Page 138 - Principles and Applications of NanoMEMS Physics

126 Chapter 3

+
ic G G c G D
1 S 0 = ¦ k − q k , (140a)
q
G E G − E G G − = ω G
k k k − q q
and

+
ic G G c G D
0 S 1 G = ¦ ' k + q ' k
q . (140b)
G E G + = ω G − E G G
' k ' k q ' k − q
Substituting (140) into (139) one obtains,

§ ·
1 2 + + ¨ 1 1 ¸
'
f [ ] = DiH,S c G G c G c G G c G − . (141)
2 ¦ k ' +q ' k ' k +q ' k ¨ E G k −E G −= ω G q E G ' k += ω G q −E G ' k −q G ¸
G G G
G
©
' k k q
¹
−q
k
Realizing that, due to energy conservation, E G − E G G = E G G − E G , (142)
' k ' k − q k+ q k
may be simplified to yield,
+ +
= ω c G G c G c G G c G
G
q
H ' ' = D 2 ¦¦ ' k + q ' k k − q ' k . (143)
2
G G G (E G − E G ) − ω G 2
q k ' k k k − q G q
Equation (140) reveals that in circumstances when (E G − E G G ) < 2 G ω , this
2
k k− q q
term is negative, thus embodying an electron-electron interaction that is
attractive, and that gives rise to the bosonic behavior mentioned previously.
Having shown that it is physically possible for a pair of electrons to attract
one another in the presence of a phonon, the next question before us is to
determine the binding energy of the pair. As usual, this is obtained from the
energy eigenvalues of Schrödinger equation, ψ = E ψ . Towards this end,
H
we begin by expressing the Hamiltonian,
G 2 G 2
G
H = p 1 + p 2 + V (r − G r ), (144)
2m 2m 1 2
G G
where the potential (rV r − ) models the interaction (143), in the center-
1 2
of-mass and relative-motion coordinates, i.e.,
G 2 G 2
G
H = P + p + V (), (145)
r
4m 2m