Simulation of H Behavior in p-GaN(Mg) at Elevated Temperatures
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energy, and Ev is the energy of the valence-band edge. The final term on the right describes driven migration due to spatial variation of the Fermi level. We assume rapid local equilibration among the H states, which is believed appropriate for the temperatures >400ºC of concern herein. One then has [6]
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H + G + \0 − E f = exp H (2) 0 kT H H0 G 0\ − − E f = exp H (3) kT H − where the quantities GH are bandgap levels associated with the electronic transitions and are defined to include degeneracy. Populations relating to the Mg acceptor are given by N GaN MgH G MgH (4) = exp Mg− H + kT Mg0 G0\ − − E f Mg = exp (5) kT Mg − where NGaN is the formula-unit density of GaN, MgH is the neutral complex, and GMgH is the nonconfigurational part of the change in Gibbs free energy arising from dissociation of the MgH complex. The hole concentration is h + = E v N (E) F (E)dE (6) h val −∞ where Nval is the density of states in the valence band and Fh is the Fermi distribution function for holes. The electron concentration is given by an analogous integral of the conduction band. The description is completed by imposing charge neutrality. Equilibrium with external H2 gas is described by the solubility equation G0 eq 1/ 2 H 0 = N P* sol (7) exp GaN gas − kT * where Pgas is the fugacity of the gas. This relationship is expressed in terms of H0 to avoid a dependence on Fermi level. The system of equations is solved numerically. Parameters in Eqs. (1)-(7) are evaluated using zero-temperature results from density functional theory for wurtzite GaN. The theoretical procedure and some results have been reported previously [4]. Using the H atom in vacuum as the reference state, we find an energy of +0.34 eV for interstitial H0, (-2.67 eV + Ef - Ev) for H+, and (+1.52 eV - Ef + Ev) for H-. The formation of the neutral Mg-H complex from Mg- and H+ has a binding energy of 0.70 eV. The ionization level for the Mg acceptor is taken from experimental studies to be Ev+0.16 eV [7]. The diffusion activation energies from density functional theory are 0.5 eV for H0, 0.7 eV for H+, and 1.6 eV for H-; the
∫
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diffusion prefactors are set equal to 0.001 cm2/s, which is representative for interstitials. The densities of states for the valence and conduction bands are evaluated from density functional theory. The quantities G in Eqs. (1)-(7) differ between 0 K, where the results of density functional theory apply directly, and the elevated temperatures of interest. Causes include the temperature-dependent thermodynamics of H2 gas and the enthalpy and entropy arising from H vibrations in the semiconductor. We treat these effects for deuterium (D), the isotope used in our experiments. The Gibbs free energy of D2 gas is taken from the literature [8]. Vibrational effects are included for the dominant D states in p-type GaN(Mg), D+ and the
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