Page 82 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Vibrating Uniform Lattice
ω k()

k ω k() k ω k() DOS
Figure 2.26. The computed phonon density of state diagram for a simplistic silicon
model, shown together with the computed phonon dispersion diagram. Notice that peaks
in the density of states diagram occur at positions where ∂ω ∂k⁄ = 0 [2.13].

2
∂ u
ρ-------- = ∇• σ = ∇• ( C:ε) (2.73)
2
∂t
T
Remember that u = u u u . It is simplest to first evaluate
1 2 3
σ = C:ε (here for a cubic-symmetry crystal) in the engineering notation
of (2.38) and (2.40)

C:ε

C 11 C 12 C 12 ε 11 C ε + C ε + C ε
12 33
11 11
12 22
C 12 C 11 C 12 ε 22 C ε + C ε + C ε
11 22
12 33
12 11
C 12 C 12 C 11 ε 33 C ε + C ε + C ε
12 11
11 33
12 22
= ⋅ =
C 2ε 2C ε
44 23 44 23
C 2ε 2C ε
44 31 44 31
C 2ε 2C ε
44 12 44 12
(2.74)
ant then to transform the result back to the usual tensor notation

Semiconductors for Micro and Nanosystem Technology 79