Page 109 - Principles and Applications of NanoMEMS Physics

3. NANOMEMS PHYSICS: Quantum Wave Phenomena 97

and σ = ( ) ↓↑, represents spin. Similarly, the corresponding energy change
is assumed to be given by,

k′
n δ
δ E = ¦ E k 0 n δ () k + 1 ¦ f ( kk ′ δ, ) ( ) ( ), (48)
n
k
k 2 Ω k k ′
where the first term represents the energy of an individual quasi-particle,
defined as,
G )
= 2 k (k − k
E = F F , (49)
k
2m *
with m * representing its effective mass, and the second term, in particular,
)
,
f ( kk ′ capturing the interaction energy between quasi-particles. Further, in
analogy with the the case of noninteracting states, the probability of a quasi-
particle occuping a state k obeys Fermi statistics,

k
n () = 1 β E , (50)
1 + e k
where β = k / 1 T and E is given by (49). In the case of the Fermi liquid,
B k
it has been found that calculations may be simplified by expressing the
interaction function as the sum of symmetric and anti-symmettric terms,
namely,
G G G G G G
′
f (k ↑, k ↑ )= f S ( kk , ′ )+ f a ( kk , ) ′ , (51a)
and

G G G G G G
′
f (k ↑, k ↓ )= f S (k , k ′ )− f a (k , k ) ′ . (51b)
Then, assuming that these interaction functions exhibit rotational symmetry,
G
G
G
and vary slowly with k , the approximation k = k = k is made, which
′
F
permits a Legendre expansion of the form [132],
G G
G G ∞ k ⋅ k ′
f s ,a ( , kk ′ )= ¦ f L s ,a P L (cos θ ) cos, θ = 2 , (52)
L = 0 k F
where P are the Legendre polynomials. Inversion of the expansion gives
L
the coefficients,