Modeling Fermi Level Effects in Atomistic Simulations
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Modeling Fermi Level Effects in Atomistic Simulations Zudian Qin and Scott T. Dunham Department of Electrical Engineering, University of Washington, Seattle, WA 98195
ABSTRACT In this work, variations in electron potential are incorporated into a Kinetic Lattice Monte Carlo (KLMC) simulator and applied to dopant diffusion in silicon. To account for the effect of dopants, the charge redistribution induced by an external point charge immersed in an electron (hole) sea is solved numerically using the quantum perturbation method. The local carrier concentrations are then determined by summing contributions from all ionized dopant atoms and charged point defects, from which the Fermi level of the system is derived by the Boltzmann equation. KLMC simulations with incorporated Fermi level effects are demonstrated for charged point defect concentration as a function of Fermi level, coupled diffusion phenomenon and field effect on doping fluctuations. INTRODUCTION Kinetic Lattice Monte Carlo (KLMC) simulations study diffusion/clustering of defects in silicon at a microscopic level [1,2]. Simulations are performed on a silicon (diamond) lattice structure with impurities and point defects mapped to lattice sites. The system evolves through transitions from one atomic configuration to the next, by virtue of point defect migration/reaction. The rates of these transitions are determined by the migration barriers combined with changes in system energy associated with transitions: − Em Ei − E f exp 2k T k BT B
ν = ν 0 exp
,
(1)
where Em is the unbiased migration barrier, Ei and E f are the system energies before and after the transition, and T is the system temperature. The system energies are calculated based on the atomic scale arrangement of impurities/defects, with parameters from ab-initio calculations and/or experimental observations. At each simulation step, one transition is chosen from the possible set based on the relative rates, and the system time is advanced by the inverse of the sum of the rates [1]. By only considering transitions (and not lattice vibrations) associated with defects and impurities present in the system, the KLMC method overcomes the time-scale limits associated with molecular dynamics to consider macroscopic systems and processing time scales. The Fermi level is of critical importance in modeling dopant diffusion in silicon, but typically has not been included in atomistic simulations. Its importance manifests in two ways [3]: (i) charged point defect concentrations vary spatially with potential, and (ii) ionized dopant atoms as well as charged point defects experience electric fields due to spatial variations in the potential. Both effects are properly modeled in continuous simulations through the quantity of local carrier concentration, which is normally calculated from dopant profiles using the charge C3.8.1
neutrality assumption. Unfortunately, dopant concentration is no longer a valid concept at the atomic scale, where dopant atoms are considered to be discrete
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