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

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along direction k; and the dots represent unique permutations of the states. The
contribution of three photon processes to x~]L(w
= 2wo) depends of several factors.
First, resonant enhancement of X~~k(w = 2wo) occurs when w :::::: Wng (two photon
resonance) or Wo :::::: Wn'g (one photon resonance). Second. contributions to '(~~k('''': =
2wo) are constrained by the matrix: elements in the numerator of equation (6.36). In
other words, the matrLx elements determine whether a process is allowed and influence
the magnitude of allowed processes. Third, the occupancy of the states requires that
the first virtual transition in the three photon process originates from an occupied
state and connects to an unoccupied state.
These constraints limit the type of process responsible for a two photon resonance
with energy 2.80 eV in X~]k(w = 2wo). Because GaN is a direct band gap semiconductor
\vith a band gap energy of 3.4 eV, the two photon resonance cannot result from
virtual transitions between valence and conduction band states only. In addition.
the requirement that the process begin with a virtual transition between an occupied
and an unoccupied state precludes any process that occurs only in the valence or
conduction bands. Thus, it is unlikely that the 2.80 eV feature results purely from
the band structure of GaN. On the other hand, it is well known that Ga~ contains
a variety of defect states [63]. Thus, the spectral feature could result from a three
photon process that involves both a defect state and the GaN bands. For example.
a bulk defect state (or defect band) with energy 2.80 eV above the valence band (a
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