Asymptotic Behavior of Reaction-Diffusion Pdes in Dopant Diffusion

The ability to modify the properties of a semiconductor through the addition of controlled amounts of impurity atoms is an important aspect of silicon device and integrated circuits manufacture. The dominant device used in the semiconductor industry today, the silicon-based metal oxide semiconductor transistor, requires spatially inhomogeneous dopant distributions. The precise shape of the dopant profile of the source, drain, and channel regions play a critical role in determining device performance, especially with scaling to smaller transistors (Packan, 2000). Improved predictive modeling of the mechanisms of dopant diffusion in silicon can be used to control the width and the shape of the diffusion band and therefore improve the fabrication of microelectronic devices (ITRS, 2004).

Atomic diffusion in the silicon crystalline lattice is mediated by intrinsic point defects and can take place through multiple mechanisms. The most important methods include the interstitial mechanism, the vacancy mechanism, the kick-out mechanism, and the Frank-Turnbull mechanism. The relative importance of these mechanisms differs for different dopants. Methods have been developed to determine the predominant mechanism experimentally by observing the effects of ambient conditions on migration frequency k_gen and migration length λ (Cowern et al, 1991). These parameters have unique responses to changes in the concentration of interstitials and vacancies under different mechanisms.

In point defect-mediated diffusion, dopant can only diffuse while in its less probable mobile form. Diffusion proceeds by a series of migration events of frequency k_gen as the dopant switches between its immobile subsitutional form and its fast-diffusing form through interactions with point defects. Because of the continual conversion, such dopant diffusion can have qualitatively different dopant profiles than that of diffusion via a single-jump process. A mathematical description taking into account of the dynamics of intermediate-species creation, motion, and annihilation has been constructed (Cowern et al, 1990). The reaction-diffusion PDEs include two kinetic parameters describing motion and exchange of the intermediate species instead of a single composite diffusion coefficient. It was observed that for sufficiently short times, diffusion profiles decay with ?exponential tails'; while for long-time diffusion, the profiles approach Gaussian or error-function forms for initial dopant profiles that are delta or step functions, respectively.

Methods have been developed for estimating the kinetic parameters of point defect-mediated diffusion from analytical expressions for the short-time and long-time decay of the initial concentration profile (e.g., Cowern et al, 1991, Vaidyanathan et al, 2006). These methods provide particularly simple analytical means for obtaining parameters connected to diffusion length and defect formation. This study thoroughly analyzes the long-time behavior where point defect-mediated dopant diffusion has similar dynamics as single-jump diffusion with an effective diffusion coefficient. More specifically, this contribution (i) derives the exact analytical expression for the effective diffusion coefficient that characterizes the long-time decay of the initial concentration profile, (ii) compares it to the expression currently used in the literature, and (iii) analyzes the approximation error in terms of the time scales of point defect-mediated diffusion. A consequence of these results is a more accurate expression for estimating kinetic parameters in point defect-mediated diffusion and specific recommendations on the estimation to avoid creating bias in the parameter estimates. A summary of some of these results are below.

The exact expression for the effective diffusion coefficient is determined by applying the method of moments to the reaction-diffusion PDEs. Comparing the zero and second moments of the reaction-diffusion PDEs with those of the effective diffusion PDE gives

D_eff=(D_mk_gen)[1-(1-e^{-(k_gen+ k_ann)t})/(( k_gen+ k_ann)t)]/(k_ann+ k_gen) (1)

which approaches

D_eff=(D_mk_gen)/(k_ann+ k_gen)(2)

for long time (e.g.t >10/( k_gen+ k_ann)) Comparing this to the expression currently used in the literature (Cowern et al, 1991, Vaidyanathan et al, 2006)

D_eff= k_gen λ²where λ= √ D_m/k_ann (3)

indicates that the literature expression (3) is only approximate. For long time, equation (3) is an accurate approximation for the exact expression (2) provided that k_genann, which was the case for the specific case of boron diffusion through silicon considered in the above references. Comparing equations (2) and (3) indicates that the error in using (3) is <1% at long time provided that the kinetic constant for the generation of mobile species is <1% of the kinetic constant for the annihilation of mobile species. Note, however, that the time-dependent term in equation (1) only decays with respect to time as 1/t. Equation (1) indicates how much time should pass before equations (2) or (3) can be applied, to avoid biasing the parameter estimates.

The term ( k_gen+ k_ann)t in equation (1) is dimensionless. Hence τ=1/( k_gen+ k_ann) is the key time scale for understanding the dynamic approach of the reaction-diffusion process to qualitative behavior like a single-jump diffusion process. Plots of the dopant concentration profile for the reaction-diffusion PDEs side-by-side with that of the effective diffusion PDE as a function of the time scales of the system gives further insights into the spatial dynamics of the reaction-diffusion mechanism.

References:

P. A. Packan; MRS Bulletin 2000, 25, 18-21

N.E.B. Cowern; K.T.F. Janssen; G.F.A. van de Walle; D.J. Gravesteijn; Phys. Rev. Lett. 1990, 65, 2434-2437

N.E.B. Cowern; G.F.A. van de Walle; D.J. Gravesteijn; C.J.Vriezema; Phys. Rev. Lett. 1991, 67, 212-215

ITRS, International Technology Roadmap for Semiconductors, 2004, http://www.itrs.net/Links/2004Update/2004Update.htm

R. Vaidyanathan; M.Y.L. Jung; R.D. Braatz; E.G. Seebauer; AICHE J. 2006, 52, 366-370