Topological Hall effect due to electron-skyrmion scattering

Technical Summary: Topological Hall Effect due to Electron-Skyrmion Scattering

Problem Statement
Electron scattering from chiral spin textures, particularly skyrmions, is a fundamental mechanism governing transport in complex magnetic systems like skyrmion crystals. While the emergent magnetic fields in these textures drive novel transport phenomena such as the topological Hall, spin Hall, and orbital Hall effects, a comprehensive understanding of the scattering process has been limited. Previous theoretical studies have predominantly focused on the weak-coupling regime, where the exchange interaction ($J$) between the itinerant electron spin and the local skyrmion spins is small compared to the electron energy. In this regime, perturbative Born approximations are valid. However, in most real materials, including many skyrmion-hosting systems, the exchange interaction is comparable to or significantly larger than the electron energy ($J \gtrsim E$), often approximated as $J \to \infty$. In this strong-coupling regime, the scattering behavior is qualitatively different, and perturbative methods fail to capture essential physics. There is a need for a non-perturbative framework valid across all coupling strengths to accurately determine scattering cross-sections and the resulting Hall conductivities.

Methodology
The authors employ a non-perturbative approach based on the Lippmann–Schwinger equation and the full Green's function (GF) formalism. This method is valid for arbitrary scattering strengths, characterized by the ratio of the exchange interaction $J$ to the electron energy $E$.

• Hamiltonian: The system is modeled using a 2D Hamiltonian where the electron moves in a spin texture $\hat{S}(r, \theta)$. The interaction is defined by the Hund's energy $J$, with the scattering potential $V(r)$ being non-zero only within the skyrmion radius $\lambda$.
• Formalism: The scattered wave function is derived from the Lippmann–Schwinger equation, $|\psi_{k\sigma}\rangle = |\phi_{k\sigma}\rangle + G_0 T |\phi_{k\sigma}\rangle$, where $T$ is the T-matrix. The T-matrix is computed by inverting the GF matrix in real space on a discrete grid covering the skyrmion region. This approach includes terms to all orders in the potential $V$, avoiding the limitations of the Born approximation.
• Spin Resolution: The formalism explicitly accounts for spin-preserving and spin-flip scattering channels. It handles the case where the incident energy $E < J$, where spin-down states cannot propagate outside the skyrmion but exist as evanescent (trapped) states within the scattering region, influencing the scattering process.
• Parameters: Numerical calculations utilize a standard skyrmion spin texture with winding number $m=1$ (unless specified otherwise), radius $\lambda = 10$ Å, and a fixed exchange energy $J = 4$ eV, while varying the incident electron energy $E$.

Key Results
The study reveals several new features in the scattering cross-section and transport properties that are absent in the weak-coupling limit:

1. Scattering Cross-Section Features:

   • Low-Energy Divergence: The total scattering cross-section diverges as the incident momentum $k \to 0$ (for spin-up) or as $E \to J$ (for spin-down). This is attributed to the electron seeing an infinite potential barrier within the skyrmion, behaving like a hard disk.
   • Intermediate-Coupling Resonances: A pronounced peak-like structure appears in the spin-up cross-section ($\sigma_{\uparrow\uparrow}$) near $E = J$. This arises from the coupling to the spin-down channel, which exhibits a divergence at $E=J$, broadened by the existence of evanescent spin-down states for $E < J$.
   • Ramsauer–Townsend Dips: For energies slightly above $J$, a dip in the scattering cross-section is observed, analogous to the Ramsauer–Townsend effect in noble gases. This results from wave interference leading to enhanced transmission. In the presence of spin-flip potentials, this dip merges with the peak at $E=J$.
   • Chiral Asymmetry: The scattering is asymmetric with respect to the incident direction. Spin-up electrons scatter predominantly to the lower half-plane ($\theta < 0$), while spin-down electrons scatter to the upper half-plane ($\theta > 0$), driven by the chiral spin structure.

2. Topological and Spin Hall Conductivities:

   • The asymmetry in scattering generates net transverse charge and spin currents.
   • Energy Regimes: The conductivities exhibit distinct behaviors depending on the incident energy:
     • $E < J$: Only spin-up electrons propagate; topological Hall conductivity ($\sigma_H$) equals spin Hall conductivity ($\sigma_{SH}$).
     • $E \gtrsim J$: Both spin channels propagate, but scattering is stronger for spin-up, leading to a large $\sigma_{SH}$ and a moderate $\sigma_H$.
     • $E \sim 2J$: Scattering is comparable for both spins, causing $\sigma_H \to 0$ while $\sigma_{SH}$ remains large. This regime allows for the suppression of transverse charge current while maintaining a pure spin Hall current.
     • $E \gg J$: Scattering becomes negligible, and conductivities vanish.

3. Higher Winding Numbers and Landau Levels:

   • For skyrmions with higher winding numbers ($m \ge 2$), the emergent magnetic field creates Landau levels within the skyrmion.
   • These levels manifest as resonant peaks in the scattering cross-section and Hall conductivities. The positions of these peaks align with the calculated Landau-level energies, particularly for larger $m$ (e.g., $m=5$).
   • The lowest Landau level ($n=0$) is often obscured by the low-energy divergence, but higher levels ($n \ge 1$) produce distinct resonant structures.

Significance and Claims
The paper claims to provide a rigorous, non-perturbative understanding of electron-skyrmion scattering that is applicable to realistic material parameters where the exchange interaction is strong. By moving beyond the weak-coupling Born approximation, the authors uncover qualitative features—such as Ramsauer–Townsend minima, intermediate-coupling resonances, and Landau-level resonances—that are critical for understanding transport in skyrmion crystals.

The work highlights that the topological and spin Hall conductivities are highly sensitive to the incident electron energy. Specifically, the authors note that tuning the electron energy can suppress the transverse charge current while preserving the spin Hall current, a feature of potential value for spintronic applications. The formalism developed is not limited to skyrmions but is applicable to other non-collinear spin textures, offering a general framework for analyzing transport in collective chiral magnetic systems.