An Analytical Model for Field-Enhanced Diffusion of Ionized Impurities in Highly Doped Si
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An Analytical Model for Field-Enhanced Diffusion of Ionized Impurities in Highly Doped Si Bogdan S. Sokolovskii, Liubomyr S. Monastyrskii, and Roman M. Kovtun Ivan Franko National University, Lviv, UA-79602, Ukraine ABSTRACT An expression for concentration dependence of diffusion coefficient of ionized impurity in semiconductors which is derived which is valid for the whole range of carrier degeneracy under the condition that the impurity concentration greatly exceeds the intrinsic carrier density. Due to reduction of impurity field screening by mobile carriers at degenerate conditions the diffusion coefficient is shown to be a monotonously increasing function of impurity concentration. It is proved that taking band gap narrowing into account results in a reduction of the diffusion coefficient in comparison with the case of unperturbed band structure. In addition, a decreasing concentration dependence of the diffusion coefficient can be realized at relatively low impurity concentrations. INTRODUCTION Diffusion of ionized impurities in semiconductors is well known to be accompanied by a redistribution of electrons and holes that causes the appearance of an internal built-in electric field enhancing the impurity migration [1]. When modeling field-enhanced impurity diffusion processes in semiconductors, for example in Si (see, for example [2]), the expression for concentration dependence of the diffusion coefficient proposed by Smits [3] is widely used, according to which at large impurity concentration the diffusion coefficient increases twice as much as its value corresponding to the case of small impurity concentration. However, in many cases of practical importance at large impurity concentration the carrier degeneracy takes place when the Smits’ expression [3] is not applicable. As far as the behavior of the impurity diffusion coefficient in degenerate semiconductors is concerned, despite of its importance for the technology of semiconductor devices little attention has been paid to this matter although the latter has a rather long history. In 1961, Shockley [4] theoretically investigated the concentration dependence of diffusion coefficient of ionized impurity for the case of a strongly degenerate semiconductor which has been also reexamined by Snapiro and Tkachenko [5]. The behavior of impurity diffusion coefficient and built-in field in a moderately degenerate material has been studied by means of numerical modeling [6,7]. As for an explicit expression which would allow to easily calculate and explain the concentration dependence impurity diffusion coefficient in the whole range of carrier degeneracy, it to our knowledge has not been available by now. In this paper, we derive a new expression for the concentration dependence of the diffusion coefficient of ionized impurity which holds true in the whole range of carrier degeneracy under the condition that the impurity concentration greatly exceeds the intrinsic carrier density. Further, this expression is generalized for the case when band gap narrowing effect t
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