Plateau and kink in P profiles diffused into Si: A result of strong bimolecular recombination?

Abstract

A simplified five-species nonequilibrium kinetic model for phosphorus diffusion in silicon is presented. The resulting system of evolution equations is of a simple reaction-diffusion form with constant diffusivities. Using first-order thermodynamic estimates for reaction rates, the phosphorus profile after a 10 min predeposition shows the expected tail. However, only when the bimolecular generation-recombination rate is significantly increased does a kink-plateau result. This suggests that recombination may be the dominant factor in producing the known nonlinearity.

Partly dissipative reaction-diffusion systems and a model of phosphorus diffusion in silicon

“…and the initial data satisfy these boundary conditions, then the solutions are bounded in where Ω is an unbounded domain such as a half-space as considered in [9,10]. The proofs can be modifed using the interior estimate ideas in [4].…”

“…For more information on the physics, we refer the interested reader to [9] and the references therein. These reactions lead to the reaction-diffusion system (1.1)…”

“…The substitutional phosphorus atom is considered immobile; therefore the equation for P in (1.1) contains no diffusion and is thus an ordinary differential equation satisfied at each x in the domain Ω where the reactions take place. Richardson and Mulvaney [9] take Ω to be the interval (0, ∞) and impose boundary conditions at x = 0: P = C * , I = I eq , V = V eq , and E x = F x = 0. Strictly speaking, it makes no sense to impose a boundary condition on P since there is no diffusion present.…”

“…The following reactions are considered by Richardson and Mulvaney [9] as a model of phosphorus diffusion in silicon:…”

On the question of global existence for reaction-diffusion systems with mixed boundary conditions

“…In [13,5] the diffusion coefficient d5 is taken to be zero, reflecting the assumption that the only significant transport mechanism for phosphorus is the combination with the vacancies and interstitials; however, for our purposes here, we will assume d5> 0 in analogy with the idea of artificial viscosity. Of physical interest for this model (cf.…”