Page 114 - Principles and Applications of NanoMEMS Physics

102 Chapter 3

2
v = v 1+ V 4 + V − V 2 2 , (66)
4
F v π 4π 2 2
F v F
and

1+ V 2π v − V 2π v
g = 4 F 2 F . (67)
1+ V 2π v + V 2π v
4 F 2 F

Kane and Fisher [140] interpret v in the case V = V > 0 , as the plasmon
4 2
velocity, which increases above v when the repulsive interactions reduce
F
the compressibility of the electron gas.

When the electron spin is included in the Hamiltonian, the interaction
becomes,
~
~
V (x − ' x ) ( ) ( ) 'xx ρρ ~ Ÿ V σσ ' (x − ' x ) ( ) ( ) 'xx ρρ σ ~ ' σ . (68)
In this case, the kinetic energy part of the Hamiltonian may be written as
follows [133].

H = v F ¦ ( ( k − k )c + c + ( k −− k )c + c )
kin F + k , s , + k , s , F − k , s , − k , s ,
k s ,
2π v , (69)
= F ¦ ρ () q ρ ( q− )
L q> ,0 α = ± s , α s , α s ,
where the substitution,

ρ ± s , () =q ¦ + k,, s c + k, , (70)
+
c
s
,
k
representing density operators for spin projections =↑,s ↓ has been made.
The potential energy, in turn, contains two types of interaction, namely,
backward scattering and for ward scattering. The backward scattering
Hamiltonian is given by,

H = 1 ¦ g c + c c c
+
−
int_ 1 1 k , + s , k , + t , , + p 2 k F + q t , , − k 2 k F − q s , , (71)
L k p , q , s , t ,