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3.2. Hydrogenic Impurities within Asymmetric and Symmetric Quantum Well Wires

In this section the energy of the ground and first excited states, the binding energy, and oscillator strengths for hydro-genic impurities confined within a cylindrical quantum well wire with a finite-height potential well are studied varia-tionally as a function of the wire radius and of the relative position of the nucleus within the quantum well wire for different barrier-height potential [77]. The trial wavefunc-tion is constructed as the product of the free wavefunction of hydrogen impurity and a simple auxiliary function that allows the appropriate boundary conditions to be satisfied.

In this context, the variational method here used constitutes a useful approach to study asymmetric and symmetric quantum systems confined by penetrable potentials.

### 3.2.1. On-Axis Hydrogenic Impurity

If the nucleus of the hydrogen impurity is located on the symmetry axis of the cylindrical quantum well wire, the model Hamiltonian for the electron within the quantum well wire can be written in atomic units (h = m* = e = 1) as

with

where r = y p2 + z2 is the electron-nucleus distance, p is the cylindrical coordinate parallel to xy plane, z is the coordinate along the wire axis, p0 is the wire radius, and V0 is the confining potential barrier.

The physical meaning of V0 in this context is to simulate, on the average, the effective potential step created by the composition difference between the quantum well wire and its surroundings.

It is well known that the Schrodinger equation for this Coulomb-type potential is not separable in cylindrical coordinates, the natural symmetry of the wire; thus we are forced to use an approximate method to calculate the ground state energy and first excited states for this system.

In order to use the variational method to solve approximately the Schrodinger equation with the Hamiltonian given by Eq. (114), we must construct a trial wavefunction fi with the basic properties listed as follows:

finite

where {a} is the set of parameters mentioned earlier.

A possible ansatz wavefunction for this problem is of the form fi ■ fi> = A(p0 - ap) exp(-ar) 0 < p < p0 (118) s fio = B exp(-ar) exp(-fip) p0 < p < x where A, B are normalization constants and a, ¡3 are the variational parameters involved in the calculation. These functions must satisfy fii\p=po dfi dP

By imposing condition (120) to the function given by Eq. (118) we have that

That is, we need to find only one variational parameter to minimize the ground state energy. Furthermore, when V0 ^ ex, a ^ 1, ¡3 ^ ex, and fio ^ 0, as expected.

The continuity condition on the boundary, Eq. (119), relates B and A:

The suitable Hamiltonian for 2p-type excited states is the same that given by Eq. (114), and the variational wave-functions for 2px, 2py, and 2pz states, when the nucleus of hydrogenic impurity is fixed on the symmetry axis of the wire, are of the form fi2px =

x exp(-yr)p cos ç o < p < Po fio = D exp(-yr)

x exp(-yr)p sin ç o < p < po fio = D exp(-yr)

where C and D are normalization constants and y, % are variational parameters to be determined and are related by

The results for the energy of the ground and 2pz states as a function of wire radius and different barrier heights are displayed in Figure 14. For a given value of the finite barrier potential the ground state (excited state) energy increases from -0.5 (-0.125) Hartrees as the wire radius is reduced. These values are characteristic of the "free" hydrogen atom.

The binding energy sb for the hydrogenic impurity is defined as the ground state energy of the system without Coulomb interaction sw, minus the ground state energy in the presence of electron-nucleus interaction s0; that is, sb = Sw - s0 (128)

The binding energy defined in this way is a positive quantity.

In Figure 15a, we display the variation of the hydrogenic impurity binding energy sb as a function of wire radius p0 for several values of finite potential barrier.

### 3.2.2. Off-Axis Hydrogenic Impurity

If the nucleus of the atom is located on the x axis, at a distance b from the axis of the wire, the electron-nucleus distance is r' = \f- bex \=y (x - b)2 + y2 + z2 = >JP2 + b2 + z2 - 2pb cos p (129)

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