My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 6. THE MODIFIED PERIODIC COULOMB INTERACTION IN
QUASI-2D SYSTEMS
The solution must not diverge as z → ±∞; solving the homogeneous version of the
equation in the three regions separately gives
⎧
Ae
⎪⎨
k ‖z
z < 0
G(z, z ′ ; k ‖ ) = Be κz + Ce −κz 0 < z < z ′
(6.46)
⎪⎩
De −κz z > z ′
where
( ) 1/3
κ 2 = k‖ 2 3n0
+ 4 . (6.47)
π
It is assumed that z ′ > 0; the solution when z ′ < 0 will be given later.
The Green’s function must be continuous at z = 0 and z = z ′ . The remaining
boundary conditions are obtained by integration of equation (6.45) over an infinitesimal
region about these points, giving the following constraints on the gradient:
( [dG(z, ] [ ] )
z ′ ; k ‖ ) dG(z, z ′ ; k ‖ )
lim
−
= 0 (6.48)
a→0 dz
z=a
dz
z=−a
( [dG(z, ] [ ] )
z ′ ; k ‖ ) dG(z, z ′ ; k ‖ )
lim
= 1. (6.49)
a→0 dz
dz
−
z=z ′ +a
z=z ′ −a
Applying these boundary conditions gives the four simultaneous equations required
to determine the four constants:
A = B + C (6.50)
k ‖ A = κ(B − C) (6.51)
Be κz′ + Ce −κz′ = De −κz′ (6.52)
κ(Be κz′ − Ce −κz′ ) + 1 = −κDe −κz′ (6.53)
so that
1
A = −
(κ + k ‖ ) e−κz′ (6.54)
B = − 1
2κ e−κz′ (6.55)
C = − (κ − k ‖)
2κ(κ + k ‖ ) e−κz′ (6.56)
D = − 1 [ ( ) κ − k‖
e κz′ + e
]. −κz′ (6.57)
2κ κ + k ‖
97