Interband Radiative Recombination Calculations in Ternary Nitride Solid Solutions
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solid solutions Ga.,All.,N , In.Alli.N and Ga Ini_,N. All our calculations were based on experimental data on energy band structures and optical absorption spectra of the nitride materials. We estimated the matrix elements for the direct optical transitions between conductivity and valence bands of these semiconductors using the experimental photon energy dependence of the fundamental absorption. In our calculations we assumed that basic parameters of a solid solution (the matrix element, effective masses of carriers and so on) could been obtained by linear interpolation between their values in the alloy components. Taking the temperature dependence of the energy gap in the form proposed by Varshni, we performed the calculations of the radiative recombination rate R in a wide interval of temperature and carrier concentrations and in the full range of the alloy composition. ENERGY SPECTRA OF THE NITRIDE SEMICONDUCTORS We consider a common and popular hexagonal phase of the nitrides. All of them belong to the crystal class C6 ,. Their conductivity bands are non-degenerate, and their electron states originate from atomic s-functions. In the r point of the Brilluine zone they transform according to Fi, the unite representation of C6,. The valence band is complicated and consists of two branches. One of them transforms according to i1, whereas the other is degenerate and forms a two-dimensional representation Fr. If spin-orbit interaction is taken into account, F6 further splits into two bands, [7 and F9. However, the latter splitting 851 Mat. Res. Soc. Symp. Proc. Vol. 482 © 1998 Materials Research Society
manifests itself only along k,2 and k., whereas it equals zero in the very r point and along k,, z being the direction of the hexagonal axis c which usually coincides with the normal to the film. We neglect this splitting and consider 16 as a degenerate band. According to the results of Refs. [1, 2, 3, 4], the order of levels in the valence band differs for various nitrides: in GaN and InN, F6 branch lies above I1-, whereas in AIN it lies below F,. The symmetry of the electron wave functions in different bands and band branches leads to the following selection rules for the radiative transitions: for a transition from Fr to Fr, only z component of the transition matrix element differs from zero, and correspondingly, an emitted photon is polarized along z axis. On the contrary, for a transition from F1 to Ir, z component of the transition matrix element equals zero, and the polarization direction of an emitted photon is perpendicular to z axis. RADIATIVE RECOMBINATION RATE: THE METHOD OF CALCULATIONS As optical absorption of the nitrides has been measured only in a small vicinity of the band edge, a calculation method is needed that would allow to express the recombination rate through the material parameters. A straightforward quantum-mechanical calculation leads to the following expression for the spontaneous radiative recombination coefficient:
B=
3 2 M 2 [27rh ] /
fi:e 2
B
2
I kBT I
Eg(T) 1 + 3kBT
1
(fr
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