Theoretical Study of Electron Initiated Impact Ionization Rate in Bulk GaN using a Wave Vector Dependent Numerical Trans
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MODEL DESCRIPTION
The details of our Monte Carlo simulator have been reported previously [8], therefore only its salient features are reviewed here. The total electron initiated impact ionization rate, quantum yield and electron energy distribution are determined using an ensemble Monte Carlo simulation, which includes the full details of the first four conduction bands of zincblende GaN. The calculation of the electronic band structure was performed within the framework of the empirical pseudopotential method using an expansion over a closed, fixed set of 113 G vectors, which produces a reasonably accurate band structure over the entire Brillouin zone. The energy eigenvalues, their gradients with respect to the k-vector, and their second derivatives are calculated at regularly spaced points within a k-space mesh and are used to calculate the electron energy during its free flight [9]. The band structure used in the calculations is shown in Fig. 1.
15
10
I2l 02
W
-5
r
X
W
L
F
K
Fig. I Bandstructure of zincblende GaN
The electron-phonon scattering mechanisms included in the simulation are polar optical, acoustic and intervalley scatterings. The rates are determined for each of these mechanisms at low energies. Polar optical scattering is included only within the central valley. The parameters used in our calculations have been reported previously [8]. At higher electron energies, deformation potential scattering is assumed to be the dominant scattering mechanism. In this regime the scattering rate has been obtained by integrating over the pseudopotential calculated final density of states including collision broadening [10]. The deformation potential is assumed to be constant and the high energy scattering rate is matched to the low energy scattering rate at a selected energy. Another scattering mechanism included in the simulator is ionized impurity scattering, calculated using the approach of Ruch and Fawcett [11]. Their approach takes into account the effect of nonparabolicity, Bloch states and a screened Coulomb potential. In addition to the above mentioned scattering mechanisms, impact ionization has to be incorporated in the simulations in the high field regime as well. The k-vector dependent
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impact ionization transition rate has been calculated using Fermi's golden rule, which applies as W~ii(n l, k0_7 E 'JA 3 226t
h(27)6
jd
ld3k'M2
2(Ef-E)
(1)
n I,2,n2
where k, ,nj are the wave vector and band index of the initiating electron, respectively, k2, n2 , k1', ni', k 2', n2 are the wave vectors and band indices of the particles after the impact ionization event (a hole and two electrons, respectively). V is the crystal volume, Ef and E, represent the final and initial energies of the particles. M stands for the matrix element of the interaction and can be expressed in terms of the direct MD and exchange ME elements as IM12 = 2IMDI2 +2IM+ I 2- (M7dtE + MDMý). (2) The matrix element is calculated based on the plane-wave expansion of the wave functions using the approach described in [12].
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