Electron-energy-loss spectrum

One of the many ways with which is possible to probe neutral excitations in a material is by injecting electrons into the sample. These are called electron-energy-loss spectroscopy experiments, where the incoming electron can create electron-hole pairs, plasmons, or even higher-order multi-pair excitations.

The incoming electron acts as an external potential, $V_\mathrm{ext}(\mathbf r', t')$, which induces a charge density in the material, $\rho_\mathrm{ind}(\mathbf r, t)$. Within linear-response theory these two quantities can be related by the reducible polarisability function, $\chi$, via a Green-Kubo relation

$\rho_\mathrm{ind}(\mathbf r, t) = \int \mathrm d^3r'\mathrm d t \chi(\mathbf r, t,\mathbf r', t')V_\mathrm{ext}(\mathbf r', t').$

If the external potential is taken as proportional to a plane-wave of momentum $\mathbf q$, then the electron energy-loss spectrum (EELS) can be taken from the imaginary part of the inverse dielectric function, $\epsilon^{-1}(\mathbf q,\omega)$, since $\epsilon^{-1} = 1 + v\chi$

$\mathrm{EELS}(\mathbf q,\omega) = -\mathrm{Im}\left[\epsilon^{-1}(\mathbf q,\omega)\right].$

Inclusion of local fields

In general, much like $\chi$, $\epsilon^{-1}$ is a function of two coordinates, i.e. $\epsilon^{-1} := \epsilon^{-1}(\mathbf r , \mathbf r', \omega)$. This has important consequences on inhomogeneous systems, where a homogeneous, constant, external electric field can induce fluctuations at the interatomic scale, and thus create microscopic fields. A direct consequence from the inhomogeneous character of the system is that $\epsilon$ has to be written as $\epsilon_{\mathbf G, \mathbf G'}(\mathbf q, \omega)$, where $\mathbf G$ is a reciprocal lattice vector. The microscopic fields are then the $\mathbf G \neq 0$ components of the tensor.

From $\epsilon^{-1} = 1 + v\chi$ it is possible to see that a problem arises when $\mathbf q \to 0$, i.e. the optical limit. In reciprocal space this equation becomes

$\epsilon^{-1}_{\mathbf G, \mathbf G'}(\mathbf q, \omega) = \delta_{\mathbf G, \mathbf G'} + \frac{4\pi}{|\mathbf q + \mathbf G|^2}\chi_{\mathbf G, \mathbf G'}(\mathbf q, \omega)$

where $v(\mathbf q + \mathbf G) = 4\pi/|\mathbf q + \mathbf G|^2$ is the Coulomb potential. At $\mathbf q= 0$, all components without microscopic fields are divergent. To circumvent this issue, the evaluation of $\epsilon^{-1}$ is replaced the Coulomb potential with

$\bar v(\mathbf q + \mathbf G) = \left\{ \begin{array}{ll} 0, & \mathbf G=0 \\ 4\pi/|\mathbf q + \mathbf G|^2, & \mathbf G\neq 0 \end{array} \right.,$

leaving the $v_0\chi_{00}$ component to be dealt with separately and then added at the end.

Micro-macro connection and relation to measured quantity

It is important to note that the actual measured quantity, $\epsilon_\mathrm{M}(\mathbf q, \omega)$, does not depend on the microscopic fields. To connect both the microscopic and macroscopic quantities, an averaging procedure is taken out, so that

$\epsilon_M(\mathbf q,\omega) = \frac{1}{\epsilon_{\mathbf G = 0, \mathbf G'=0}^{-1}(\mathbf q,\omega)}.$

Since VASP computes the macroscopic function, the final result can be linked to EELS via

$\mathrm{EELS}(\omega) = -\mathrm{Im}\left[\frac{1}{\epsilon_M(\mathbf q,\omega)}\right]$

Note that the inclusion of local fields and the connection to the macroscopic observable must be considered regardless of the level of approximation to the polarisability function, $\chi$. Within VASP, this can be done at several levels of approximation, which are discussed in the next section.

Accounting for electron-hole interaction

EELS from density functional theory

The simplest way to compute the dielectric function is by