Page 84 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Vibrating Uniform Lattice
kx
Af τ()
t –
, with τ =
u =
----------
c ⋅ (2.78)
k
and unit vector pointing in the direction of wave propagation, so that
(2.77a) becomes an eigensystem
2
( Γ – ρc ) A⋅ = 0 (2.79)
Γ
with the tensor defined for cubic materials such as silicon as
2 2 2
C k + C ( k + k ) ( C 12 + C )k k ( C 12 + C )k k
44
44
1 3
3
2
11 1
44
1 2
2
2
2
Γ = ( C 12 + C )k k C k + C ( k + k ) ( C 12 + C )k k
44
1 2
11 2
1
2 3
44
3
44
2
2
2
( C 12 + C )k k ( C 12 + C )k k C k + C ( k + k )
44
11 3
2 3
1
2
44
44
1 3
(2.80)
Equation (2.79) is also known as the Christoffel equation, and finding its
eigenvalues using material parameters for silicon, over a range of angles,
results in the slowness plots of Figure 2.27 [2.11]. Along the 100[ ] fam-
ily of silicon crystal axes, the principal wave velocities are
⁄
c = C ρ ⁄ = 8433ms , (2.81a)
1 11
⁄
c = c = C ρ ⁄ = 5845ms (2.81b)
2 3 44
Along the 110[ ] direction, we obtain the following results
⁄
c = C 44 ρ ⁄ = 5845ms (2.82a)
1
⁄
⁄
c = ( C – C ) 2ρ = 4674ms (2.82b)
2 11 12
⁄
⁄
c = ( C + C + 2C ) 2ρ = 9134ms (2.82c)
3 11 12 44
2.4.2 Phonons, Specific Heat, Thermal Expansion
The behavior of certain material properties, such as the lattice heat capac-
ity at low and medium temperatures, can only be explained if we assume
that the acoustic energy of lattice vibrations is quantized, a result indi-
Semiconductors for Micro and Nanosystem Technology 81