Solution of the nonstationary diffusion equation for interstitial impurity atoms by the method of Green functions

TL;DR

This paper derives analytical solutions for the nonstationary diffusion of interstitial impurity atoms using Green functions, enabling better modeling and verification of impurity migration during semiconductor thermal treatments.

Contribution

It introduces new analytical solutions for impurity diffusion equations with specific boundary conditions, applicable to modeling impurity redistribution in semiconductors.

Findings

Analytical solutions match experimental boron profiles after annealing.

Solutions are useful for verifying numerical models of impurity diffusion.

The method applies to nonstationary impurity migration in semiconductors.

Abstract

On the basis of the Green function method, analytical solutions of the diffusion equation which describes nonstationary migration of nonequilibrium interstitial impurity atoms have been derived. It is supposed that the initial distribution of nonequilibrium impurity interstitials is formed due to ion implantation and, therefore, is described by the Gaussian function. The condition of the constant concentration of impurity interstitials (the Dirichlet boundary condition) or reflecting boundary condition was imposed on the surface of a semiconductor. The Dirichlet boundary condition was also enforced for the concentration of impurity interstitials in the infinity, i.e., in the bulk of a semiconductor. On the basis of the solutions derived the redistribution of ion-implanted boron in silicon substrate during low-temperature thermal treatment has been simulated. The calculated profile of boron atoms after annealing agrees well with experimental data. It means that the analytical solutions derived can be used both for verifying the numerical results and for modeling the long-range migration of nonequilibrium impurity interstitials during low-temperature thermal treatments.