10. Appendix

610 Appendix B
Note that in real situations one often has to diagonalize the Pikus-Bir
Hamiltonian by assuming that the unperturbed states include both the J 3/2
and J 1/2 states. In this case one can use the large spin-orbit coupling approximation
only for low stress to obtain the linear stress-induced splitting
2b(S11 S12)X. For high uniaxial stress the mJ ±1/2 states of both the
J 3/2 and J 1/2 states will be coupled by the stress Hamiltonian and one
obtains a nonlinear stress dependence of their energies. Students are urged to
derive the 6×6 matrices for the two cases of a [100] and a [111] uniaxial stress
and then diagonalize these matrices to obtain the stress dependent energies of
the J 3/2 and 1/2 states.
Solution to Problem 3.9
From (2.84a) and (2.94b) we can express the zone-center conduction and valence
band energies Ec and Ev as:
Ec Es Es |Vss| and Ev Ep Ep |Vxx| .
If we assume that both |Vss| and |Vxx| depend on the nearest neighbor distance
d as d 2 and d is related to the lattice constant a by d (a/4) √ 3 then
¢Vss
Vss
¢Vxx
Vxx
2 ¢a
a 2
3
¢V
V
where ¢V
V
is the volume dilation.
It can be expressed in terms of the trace of the strain tensor e : ¢V/V
Trace(e). We can express the above results as:
¢Ec
2
3
|Vss|Trace(e); and ¢Ev
2
3
|Vxx|Trace(e).
The relative deformation potential (ac av) is then given by: ¢Ec ¢Ev
(ac av)Trace(e) (2/3)[|Vss| |Vxx|]Trace(e). Hence the relative deformation
potential (ac av) is given by: (2/3)[|Vss| |Vxx|].
In the following table we compare the values of (ac av) inC,SiandGe
obtained from the tight binding parameters in Table 2.26 and compare them
with experimental values (all energies in units of eV).
|Vss| |Vxx| (ac av)theoretical (ac av)experimental
C 15.2 3.0 12.1
Si 8.13 3.17 7.53 10
Ge 6.78 2.62 6.07 12
While this approach gives the right order of magnitude and sign for the deformation
potentials, it is not accurate enough. As seen in the cases of Si and Ge
the values obtained from the model are smaller than the experimental values
by about a factor of two.