Route to Topological Superconductivity via Magnetic Field Rotation

Abstract

The verification of topological superconductivity has become a major experimental challenge. Apart from the very few spin-triplet superconductors with p-wave pairing symmetry, another candidate system is a conventional, two-dimensional (2D) s-wave superconductor in a magnetic field with a sufficiently strong Rashba spin-orbit coupling. Typically, the required magnetic field to convert the superconductor into a topologically non-trivial state is however by far larger than the upper critical field H(c2), which excludes its realization. In this article, we argue that this problem can be overcome by rotating the magnetic field into the superconducting plane. We explore the character of the superconducting state upon changing the strength and the orientation of the magnetic field and show that a topological state, established for a sufficiently strong out-of-plane magnetic field, indeed extends to an in-plane field orientation. We present a three-band model applicable to the superconducting interface between LaAlO3 and SrTiO3, which should fulfil the necessary conditions to realize a topological superconductor.

Figure 1. Energy bands (pink) and (blue) and Fermi surfaces with α > 0 and magnetic filed |H| < H_t.

(a) For H = 0, the two bands touch at k = 0. (b,c) For |H| > 0, the band splitting at k = 0 is equal to the Zeeman splitting 2μ_B|H|. The centers of the shifted Fermi surfaces in (c) are at the momenta momenta q⁺/2 = (q⁺/2, 0) and q⁻/2 = (q⁻/2, 0), respectively. Although q⁺ ≈ −q⁻, their absolute values are in general different. In (c), the spins on the k_x-axis orient according to the magnetic field rather than according to the SOC, if H_y > α|sin k_F|. Note that the Fermi energy is somewhat larger than |ε₀| because of the SOC induced band splitting.

Figure 2. z-component of the Berry curvature Ω(k) for an out-of-plane magnetic field H_z in the SC state.

Ω(k) is finite within the window Δ below the Fermi energy. (a) In the topologically trivial state, (here: , and Δ is fixed to 0.1 t), the total Berry curvature integrates to zero over the Brillouin zone. (b) In the topological situation (B) (see main text, and ), the Berry curvature integrates to 2πC = 4π over the Brillouin zone.

Figure 3. Self-consistent solutions of the SC order parameter for three magnetic-field directions and V = 4 t, α = 0.5 t and a constant electron density n = 0.05.

This value of n corresponds to . For such low densities, a large interaction strength V is required to obtain a reasonably large order parameter. For each value of |H|, q⁻ = (q, 0) is obtained by minimizing the free energy. The red circles indicate the magnetic field strength above which a finite COMM q ≠ 0 is present.

Figure 4. Phase diagram showing the topologically different SC states as a function of out-of-plane magnetic field H_z and in-plane magnetic field H_y, not including orbital coupling to the magnetic field.

The blue circles (a–e) mark the H_y − H_z-points for which the energy spectra are shown in Fig. 5. The dashed lines indicate the transition from zero COMM to finite COMM pairing.

Figure 5. Energy spectra E_n(k_x) for a stripe geometry with 600 × 100 sites, open boundary conditions and in-plane magnetic field component in y-direction, and parameters V, α, and n as in Fig. 3.

(a–c) The evolution of the edge modes (green line: upper edge, red line: lower edge) upon rotating the magnetic field is shown for (a) , (b) , (c) , and . The self-consistently calculated order parameters and COMMs q⁻ are (a) , (b) , and (c) . (d,e) illustrate the crossover regime : (d) and , and (e) and . The black arrows in (d) indicate the partial occupation of states originating from the -band. The opacity of each point encodes the weight with which the corresponding state contributes to the density of states.

Figure 6. Band structure of the three-band model for k_y = 0.

In order to ensure (red dashed line), the Fermi energy should be at the degeneracy point of the upper, Rashba-like doublet. The parameters are here: , Δ₀ = t, .