Page 250 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Electron-Phonon
does not correspond to the real situation. There is a permanent motion of
the ions that form the crystal lattice. Thus we analyze the periodic poten-
tial in which electrons moves V x() = ∑ V x –( R ) with respect to
i
x
small displacements of the ions, where is the position of the electron.
0
R = R + s t() is the time dependent position of the ion as described in
i i i
Sections 2.3 and 2.4 and the s t() are longitudinal vibrations of the crys-
i
tal lattice. The periodic potential structure results in a certain energy E
c
of the conduction band edge. In an effective mass description the varia-
tion of the conduction band energy is to lowest order given by the change
δE in the conduction band edge energy. Longitudinal acoustic phonons
c
are compression waves s r t,( ) for the set of ions in the solid with a rela-
,
(
tive volume change given by ∆ r t,( ) = ∇ s r t) = δVV . This leads to
⁄
a change in the lattice constant and therefore in the valence band edge
energy. Thus we write
∂E c
(
(
,
,
δE = V ∇ s r t) = Ξ ∇ s r t) (7.31)
c ∂ V 1
where Ξ 1 is called the deformation potential which takes a characteristic
material dependent value. Let us write the longitudinal vibrations in
terms of normal coordinates and look at one Fourier component for the
wavevector q

⁄
—
+ –
(
,
s r t) =  --------------------  12 ( b e iqr + b e iqr )e e i – ω q t (7.32)
q  2MNω  q q q
q
where N is the total number of atoms in the lattice, M is the total mass
and e is the polarization of the lattice vibration. Thus (7.31) reads
q
Ξ 12
⁄
e – ph 1  —  + qr
,
(
t
δE = V ( r t) = --------------  ∑ --------- iqe b t() + b ())e (7.33)
c def 2ω  q q – q
MN q
q
Let us use the fermion and boson number representations to write down
the interaction operator, where the fermions are expanded in terms of
Bloch waves (3.92)
Semiconductors for Micro and Nanosystem Technology 247