My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 8. APPLYING THE PLASMON NORMAL MODE THEORY TO
SLAB SYSTEMS
Making the approximation (once again) that ¯n is constant within the slab then gives
( )
me e 2 3
χ(r) = χ bulk (r) +
4πɛ 0 2 16 n 0L 4 cπ √ πe k2 c L 2 c/4
{ ( z
× − + k cL c
+ 1 ) ( z
erfc + k )
cL c
L c 2 k c L c L c 2
( s − z
− + k cL c
+ 1 ) ( s − z
erfc + k )
cL c
L c 2 k c L c L c 2
( z − a(z)
+ + k cL c
+ 1 ) ( z − a(z)
erfc
L c 2 k c L c L
( c
a(z) − z
+ + k cL c
+ 1 ) ( a(z) − z
erfc
2 k c L c
L c
L c
+ 1 √ π
e −(z/Lc+kcLc/2)2 + 1 √ π
e −[(s−z)/Lc+kcLc/2]2
+ k )
cL c
2
)
+ k cL c
2
− 1 √ π
e −([z−a(z)]/Lc+kcLc/2)2 − 1 √ π
e −([a(z)−z]/Lc+kcLc/2)2 }
(8.119)
where
⎧
0 when z < 0
⎪⎨
a(z) = z when 0 < z < s
⎪⎩
s otherwise.
(8.120)
This version of the modification to χ is plotted in figure 8.13; the relative size of
the correction is small (∼ 1% in the centre of the slab). Fortunately, the correction
is cuspless. The modification obtained by using the true density and integrating
equation (8.118) is very similar to that shown in figure 8.13 for the unbounded 9
slab; the difference is slightly more noticeable for the bounded slab.
The sum containing u pl in equation (8.112) is unrestricted: it includes terms
with i = j. It is perhaps clearer to organise one- and two-body functions separately,
so that the full plasmon wave function (with appropriate cusps, and without
9 See chapter 5 for a description of these two versions of the jellium slab.
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