My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 6. THE MODIFIED PERIODIC COULOMB INTERACTION IN
QUASI-2D SYSTEMS
Choosing the origin so that the external charge is at (r 0 ) ‖ = 0 gives the Fourier
components as
1
k
δ ˜ρ ext (k ‖ , z) = −
2 L 2√ 2πσ 2 e−σ2 ‖ /2 e −(z−z 0) 2 /2σ 2 + 1
L 2 s Θ(z)Θ(s − z)δ k ‖ ,0. (6.39)
This reduces equation (6.36) to the form
[ˆL(z) − k
2
‖
]
δ ˜φ tot (k ‖ , z) =
with
and
4π
k 2
L 2√ 2πσ 2 e−σ2 ‖ /2 e −(z−z 0) 2 /2σ 2 − 4π
L 2 s Θ(z)Θ(s − z)δ k ‖ ,0
(6.40)
ˆL(z) = d2
− f(z) (6.41)
dz2 ( ) 1/3 3n(z)
f(z) = 4
. (6.42)
π
This nonhomogeneous equation may be solved by computing the appropriate Green’s
function, which satisfies the following equation:
[ˆL(z) − k
2
‖
]
G(z, z ′ ; k ‖ ) = δ(z − z ′ ). (6.43)
In order to proceed further, an analytic expression for the electron density is
required. In the case of a realistic slab, this is not available. However, it is possible
to analyse a more simple case: a sharp surface, where the density profile is
n(z) = n 0 Θ(z). (6.44)
The additional positive charge density which was added to maintain the neutrality
of the cell disappears in this limit, which corresponds to letting s → ∞. Some of
the essential surface physics may be lost because of the sharp boundary: the true
unbounded slab will be treated later using computational methods.
For the simplified surface, the differential equation for the Green’s function is
[
( )
d 2
1/3
dz − 3n0
2 k2 ‖ − 4 Θ(z)]
G(z, z ′ ; k ‖ ) = δ(z − z ′ ). (6.45)
π
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