## Physical Effects Of Impurity States And Atomic Systems Confined In Semiconductor Nanostructures

In contrast to bulk materials, the nanostructures are characterized by a lack of translational invariance along the confinement direction. Thus, the impurity binding energy explicitly depends on the precise location of the impurity. It is important to know where the impurity atom is placed in the nanostructure, and the binding energy of the donor state associated with impurity atom will depend on whether the impurity atom sits at the center or the axis of the nanostructure.

The second important feature of Coulombic problems in nanostructures is the variation of the impurity binding energy with the characteristic dimension of the well. When the nanostructure confinement dimension (confinement size) decreases, the impurity binding energy increases, as long as the penetration of the unperturbed nanostructure wavefunction in the barrier remains small. This may seem surprising at first sight since we intuitively associate extra kinetic energy with the localization of a particle in a finite region of space. Moreover, it is actually true that the energy of the ground bound state of the impurity increases as the confinement size decreases, when it is measured from some fixed reference. However, the onset of the impurity continuum, which (for donors) coincides with the energy of the nanostructure bound state, also moves up with decreasing confinement size. The binding energy finally increases since, in the bound impurity states, the carrier is kept near the attractive center by nanostructure walls and thus experiences a larger potential energy than in the absence of a nanostructure. On the other hand the onset of the impurity continuum does not benefit from any extra potential energy gain, as the continuum states are hardly affected by the Coulombic potential.

When the Schrodinger equation for a given quantum system is nonseparable in nature, then some approximation techniques must be used to study its solution. In a similar situation for a confined quantum system, the varia-tional method might constitute an economical and physically appealing approach [65]. It has been shown [66] that the variational method is useful for studying this class of systems when their symmetries are compatible with that of the confining boundaries. In general, in the problems related to impurity and atoms confined in the different nanostructures, these considerations are fulfilled.

3.1. Hydrogenic Impurity in Quasi-Two-Dimensional Systems

In Figure 8, we sketched the shape of the impurity wave-function, keeping the in-plane distance between the carrier and the attractive center p(p = ^Jx2 + y2) equal to zero. Several quantum well thicknesses (upper part of Fig. 8) and several impurity positions in a thick well (lower part of Fig. 8) have been considered. For thick wells (L » a*B) the on-center impurity wavefunction resembles that of the bulk 1i states. On the other hand, the on-edge impurity

Figure 8. Evolution of the shape of the impurity wave function in a quantum well of decreasing well thickness for an on-center impurity (upper part) and with the impurity position in a thick well (lower part).

wavefunction approaches the shape of a 2pz wavefunction if (L/a*B) » 1. This is because the barrier potential forces the impurity wavefunction to almost vanish at z = ±L/2. If L/a*B ^ 1 and if the barrier height is very large, the electron z motion becomes forced by the quantum well potential. Thus, along the z axis the impurity wavefunction looks like the ground state wavefunction of the well ^(z).

3.1.1. Approximate Solutions of the Hydrogenic Impurity Problem

In all that follows we will consider donorlike impurities unless otherwise specified. The conduction bands of both host materials of the quantum well are assumed to be iso-topic and parabolic in k. We neglect the effective mass jumps at the interfaces as well as the differences in the relative dielectric constants of the two host materials. The impurity envelope functions are the solutions of the effective Hamiltonian

where V0 is the barrier height and Y(x) is the step function [Y(x) = 1 if x > 0; Y(x) = 0 if x < 0]. The impurity position along the growth axis is zt. The (x, y) origin is at the impurity site because all the impurity positions are equivalent in the layer plane; p is the projection of the electron position vector in the layer plane [p(x, y)].

In the absence of the impurity the eigenstates of H0 are separable in (x, y) and z,

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