My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 8. APPLYING THE PLASMON NORMAL MODE THEORY TO
SLAB SYSTEMS
0.4
0.3
z’ = -2.0 (unsmoothed)
z’ = -2.0 (smoothed)
z’ = 2.0 (unsmoothed)
z’ = 2.0 (smoothed)
z’ = s/2 (unsmoothed)
z’ = s/2 (smoothed)
u pl
(∆r ll
=0, z, z’)
0.2
0.1
0
-2 0 2 s
z
Figure 8.10: Removing the cusps from u pl at the slab boundaries. In each plot, one electron is fixed
while the other is scanned along a line in the z-direction; the x- and y-coordinates are chosen to
be the same, so that ∆r ‖ = 0. The smoothed and unsmoothed curves are almost indistinguishable
when the electrons are far from the slab edges; the most pronounced difference appears when one
electron is just inside the slab edge (the black curves). This and the following plots were calculated
for a cell containing 600 electrons, with s = 17.64248 and r s = 2.07 (in Hartree atomic units).
The free parameter α σi σ j
in the electron-electron cusp function u cusp determines the
distance at which the short-range cusp-dominated behaviour is replaced by the longrange
plasmonic form. Following the discussion in the previous section, this distance
should be ∼ k −1
c , which means that α σi σ j
= k c . The remaining parameter, β σi σ j
is
determined by equation (B.21).
There is one further subtlety related to the use of periodic boundary conditions.
The plasmon two-body function, by construction, is periodic in the xy-plane;
however, the cusp function is not. It is therefore important to ensure that u cusp
is effectively zero before the electron-electron separation reaches a maximum; 7 not
7 The electron-electron distance has a maximum because all electron-electron interactions (in-
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