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Schematic representation of the effect of donor position on the wave function
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Methods of solution of the Schrodinger equation have centred around two basic approaches. The first of these involves expanding the electron wave function as a linear combination of Gaussian functions [103,104]. While this technique has been successful in calculating the properties of donors in simple quantum well structures, the generalisation to more complex structures, including graded gap materials and systems where piezo electric fields are present, is non-trivial. The other category of approach is based on the variational principle. In this method, a trial wave function is chosen whose functional form may contain one or more unknown parameters. These parameters are varied systematically and the expectation value of the energy calculated for each set. The variational principle [3,29] states that the lowest energy obtained is the closest approximation to the true state of the system. The success of variational approaches centres around the general choice of the trial wave function. A common choice is a product of two terms [105], i.e.
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where r' is the electron donor separation and A is a variational parameter; thus the second factor is a simple hydrogenic wave function. The function u(z) is the uncorrelated eigenfunction, i.e. the straightforward wave function calculated in s 2 and 3, of the electron in the quantum well without the donor [106,107].
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BINDING ENERGY IN A HETEROSTRUCTURE
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The latter restriction can be removed and a much more general choice of the donor wave function u can be made. The motivation being generality, the problem is recast in a form which is suitable for numerical solution as described in 3, thus making it applicable to any quantum well structure, be it a double quantum well, a diffused quantum well, a graded gap quantum well, etc. With this aim, the trial wave function to be employed here is influenced by the above, but for now will be written as follows: The one-dimensional envelope function x(z) is yet to be determined and the function (z, y, z rd) is expected to be a hydrogenic type factor which is a function of the spatial coordinates and one or more variational parameters. It can be seen that forming the Schrodinger equation with the Hamiltonian of equation (5.4) and the trial wave function w of equation (5.7) will lead to a term V 2 W ; it is worthwhile to derive this now. Consider:
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and similarly for y. Consider now
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Hence where the subscript z implies derivatives with respect to z only. Then, forming the Schrodinger equation with the Hamiltonian of equation (5.4) gives the following:
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where E is the total energy of the system. The donor binding energy ED0 is equal to the difference between this and the standard confinement energy of the electron in the heterostructure without the donor present, i.e. using the notation of 2, the ground state energy for the electron in the well without the donor would be E1, and hence
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In order to proceed further, it is necessary to be more specific about the form of the hydrogenic factor . In a free hydrogen atom the ground state wave function would be spherically symmetric and given by:
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where A is known as the Bohr radius. Indeed this is the form also used for donors in bulk materials. However, in heterostructures there is a loss of translational symmetry along the growth (z-) axis and hence it cannot be assumed that the spatial co-ordinate is spherically symmetric. For now, the hydrogenic term can be taken as:
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where the spatial coordinate r" is, of course, a function of x, y and (z r d ). Again, A is referred to as the Bohr radius, but now it will be employed as a variational parameter in order to minimise the total energy of the system. Substituting for into equation (5.13) gives:
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r" Multiplying by e * and integrating over the x-y plane, leads to an equation of the form:
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where the integrals Ij,(j = 1,2,3,4) are defined as follows:
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Returning to the Schrodinger equation (equation (5.18)) then:
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