Page 260 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Electron-Phonon
From (7.58a) we see that the electric field and the stress cause a strain ε
to appear in the material. Alternatively, from (7.58b), the stress and the
electric field cause a dielectric displacement D . For further analysis we
– 1
now multiply (7.58a) by the mechanical stiffness tensor E = s
EHT EHT
to obtain
– 1
E :ε = s :s :σ + E :d ⋅ E or (7.59a)
EHT EHT EHT EHT HT
–
σ = E :ε e ⋅ E (7.59b)
EHT HT
For (7.58b) we use the fact that σ = E :ε
EHT m
D = d :E :ε + κ ⋅ E = e T :ε + κ ⋅ E or (7.59c)
HT EHT m σHT HT m σHT
E = κ – 1 ⋅ D κ – 1 e ⋅ T :ε (7.59d)
–
σHT σHT HT m
We now have the constitutive equations in the most convenient format,
and can insert them into the mechanical and the electromagnetic equa-
tions of motion, which we first restate
2
∂ u
⋅
ρ-------- = ∇σ – f (7.60a)
2
∂t
2
∂ E
∇ × ∇ × E = – µκ--------- (7.60b)
∂t 2
Taking the divergence of equation (7.59b)

⋅
∇σ = ∇ (⋅ E :ε e ⋅ E)
–
EHT HT
(7.61)
(
⋅
⋅
⋅
⋅
⋅
= ( ∇ E ):ε + E : ∇ε) ( ∇ e ) E e ( ⋅ ∇ E)
–
–
EHT EHT HT HT
If the material properties are piece-wise constant, the above equation
simplifies greatly, because then the material property divergences are
nonzero only across material interfaces and appear therefore as jump
conditions.
Semiconductors for Micro and Nanosystem Technology 257