Concentration-Dependence of Self-Interstitial and Boron Diffusion in Silicon
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1070-E06-08
Concentration-Dependence of Self-Interstitial and Boron Diffusion in Silicon Wolfgang Windl Materials Science and Engineering, The Ohio State University, Columbus, OH, 43210-1178 ABSTRACT In this paper, we discuss the accuracy of ab-initio calculations for self-interstitial and boron diffusion in silicon in light of recent experimental data by de Salvador et al. and Bracht et al. Mapping the experimental data onto the activation energy vs. Fermi level representation commonly used to display ab-initio results, we show that the experimental results are consistent with each other. While the theoretical LDA value for the boron activation energy as a function of the Fermi level agrees well with experiment, we find for the self-interstitial in line with other calculations an underestimation of the experimental values, despite using total-energy corrections. INTRODUCTION The diffusion mechanism of boron has not been understood for a very long time despite considerable research interest [1]. It is widely accepted by now that boron diffuses nearly exclusively with the help of Si self-interstitials (Is) [2], i.e., the mobile entity is thought to be a B atom paired with an I. As concerning the diffusion mechanism, first ab-initio modeling for neutral systems using the drag method had suggested that a kick-out mechanism with long-range low-barrier interstitial migration would be the dominant mechanism [3,4], in contrast to previous perception. Later, we used the nudged elastic band method (NEBM) [5] implemented into VASP [6] to reexamine the minimum-energy barrier diffusion path for I-assisted, charge-state dependent B diffusion within both LDA and GGA, using 64-atom supercells [7] with new results (see below). Although this work has been performed several years ago, very recent experiments by De Salvador et al. [8] and Bracht et al. [9] have re-examined the charge states of both the boron atom and interstitials involved, which helps to re-evaluate the simulation results. DIFFUSION EQUATIONS Especially interesting is the simultaneous determination of the interaction rate g between B and I and the mean free path λ in [8], from which the diffusivity D can be calculated, D = gλ2 [10,11]. Under thermodynamic equilibrium conditions in a homogenous material, the impurity transport can be described by [10] ∂CBI ∂t = DBI ∇ 2CBI − rCBI + gCBs ,
∂(CBs + CBI ) ∂t = DBI ∇ 2CBI ,
(1)
where CBs and CBI are the concentrations of the ground state, substitutional boron Bs, and the mobile boron interstitial pair, BI. Since for B diffusion in Si the Frank-Turnbull mechanism (Bs BI + V) is energetically much less favorable than the kick-out mechanism (Bs + I BI), the generation rate g is given by the forward reaction rate [12] of the kick-out mechanism times selfinterstitial concentration,
→
→
g = 4πacBI DI CI ,
(3)
where acBI is the B-I capture radius. g is thus directly proportional to the self-interstitial transport coefficient, DICI. λ = DBI r [10]. With β = 1 / (k BT ) , The mean free path can be calculated from
(
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