Page 136 - Principles and Applications of NanoMEMS Physics

124 Chapter 3

The third term is the familiar electron-phonon interaction [159], in which an
acoustic phonon distorts the lattice and, as a consequence, produces a grating
in the band edges which, in turn, causes electrons to scatter off of it. This
interaction is captured by the interaction potential for acoustic phonons
given by,
G G G
U ( ) Dt,r = ∇ u ( ) t,r , (130)
AP
where D is the deformation potential and,

G = [ r ω () ] i − [ r ω () ] )
G
G
G
G
G
G
⋅
−
⋅
−
u ( ) t,r = G ( ea G i q q t + a G * e q q t , (131)
2ρ Vω () q q q
the lattice displacement. The pertinent energy of interaction is,
G G G G
H = ³ d rΨ + ()Ur () () rr Ψ , (132)
ep AP
where,
G G G G
Ψ () =r ¦ c G e k i ⋅r φ G () r , (133)
G k k
k
is the unperturbed one-electron Block state. With these definitions, the first-
order electron-phonon interaction may be written as,

H ' = iD ¦ c + G G c G (a G − a G * )
G G k + σ , q σ , k q q . (134)
, k σq

The Hamiltonian describing the electron-phonon system, then, is given by,

+
a
H = ¦ E G c + G c G + ¦ = ω G a G +iD c G + Gc G ( −aa G * G )
G
G G
G σ , k σ , k σ , k G q q q ¦ k +q k q q . (135)
σ , k q q k
Now, to determine the nature of the electron-electron interaction, we have to
transform (135) into a Hamiltonian that does not contain the O(D) term, i.e.,
in which the phonon coordinates are eliminated and only electron-electron
interaction terms are present. This is accomplished by transforming (135)
~
~
into a new Hamiltonian given by H = e − S He , and so choosing S that H
S