeled and color coded). Red symbols: H _ts(T) found from AC susceptibility measurements in (b). Error bars: standard deviations. The black curve shows the standard BCS dependence H _c1(T) ∝ 1 − (T/T _c)². Brown curve: best fit to H _c2(T) using the two‐band model of superconductivity (Supporting Information). Yellow curve: guide to the eye. The H _c3/H _c2 ratio changes from 2.0 at 2 K to 1.7 at 5.6 K, as expected (see text).

Figure 3

Effect of surface quality. a) DC magnetization for a sample with a rough surface shown in the photo (scale bar: 0.5 mm). Hysteresis in M(H) between increasing and decreasing field remains small, comparable to our best crystals (cf. Figure 1c). This indicates that the surface roughness did not affect the quality of the bulk. b) Comparison of AC susceptibility for crystals with comparable bulk pinning but smooth and rough surfaces (black and red curves, respectively). In both cases, h _ac = 0.1 Oe. See also Figure S3, Supporting Information, for similar data on a spherical crystal before and after intentional surface degradation.

Figure 4

Band structure of In₂Bi and experimental evidence of multiband superconductivity. a) Calculated Fermi surface of In₂Bi. b) Temperature evolution of DC magnetization. The curves are for T = 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, and 5.6 K. Inset: Temperature dependence of the extracted Maki parameter κ₂. Error bars: standard deviations. c) Band structure of an In₂Bi ribbon near the H (H′) points (see “Topological surface states” in Supporting Information for details). Bands due to In chains are omitted for clarity. Four pairs of counter‐propagating edge states cross within the bulk bandgap. Two of the pairs connect bands split by the spin–orbit gap (Δ_SO) between Bi‐derived bands, while the other two are within a smaller gap of In‐derived bands. In these calculations, hopping amplitudes that break particle–hole symmetry were not included. d) Comparison of the observed AC response (symbols) with the theory for conventional surface superconductivity (blue curve) and our model that includes proximitized surface states (red).