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but r'2 = r2 + z'2, and therefore r'dr' = r_L.dr_L. Using this substitution and changing the limits of integration, then obtain:
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and finally:
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Writing I3 in plane polar coordinates with the three-dimensional form for r", then:
In order to proceed, it is necessary to evaluate the differential. Consider:
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which, because of the isotropy of the exponential term in this case, yields, in the same manner as above in equation (5.73):
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Differentiating again:
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and similarly for V2ye r/t. The same is also true for V2ze r/t and can be followed through by noting, however, that:
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Gathering all of the terms together, obtain:
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Figure 5.10 Difference in total energy for the two- and three-dimensional cases, as a function of donor position rd across the 60 A CdTe well
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Figure 5.11 The electron wave function u, as in equation 5.7, for the donor positions given on the right hand axis
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Using this form in equation (5.78) then:
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which, using equation (5.64) gives the following:
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Again, substituting r' for r , then gives:
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Using the final form for I1, as in equation (5.69), then:
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Figure 5.12 Comparison between the total electron wave function u(z) and the numerically determined envelope x(z) for three different donor positions Finally, for the three-dimensional case, I4 becomes:
which, on changing the variable to r', becomes trivial, i.e.
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and therefore:
Figure 5.13 Magnitude of the donor binding energy EDo in GaAs/Ga1 _x Alx As as a function of quantum well width, for a variety of barrier alloys x
Looking upon these results from a computational viewpoint, it can then be seen that this 3D trial wave function has an immediate advantage over the previous, seemingly simpler, 2D case, namely that all of the integrals I1, I2, I3 and I4 have analytical expressions. Indeed, evaluation of the integrals and the minimisation of the energy is computationally much less demanding than previously. Figure 5.10 shows the change in the total energy E of the electron between the previous 2D trial wave function and the more complex 3D case presented in this section. It is clear that the energy E is lower for all donor positions across this (a typical) quantum well. The difference between the two trial wave functions is smallest when the separation between the donor and the electron is larger, i.e. when the donor is deep in the barrier (rd = 0 A) and the electron is, as always, centred in the well (z = 230 A). Thus it might be concluded that the 2D wave function is a reasonable approximation when the donor is in the barrier, although as the graph shows, for donors in the well, considerably lower energies can be obtained by using a spherical hydrogenic term. Recalling the variational principle, then the lower energies obtained imply that the 3D approximation to the wave function is a more accurate representation than the 2D case. When coupled together with the computational advantage, as mentioned above, then the argument in favour of the 3D trial wave function is clear. Figure 5.11 displays the total wave function, u = x(z)e r/t for the range of donor positions across the quantum well. It can be seen that the wave function u
Figure 5.14 Magnitude of the donor binding energy EDo for donors at the centre of GaAs wells of large width, surrounded by Ga 0.9 Al 0.1 As barriers
resembles the one-particle wave function u for an electron without a donor present, for donors in the barrier, i.e. rd <160 A. As the donor approaches the electron wave function, i.e. nears the barrier-well interface at z = 200 A, then the electron is drawn r/ distinctly to the left towards the donor. The influence of the hydrogenic factor e r/t can be seen for the donor positions rd = 220 and 230 A within the well. In the 2D case, the total wave function was given by:
and hence the z-dependence is merely u ( z ) = x(z)- Furthermore, it was found that the numerically determined envelope X(z) was a very close approximation to the electron wave function u(z) without the donor present. In this case, as is clear from Fig. 5.11, u ( z ) # x(z); this is illustrated more clearly in Fig. 5.12. Moving on to the GaAs/Ga1_xAlxAs material system and employing the bulk values of m* = 0.067mo and e = 13.18, Fig. 5.13 shows the effect of well width on the neutral donor binding energy, for donors at the centre of the well, for a variety of barrier compositions. As would be expected from earlier results, EDo peaks at a narrow well width and then tails off towards the bulk value. This important limit is explored further in Fig. 5.14; it is clear from this figure that the convergence is very close, and this helps give justification to the methods developed. The variation in the Bohr radius is displayed in Fig. 5.15, and it too converges well to a value of 104 A at very large well widths. This compares admirably with the value deduced from the simple bulk hydrogenic model of 103 A, calculated at the beginning of the chapter. The very small difference could arise from the finite A increment employed of 1 A.