Page 77 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Crystal Lattice System
,
(
ω k k )
x
y
k
y

 π π 
,
--- ---
Σ  a a
M k y
Γ
 π , π  ∆ Z
 – --- – --- X
a 
a
k k x
y
 π π  k x
(a)  --- –, --- (b)
a 
a
Figure 2.22. Acoustic dispersion branch surfaces ω k k,( ) for a monatomic lattice with
x y
nearest and next-nearest-atom linear elastic interaction. (a) One quarter of the surfaces
are omitted to illustrate the inner structure. (b) The contours of the upper and lower sur-
faces. The inter-atom forces have a ratio of β = 0.6 .
By now the general method should become clear: the equations of
motion, (2.63), written in terms of the discrete inter-atomic forces
∞
0
F = – ∑ Γ ⋅ u acting on a typical lattice atom , with a harmonic
0 ∞ p p
[
⋅
ansatz for the motion u k() = Aexp { j kr – ωt]} , leads to an
p p
eigenvalue equation. For a many-atom unit cell we obtain
2
⋅
ω B = D B (2.71)
⁄
12
for the amplitude eigenvectors B j() = m j A j() and eigenfrequencies
ω . How to compute the dynamical matrix D is described, for the carbon
lattice, in Box 2.4. Solving (2.71) for different values of enables us to
k
compute the phonon dispersion diagram shown in Figure 2.26 (but also
see Figure 2.4). For Figure 2.4, a 6 × 6 low order approximation of the
dynamical matrix was used that could easily be fitted with the elastic

74 Semiconductors for Micro and Nanosystem Technology