My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 8. APPLYING THE PLASMON NORMAL MODE THEORY TO
SLAB SYSTEMS
An improved one-body function may be obtained by replacing u bulk in equation
(8.114) with the full two-body term, giving
χ(r) = χ bulk (r) + χ cusp (r) (8.115)
where
∫
χ cusp (r) = 1 2 V
(
me e 2
=
[
]
u cusp (r ↑, r ′ ↑) + u cusp (r ↑, r ′ ↓) ¯n(z ′ ) d 3 r ′ .
4πɛ 0 2 ) 3
8k c
∫V
¯n(z ′ )e −kc|r−r′ |−(r−r ′ ) 2 /L 2 c d 3 r ′ .
(8.116)
This step is not rigorously justified; the relationship (8.114) has only been shown
to apply to the plasmon part of the Jastrow factor. However, this relationship has
proven to be accurate in the past. The surface plasmon term u surf is not included;
the in-plane integral is zero 8 because there is no surface plasmon with k ‖ = 0.
The integral in equation (8.116) is over the cell. However, the factor of e −(r−r′ ) 2 /L 2 c
in u cusp is designed to ensure that u cusp becomes zero well before |r − r ′ | approaches
the size of the cell. Conveniently, this means that the integral may equally well be
evaluated over the entire xy-plane. Switching to cylindrical polar coordinates gives
∫
∫ ∞ ∫ ∞
(
√
¯n(z ′ )e −kc|r−r′ |−(r−r ′ ) 2 /L 2 c d 3 r ′ = 2π
¯n(z ′ ) exp −k c ρ
′2
+ (z − z ′ ) 2
V
z ′ =−∞
ρ ′ =0
− ρ′2 + (z − z ′ ) 2
L 2 c
There is no dependence on the in-plane components of r.
The integral is solved in appendix C.
)
ρ ′ dρ ′ dz ′ .
(8.117)
Without specifying the form of ¯n, the
result is
( )
me e 2 3πL
2
∫ {
∞
( [
χ(r) = χ bulk (r) +
c
e k2
4πɛ 0 2 c L2 c
|z − z /4 ¯n(z ′ ′ |
) exp −
8k c
z=−∞
L c
− k √ (
cL c π |z − z ′ |
erfc + k ) }
cL c
dz ′ .
2
2
L c
+ k ] 2 )
cL c
2
(8.118)
8 See equation (8.92).
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