My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 8. APPLYING THE PLASMON NORMAL MODE THEORY TO
SLAB SYSTEMS
The cut-off wave vector k c is related to the electron density [73]:
k c ∼ 1 √
rs
. (8.86)
In a homogeneous system, plasmons with wave vector greater than k c are no longer
well-defined; at k ∼ k c , the (almost flat) plasmon dispersion curve merges with the
particle-hole continuum [17]. In general, it is not obvious how to apply the cut-off
in an inhomogeneous system. However, in the slab system under consideration here,
the electron density is approximately constant within the slab and zero outside; it is
therefore reasonable to apply the same cut-off as would be used in a homogeneous
system of equivalent density.
Substituting these functions in place of the {f i } in equation (7.81) gives
χ bulk (r) =
e2 ∑(
∫
φ 1k (r) φ 1k (r ′ )¯n(z ′ ) d 3 r ′
ω p ɛ 0
k
V
∫
)
+ φ 2k (r) φ 2k (r ′ )¯n(z ′ ) d 3 r ′
= e2
ω p ɛ 0
∑
k z
4
k 2 zs sin k zz
V
∫ s
while substitution in equation (7.82) gives
k
0
¯n(z ′ ) sin k z z ′ dz ′ Θ(z)Θ(s − z) (8.87)
u bulk (r, r ′ ) =
e2 ∑(
)
φ 1k (r)φ 1k (r ′ ) + φ 2k (r)φ 2k (r ′ )
ω p ɛ 0
= e2
ω p ɛ 0 V
∑
k
4
k 2 cos k ‖ · (r ‖ − r ′ ‖) sin k z z sin k z z ′
× Θ(z)Θ(s − z)Θ(z ′ )Θ(s − z ′ ). (8.88)
The sum is understood to include the term with k ‖ = 0. The Heaviside functions
appear because the bulk plasmons have zero amplitude outside the slab. Note that
only the modes with k ‖ = 0 contribute to the χ-function; all others integrate to
zero. Taking the density to be constant (to be consistent with the derivation of the
plasmon modes), ¯n(z) = n 0 , giving
χ bulk (r) =
e2 ∑ 4n 0
ω p ɛ 0 k 3 k z
zs sin k zz(1 − cos k z s)Θ(z)Θ(s − z). (8.89)
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