My PhD thesis - Condensed Matter Theory - Imperial College London

APPENDIX A. THE QUASI-2D EWALD SUM
The second contribution comes from a reciprocal space sum; the charge distribution
is therefore rewritten as
ρ 2 (r) = e −z2 /σ 2 ∑ k
ρ k e ik·r ‖
(A.7)
where k is the set of 2D reciprocal lattice vectors and the ρ k are Fourier coefficients.
The sum excludes k = 0, because ρ k=0 = 0 by design. The other coefficients are
given by
ρ k = 1 A
∫
cell
∑
R
1
√ ππσ
3 e−(r ‖−R) 2 /σ 2 e −ik·r ‖
dr ‖
1
=
A √ e
ππσ
∫space
−r2 3 ‖ /σ2 −ik·r ‖
dr ‖
= 1
A √ σ 2 /4
πσ e−k2 .
The desired potential is expressed as a similar series:
(A.8)
φ 2 (r) = ∑ k
φ k (z)e ik·r ‖
.
(A.9)
These expressions may then be substituted into Poisson’s equation,
∇ 2 φ 2 (r) = −4πρ 2 (r),
to give an equation for the coefficients φ k (z):
( ) d
2
dz − 2 k2 φ k (z) = − 4√ π /σ 2 −k 2 σ 2 /4
σA e−z2 .
(A.10)
(A.11)
This may be solved with the Green’s function
G k (z, z ′ ) = − 1
2k e−k|z−z′ |
to give
∫ ∞
(
1
|
φ k (z) = −
−∞ 2k e−k|z−z′ − 4√ )
π /σ 2 −k 2 σ 2 /4
σA e−z′2 dz ′ ,
which, after integration, yields the potential
φ 2 (r) = π A
∑
k
1
k
[
e −kz erfc
( σk
2 − z ) ( σk
+ e kz erfc
σ
2 + z σ
)]
e ik·r ‖
.
(A.12)
(A.13)
(A.14)
186