Transverse Transport in Two-Dimensional Relativistic Systems with Nontrivial Spin Textures

Using multiple scattering theory, we show that the generally accepted expression of transverse resistivity in magnetic systems that host skyrmions, given by the linear superposition of the ordinary, the anomalous, and the topological Hall effect, is incomplete and must be amended by an additional term, the “noncollinear” Hall effect (NHE). Its angular form is determined by the magnetic texture, the spin-orbit field of the electrons, and the underlying crystal structure, allowing us to disentangle the NHE from the various other Hall contributions. Its magnitude is proportional to the spin-orbit interaction strength. The NHE is an essential term required for decoding two- and three-dimensional spin textures from transport experiments.

Figure 1

Evolution of the scattered current for a magnetic trimer in an equilateral geometry as a function of the opening angle ${\theta }_M$ of the magnetic moments and of the spin-orbit strength $\mathcal{R}$. (a) The THE contribution to the current determined by the scalar spin chirality. (b) NHE current originating from relativistic effects, its nontrivial angular dependence is given in Eq. (2). (c) Total Hall current emerging from three spin processes, which present a nonsymmetric dependence for ${\theta }_M\to \pi -{\theta }_M$.

Figure 2

The chart in the left panel describes the size of the topological (blue) and noncollinear (red) Hall current for six topological spin textures calculated for Rashba-type SOC with spin-orbit strength of $\mathcal{R}=25\%$. The latter are shown in the right panel: (a) Néel-type skyrmion, (b) anti-skyrmion, (c) bi-skyrmion, (d) anti-bi-skyrmion, (e) Bloch-type skyrmion, (f) meron. We consider clusters containing 91 magnetic sites with a nearest neighbor distance of $d=3\AA$. Smearing $\mathrm{\Gamma }=0.2\mathrm{eV}$ is used in the vicinity of the Fermi energy to account for eventual temperature or disorder effects.