Page 254 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Electron-Phonon
(7.45)
–
δE =
ε EP
∞
where ε ∞ is the optical or high frequency dielectric constant. For the
induced electric field we have to take into account that the respective
electrostatic potential is due to screening effects because of the movable
charges present and thus will be of the form as given in (7.20) or (7.21)
and therefore (7.45) reads
⁄
NMω 2 opt 1 1  12 q 2 iqr + – iqr i – ωt

δE = --------------------- ----- – ---- ----------------- b e[ + b e ]e e (7.46)
4πV  ε ∞ ε  q + λ 2 q q q
2
0
7.3.3 Piezoelectricity

Simple Consider the 1D lattice with a basis of Section 2.4.1 (also see
Piezoelectric Figure 2.20), but now endow the two atoms each with equal but opposing
Model
charges. The kinetic co-energy is added up from the contributions of the
individual ions

,
n 2
1 2
T ∗ = --- ∑ m u˙ (7.47)
α iα
2
iα
The index α counts over the ions in an elementary basis cell, and the
index counts over the lattice cells. Similarly, the harmonic bond poten-
i
tial energy is dependent on the stretching of the inter-ion bonds
,,,
n 2 n 2
1 E αiβj 2
U = --- ∑ ------------ u –( iα u ) (7.48)
jβ
4 a
,
iαβj
to which we now add the ionic energy due to the nearest-neighbor charge
interaction
1 q α
U = ---------------- (7.49)
q
4πε r
0 α

Semiconductors for Micro and Nanosystem Technology 251