10. Appendix

Solution to Problem 3.16 613
(1) All the components containing two identical indices, such as (em)xxy and
(em)yyy, must be zero by applying the C2 rotations.
(2) This leaves as the only non-zero elements to be those where all the three
indices are different, such as (em)xyz (or (em)14 in the contracted notation).
By applying C3 operations then all the non-zero elements can be shown to
be identical.
The form of the electromechanical tensor in the zincblende crystal can be expressed
as a 3 × 6 matrix of the form:
⎛
⎝ 0 0 0 (em)14 0 0
0 0 0 0 (em)14 0
0 0 0 0 0 (em)14
⎞
⎠
(b) Repeating the calculation by applying the symmetry operations of the
wurtzite structure (assuming that the c-axis of wurtzite is chosen as the z-axis)
we can show that the components (em)ijk containing x and y as indices satisfy
the same constraint as in the zincblende crystal. This means: (em)xxx and
(em)yyy are both zero but not (em)zzz ( (em)33). Similarly (em)xxy and (em)yyx
are both zero but not (em)xxz and (em)yyz (these two components are both
equal to (em)15) while (em)zxx and (em)zyy are also not zero (both equal to
(em)31). However, there is now a new symmetry operation involving reflection
onto the zy plane. This symmetry operation will change the sign of x while
leaving those of y and z unchanged. As a result of this symmetry operation,
the components (em)xyz etc., are now all zero. Thus (em) in wurtzite crystals
can be expressed as:
⎛
⎝ 0 0 0 0 (em)15
⎞
0
0 0 0 (em)15 0 0⎠
(em)31 (em)31 (em)33 0 0 0
Solution to Problem 3.16
Using the result of Prob. 3.15 we will assume that the non-zero and linearly
independent elements of the electromechanical tensor em in wurtzite crystals
have the contracted form:
⎛
⎞
em
⎝ 0 0 0 0 e15 0
⎠
0 0 0 e15 0 0
e31 e31 e33 0 0 0
In the contracted notation the strain tensor e corresponding to a phonon with
displacement vector u and wavevector q is given by: