10. Appendix

Solution to Problem 7.12 657
or scattered photon is resonant with the electric-quadrupole allowed yellow
exciton. An example of the application of this resonant Raman process involving
the odd-parity ° 3 (or ° 12 ) phonon mode to map out the yellow excitonic
series in Cu2O is shown in Fig. 7.34.
Solution to Problem 7.12
For a [100] uniaxial stress of magnitude X the strain tensor is given by (as
shown in Problem 3.4):
⎛
eij ⎝ S11
⎞
0 0
0 S12 0 ⎠ X
0 0 S12
In a diamond-type semiconductor the zone-center optical phonons are threefold
degenerate at X 0. Under the [100] uniaxial stress the crystal symmetry
is lowered to tetragonal so we expect the phonon to be split into a singlet and
doublet as discussed also in Problem 6.13. To obtain the magnitude of the
stress-induced shift in the phonon frequencies we will first substitute the elements
of the above strain tensor into the determinant of the secular equation
given in Problem 6.23. We obtain:
(pS112qS12)X Ï 0 0
0 [pS12q(S11S12)]X Ï 0
0 0 [ ( pS12
... S12)]X
The three solutions of the corresponding secular equations consist of a nondegenerate
(singlet) solution:
Ï1 ˆ 2 1 ˆ2 0 X(pS11 2qS12)
and a doubly degenerate (doublet) solution:
Ï2 ˆ 2 2 ˆ2 0 X[pS12 q(S11 S12)].
Under the small strain (and therefore low stress) condition we have assumed,
we can approximate Ï ˆ 2 ˆ 2 0 by (ˆ ˆ0)(ˆ ˆ0) ∼ 2ˆ0(ˆ ˆ0). The
shifts of the singlet and doublet optical phonons are thus given, respectively,
by:
¢ˆs X
[pS11 2qS12]
2ˆ0
and
¢ˆd X
[pS12 q(S11 S12)]
2ˆ0
The average of the three phonon modes: ¢ˆH (ˆs 2ˆd)/3 shifts with stress
as: ¢ˆH (X/6ˆ0)(p 2q)(S11 2S12) while the splitting between the two