Hydrogen-like Impurity States in Axially Symmetric Crystals [Letter to the Editor

Abstract

The ionization energies of donors and acceptors in germanium and silicon have been successfully interpreted on the basis of hydrogen-like models. In its simplest form such a model merely changes the potential energy from ( −q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> /r) to ( −q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> /Kr), where K is the dielectric constant, andr eplaces the electron mass by an effective mass. <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> Several authors have extended the hydrogen-like model to the case of an effective mass tensor which is anisotropic but has ana xis of rotational symmetry. <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2–4</sup> Explicit results were given only for cases in which the longitudinal effective mass M <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</inf> is greater than the transverse effective mass M <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> . I n addition, the method habse en applied onlyt o cubic crystals, in whicthh e dielectric constant,a nd consequently, the potential of a point charge, are isotropic. However, interest has recently developed in cases in which the crystal has only one axis of three- or four-fold rotational symmetry. These are CdAs <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> , a tetragonal crystal, and p-type germanium and silicon elastically strained along a symmetry axis. In these cases the dielectric constant tensor K has the form of equation where coordinate z is the symmetry axis. They differ in one other matter of detail from those considered in the references <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2–4</sup> : although the effective mass tensors have an axis of rotational symmetry, the transverse mass is greater than the longitudinal mass. The purpose of this note is to extend the method of the references to take account of the anisotropy of the dielectric constant tensor and to present numerical results for the case in which the transverse mass is greater than the longitudinal mass.