Theory of Momentum-Resolved Electron Energy-Loss Spectra of Coupled Phonon and Magnon Excitations

Abstract

We develop a theory of momentum-resolved electron energy-loss spectra in the scanning transmission microscope (STEM-EELS) that captures the effects of coupled phonon and magnon excitations within a unified formalism, and apply it to body-centered cubic iron at 300 K. By advancing the Time Autocorrelation of Auxiliary Wavefunctions (TACAW) method to incorporate atomistic spin-lattice dynamics (ASLD), we simulate the EELS signal, including phonon-magnon interaction effects, dynamical diffraction, and multiple scattering. Our results reveal non-additive spectral features arising from phonon-magnon coupling, hybridization, and energy shifts, and further allow estimation of the electron dose required to detect magnon scattering under optimized detector conditions.

Figures (2)

• Figure 1: Momentum-resolved EELS simulations from ASLD–TACAW for bcc Fe at 300 K in the [001] zone axis.(a) Total (elastic+inelastic) diffraction intensity, $\int I(\mathbf{q},E)\,dE$, on a logarithmic scale, showing Bragg peaks and diffuse scattering. (b) CPLED: fully coupled ASLD–TACAW momentum-resolved EELS, $I(\mathbf{q},E)$, including phonons, magnons, and their mutual coupling. (c) MG-ASLD: magnon-only $I(\mathbf{q},E)$ obtained by freezing atomic positions in the ASLD trajectory, showing a broad magnon band extending up to $\sim 500$ meV (adiabatic magnon dispersion overlaid in white). (d) PH-ASLD: phonon-only $I(\mathbf{q},E)$ obtained by freezing magnetic moments, revealing LA and TA branches below $50$ meV (adiabatic dispersions overlaid in white). (e) DIFF-PH = CPLED $-$ PH-ASLD (signed residual), plotted for both energy-loss and gain regions (negative energies) on a symmetric diverging scale, highlighting coupling-induced redistributions relative to the phonon background. (f)$|\text{DIFF-PH}|$, approximating a background-subtracted experiment where a phonon baseline is removed to isolate the magnetic component and coupling effects. (g) DIFF-UNC = CPLED $-$ UNCOUPLED, where UNCOUPLED is the additive sum of MD-TACAW phonon and ASD-TACAW magnon spectra, showing sign-alternating residuals that track the magnon dispersions from $\sim 150$ meV to $\sim 500$ meV. (h) DIFF-UNC in the phonon energy range, exhibiting sign-alternating lobes localized around the LA branch.
• Figure 2: Magnetic signal detectability from detector-integrated EELS.(a) Energy-integrated angular distribution of the MG-ASLD (magnon-only ASLD) signal in the diffraction plane, highlighting an ADF detector with inner and outer collection angles of 2 mrad and 7 mrad, following the optimal detection geometry reported in Ref. castellanos_unveiling_2023. (b) Momentum-resolved EELS spectra, $I(\mathbf{q},E)$, integrated over the ADF detector, showing CPLED, PH-ASLD, MG-ASLD, and $|\text{DIFF-PH}|$, together with the uncoupled reference spectra PH-MD (phonons from MD-TACAW) and MG-ASD (magnons from ASD-TACAW) that define UNCOUPLED. The inset magnifies the 200--280 meV magnetic window, where small but systematic meV-scale energy shifts between the coupled (MG-ASLD) and uncoupled (MG-ASD) spectra become visible. (c) Required beam current versus acquisition time to achieve SNR = 3 and SNR = 5 for the magnon signal integrated either over the magnon energy window (280--420 meV, red curves) or over the full simulated energy range (black curves). Gray lines indicate the corresponding estimates from Ref. castellanos_unveiling_2023.