Partial vibrational density of states for amorphous solids from inelastic neutron scattering

We introduce coherent inelastic neutron scattering with isotope substitution as a technique for measuring site-specific information on the vibrational dynamics of amorphous solids. The technique is used to extract experimentally the partial vibrational density of states $G_{\alpha }\left(E\right)$ for both Ge and Se in the prototypical network-forming glass ${\mathrm{GeSe}}_2$, where $\alpha$ specifies the chemical species and $E$ denotes the energy. The efficacy of the approximations used in interpreting the experimental data is validated by using a first-principles model, for which the set of true partial vibrational density of states is directly accessible. Our approach offers an opportunity for exploring accurately the vibrational dynamics of disordered materials.

Figure 1

The total differential cross section ${d\sigma /d\mathrm{\Omega }|}_{\mathrm{tot}}=F\left(Q\right)+{\sigma }_{\mathrm{tot}}/4\pi$ for different isotopically enriched samples of vitreous ${\mathrm{GeSe}}_2$ as obtained (i) from $I_{\mathrm{s}}(Q,E)$ at 5 K by integrating over the energy range accessible to the MERLIN spectrometer $(-50\text{meV}\le E\le 50$ meV) at a given value of $Q$ (black curves with vertical error bars) or (ii) from neutron diffraction experiments on the same samples at 299(1) K (solid red curves) [21, 22]. For clarity of presentation, the curves for ${}^{70}\mathrm{Ge}{}^{\mathrm{N}}\mathrm{Se}_2$ and ${}^{73}\mathrm{Ge}{}^{76}\mathrm{Se}_2$ are shifted vertically by 1 and 2 barns, respectively.

Figure 2

Contour plots of the measured $G_1(Q,E)$ functions for vitreous (a) ${}^{70}\mathrm{Ge}{}^{\mathrm{N}}\mathrm{Se}_2$, (b) ${}^{\mathrm{N}}\mathrm{Ge}{}^{\mathrm{N}}\mathrm{Se}_2$, and (c) ${}^{73}\mathrm{Ge}{}^{76}\mathrm{Se}_2$.

Figure 3

The $G_1\left(E\right)$ functions measured using CINS for vitreous ${}^{\mathrm{N}}\mathrm{Ge}{}^{\mathrm{N}}\mathrm{Se}_2$ in the present work at 5 K (solid black curve with vertical error bars), in the work by Walter et al. [29] at 13(2) K (dash-dotted green curve), and in the work by Sinclair et al. [30] at 18(1) K (solid red curve). The results are compared to the sum $G_{\mathrm{tot}}\left(E\right)=G_{\mathrm{Ge}}\left(E\right)+G_{\mathrm{Se}}\left(E\right)$ (dashed blue curve with vertical error bars).

Figure 4

Contour plots of the measured $G_{\alpha }(Q,E)$ functions for (a) $\alpha =\mathrm{Ge}$ and (b) $\alpha =\mathrm{Se}$.

Figure 5

The $〈G_{\alpha }(Q,E)〉$ functions for vitreous ${\mathrm{GeSe}}_2$, where the average extends over a range of energies corresponding to (a) and (b) the first peak, (c) and (d) the second peak, and (e) and (f) the third peak in $G_1\left(E\right)$ (Fig. 3). The measured functions (points with vertical error bars) are compared to those obtained from a first-principles model (solid red curves). Peaks in the modeled functions are shifted to smaller energies compared to experiment, so the integration ranges are 8–18, 23–28, and 32–40 meV for experiment versus 6–16, 20–25, and 28–36 meV for the model.

Figure 6

The $G_{\alpha }\left(E\right)$ functions for (a) Ge and (b) Se for vitreous ${\mathrm{GeSe}}_2$ at 5 K as measured using CINS (solid black curves with vertical error bars) or calculated using a first-principles model (dashed red curves). From experiment, there are peaks in $G_{\mathrm{Ge}}\left(E\right)$ at 11.7(2), 24.7(1), and 38.0(1) meV and a shoulder at 35(2) meV, and there are peaks in $G_{\mathrm{Se}}\left(E\right)$ at 9.9(1) and 26.8(1) meV followed by a broad feature with peaks at 33.8(1), 35.5(1), and 38.4(1) meV. The inelastic contribution to the measured signal could not be separated from the resolution-broadened elastic scattering at $E<7.5\mathrm{meV}$. From the model, $G_{\alpha }\left(E\right)\simeq Z_{\alpha }\left(E\right)$, where the $Z_{\alpha }\left(E\right)$ functions are given in (a) and (b) by the dash-dotted blue curves.