Efficient First-Principles Methodology for the Calculation of the All-Phonon Inelastic Scattering in Solids

Inelastic scattering experiments are key methods for mapping the full dispersion of fundamental excitations of solids in the ground as well as nonequilibrium states. A quantitative analysis of inelastic scattering in terms of phonon excitations requires identifying the role of multiphonon processes. Here, we develop an efficient first-principles methodology for calculating the all-phonon quantum mechanical structure factor of solids. We demonstrate our method by obtaining excellent agreement between measurements and calculations of the diffuse scattering patterns of black phosphorus, showing that multiphonon processes play a substantial role. The present approach constitutes a step towards the interpretation of static and time-resolved electron, x-ray, and neutron inelastic scattering data.

Figure 1

(a) Schematic of a multiphonon scattering process. ${\mathbf{K}}_i$ and ${\mathbf{K}}_f$ denote the beam wave vectors before and after inelastic scattering. The momentum transfer to the crystal is $\hslash \mathbf{Q}=\hslash ({\mathbf{q}}_1+{\mathbf{q}}_2+{\mathbf{q}}_3)$, where $\mathbf{Q}$ and $\mathbf{q}$ are the scattering and reduced phonon wave vectors. (b) Left: raw scattering pattern of bP as collected on the detector. $\mathbf{G}$ represents a Bragg peak vector. The Brillouin zone with the $\mathrm{\Gamma }$, $A$, and $X$ high-symmetry points together with the armchair and zigzag directions are indicated. Right: difference scattering pattern, $\mathrm{\Delta }I(\mathbf{Q},t=50\mathrm{ps})$, described in the main text.

Figure 2

(a) Zero-plus-one-phonon ($\mathrm{\Delta }{I}_0+\mathrm{\Delta }{I}_1$), (b) multiphonon ($\mathrm{\Delta }{I}_{\text{multi}}$), and (c) all-phonon ($\mathrm{\Delta }{I}_{\mathrm{all}}$) difference scattering patterns of bulk bP calculated as $\mathrm{\Delta }I(\mathbf{Q})=I(\mathbf{Q},300\mathrm{K})-I(\mathbf{Q},100\mathrm{K})$ to match the experimental conditions [32, 33]. (d) Experimental difference scattering pattern of bulk bP measured at 50 ps from FEDS. Data are divided by the maximum count due to elastic scattering. (e) All-phonon difference scattering pattern of bulk bP showing three vertical paths P1, P2, and P3 at $Q_x=4.83$, 6.21, and $7.59{\AA }^{-1}$. Paths pass through several high-symmetry $X$ points in the zigzag direction. (f)–(h) $\mathrm{\Delta }I(\mathbf{Q})$ as a function of $Q_y$ along P1, P2, and P3. Zero-plus-one-phonon, all-phonon, and experimental data are represented by green, red, and blue. Vertical dashed lines indicate positions of high-symmetry $X$ points. All calculated intensities were divided by the Bragg intensity at the zone center, i.e., $I_0(\mathbf{Q}=\mathbf{0})$, and multiplied by the same scaling factor to facilitate comparison. The Brillouin zone was sampled using a $50\times 50\times 50$ $\mathbf{q}$-grid and full patterns were obtained by a fourfold rotation.

Figure 3

(a)–(c) Colored maps showing the percentage contribution of multiphonon interactions to thermal diffuse scattering, $\mathcal{P}$, across a wide range in reciprocal space of bP at 100, 300, and 500 K. (d) ${\mathcal{P}}_E$ calculated within the Einstein model for thermal diffuse scattering at 300 K. Rectangles represent different Brillouin zones centered at $\mathrm{\Gamma }+\mathbf{G}$.