Exact analytical solutions for shallow impurity states in symmetrical paraboloidal and hemiparaboloidal quantum dots

The problem of a shallow donor impurity located at the centre of a symmetrical paraboloidal quantum dot (SPQD) is solved exactly. The Schrödinger equation is separated in the paraboloidal coordinate system. Three different cases are discussed for the radial-like equations. For a bound donor, the energy is negative and the solutions are described by Whittaker functions. For a non-bound donor, the energy is positive and the solutions become coulomb wave functions. In the last case, the energy is equal to zero and the solutions reduce to Bessel functions. Using the boundary conditions at the dot surfaces, the variations of the donor kinetic and potential energies versus the size of the dot are obtained. The problem of a shallow donor impurity in a Hemiparaboloidal Quantum dot (HPQD) is also studied. It is shown that the wave functions of a HPQD are specific linear combinations of those of a SPQD.