Page 65 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Crystal Lattice System
1
T
T
2∫
3
h
2∫
•
--- [
--- [
d
U =
d
–
V e • CR() e] R = – 1 V e • s] R 3 (2.39)
The further reduction of the number of independent elastic coefficients
now depends on the inherent symmetries of the underlying Bravais lat-
tice. Silicon has a very high level of symmetry because of its cubic struc-
ture. The unit cell is invariant to rotations of 90° about any of its
coordinate axes. Consider a rotation of the x-axis of 90° so that x → , x
y → z and z → – y . Since the energy will remain the same, we must have
that C 22 = C 33 , C 55 = C 66 and C 21 = C 31 . By a similar argument for
rotations about the other two axes, we obtain that C 11 = C 22 = C 33 ,
C 21 = C 31 = C 23 and C 44 = C 55 = C 66 . Since the other matrix
entries experience an odd sign change in the transformed coordinates, yet
symmetry of is required, they must all be equal to zero. To summarize,
C
Si and other cubic-symmetry crystals have an elasticity matrix with the
following structure (for coefficient values, consult Table 2.1) but with
only three independent values:
C C C
11 12 12
C C C
12 11 12
C 12 C 12 C 11
C = (2.40)
C
44
C 44
C 44

C can be inverted to produce with exactly the same structure and the
S
following relation between the constants:
– 1 – 1
S 44 = C 44 ( S – S ) = ( C 11 – C ) 
12
12
11
 (2.41)
( S + 2S ) = ( C 11 + 2C ) – 1 
11
12
12
The bulk modulus B and compressibility K of the cubic material is
given by

62 Semiconductors for Micro and Nanosystem Technology