Page 83 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Crystal Lattice System
C:ε
C ε + C ( ε + ε ) 2C ε 2C ε
11 11 12 22 33 44 12 44 31
= 2C ε C ε + C ( ε + ε ) 2C ε
11 22
11
33
12
44 12
44 23
2C ε 2C ε C ε + C ( ε + ε )
44 31 44 23 11 33 12 11 22
(2.75)
We take the divergence of ∇• ( C:ε) , to obtain a first version of the wave
equation
2
(
∂ u 1 ∂ε 11 ∂ε + ε )  ∂ε 12 ∂ε 31
22
33
ρ---------- = C ---------- + C ---------------------------- + 2C 44  ---------- + ---------- (2.76a)
∂x 
12
11
2
∂t ∂x 1 ∂x 1 ∂x 2 3
2
(
∂ u 2 ∂ε 22 ∂ε + ε )  ∂ε 12 ∂ε 23
11
33
ρ---------- = C ---------- + C ---------------------------- + 2C 44  ---------- + ---------- (2.76b)
∂x 
11
12
2
∂t ∂x 2 ∂x 2 ∂x 1 3
2
(
∂ u 3 ∂ε 33 ∂ε + ε )  ∂ε 31 ∂ε 23
22
11
ρ---------- = C ---------- + C ---------------------------- + 2C 44  ---------- + ---------- (2.76c)
∂x 
11
12
2
∂t ∂x 3 ∂x 3 ∂x 1 2
With the definition of strain (2.31) we obtain the final form of the bulk
lattice wave equation system for a cubic-symmetry crystal
2 2 2 2 2 2
∂ u 1 ∂ u 1  ∂ u 2 ∂ u   ∂ u 1 ∂ u 
1
3
44 
ρ---------- = C ---------- + ( C 12 + C ) ----------------- + ----------------- + C  ---------- + ----------
11
44
2
2
2
2
∂t ∂x 1  ∂x ∂x 2 ∂x ∂x 3  ∂x 2 ∂x 
1
1
3
(2.77a)
2 2 2 2 2 2
∂ u 2 ∂ u 2  ∂ u 1 ∂ u   ∂ u 2 ∂ u 
2
3
44 
ρ---------- = C ---------- + ( C 12 + C ) ----------------- + ----------------- + C  ---------- + ----------
44
11
2
2
2
2
∂t ∂x  ∂x ∂x 1 ∂x ∂x 3  ∂x ∂x 
2
2
2 1 3
(2.77b)
2 2 2 2 2 2
∂ u 3 ∂ u 3  ∂ u 1 ∂ u   ∂ u 3 ∂ u 
3
2
44 
ρ---------- = C ---------- + ( C 12 + C ) ----------------- + ----------------- + C  ---------- + ----------
11
44
2
2
2
2
∂t ∂x  ∂x ∂x 1 ∂x ∂x 2  ∂x ∂x 
3
3
3 1 2
(2.77c)
To find the principal bulk modes, we assume a harmonic plane wave
solution of the form
80 Semiconductors for Micro and Nanosystem Technology