Interaction of Impurities and Mobile Carriers in Semiconductors

Abstract

The interaction of mobile carriers in semiconductors with impurity atoms and ions tends to reduce the impurity activation energy. This effect is studied using the familiar model of the semiconductor as a uniform medium of dielectric constant $K$ in which randomly distributed impurity ions appear as fixed unit charges and the mobile carriers as charges with opposite sign and effective mass $m^{*}$. The treatment is based on the solution of Hartree equations for nonlocalized orbitals ${\psi }_i$ describing mobile carriers and localized orbitals ${\varphi }_j$ describing electrons trapped in the neighborhood of impurity ions. Determination of the individual nonlocalized orbitals is made unnecessary by a method that expresses the fluctuation in mobile carrier density approximately as a linear functional of the fluctuations in electronic potential due to impurity atoms and ions. On use of this relation, Poisson's equation becomes a linear integro-differential equation for the electronic potential energy, which can be solved in terms of integrals involving the localized orbitals ${\varphi }_j$. All localized orbitals are taken to have the same form $\varphi$, satisfying an integro-differential equation obtained by averaging the potential energy for a trapped electron over all configurations of the other impurities; this is solved by a variational procedure. All orbitals in the theory depend on temperature $T$, the Fermi level $\zeta$, and the impurity density $\frac{N}{V}$, since the distribution of qauntum numbers of the occupied orbitals depends on these quantities. The free energy $F$ of the system, first expressed in terms of the orbitals, $T$, and $\zeta$, is then reduced to a function of $\zeta$, $T$, and $\frac{N}{V}$. For given $T$ and $\frac{N}{V}$, the physically significant $\zeta$ is determined as that which minimizes $F$; the carrier density and the effective impurity activation energy are then computed as functions of $T$ and $\frac{N}{V}$. The theory differs from all others in predicting a marked $T$-dependence of the activation energy, especially for high impurity concentrations. This appears because the polarizability of the mobile carrier distribution, which has an important effect on the interaction of impurities and mobile carriers, is temperature dependent in the present theory, and is completely ignored in earlier theories. For moderately high $T$ the reduction in impurity activation energy predicted by the present theory is of the order of that predicted by Shifrin, and by Pearson and Bardeen; at low temperatures it is much less. Existing data on germanium and silicon suggest that the theory underestimates the reduction in activation energy at high impurity concentrations.