My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 8. APPLYING THE PLASMON NORMAL MODE THEORY TO
SLAB SYSTEMS
In addition, it is desirable that T (0) = 1 ; then if f(x) is continuous, the original
2
value of f on the boundary is preserved. A function with these characteristics is
T (x) = 1 − tanh k cx
. (8.108)
2
This function has a transition region of size ∆x ∼ kc
−1 , which is the shortest lengthscale
available for plasmons with a cut-off of k c in reciprocal space.
Replacing Θ with T in equations (8.88) and (8.89) gives the corrected bulk
plasmon formulae:
χ s bulk(r) =
e2 ∑ 4n 0
ω p ɛ 0 k 3 k z
zs sin k zz(1 − cos k z s)T (z)T (s − z) (8.109)
u s bulk(r, r ′ ) =
e2 ∑ 4
ω p ɛ 0 V k cos k 2 ‖ · (r ‖ − r ′ ‖) sin k z z sin k z z ′
k
× T (z)T (z ′ )T (s − z)T (s − z ′ ). (8.110)
The cusps of the surface plasmon two-body term are contained in the function
F k‖ (z, z ′ ) (given in table 8.1). The smooth version of F k‖
F s k ‖
(z, z ′ ) = F k‖ (z < 0, z ′ < 0)T (z)T (z ′ )
is
+ F k‖ (z < 0, 0 < z ′ < s)T (z)T (−z ′ )T (z ′ − s)
+ F k‖ (z < 0, z ′ > s)T (z)T (s − z ′ )
(8.111)
+ · · · .
Replacing F k‖ with Fk s ‖
in equation (8.90) renders u surf cusp-free.
Figures 8.10 and 8.11 illustrate the effect of removing the cusps from χ bulk and
u pl ; some detail is lost when the electrons are close to the slab edges.
8.3.2 Applying the electron-electron cusps
Having removed the undesirable cusps in the plasmon wave function, the next step
is to insert the desirable ones! This proceeds as indicated in equation (8.104):
[
Ψ = exp − 1 ∑
u cusp (x i , x j ) − 1 ∑
u s
2
2
pl(r i , r j ) + ∑ ]
χ s bulk(r i ) D ↑ (R ↑ )D ↓ (R ↓ ).
i≠j
i,j
i
(8.112)
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