Quantum Dynamics of Electron Scattering from Skyrmions

Abstract

Scattering of electrons from chiral spin textures such as the skyrmions is an emerging research area due to its richness in topological quantum transport, which is significant for spintronic devices. We study the dynamical process of scattering of the spin-$\frac{1}{2}$ particles in the form of Gaussian wavepackets from skyrmions with the aid of the non-relativistic time-dependent Schrödinger equation. The scattering cross section shows a rich angular dependence and is deterministically influenced by the iterative flipping of the spin state inside the skyrmion. The latter leads to a set of non-trivial outcomes which include finite transmission and reflection probabilities irrespective of interaction strength, formation of secondary wavefronts associated with back-converted spin components, and a long-lived quasi-bound state at the scattering center. In addition to the rich and intriguing physics, the numerical recipe developed here can be easily adopted for any arbitrary spin texture, which will prepare a playground to explore tunable spin transport.

Figures (9)

• Figure 1: Electron scattering from an isolated skyrmion obtained from the solution of the time-dependent Schrödinger equation. The color coding indicates the scattered wave packet at a fixed time after the collision.
• Figure 2: (a) The contour map of the potential components for 2D skyrmion based on Eq. \ref{['Eq:sky pot in']}. The color gradient indicates the potential strength. The skyrmion is centered at $(0.5L,0.5L)$ with a radius of $0.05L$ where $L$ is the length of the box. The off-diagonal terms are complex with $V^{\uparrow \downarrow} = V^{\downarrow\uparrow*}$. (b) A 3D schematic diagram for a 2D skyrmion. The marked rectangle indicates a 1D cross-section of the 2D skyrmion, which we defined as 1D skyrmion. (c) Potential components of this 1D skyrmion centered at $0.5L$ with length $0.1L$.
• Figure 3: The flow chart of the numerical methods described in Sec. \ref{['Sec:Theoretical Formulation']} to solve the TDSE of the propagating electron in any arbitrary 2D potential. A finite difference method is followed, and a hybridized technique of operator-splitting with forward elimination and back substitution is employed. This approach reduces the computational complexity and enhances the numerical stability as explained in Appendix-C, which provides an improvement over the conventional matrix inversion method 10.1119/1.13811.
• Figure 4: The snapshots of probability amplitudes at different timesteps($n$) of the scattering of spin-up (magenta) and spin-down (orange) electron wave packets from a 1D skyrmionic potential as shown in Fig. \ref{['Fig:sky_pot']}(a). The potential region is centered at ($0.5L,0.5L$) with a length of $0.04L$. The results are shown for different values of the dimensionless parameter $\beta$ ($< 1, =1, >1$). The first row in each panel shows the time evolution of the probability density of both spin-up and spin-down electrons with initial spin-up polarization, whereas the second row in each panel represents the same for initial spin-down polarization. The skyrmion region is indicated in teal colored spike. The regions of significance are zoomed in on the insets. In (c) and (f) plots, they highlight the dominance of reflection over transmission in the spin-flip channel when $\beta<1$. In the (i) and (o) plots, the yellow spikes inside the skyrmion region indicate the presence of the metastable spin-down state.
• Figure 5: Different plots of the transmission coefficient ($\mathrm{T}$) and the reflection coefficient ($\mathrm{R}$) of $\uparrow \uparrow$ and $\uparrow \downarrow$ channels between a skyrmion potential (a) and a step potential (with all $V^{\uparrow \uparrow}$, $V^{\downarrow \downarrow}$, $V^{\uparrow \downarrow}$ and $V^{\downarrow \uparrow}$ non-zero values potential) (b) and the plots show the variation of $\mathrm{T}$ and $\mathrm{R}$ as a function of $\beta$. (c) The saturation value of the different transmission and reflection coefficients ($\mathrm{T}_s$ and $\mathrm{R}_s$)is plotted as a function of skyrmion size $a$, which is in the units of $L$.