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]

So the computation of EELS is now reduced to the evaluation of the inverse dielectric function with VASP. This can be done at different levels of approximation which are described below.