Understanding phonon selection and interference in momentum-resolved electron energy loss spectroscopy

TL;DR

This work addresses how phonon signals in momentum-resolved EELS are shaped by interference across crystal basis and by scattering selection rules. It introduces the interferometric Brillouin zone and direction-selectivity as key concepts, and formalizes coherent versus incoherent phonon treatments to enable efficient simulations with SED and LD. The authors demonstrate that SED and LD can reproduce the main q-EELS features, including zone unfolding and polarization-sensitive vibrational density of states, while highlighting the roles of convergence angle and sample thickness. They provide practical guidance for using SED/LD to simulate q-EELS and interpret experimental data, and discuss depth sensitivity and potential spurious through-plane modes that can arise in thicker samples.

Abstract

As momentum-resolved Electron Energy Loss Spectroscopy (q-EELS) becomes more widely used for phonon measurements, better understanding of the intricacies of the acquired signal is necessary. Selection rules limit the allowed scattering, which may prohibit the appearance of specific phonon branches for certain measurements. Simultaneous sampling of the lattice across all basis indices also warrants a coherent treatment of phonons, which yields a larger repeating unit in reciprocal space. We thus introduce the concept of the ``interferometric Brillouin zone'', which is closely related to the Dynamic Structure Factor. Both effects determine where phonon modes may be observed. Through a rigorous understanding of both, we introduce a new efficient method for simulating scattering experiments via Spectral Energy Density (SED) and/or Lattice Dynamics (LD) calculations and results are compared to established q-EELS simulation methods. Finally, we demonstrate the use of scattering selection rules on well-studied systems and explore the acquisition of a polarization-selective vibrational density of states.

Figures (3)

• Figure 1: SED calculations are performed for silicon (diatomic basis) using either the traditional incoherent summing (a,b), or coherent summing (c,d). Results are shown as a dispersion (a,c), with color used to denote eigenvector polarization (red, yellow, and blue denote eigenvector magnitudes along or perpendicular to the path, or through plane, respectively. this roughly corresponding to longitudinal, transverse in-plane, and transverse through-plane modes). RYB color mixing denotes degeneracy or mixed-polarization branches. Coherent LD is shown in dotted white in the $\Gamma$-$X$ direction in panel c. Iso-energy slices in reciprocal space are shown at 6 THz (b,d). In the coherent case, Brillouin zones are no longer identical, and we refer to the new larger minimum repeating unit in reciprocal space as an "interferometric Brillouin zone". The interferometric Brillouin zone is shown in dotted white in (d), and its size depends on the interatomic spacing as opposed to the size of the primitive cell. We also differentiate between non-equivalent $\Gamma$ and $K$ points with the $\Gamma'$ and $K'$ notation.
• Figure 2: (a) phonon dispersions can be generated from q-EELS by collecting energy spectra along a reciprocal space path. Here the sampled path is identical to that shown in \ref{['fig:coherence']}.b, but centered on the [220] $\Gamma$ point so as to avoid suppression of transverse branches. Direction selectivity according to $\vv{\bm{q}}$$\bullet$$\vv{\bm{\varepsilon}}$ implies only phonons with eigenvectors $\vv{\bm{\varepsilon}}$ in the direction of the electron scattering vector $\vv{\bm{q}}$ will be visible. This is shown for the 5 THz energy slice (b) where coherent SED (without the application of selection rules) fails to replicate q-EELS. Upon the application of $\vv{\bm{q}}$$\bullet$$\vv{\bm{\varepsilon}}$ in SED however, the primary features from q-EELS can be replicated. Purely longitudinal and purely transverse branches appear as crescents, since the intensity fades to zero where atomic displacements $\vv{\bm{\varepsilon}}$ are perpendicular to $\vv{\bm{q}}$. There is also a near-complete suppression of modes comprised soley of through-plane vibrations (e.g. TO$_\perp$, blue branches from Fig. \ref{['fig:coherence']}). Additional energy-resolved diffraction images are shown for 7 THz (c), 14 THz (d) and 16 THz (e), generated via q-EELS, SED, and LD. The interferometric Brillouin zone is shown in dotted white, and the traditional Brillouin zone is shown in (c) in solid white. "Unfolding" behavior is also clearly visible: optical branches form crescents or circles about $\Gamma'$ at high frequencies, and there are ellipses about $K'$ points (and not $K$ points) at 7 THz.
• Figure 3: AlN in the [010] plane is used to examine polarization selection and the effects of a convergent beam. (a) an energy-resolved diffraction image is shown, which is used to inform selection of Brillouin zones and dark-field mask diameter. (b) circular masks are applied (with radius 1/$c$), centered on several $\Gamma$ points, shown here in the 3 mrad diffraction image. (c) We compare the DOS acquired from the central, right, and upper Brillouin zones (black, green, and blue, respectively) between parallel-beam q-EELS (solid) and calculations from SED (dashed). (d) there is reduced sensitivity at outer Brillouin zones, likely due to dynamic scattering effects. (e,f) Direction selectivity is maintained for a convergent probe, however selectivity is reduced at large angles. (g) The total q-EELS signal intensity is simulated as a function of depth (with finer resolution steps shown in the inset). (h) The signal on a per-layer basis is taken via the integral of (g). In both cases, the highest signal comes from the upper layers of the sample. Dynamic effects (Pendellösung oscillations) can also be seen.