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

14
2.3
(2)
Xijk and Symmetry
In general Xm(w = 2wo) has 18 unique elements. Group theory can be used to reduce
this complexity further by determining relationships among the nonzero elements.
The present discussion focuses on the nonlinear susceptibility elements of Ga:\". but
the general application of group theory to determine relationships among the elements
of a general response function follows similarly.
The strength of group theory for determination of the relationships among the
susceptibility elements arises from Neumann's principle. Neumann's principle states
that any macroscopic physical property has at least the symmetry of the point group
of the space group of that medium [36]. By applying the symmetry operators of the
appropriate point group to X~~k,
which transforms as a three component Cartesian
tensor. relationships among the symmetry elements are determined.
GaN has C 6v or 6mm symmetry, which is the symmetry of a regular hexagon. The
generators of this group are
(2.9)
Definitions of these symbols are found in reference [36].
Note, C 6v contains only
symmetry operators in a plane. Therefore for GaN [0001], which is the cut of our
samples, the surface and interface have the same nonzero nonlinear susceptibility
elements as the GaN bulk. Because far fewer atoms participate in a nonlinear process
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