Page 253 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Interacting Subsystems
Again the macroscopic deformation field parameter D
depends on the
respective material properties. We see that the interaction itself is not q-
dependent as in the case of acoustic phonons. Since the optical phonons
are rather energetic the high temperature limit lies far above 300K and is
not to be applied as in the acoustic case. The relative displacement field
Fourier component with wavevector q is given similar to (7.32) by
2
—D iqr + – iqr i – ωt
[
w = -------------------- b e + b e ]e e (7.42)
q 2MNω q q q
q
– 1 – 1 – 1 – 1
Note the difference in the pre-factor, where M = M + M . M is
+ - +
the mass of the sub-lattice with displacement s in positive direction and
+
– 1
M the respective masses of the sub-lattice with displacement s in
- -
negative direction. We may think of the total displacement as
w = s – s . Thus the respective matrix element has the same structure
q + -
as (7.38), it reads for the optical phonon scattering
2
—D
(
,
V kk' q) = -------------------- ×
;
2MNω q (7.43)
(
[ n δ E –( E + —ω ) + n + 1δ E – E – —ω )]
q k' k + q q q k' k + q q
Polar In polar crystals like GaAs a very effective interaction mechanism is
Materials present due to the opposite polarization of the two sub-lattices. The dis-
placement of positive and negative ions s ± with the effective charges
± e∗ leads to a local dipole moment p = e∗ s –( s ) from which we
+ -
derive a macroscopic polarization of the form

⁄
NMω 2 opt 1 1  12

P = --------------------- ----- – ---- w q (7.44)
4πV  ε ∞ ε 
0
where w is the relative displacement of positive and negative charges as
q
– 1 – 1 – 1
in (7.42) and M = M + M is given by their respective masses.
+ -
ω is the optical phonon frequency. The interaction energy in this case
opt
is given by the product of the induced electric field E and the polariza-
tion P

250 Semiconductors for Micro and Nanosystem Technology