Modeling energy-loss spectra due to phonon excitation

B. D. Forbes and L. J. Allen

Phys. Rev. B 94, 014110 – Published 13 July 2016

Abstract

We discuss a fundamental theory of how to calculate the phonon-loss sector of the energy-loss spectrum for electrons scattering from crystalline solids. A correlated model for the atomic motion is used for calculating the vibrational modes. Spectra are calculated for crystalline silicon illuminated by a plane wave and by an atomic-scale focused coherent probe, in which case the spectra depend on probe position. These spectra are also affected by the size of the spectrometer aperture. The correlated model is contrasted with the Einstein model in which atoms in the specimen are assumed to vibrate independently. We also discuss how both the correlated and Einstein models relate to a classical view of the energy-loss process.

Received 23 March 2016
Revised 11 May 2016

DOI:https://doi.org/10.1103/PhysRevB.94.014110

Physics Subject Headings (PhySH)

Optical phonons

Electron energy loss spectroscopy

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

B. D. Forbes and L. J. Allen^*

School of Physics, University of Melbourne, Parkville, Victoria 3010, Australia

^*lja@unimelb.edu.au

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Issue

Vol. 94, Iss. 1 — 1 July 2016

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Images

Figure 1
(a) Transition probability in real space $| H_{n 0} {(r) |}^{2}$ projected along [001] for a longitudinal optical phonon with wave vector $k = (- 0.2, 0, 0)$ , for $m = 1$ and corresponding to an energy loss of 63.6 meV. The image has been blurred for the purposes of visualisation using a Gaussian kernel of FWHM 0.33 Å. The inset is a blowup of the area indicated (encompassing one atom, position indicated), without blurring. (b) Transition probability in reciprocal space $| H_{n 0} {(q) |}^{2}$ for the phonon mode in panel (a), blurred with a FWHM of 0.11 Å $^{- 1}$ . (c) Transition probability in real space for a longitudinal acoustic phonon with wave vector $k = (- 0.2, - 0.4, 0)$ , for $m = 1$ and corresponding to an energy loss of 25.2 meV, blurred as in panel (a). The inset is a blowup of the area indicated (encompassing one atom, position indicated), without blurring. (d) Transition probability in reciprocal space for the phonon mode in panel (d), blurred as in panel (b). The projected structure of the conventional cubic unit cell is overlaid on panel (a). The scale bar on panel (c) also applies to panel (a) and that on panel (d) also to panel (b).
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Figure 2
Phonon-loss spectra for 300-keV plane-wave electrons incident on silicon down [001], calculated at room temperature. (a) Spectrum obtained for single-phonon excitation in both the correlated and Einstein models, which is dominated by scattering in the forward direction. (b) Spectrum in the strictly backwards direction for the Einstein model, taking into account multiphonon excitation, and a classical model incorporating Doppler broadening. The peak value occurs at the classical recoil energy $E_{R}$ , indicated by the solid line. The Einstein and classical model results in panel (b) have been normalized to each other and are almost indistinguishable.
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Figure 3
Spectra (in arbitrary units) as a function of energy loss in meV, for the four equally spaced probe positions as indicated in the schematic of the cubic unit cell (inset, viewed along [001]) and presented in the order indicated for purposes of visualization. All spectra are on the same absolute scale. Besides variations in magnitude, variations in shape are evident.
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Figure 4
Differential cross section as in Eq. (28) for a plane wave of incident energy 300 keV as a function of energy loss and scattering angle, calculated for a silicon atom in an isotropic harmonic oscillator potential with trap frequency $ω = 4.6 \times 10^{14} s^{- 1}$ . The overlaid (yellow) line is the classical relationship in Eq. (30).
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