William Angerer - Department of Physics and Astronomy - University ...

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Let us examine this statement from a different view point. Since all of the symmetry
operations of the crystal P(Xd commute with H. the operation P(Xd on 1/Jnk produces
another function with energy En. This function must be a linear combination of the
degenerate functions l/Jnj,
I
P(XdWnk = L r n(}(djk1fJnj' (5.3)
j=l
By applying equation (5.3) to all symmetry operations Xl' a set of matrices, r n, are
generated that form an irreducible representation of H. Thus the eigenstateslpnj of En
transform as r n' This leads naturally to selection rules and splitting of states under
external perturbations. vVe will be concerned primarily with the former, specifically
to determine contributions from various valence and conduction band states to the
resonant components of x~JL(2w = wo). Tables 5.4 and 5.5 are the character table and
multiplication table for C 6v symmetry, respectively.
5.4.1 Second-Order Nonlinear Optical Processes at k = 0
\Ve will now explicitly calculate nonzero contributions to x~JL(w
= 2wo) at k=O.
In general, the nonlinear susceptibility depends on all possible direct three photon
processes in GaN. As stated in section 5.2.1, it is sufficient to use the point group
representations of the band symmetries without spin; since spin orbit splitting is so
small in GaN, it does not alter the symmetries of the bands significantly.
For spectroscopic measurements of GaN with a two photon energy near the band
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