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, [math]\displaystyle{ V_\mathrm{ext}(\mathbf r', t') }[/math], which induces a charge density in the material, [math]\displaystyle{ \rho_\mathrm{ind}(\mathbf r, t) }[/math]. Within linear-response theory these two quantities can be related by the reducible polarisability function, [math]\displaystyle{ \chi }[/math], via a Green-Kubo relation

[math]\displaystyle{ \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'). }[/math]

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

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

Inclusion of local fields

In general [math]\displaystyle{ \epsilon^{-1} }[/math] is a function of two coordinates, i.e. [math]\displaystyle{ \epsilon^{-1} := \epsilon^{-1}(\mathbf r , \mathbf r', \omega) }[/math]. 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 [math]\displaystyle{ \epsilon }[/math] has to be written as [math]\displaystyle{ \epsilon_{\mathbf G, \mathbf G'}(\mathbf q, \omega) }[/math], where [math]\displaystyle{ \mathbf G }[/math] is a reciprocal lattice vector. The microscopic fields are then the [math]\displaystyle{ \mathbf G \neq 0 }[/math] components of the tensor.

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

[math]\displaystyle{ \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) }[/math]

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

[math]\displaystyle{ \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., }[/math]

leaving the [math]\displaystyle{ v_0\chi_{00} }[/math] 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, [math]\displaystyle{ \epsilon_\mathrm{M}(\mathbf q, \omega) }[/math], does not depend on the microscopic fields. To connect both the microscopic and macroscopic quantities, an averaging procedure is taken out, so that

[math]\displaystyle{ \epsilon_M(\mathbf q,\omega) = \frac{1}{\epsilon_{\mathbf G = 0, \mathbf G'=0}^{-1}(\mathbf q,\omega)}. }[/math]

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

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

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, [math]\displaystyle{ \chi }[/math]. 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 (DFT)

The simplest way to compute the dielectric function is by performing a ground state calculation using DFT, with the tags NBANDS and LOPTICS.

SYSTEM = Si
NBANDS = 12
ISMEAR = 0 ; SIGMA = 0.05
ALGO = N
LOPTICS = .TRUE.
KPAR = 4