Note on the potential of ionized impurities in semiconductors

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Abstract
The theory of the potential of an impurity ion in a semiconductor is reconsidered by extending the original treatment of Dingle and Mansfield which resulted in a screened Coulomb potential. This simple form was obtained by using an approximation; namely, first expanding the Fermi-Dirac integral in terms of the ion potential and then discarding terms higher than linear. In the present treatment a modified ion potential is derived by retaining also the quadratic term in the expansion of the Fermi-Dirac integral. The modified potential is expressed as a product of the Dingle-Mansfield potential and a correction factor. At any distance from the impurity ion the correction factor is found to have a maximum as a function of charge carrier concentration. It is concluded that the screened Coulomb potential approximates well the potential of an impurity ion in the semiconductor since, in that range of distance from the ion which is important for scattering problems, its magnitude is only a few percentage smaller than that of the more exact modified potential. From this fact it is also concluded that consideration of higher than quadratic terms in the expansion of the Fermi-Dirac integral would lead to negligible further corrections in the magnitude of the modified potential.

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