Page 104 - Principles and Applications of NanoMEMS Physics

92 Chapter 3

L, and that the wavefunction fulfills periodic boundary conditions at each of
its faces, namely,
G G G G
ψ (r + ˆ ) x = ψ (r + Lˆ ) y = ψ (r + ˆ ) z L = ψ ( ) r , (31)
L

These assumptions yield solutions of the form
G k i G r ⋅
G
φ k σ () r = e χ , (32)
σ
V
where σ = ± 1 2 represents the two values of electron spin and χ σ
represents the two spin functions,

ª1 º ª0 º
χ 1 2 = « » , χ − 2 = « » . (33)
1
¬ 0 ¼ ¬ 1 ¼
Because of the periodic boundary condition, the wave vector is defined by,
k = 2 π n , k = π 2 n , k = π 2 n , (36)
x x y y z z
L L L

2
2
2
2
where n x , n y , n z = , 0 ± , 1 ± 2 ,..., k = k + k + k . The energy
z
y
x
eigenvalues of (32) are given by,
E kσ = E = = 2 k 2 . (37)
k
2 m
The salient properties of the electron gas as a whole are captured by its
wave function, its total energy, and various quantities such as its specific
heat, and its magnetic susceptibility. The wave function is given by the
Slater determinant [132],

φ () 1 φ ( ) 2 ... φ ( )
N
ν 1 ν 1 ν 1 ( )
ψ ... ν ( 2,1 3 , ,... N ) = 1 φ ν 2 () 1 φ ν 2 ( ) 2 ... φ ν 2 N , (38)
ν 1 ν 2 ν 3 N ... ... ... ...
N!
φ ν () φ1 ν ( ) ...2 φ ν ( )
N
N N N
which ensures that the Pauli exclusion principle is obeyed, i.e., if two of the
G
one-particle states ν are the same, then ψ ... ≡ 0 . With () 0=rU , the
i ν 1 ν 2 ν N