Efficient calculation of k -integrated electron energy loss spectra: Application to monolayers of MoS2, hBN, and graphene

Abstract

The theoretical scattering cross section of electron energy loss spectroscopy (EELS) is essentially given by $\ensuremath{-}\text{Im}\phantom{\rule{0.16em}{0ex}}{\ensuremath{\varepsilon}}^{\ensuremath{-}1}(\mathbf{k},\ensuremath{\omega})$ with the energy loss $\ensuremath{\hbar}\ensuremath{\omega}$ and the momentum transfer $\ensuremath{\hbar}\mathbf{k}$. The macroscopic dielectric function $\ensuremath{\varepsilon}(\mathbf{k},\ensuremath{\omega})$ can be calculated from first principles using time-dependent density-functional theory. However, experimental EELS measurements have a finite $\mathbf{k}$ resolution or, when operated in spatial resolution mode, yield a $\mathbf{k}$-integrated loss spectrum, which deviates significantly from EEL spectra calculated for specific $\mathbf{k}$ momenta. On the other hand, integrating the theoretical spectra over $\mathbf{k}$ is complicated by the fact that the integrand varies over several (typically six) orders of magnitude around $k=0$. In this article, we present a stable technique for integrating EEL spectra over an adjustable range of momentum transfers. The important region around $k=0$, where the integrand is nearly divergent, is treated partially analytically, allowing an analytic integration of the near divergence. The scheme is applied to three prototypical two-dimensional systems: monolayers of ${\mathrm{MoS}}_{2}$ (semiconductor), hexagonal BN (insulator), and graphene (semimetal). Here, we are confronted with the added difficulty that the long-range Coulomb interaction leads to a very slow supercell (vacuum size) convergence. We address this difficulty by employing an extrapolation scheme, enabling an efficient reduction of the supercell size and thus a considerable speedup in computation time. The calculated $\mathbf{k}$-integrated spectra are in very favorable agreement with experimental EEL spectra.