Topological Hall and spin Hall effects in disordered skyrmionic textures

We carry out a thorough study of the topological Hall and topological spin Hall effects in disordered skyrmionic systems: the dimensionless (spin) Hall angles are evaluated across the energy-band structure in the multiprobe Landauer-Büttiker formalism and their link to the effective magnetic field emerging from the real-space topology of the spin texture is highlighted. We discuss these results for an optimal skyrmion size and for various sizes of the sample and find that the adiabatic approximation still holds for large skyrmions as well as for nanoskyrmions. Finally, we test the robustness of the topological signals against disorder strength and show that the topological Hall effect is highly sensitive to momentum scattering.

Figure 1

(a) The four-terminal setup is made up of a central skyrmion scattering region attached to ferromagnetic leads L, T, R, B at chemical potentials ${\mu }_{L,T,R,B}$. A constant voltage bias is applied between L and R while the induced transverse charge-spin voltages are probed in T and B. (b) The magnetic field emerging from the skyrmionic texture.

Figure 2

Different radial distributions of the skyrmion polar angle ${\theta }_i$ ($i$=1,2,3,4). Inset: The corresponding out-of-plane magnetization spatial profile.

Figure 3

(a) The spin polarization and (b,c) topological Hall angles for different transport energies ${\epsilon }_{\mathrm{tr}}$ for a sample size of $L=W=96a_0$ and a skyrmion radius of $10a_0$. The value of the exchange coupling $\mathrm{\Delta }$ defines the boundaries and affects the magnitudes of ${\theta }_{\mathrm{TH}}$ and ${\theta }_{\mathrm{TSH}}$.

Figure 4

The nonzero topological Hall angles ${\theta }_{\mathrm{TH}}$ (blue) and ${\theta }_{\mathrm{TSH}}$ (red) as a function of the skyrmion radius for three different energies, ${\epsilon }_{\mathrm{tr}}=-3.7, -2.0$, and 0. Here, the sample size is $128\times 128a_0^2$ and the exchange coupling is $\mathrm{\Delta }=\frac{2t}{3}$. The vertical lines indicate the minimal and the optimal skyrmion radius we use in the rest of the paper.

Figure 5

The normalized ${\theta }_{\mathrm{TH}}^{\mathrm{gm}}$ (blue) and ${\theta }_{\mathrm{TSH}}^{\mathrm{gm}}$ (red) as a function of the energy ${\epsilon }_{\mathrm{tr}}$. The skyrmion radius is ${\mathrm{r}}_{\mathrm{sk}}$ = $10a_0$. For all widths $W=L=64a_0, 96a_0$, and $128a_0$, the normalized signals are superimposed and give exactly the same value. For reference, the exchange coupling is $\mathrm{\Delta }=\frac{2t}{3}$.

Figure 6

(a) Spatial variation of the magnetization in a skyrmion crystal $N=16$ for the $128\times 128a_0^2$ sample and a skyrmion radius of $10a_0$. (b,c) The dependence of the (b) THE and (c) TSHE with the number of skyrmions $N$, at the transport energies ${\epsilon }_{\mathrm{tr}}=-2.0$ and 0.0, respectively.

Figure 7

Energy dependence of (a) ${\theta }_{\mathrm{TH}}$ and (b) ${\theta }_{\mathrm{TSH}}$ in the clean limit and for $V_{\mathrm{imp}}=0.5, \mathrm{\Delta }=\frac{2t}{3}$, and $W=L=64a_0$. The signal is reduced and smeared out.

Figure 8

(a) Normalized conductance of the metallic layer as a function of its length for different impurity strengths. (b) Extraction of the mean free path $\lambda$ for the first transport energy ${\epsilon }_{\mathrm{tr}}=-3.7$. Note that changing the transport energy modifies the correspondence between $V_{\mathrm{imp}}$ and $\lambda$. (c,d) The nonvanishing ${\theta }_{\mathrm{TH}}$ (blue) and ${\theta }_{\mathrm{TSH}}$ (red) for a sample size of $W=64a_0$ and for the three different energies, (c) as a function of impurity strength $V_{\mathrm{imp}}$ and (d) as a function of the equivalent mean free path $\lambda$.