Frequency-resolved frozen phonon multislice method and its application to vibrational electron energy loss spectroscopy using parallel illumination

We explore the capabilities of the frequency-resolved frozen phonon multislice method introduced in [Phys. Rev. Lett. 124, 025501 (2020)] to model inelastic vibrational scattering in transmission electron microscopy. We review the method in detail and discuss advantages of using a so-called hotspot thermostat instead of the $\delta$ thermostat used in our first report. We apply the method to simulate vibrational electron energy loss spectra of hexagonal boron nitride under plane wave illumination. Simulated spectroscopic information well represents the theoretical phonon band structure of the studied material, both in terms of energies as well as polarization vectors of individual phonon modes.

Figure 1

Plot of the phonon dispersion and PDOS of bulk hBN. The inset shows the path in the first Brillouin zone, along which we plot the phonon dispersion. The contribution of the $x/y$- and $z$-projected PDOS to the total PDOS is separately indicated. These results are computed from the second-order force constants of the MD force field using the phonolammps and phonopy software packages.

Figure 2

Calculated PDOS and VPS of hBN. Panel (a) shows a comparison of bulk hBN PDOS with the VPS of a $NVE$ MD simulation. In panel (b), we compare the VPS of the $NVE$ simulation (broadened by a Lorentzian of 1.2 meV FWHM) with the VPS of a $NVT$ simulations using a Langevin thermostat with damping parameters $\tau$ of 0.5 ps and 0.05 ps. In panel (c) we compare the $NVE$ VPS (broadened by a Lorentzian of 12 meV FWHM) and the VPS obtained from a simulation with Langevin thermostat with damping 0.05 ps. Note that the broadened VPS and the VPS from a Langevin thermostat calculation with $\tau =0.5$ ps are practically indistinguishable in panel (b). All spectra are normalized to unit integral between 0 and 220 meV. Arbitrary units (a.u.).

Figure 3

Comparison of the VPSs of $\delta$ thermostat and hotspot thermostats with the broadened $NVE$ VPS. The black curves correspond to the VPSs for the different energy bins. The broadened $NVE$ VPSs have already been shown in Fig. 2. The “trace of maxima” is a guide for the eye to facilitate comparison with the reference VPS and indicates the maximum value of the VPS for each energy bin. The overall scaling of individual VPSs of $\delta$ and hotspot thermostats is such that the integral of their trace of maxima is one between 48 and 215 meV in panels (a) and (b) and 20 and 215 meV in panel c). Arbitrary units (a.u.).

Figure 4

Comparison of VPS with an off-axis EELS spectrum calculated with a plane-wave electron beam. The collection semiangle is 22 mrad, displaced by 61 mrad along the ${\theta }_x$ direction (parallel to the $\mathrm{\Gamma }$-M line), as is illustrated in the inset. The light gray spectrum is a raw FRFPMS EELS spectrum. The orange and blue filled spectra correspond to the raw spectrum multiplied by energy or the square of the energy, respectively. The dark gray areas mark the energy range, which is not covered by FRFPMS calculation. Total VPS and raw spectra are normalized to unit integral between energies of about 40 and 215 meV. The $x/y$-projected VPS is normalized such that it matches the peak at 190 meV of the total VPS. Arbitrary units (a.u.).

Figure 5

Spectral variations in FRFPMS EELS as a function of detector position in the diffraction plane at a thickness of about 2.5 nm. Incoming beam is a 60 kV plane wave. Panels (a) and (b) show a comparison of energy-multiplied EELS spectra for a detector entrance aperture centered around M and K points, respectively, in the diffraction plane. The collection semiangle of the detector is in all cases 3 mrad. The light gray dashed lines indicate the positions of the modes indicated at the top of the figure, which we expect from phonon band structure, see Fig. 1. The inset shows the detector position in the diffraction plane corresponding to the spectrum of the same color. The black arrows in the inset of panel (a) indicate the direction of the phonon polarization vectors of the longitudinal phonon modes at the M point and white arrows indicate the direction of the momentum transfer. Light gray hexagons indicate boundaries of reciprocal space Wigner-Seitz unit cells.

Figure 6

The same as Fig. 5 for sample thickness of 15 nm.

Figure 7

Comparison of the raw vibrational EELS signal, calculated for a 60 kV plane wave electron beam at scattering angles along selected directions in the diffraction pattern with the phonon dispersion relation (white lines). The directions lie in (a) and (b) along the $\mathrm{\Gamma }\text{−}\mathrm{M}\text{−}\mathrm{\Gamma }$ and $\mathrm{\Gamma }$-K-M-K-$\mathrm{\Gamma }$ line, respectively. Both paths are indicated relative to the diffraction pattern in (c). Paths symmetric around $q=0{\AA }^{-1}$ were chosen in order to show both the agreement between phonon bands and peaks of calculated spectra (left side) as well as the spectra unobscured by the phonon band structure (right side). The logarithmic color scale ranges from dark blue for small values of the EELS signal over blue, turquoise, and green to yellow for large values. The EELS signal was not calculated for energies corresponding to the gray area.