Temperature-dependent vibrational EELS simulations with nuclear quantum effects

Abstract

The Time Autocorrelation of Auxiliary Wave (TACAW) method has established a framework for modeling angle-resolved electron energy loss spectroscopy (EELS) of phonons and magnons by deriving scattering intensities from the time autocorrelation of the beam wavefunction. This approach enables efficient computation of scattering intensities while naturally accounting for dynamical diffraction and multiple-scattering effects. In the cryogenic regime, vibrational spectra are dominated by nuclear quantum effects, notably zero-point motion. To capture these effects in low-temperature vibrational EELS, we incorporate thermostatted ring polymer molecular dynamics (TRPMD) into the TACAW formalism. Our results demonstrate that nuclear quantum effects lead to significant deviations from classical molecular dynamics predictions in the vibrational spectra of silicon at low temperatures and correctly predict the nearly temperature-independent optical phonon peak intensities in silicon, consistent with the first Born approximation. The TRPMD-TACAW method provides a robust theoretical tool for probing the low-temperature limit of vibrational EELS, offering a necessary benchmark for the quantitative analysis of emerging cryogenic scanning transmission electron microscopy experiments.

Figures (5)

• Figure 1: Numerical simulation of TRPMD for silicon were performed at several temperatures. After the thermalization, the emsamble average square of radius of gyration of ring polymer obtained from TPRMD is shown in panel (a). The ensemble averaged MSD computed evaluated after thermalization from classical MD and TRPMD simulations, are shown in the panel (b). In the panel (c), the ensemble average MSD against squared of radius of gyration is plotted, together with a linear fit.
• Figure 2: The FVACF for silicon at various temperatures is computed from centroid velocity of ring-polymer. The inset shows the peak energy of the FVACF spectrum as a function of temperature.
• Figure 3: The temperature dependent angle resolved phonon EELS of silicon were obtained from the scattering intensity $I(q_x,q_y=0,E)$ is computed using TRPMD and classical molecular dynamics. The calculations span the temperature range from 1000 K down to 10 K. The TRPMD and classical results are shown in the first and the second row, respectively. The bottom row displays the ratio between the two angle-resolved spectra, highlighting the temperature-dependent deviations arising from nuclear quantum effects in low temperature limit.
• Figure 4: The EELS spectra at different temperatures were obtained from the scattering intensity $I(\boldsymbol{q}, E)$ on a logarithmic scale by placing the detector at the center with a collection semi-angle of 35 mrad. The first panel shows the EELS spectra for different temperatures computed using TRPMD and classical MD, represented by solid and dashed lines, respectively. The second panel displays the difference ratio the two spectra.
• Figure 5: The EELS spectra at temperature 300 K obtained from TRPMD using three different thermostats: Langevin, PILE-L and PILE-G, with thermostat relaxation time set to 200 fs. The scattering intensity $I(\boldsymbol{q}, E)$ is plotted on a logarithmic scale. The same detector as in Fig. \ref{['fig:EELS 35mrad']} was assumed: collection semi-angle of 35 mrad around the center of diffraction pattern.