120
5 Mechanical Properties
Table 5.2 Elastic constants (in GPa) of some cubic semiconductors at room temperature. I K refers to the Keating
criterion (5.59)
Material
C 11
C 12
C 44
I K
C
1076.4
125.2
577.4
1.005
Si
165.8
63.9
79.6
1.004
Ge
128.5
48.3
66.8
1.08
BN
820
190
480
1.11
GaAs
119
53.4
59.6
1.12
InAs
83.3
45.3
39.6
1.22
AlAs
120.5
46.86
59.4
1.03
ZnS
104.6
65.3
46.3
1.33
MgO
297
156
95.3
0.80
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
S 11 S 12 S 12 0
0
0
S 12 S 11 S 12 0
0
0
S 12 S 12 S 11 0
0
0
0
0
0 S 44 0
0
0
0
0
0 S 44 0
0
0
0
0
0 S 44
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,
(5.56)
with the stiffness coefficients in this notation given by
S 11 =
C 11 + C 12
(C 11 − C 12 ) (C 11 + 2C 12 )
(5.57a)
S 12 =
C 12
−C
2
11 − C 11 C 12 + 2C
2
12
(5.57b)
S 44 =
1
C 44
.
(5.57c)
We emphasize that in this convention (also called the engineering convention), e.g. e 1 = xx and
e 4 = 2 yz . There is also another convention (the physical convention) without this factor of two; in
this case the matrix in (5.55) contains the elements 2C 44 . We introduce
C 0 = 2 C 44 + C 12 − C 11 ,
(5.58)
and note that C 0 = 0 for an isotropic material. The relation
I K =
2 C 44 (C 11 + C 12 )
(C 11 − C 12 ) (C 11 + 3C 12 )
= 1
(5.59)
known as the Keating criterion [396, 397], stems from the consideration of bending and stretching
of the tetrahedral bonds in the valence force field (VFF) model. It is closely fulfilled (Table 5.2)
for many tetrahedrally bonded semiconductors, in particular for the covalent ones. For MgO, the
Keating criterion is not fulfilled because it has (six-fold coordinated) rocksalt structure and is thus not
tetrahedrally bonded.
The Young’s modulus Y ,
σ nn = Y (n) ) nn ,
(5.60)