Exact and variational calculations of a hydrogenic impurity binding energy in a multilayered spherical quantum dot

The binding energy of a central or an off-central impurity located in a multilayered spherical quantum dot has been calculated. Different effective masses and dielectric constants in the different regions of the MSQD are taken into consideration. New analytical forms for the exact solution of the radial Schrödinger equation in the absence of the impurity and in the presence of a central impurity have been introduced. The analytical forms in the presence of the central impurity have the advantage that they are valid for any electron energy negative or positive unlike the forms used hitherto. The variational method has also been applied for both central and off-central impurities. The trial wavefunction has been taken in a new form in addition to the conventional form used in previous treatments. The new form has the advantage of satisfying the boundary conditions at the interfaces in the case of different masses. It further raises the results of binding energy so that they approach the exact solution from below. The two forms have yielded new analytical expressions for the energy expectation value and binding energy.