A general model for motion bound to an impurity in an anisotropic semiconductor (Q1174815)

The classical motion of an electron bound to an impurity in an anisotropic semiconductor is investigated. In a crude approximation the motion can be described as a planar motion in a central potential \(V(r)\) (which is vanishing at the infinity and at the origin has the behaviour \(r^ x\), \(x\in (0,2)\)) and whose kinetic energy term is anisotropic. This generalized anisotropic model (GAM) as a special case includes the anisotropic Kepler model (AKM) and its different variations due to the radial variations of dielectric constant, or due to the metallic screening. It is shown that the flow on the collision manifold has for GAM the same qualitative behaviour as for AKM (only the winding number of orbits depends on \(x\)). The conditions for chaotic behaviour are formulated, provided the anisotropy is large enough. The qualitative features of those chaotic orbits depend only on the parameter \(x\), and can be described within the framework of symbolic dynamics. The known results for AKM are extended and improved. The quantum aspects of the impurity problem are briefly discussed.