Manipulation of the Majorana Ising spin via Rashba–Dresselhaus spin–orbit coupling

Majorana surface states in time-reversal invariant (TRI) topological superconductors (TSCs) typically exhibit a highly anisotropic magnetic response, a phenomenon termed “Majorana Ising spins.” This Ising character is governed by the crystalline symmetries protecting the topological phase. In this work, we investigate the orientation and tunability of Majorana Ising spins within TRI TSCs engineered in two-dimensional spin–orbit coupled systems proximitized to an extended s-wave superconductor. We demonstrate that the interplay between Rashba and Dresselhaus spin–orbit couplings (SOC) plays a decisive role in determining the Ising spin orientation. In the limit of pure Rashba SOC, the Ising spin aligns along the x-axis, protected by mirror symmetry M_x, whereas for pure Dresselhaus SOC, it orients along the y-axis, protected by the rotational symmetry C_2y. Crucially, we reveal that when both Rashba and Dresselhaus interactions coexist, the Ising spin direction becomes continuously tunable within the basal plane. By adjusting the relative strengths of the SOC parameters — experimentally accessible via gating in semiconductor heterostructures — any orientation between the x- and y axes can be achieved. We validate these findings by calculating the topological winding number W and elucidating the symmetry-protection mechanism for the tunable phases. Our results propose a pathway for manipulating Majorana fermions in quantum devices through purely electrical means, bridging the gap between symmetry-protected topology and functional spintronic applications.