First-principles calculation of electron-phonon spectral functions for defects using phonon interpolation
Point defects in wide-band-gap semiconductors exhibit optical spectra strongly shaped by electron-phonon coupling, but direct first-principles calculation of the corresponding phonon sidebands is often limited by the coarse vibrational spectrum of the largest defect supercells accessible by ab-initio electronic structure codes. In this work, we present a phonon-interpolation method for Huang-Rhys spectral densities and optical lineshape functions on hypercells created by extending the defect-containing supercells on arbitrarily dense phonon $q$-point grids. The method is based on reconstructing the transition-induced force associated with the optical excitation and using this localized force source to couple the defect transition to a densely sampled vibrational continuum. In this formulation, the local defect physics is obtained from ab initio supercell calculations, while the long-wavelength acoustic modes and the detailed structure of the host phonon spectrum are recovered by diagonalizing interpolated dynamical matrices in large hypercells. We demonstrate the method on the negatively charged nitrogen-vacancy centre in diamond between its ground $^{3}A_{2}$ and excited $^{3}E$ states. The transition-force is shown to be strongly localized around the defect, with converged localization measures obtained in a $4\times4\times4$ supercell accessible by density functional theory calculations. We interpolate the electron-phonon coupling on hypercells up to $32\times32\times32$ corresponding to approximately 17 million atoms, thereby recovering smooth, continuous Huang-Rhys spectral densities with ultrafine spectral resolution. The dominant coupling band is found near 63~meV; the low-energy acoustic contribution follows the expected linear scaling; and we recover the finer van-Hove-related structures in the optical phonon regime observed in experiments.