My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 8. APPLYING THE PLASMON NORMAL MODE THEORY TO
SLAB SYSTEMS
Several standard integrals involving Bessel functions have been used.
In the limit s → ∞, the surface term simplifies to
u s→∞
surf (r, r ′ ) =
√
2e
2
ɛ 0 ω p L 2 ∑
k ‖
1
k ‖
cos k ‖ · ∆r ‖ e −k ‖(|z|+|z ′ |) . (8.98)
When the limit L → ∞ is also taken, the prescription of equation (8.95) for converting
the summation to an integral gives
u ∞ surf(r, r ′ ) =
=
=
=
√
2e
2
∫ ∞ ∫ π
k dk dθ e−k ‖(|z|+|z ′ |)
cos k
4π 2 ‖ · ∆r ‖
ɛ 0 ω p 0
0 k
√
2e
2
∫ ∞
∫ π
dk e −k ‖(|z|+|z ′ |)
dθ cos ( k
4ɛ 0 π 2 ‖ ∆r ‖ cos(θ − φ) )
ω p 0
0
√
2e
2
∫ ∞
( )
dk e −k ‖(|z|+|z ′ |) πJ
4π 2 0 k‖ ∆r ‖
ɛ 0 ω p 0
√
2e
2
√
4πɛ 0 ω p (|z| + |z′ |) 2 + (∆r ‖ ) . (8.99)
2
The full plasmon two-body function in this limit is therefore
[ (
)
u ∞ pl (r, r ′ e 2
) = Θ(z)Θ(z ′ 1
)
4πɛ 0 ω p |r − r ′ | − 1
√
(z + z′ ) 2 + (∆r ‖ ) 2
√ ]
2
+ √ .
(|z| + |z′ |) 2 + (∆r ‖ ) 2
(8.100)
This function is plotted in figures 8.6, 8.7, 8.8 and 8.9. The contribution from the
bulk plasmons is only relevant when both electrons are inside the metal; when the
electrons are deep inside, u ∞ pl tends to the expected homogeneous electron gas form.
The correlation is boosted for electrons closer to the boundary.
The singularities present in equation (8.100) are a feature of the approximations
which have been made; they do not appear in the original expressions relevant to
a finite slab and cell. In any case, the plasmon theory is not expected to predict
electron-electron correlation accurately at short range; the cusp conditions described
in appendix B and discussed in the next section give more information about this
regime.
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