10. Appendix

u3. The above equation can be reduced to:
Solution to Problem 3.8 (a, c and d) 607
ρ 2u1 X11
t2 x1
X12
x2
X13
x3
Using the results of part (b) the stress tensor can be expressed in terms of the
strain tensor eij (or ei in the contracted notation) and the stiffness tensor Cijkl
(or as Ckl in the contracted notation) as:
⎛
⎞
C11e1 C12e2 C13e3
⎜ C12e1 C11e2 C13e3 ⎟
⎜
⎟
⎜ C13e1 C13e2 C33e3 ⎟
Xij ⎜
⎟
⎜ C44e4 ⎟
⎝
⎠
C44e5
C11C12
2
e6
Substituting these stress tensor elements into the equation of motion we obtain
the following equation:
ρ 2u1
C11
t2 2u1 x2
1
C11 C12
2
2u1 x2
C44
2
2u1 x2
3
C11 C12
2
(C13 C44) 2 u3
x1x3
with two similar differential equations for u2 and u3.
Solution to Problem 3.8 (a, c and d)
2 u2
x1x2
(a) In Prob. 3.4 (b) it was shown that a tensile uniaxial stress X applied along
the [100] axis of a zincblende crystal will induce a strain tensor of the form:
⎛
˜e ⎝ S11X
⎞
0 0
0 S12X 0 ⎠ .
0 0 S12X
This matrix can be decomposed into its irreducible components consisting of
two matrices:
a diagonal matrix (belonging to the °1 irreducible representation) of the form:
⎛ ⎞
1 0 0
˜ehydrostatic ⎝ 0 1 0⎠
(S11 2S12)(X/3)
0 0 1
which represents a hydrostatic strain; and the traceless matrix (belonging to
the °3 irreducible representation):
⎛
⎞
2 0 0
˜eShear ⎝ 0 1 0 ⎠ (S11 S12)(X/3) .
0 0 1
This decomposition allows us to simplify the evaluation of the strain Hamltonian
HPB in (3.23). [note that there is a difference of a factor of 3 in the second