Role of Defects in the Dopant Diffusion in Si

Peter Pichler1    Technology Simulation, Fraunhofer Institute for Integrated Systems and Device Technology IISB, Erlangen, Germany
University of Erlangen-Nuremberg, Erlangen, Germany
1 Corresponding author: email address: peter.pichler@iisb.fraunhofer.de

1 Introduction

In semiconductors, dopants reside predominantly on substitutional sites where they either provide free electrons (donors) or bind them (acceptors) to complete the valence-bond structure. The most successful concepts developed to describe dopant diffusion assume that the substitutional dopants form mobile pairs with the intrinsic point defects, i.e., vacancies and self-interstitials. These models allow to explain a variety of phenomena like different profile shapes observed for short and long diffusion times; the dependence of the profile form on the concentration of dopants; enhanced dopant diffusion below regions with high dopant concentration; nonequilibrium effects caused by chemical reactions like oxidation or nitridation at surfaces; immobilization and reduced electrical activation of dopants via the formation of impurity phases, small clusters and complexes with other impurities; and, finally, the pile-up of dopants at interfaces and surfaces. Due to the limited space, citation can be only exemplary. For a more extensive account of diffusion phenomena, the interested reader is referred to specific reviews in this field (Fahey et al., 1989; Pichler, 2004).

This chapter is structured as follows: In the first section, a methodology is explained which is commonly used in continuum simulation to describe the diffusion of dopants, intrinsic point defects, and other impurities as well as their interactions via coupled systems of continuity equations. In the following section, the diffusion of dopants via intrinsic point defects is discussed. This includes a review of the basic diffusion mechanisms, a derivation of the diffusion equations on the basis that dopant diffusion proceeds via a pair diffusion mechanism, and a discussion of the system behavior in terms of diffusion phenomena and diffusion profiles to be expected. The current state of knowledge about the actual diffusion mechanisms of dopants in silicon is summarized thereafter. In the subsequent section, processes are outlined that perturb the intrinsic point defects and lead to a variety of diffusion phenomena. Thereafter, the formation of impurity phases, clusters and complexes as well as associated effects on the intrinsic point defects are discussed. The chapter ends with an outline of interface segregation, a phenomenon that may lead to the loss of a substantial fraction of the dopants in a sample.

2 The Framework of Diffusion–Reaction Equations

While pairing and dissolution reactions as well as migration of all kinds of point defects can be implemented directly in kinetic Monte Carlo approaches (see, e.g., Jaraiz, 2004), an indirect approach is required for continuum simulation. One such approach is to consider a number of point-like species, their diffusion, and possible reactions between them. Species in this sense refers to simple point defects like vacancies and self-interstitials as intrinsic point defects as well as dopant atoms on substitutional sites or other impurity atoms, but also to complexes between dopants and impurities with intrinsic point defect as well as clusters comprising dopants, intrinsic point defects, and other impurities. In the following, the framework of diffusion–reaction equations is briefly outlined. This framework is used in the subsequent sections to explain phenomena associated with the diffusion of dopants and typical forms of diffusion profiles. For a full account, the interested reader is referred to more extensive reviews in the field (e.g., Pichler, 2004, section 1.5).

Within the framework of diffusion–reaction equations, for each of the species considered, a continuity equation is solved. For the diffusion and reaction of species A, as an example, it would read

CA∂t=-divJA+RA

with the flux JA given for diffusion in an electrostatic field E by

A=-DA∙gradCA-zA∙μA∙CA∙E.

The terms t, CA, RA, DA, and μA stand for time, concentration, a reaction term accounting for generation and loss due to quasi-chemical reactions, the diffusion coefficient and the mobility of the species, and div and grad are the divergence and gradient operators. The mobility is related to the diffusion coefficient by the Einstein relation A/μA=k∙T/q with k and q denoting Boltzmann's constant and elementary charge, respectively. In the tradition of early reviews in this field (e.g., Fair, 1981; Fichtner, 1983; Tsai, 1983; Willoughby, 1981), the charge state zA has been defined here as the number of electrons associated (e.g., + 1 for a singly negatively charged defects like ionized acceptors, −1 for a singly positively charged defect like an ionized donor, −2 for a doubly positively charged defect). It should be noted, though, that an association of the charge state with positive charges is likewise common (e.g., Fahey et al., 1989) and would manifest itself in a positive sign of the field term. While the definition of the charge state may not always be immediately apparently, it is easy to find it out from the equality (number of negative charges) or inequality (number of positive charges) of the signs of diffusion and field term. Written in terms of the electrostatic potential Ψ related to the electric field by =-gradΨ, the diffusion flux (2) takes the familiar form

A=-DA∙gradCA+zA∙DA∙CA∙gradΨUT

with the thermal voltage UT introduced as abbreviation for T=k∙T/q.

The effects of the quasi-chemical reactions between the species considered are comprised in the reaction term RA. In the following, to illustrate how quasi-chemical reactions between species can be taken into consideration within the framework of diffusion–reaction equations, let us consider reactions in the form

AA+νBBk→⇌k←νCC+νDD

with the forward and backward reaction rates denoted by → and ←, respectively. The stoichiometric numbers ν denote how many of the respective species participate in the reaction. By definition, stoichiometric numbers appearing on the left-hand side are negative. Therefore, their absolute values were used in (4) for the sake of consistency. In chemistry, the concentrations of the species involved are usually given in the form of mole fractions. In crystals, it is more convenient to use site fractions =C/CS defined as concentration C divided by the concentration of sites CS for this defect in the lattice. For vacancies, as an example, CS corresponds to the concentration of lattice sites CSi. For bond-centered interstitial defects, as another example, the concentration of possible sites is twice that of lattice sites since there are four around each lattice atoms, which are shared among two neighboring atoms. Assuming ideally dilute concentrations so that the respective activity coefficients are unity, the site fractions of the defects are related to each other in equilibrium via the law of mass action

=∏ixiνi=xCνC∙xDνDxAνA∙xBνB=∏iθi∙exp-1k∙T∙∑iνi∙Gif

with K denoting the equilibrium constant of the reaction. The θi stand for the—often...