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In conclusion, the theoretical donor binding-energy calculations outlined above have been shown to be of direct relevance to simulating spin-flip Raman spectroscopy in diluted magnetic semiconductors. This allows the spatial distribution of donors to be investigated, although the importance of high quality samples has been shown to be significant. It has been proposed that the magnetic behaviour of the first few monolayers of a dilute magnetic semiconductor adjacent to an interface with a non-magnetic semiconductor, could be significantly different from that of the bulk [121-123]. One contribution to this effect arises from a reduction in the number of antiferromagnetically coupled pairs due to a decrease in the number of nearest neighbour magnetic ions [121]. A single layer of donors,s-doped into a quantum well structure in the region of the well-barrier interface could provide a useful probe of the magnetism via the observations made from spin-flip Raman spectroscopy. 5.12 ALTERNATIVE APPROACH TO EXCITED IMPURITY STATES The earlier method of calculating the energy levels of impurities within semiconductor heterostructures requires major mathematical analysis in order to extend to calculate excited states. In this section, a simpler, more general approach is developed that can handle the ground and excited states as well as the spatially degenerate states (2s, 2px;, 2pz) within a general heterostructure potential, that might also include an electric field (bias). As in equation (5.4) the Hamiltonian of an impurity atom within a semiconductor heterostructure under the single-band effective mass and envelope function approximations is: where again m* is the effective mass of the charge carrier, V(z) is the electrostatic potential which defines the heterostructure (and may include a bias eFz) and r is the distance between the impurity and the charge carrier. Placing the x and y origins on the impurity atom, which is at a position ri, then:

<= =

Taking a trial wave function of a charge carrier (electron or hole):

nk(*) =

addition is monotonic } by the principle of strong induction, and the fact that n ^ 2, we may assume that G n ~ 2 ^ F.n ^ G"-1 and G n ~ 3 ^ F.(n-l) ^ G""2 }

where w ( z ) is the wave function of the electron (or hole) in the same heterostructure but without the impurity present and e(r) is a hydrogenic like term describing the interaction between the electron (or hole) and the donor (or acceptor) ion. For the 1s (ground) state of the impurity then the interaction term would be given by:

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Figure 5.25 Comparison of the results of calculations using the approach developed in this section (open symbols) with the earlier method (closed symbols), presented in Fig. 5.13. The system is a donor at the centre of a GaAs quantum well surrounded by Ga1 x AlxAs barriers.

true . Look again at this proof to ensure that you have understood the use of strong induction. Its use, rather than simple induction, was necessary in the very last step above. It is also worth looking again at the warnings we made before establishing the basis of the proof. The real significance of these warnings only becomes apparent in the last step. Satisfy yourself that the step is valid and why it would be invalid if the basis had excluded the case n = 2. D Exercise 12.13. Prove that F.(n+l) xF.(n-l) - (F.n)2 = (-1) M for all n,n^l. Does your proof use strong induction or simple induction D Exercise 12.14. Here is a proof that F.n = 0 for all n. What is wrong with it Basis. F.O = 0 by definition. Induction step. We have, for all n, n ^ 0, { { definition } by the principle of strong induction

Usually, 0.9 < 7 < 1. The above can be adapted to allow the same learning rate for all the weights in the following way:

where A is a variational parameter. The variational calculation is implemented by adjusting A in order to minimise the expectation value of the Hamiltonian operator (the total energy):

we may assume that F. (n- 1 ) = 0 and F.n = 0 }

The energy E is evaluated for different values of A by direct numerical integration of the numerator and denominator in the above equation. For example,

arithmetic {

Sanger developed another principal components learning algorithm, similar to that of Oja, referred to as generalized Hebbian learning [Sanger 1989]. The only difference is the inclusion of more feedback information and a decaying learning rate n(t): ,p -ok,p

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