My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 6. THE MODIFIED PERIODIC COULOMB INTERACTION IN
QUASI-2D SYSTEMS
The solution is therefore
⎧
2κ
G(z, z ′ ; k ‖ ) = − 1 ⎪⎨ e k ‖z−κz ′ z < 0
κ + k ‖ ( )
2κ
κ − k‖
⎪⎩ e −κ|z−z′| + e −κ(z+z′ )
z > 0
κ + k ‖
An identical method gives the Green’s function when z ′ < 0:
( )
⎧⎪
G(z, z ′ ; k ‖ ) = − 1 ⎨ e −k ‖|z−z ′| k‖ − κ
+ e −k ‖(z+z ′ )
z < 0
k ‖ + κ
2k ‖ ⎪
2k ‖
⎩
k ‖ + κ ek ‖z ′ −κz
z > 0
(z ′ > 0). (6.58)
(z ′ < 0). (6.59)
The Green’s function has the required symmetry property G(z, z ′ ; k ‖ ) = G(z ′ , z; k ‖ ).
Once G(z, z ′ ; k ‖ ) is known, the potential may be obtained by integration:
δ ˜φ tot (k ‖ , z) =
∫ ∞
−∞
(
)
G(z, z ′ 4π
k
; k ‖ )
2 L 2√ 2πσ 2 e−σ2 ‖ /2 e −(z′ −z 0 ) 2 /2σ 2 dz ′ . (6.60)
Since the aim is to calculate the induced change in the charge density (which is
proportional to the original density, and hence zero outside the slab), it is only
necessary to consider the region z > 0. The integration then gives
(
) [ ∫
δ ˜φ 4π
0
k
tot (k ‖ , z) = −
2 L 2√ 2πσ 2 e−σ2 ‖ /2
+ 1
2κ
+ 1
2κ
= − π
L 2 κ
−∞
1
κ + k ‖
e k ‖z ′ −κz e −(z′ −z 0 ) 2 /2σ 2 dz ′
( ) ∫ κ − ∞ k‖
e −κ(z+z′) e −(z′ −z 0 ) 2 /2σ 2 dz ′
κ + k ‖
∫ z
0
[
0
e −κ(z−z′) e −(z′ −z 0 ) 2 /2σ 2 dz ′ + 1 ∫ ]
∞
e −κ(z′ −z) e −(z′ −z 0 ) 2 /2σ 2 dz ′
2κ z
( ( ))
2κ
e −κz+k z0
‖z 0
+ k ‖ σ 2
1 − erf √
κ + k ‖ 2σ
+ κ − k ‖
(κ + k ‖ ) e(κ2 −k 2 ‖ )σ2 /2−κz−κz 0
(
1 + erf
+ e (κ2 −k 2 ‖ )σ2 /2−κz+κz 0
(
erf
(
z0 − κσ 2
√
2σ
))
( ) ( ))
z − z0 − κσ 2 z0 + κσ 2
√ + erf √
2σ 2σ
( ))
+ e (κ2 −k‖ 2 )σ2 /2+κz−κz 0
(1 ] z − z0 + σ 2 κ
− erf √ .
2σ
(6.61)
98