My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 6. THE MODIFIED PERIODIC COULOMB INTERACTION IN
QUASI-2D SYSTEMS
Since the induced charge density is created entirely by rearrangement of electrons,
equation (6.30) becomes
( ) 1/3 3n
δρ ind = − δφ
π 4 tot . (6.32)
Substitution in equation (6.29) leads to the following equation for δφ tot :
[ ( ) ] 1/3 3n
∇ 2 − 4 δφ tot = −4πδρ ext . (6.33)
π
In order to solve this equation, the functions are first expanded in Fourier series
in the xy-plane:
δφ tot (r) = ∑ k ‖
δ ˜φ tot (k ‖ , z)e −ik ‖·r ‖
(6.34)
δρ ext (r) = ∑ k ‖
δ ˜ρ ext (k ‖ , z)e −ik ‖·r ‖
. (6.35)
This step has introduced in-plane periodicity into the problem, which is desirable,
since the aim is to investigate the effect of cell size on the hole. The equation to be
solved is now [ ( ) ]
d 2
1/3 3n(z)
dz − 2 k2 ‖ − 4
δ
π
˜φ tot (k ‖ , z) = −4πδ ˜ρ ext (k ‖ , z). (6.36)
At this stage, it is useful to supply the form of the external charge. Since the
idea is to investigate the hole around an electron, the appropriate form is
δρ ext (r) = − ∑ R
δ(r − r 0 − R) + 1 Θ(z)Θ(s − z). (6.37)
L 2 s
The external charge must be periodic if it is to be expanded in a Fourier series, as
in equation (6.35); this is ensured by the sum over the in-plane lattice vectors R.
The positive charge (which is uniform over the slab) has been added to ensure that
the cell remains charge-neutral.
The problem with this charge density is that it leads to a potential δφ tot which
is divergent at r = r 0 . Instead, it is convenient to smear out the charge distribution
slightly:
δρ ext (r) = − ∑ R
1
(2πσ 2 ) 3/2 e−(r−r 0−R) 2 /2σ 2 + 1 Θ(z)Θ(s − z). (6.38)
L 2 s
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