Magnetically assisted spin-resolved electron diffraction: Coherent control of spin population and spatial filtering

Abstract

Electron diffraction from nanogratings provides a platform for free-electron interferometry, yet controlled manipulation of electron spin in such geometries remains largely unexplored. In particular, the role of the self-generated magnetic field arising from electron motion and the feasibility of coherent spin control without disrupting diffraction coherence have not been quantitatively investigated. In this article, a self-consistent Maxwell-Pauli framework is developed to study spin-resolved electron diffraction from nanogratings in the presence of magnetic fields. The model incorporates geometric confinement, image-charge interactions, self-generated magnetostatic fields, and externally applied magnetic fields. Numerical simulations show that the intrinsic magnetic self-field produced by the electron probability current is several orders of magnitude too weak to induce measurable spin mixing, demonstrating that nanogratings act as spin-conserving beam splitters under field-free conditions. When a uniform magnetic field is applied upstream of the nanograting, coherent Larmor precession enables controlled spin rotation without modifying the diffraction geometry or degrading coherence. The magnetic field required for a $π$ spin rotation scales inversely with the interaction length and electron de Broglie wavelength $λ_{dB}$. Furthermore, a downstream nonuniform magnetic field applied after the nanograting imparts a spatially varying Zeeman phase, producing opposite transverse momentum shifts for the two spin components. The spin-dependent transverse dynamics is analyzed using Husimi Q-function phase-space maps, which visualize spin-dependent population redistribution and momentum separation. The proposed approach enables tunable spatial separation of spin-resolved free electron beams and establishes an all-magnetic route for coherent spin rotation, control, and interferometry.

Figures (13)

• Figure 1: (a) Schematic diagram of the experimental system. WG: waveguide; MC: multicusp; PLE: plasma electrode; EL$_1$: Einzel lens system; BL$_1$ and BL$_2$: beam collimating apertures; DG: diffraction grating; SS: scanning slit; and D: detector. The electron beam is extracted from the plasma. (b) Schematic of the simulation setup. Spin-polarized electrons with kinetic energy $E=20$ eV, initially aligned along the $z$ axis, propagate along $x$ and encounter a nanograting of period $d=50$ nm and thickness $h=25$ nm placed in the $yz$ plane. Upstream of the grating, electrons pass through a uniform magnetic field $\mathbf{B}_{1} = B_{1}\hat{\mathbf{x}}$ over a length $L_{B1}$, followed downstream by a nonuniform field $\mathbf{B}_{2} = zG_{2}\hat{\mathbf{y}} + yG_{2}\hat{\mathbf{z}}$ extending over $L_{B2}$. The far-field diffraction pattern is recorded on a screen at a distance $L_{GS}$ using a scanning slit and detector. (c) Electron trajectory simulation results obtained using AXCEL-INP with plasma parameters as input. The given electrode configuration and applied voltages produce a well-collimated electron beam.
• Figure 2: Spatial maps of (a) the geometric potential energy $V_g(x,y)$ and (b) the image-charge potential energy $V_{\mathrm{image}}(x,y)$ for an electron in a multi-slit grating.
• Figure 3: Cross-sectional view of the diffraction grating. ABCD represents a slit opening, with AB and CD denoting the planes of two neighboring bars. The distances from a point $P(x,y)$ inside the slit to the AB and CD planes are labeled $d_1$ and $d_2$, respectively.
• Figure 4: Simulation results for a spin-up polarized state ($\alpha_0=1$, $\beta_0=0$) in the absence of external magnetic field $\mathbf{B^{ext}}$. Panels (a–d) show the probability density $|\psi_{\uparrow}(x,y,t)|^{2}$ immediately before the grating and the associated self-generated fields: (b) $A_{y}(x,y,t)$, (c) $A_{x}(x,y,t)$, and (d) out-of-plane magnetic field $B^{self}_{z}(x,y,t)$ along $\bf{\hat{z}}$. Positive values of the magnetic field indicate that the field is oriented along the $\hat{\mathbf{z}}$ direction, whereas negative values indicate orientation along the $-\hat{\mathbf{z}}$ direction. Panels (e–h) display the corresponding quantities immediately after diffraction: (e) $|\psi_{\uparrow}(x,y,t)|^{2}$, (f) $A_{y}(x,y,t)$, (g) $A_{x}(x,y,t)$, and (h) $B^{self}_z(x,y,t)$.
• Figure 5: (a) Transverse probability density profiles along the $y$ direction for the initial spinor components $|\psi_{\uparrow}(x,y,0)|^{2}$ and $|\psi_{\downarrow}(x,y,0)|^{2}$, corresponding to the coefficients $\alpha_{0}=\sqrt{3}/2$ and $\beta_{0}=1/2$. These values yield initial spin-up and spin-down populations of 75% and 25%, respectively. The profiles are normalized to the peak value of the spin-up component. (b) Spin-resolved far-field diffraction intensities $I_{\uparrow}(y)$ and $I_{\downarrow}(y)$, recorded at a detection screen placed $L_{\mathrm{GS}}=50~\mathrm{cm}$ downstream of the nanograting.