Interplay of magnetic textures with spin-orbit coupled substrates

TL;DR

This work addresses how magnetic textures such as skyrmions couple to intrinsic spin–orbit coupling on substrates. It develops a spintronic gauge theory to show that texture-induced SOC and substrate SOC are generally non-additive, supported by a local current marker and tight-binding simulations, and connected to mesoscopic transport via a four-terminal topological Hall setup. The study reveals frame-dependent induced SOC, nonlinear dependence on intrinsic SOC, and symmetry relations between Rashba and Dresselhaus couplings across skyrmion types, with broad implications for spintronic devices and potential routes to topological superconductivity. The multi-scale approach provides a coherent framework linking microscopic gauge fields to global transport signatures, offering experimentally accessible markers and guidance for engineering SOC-skyrmion interactions.

Abstract

Magnetic textures such as skyrmions in thin films grown on substrates possess significant technological potential. Inhomogeneous magnetic structures can be described as homogeneous ferromagnetic order in the presence of anisotropic spin-orbit coupling (SOC). It remains unexplored, however, how this {\it induced} SOC stemming from the magnetic textures interacts with the SOC of the substrate. Here we show that these two contributions to SOC are in general {\it not} additive. We demonstrate this by employing a spintronics gauge theory. We further compute local currents which, when considered in the proper frame, match the spintronics gauge theory results. Finally, we analyze global transport quantities and show that they substantiate our previous results quantitatively. The implications for skyrmionics as well as topological superconductivity are discussed.

Figures (6)

• Figure 1: Induced magnetic field of a $R=10 a_{0}$ Néel skyrmion on a $40 a_0 \times 40 a_{0}$ square system for different intrinsic SOC strengths. Left: $\lambda_{\text{R}}=\lambda_{\text{D}}=0$. Center: $\lambda_{\text{R}}=0.2$ eV, $\lambda_{\text{D}}=0$. Right: $\lambda_{\text{R}}=0$, $\lambda_{\text{D}}=0.2$ eV. Note that both $\lambda_{R,D}$ are related to the continuum SOC parameters via $\lambda_{R}=\frac{\alpha_{0}}{a_{0}}$ and $\lambda_{D}=\frac{\beta_{0}}{a_{0}}$ respectively.
• Figure 2: Induced Rashba and Dresselhaus SOC strengths in the $p_{x}$ and $p_{y}$ directions of a $R=10 a_{0}$ Néel skyrmion on a $40 a_0 \times 40 a_{0}$ square system for different intrinsic SOC strengths. Top Row: No SOC. Middle Row: $\lambda_{\text{R}}=0.2 \text{ eV}$. Bottom Row: $\lambda_{\text{D}}=0.2 \text{ eV}$. Note that both $\lambda_{R,D}$ are related to the continuum SOC parameters via $\lambda_{R}=\frac{\alpha_{0}}{a_{0}}$ and $\lambda_{D}=\frac{\beta_{0}}{a_{0}}$ respectively.
• Figure 3: Induced Rashba and Dresselhaus SOC markers for different intrinsic SOC strengths. First row: No SOC in the laboratory frame. Second row: No SOC in the magnetic frame. Third row: $\lambda_{\text{R}}=0.2 \text{ eV}$ in the magnetic frame. Fourth row: $\lambda_{\text{D}}=0.2 \text{ eV}$ in the magnetic frame.
• Figure 4: Schematic diagram of a four-terminal Hall experiment with a scattering region containing a magnetic Bloch skyrmion, in the presence of SOC, attached to four ferromagnetic leads: left (L), top (T), right (R), and bottom (B). With the system experiencing a longitudinal voltage bias of eV inducing a current to flow in from the L lead and get scattered thus giving a Hall response.
• Figure 5: Topological Hall effect for varying SOC with different skyrmions. Top: Rashba Profile. Bottom: Dresselhaus profile. NS: Néel skyrmion. BS: Bloch skyrmion. NA: Néel antiskyrmion. BA: Bloch antiskyrmion.