Polarizability models for simulations of finite temperature Raman spectra from machine learning molecular dynamics

Raman spectroscopy is a powerful and nondestructive method that is widely used to study the vibrational properties of solids or molecules. Simulations of finite-temperature Raman spectra rely on obtaining polarizabilities along molecular-dynamics trajectories, which is computationally highly demanding if calculated from first principles. Machine learning force fields (MLFF) are becoming widely used for accelerating molecular-dynamics simulations, but machine-learning models for polarizability are still rare. In this work, we present and compare three polarizability models for obtaining Raman spectra in conjunction with MLFF molecular-dynamics trajectories: (i) a model based on projection to primitive cell eigenmodes, (ii) a bond polarizability model, and (iii) symmetry-adapted Gaussian process regression (SA-GPR) using a smooth overlap of atomic positions. In particular, we investigate the accuracy of these models for different systems and how much training data are required. Models are first applied to boron arsenide, where the first- and second-order Raman spectra are studied as well as the effect of boron isotopes. With ${\mathrm{MoS}}_2$ we study the applicability of the models for highly anisotropic systems and for simulating resonant Raman spectra. Finally, inorganic halide perovskites are studied with a particular interest in simulating the spectra across phase transitions. All models can be used to efficiently predict polarizabilities and are applicable to large systems and long simulation times, and while all three models were found to perform similarly for BAs and ${\mathrm{MoS}}_2$, only SA-GPR offers sufficient flexibility to accurately describe complex anharmonic materials like the perovskites.

Figure 1

(a) Schematic of the workflow used in this work. Red panels represent DFT level calculation, blue ones GPUMD calculations, and green ones postprocessing steps. (b),(c) Phonon dispersion curves of BAs and ${\mathrm{MoS}}_2$, respectively. Results from vasp-MLFF (in red) and NEP (in yellow) potentials are compared with DFT (in black) for benchmarking.

Figure 2

(a) Convergence of BPM RMSE with training set size for different expansion orders. (b) Convergence of SA-GPR with training set size for different numbers of radial and angular functions [$n$ and $l$ in Eq. (10)]. (c) Comparison of DFT polarizabilities with those obtained from the three final models.

Figure 3

(a) First-order Raman spectra of BAs at different temperatures. Black lines show Lorentzian fits. Spectra are obtained using SA-GPR (see Fig. S3 for a comparison between methods). (b),(c) Position and width of the first-order peak with temperature. Results from SA-GPR are compared with experimental results from Ref. [73]. Frequencies and width are shifted, respectively, by $-13$ and $-4{\mathrm{cm}}^{-1}$ for better visual comparison.

Figure 4

Second-order Raman for different size of supercells using (a) RGDOS, (b) BPM, and (c) SA-GPR polarizability models.

Figure 5

First-order (left) and second-order (right) Raman spectra of ${\mathrm{B}}_x^{11}{\mathrm{B}}_{1-x}^{10}\mathrm{As}$ with different isotope content $x$. Gray shaded areas show experimental measurements from Ref. [73]. Spectra are obtained using SA-GPR (see Fig. S4 for a comparison between methods).

Figure 6

(a),(b) Comparison of in-plane and out-of-plane polarizabilities from models with DFT in the nonresonant case. (c),(d) Real and imaginary part of the dielectric function from a randomly selected structure. Different models are compared with DFT results (black lines). (e),(f) Errors on the real and imaginary part of the dielectric function as a function of energy. Black lines show the standard deviation of polarizabilities from DFT. The legend for the different models is given at the top and is used for all panels. In panels (c)–(f), only $xx$ components of the dielectric function are considered.

Figure 7

(a) Different polarizations of the first-order Raman spectrum using all three polarizability models. Dashed vertical lines show the experimental frequencies from Ref. [85]. Second-order Raman spectra at different excitation energies using BPM and SA-GPR models, respectively. (d) Second-order Raman spectra at selected excitation energies. For both energies, BPM and SA-GPR are compared. The gray area shows the experimental resonant Raman spectrum at 532 nm (2.33 eV) from Ref. [85].

Figure 8

(a),(b) Lattice constants with temperature during heating for ${\mathrm{CsPbBr}}_3$ and ${\mathrm{CsSnBr}}_3$, respectively. The different colors correspond to the different lattice constants. (c),(d) Comparison between polarizabilities from DFT and SA-GPR for ${\mathrm{CsPbBr}}_3$ and ${\mathrm{CsSnBr}}_3$, respectively. Three temperatures are compared to show all three phases.

Figure 9

Raman spectra at different temperatures for (a) ${\mathrm{CsPbBr}}_3$ and (b) ${\mathrm{CsSnBr}}_3$. Temperatures are selected so that all three phases are included. For ${\mathrm{CsPbBr}}_3$, experimental results from Ref. [93] are represented with gray lines. (c) Visualization of ${\mathrm{CsPbBr}}_3$ phonon modes. Cs is in green, Pb in black, and Br in brown.

Figure 10

Raman spectra at different temperatures during the orthorhombic to tetragonal phase transition for (a) ${\mathrm{CsPbBr}}_3$ and (b) ${\mathrm{CsSnBr}}_3$. Insets shows the intensity of the central peak with temperature for the respective materials.

Figure 11

Schematic of (a) energy surface of the octahedral tilts and (b) resulting Raman peaks. Temperatures below, at, and above the phase transition are shown to explain the behavior of the central peak. (c)–(j) Histogram of the bromide polar coordinates at various temperatures. Coordinates are mirrored so that the starting point of every atom is in the top right quadrant. (k) Autocorrelation function of the bromide positions with time at the same temperatures as in (c)–(j).