Multi-phonon diffuse scattering in solids from first-principles: Application to 2D MoS$_2$, bulk MoS$_2$, and black Phosphorous

Time-resolved diffuse scattering experiments have gained increasing attention due to their potential to reveal non-equilibrium dynamics of crystal lattice vibrations with full momentum resolution. Although progress has been made in interpreting experimental data on the basis of one-phonon scattering, understanding the role of individual phonons can be sometimes hindered by multi-phonon excitations. In Ref. [arXiv:2103.10108] we have introduced a rigorous approach for the calculation of the all-phonon inelastic scattering intensity of solids from first-principles. In the present work, we describe our implementation in detail and show that multi-phonon interactions are captured efficiently by exploiting translational and time-reversal symmetries of the crystal. We demonstrate its predictive power by calculating the diffraction patterns of monolayer molybdenum disulfide (MoS$_2$), bulk MoS$_2$, and black phosphorus (bP), and we obtain excellent agreement with our measurements of thermal electron diffuse scattering. Remarkably, our results show that multi-phonon excitations dominate in bP across multiple Brillouin zones, while in MoS$_2$ play a less pronounced role. We expand our analysis for each system and we examine the effect of individual atomic and interatomic vibrational motion on the diffuse scattering signals. Our findings indicate that distinct features are explained by the collective displacement of MoS and specific pairs of P atoms. We further demonstrate that the special displacement method reproduces the thermally distorted configuration which generates precisely the all-phonon diffraction pattern. The present methodology opens the way for high-throughput calculations of the scattering intensity in crystals and the accurate interpretation of static and time-resolved diffuse scattering experiments.