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.

The inclusion or not of local fields is something that has to 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