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The application a magnetic field to a heterostructure leads to a Zeeman splitting of the confined energy states, where in the case of an n-type material (i.e. material with donors present), two electron states are formed with the spin-up states having a higher energy than the spin-down states. At low temperatures, the carriers populate the lowest energy state, i.e. the spin-down, but can be excited by photons into the higher energy state. This excitation can be detected as a Raman shift [119], as illustrated schematically in Fig. 5.21. This is a very powerful technique because the spectroscopy depends only upon one carrier type and theoretical modelling of the experimental data requires the parameter set of only one band. In this case, the parameters consist of the relative permittivity of the material, and the electron effective mass from bulk, together with the conduction band offset between the non-magnetic CdTe well and the magnetic Cd1_xMnxTe barriers [120]. The Raman shift is simply the energy difference between the two electron-donor spin states, Fig. 5.22 displays the energy of these two spin states for a lightly n-doped CdTe/Cd1_xMnxTe system, calculated using the 3D trial wave function. The magnetic ion concentration in the barriers has been chosen as 15 % in order to maximize the energy difference between the states. Fig. 5.23 displays the spin-flip energy for this system, again as a function of donor position. If the donors are assumed to be uniformly distributed across the entire heterostructure, it is possible to represent the data of Fig. 5.23 in the form of an intensity / versus spin-flip energy E Raman spectrum, by assigning a Gaussian distribution to

For n = 2, we have G 2 ^ 2 ^ F.2 ^ G2"1 {

Figure 5.22 Energy of the spin-up (solid circles) and spin-down (open circles) states at a magnetic field of 8 T, for a range of donor positions across a 60 A CdTe well surrounded by Cdo.ssMno.isTe barriers

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The first term corresponds to standard Hebbian learning (refer to equation (4.2)), while the second term is a forgetting factor to prevent weight values from becoming unbounded. The value of the learning rate, 77, above is important to ensure convergence to a stable state. If n is too large, the algorithm will not converge due to numerical unstability. If n is too small, convergence is extremely slow. Usually, the learning rate is time dependent, starting with a large value which decays gradually as training progresses. To ensure numerical stability of the algorithm, the learning rate rjk(t] for output unit ok must satisfy the inequality:

arithmetic, definition of F.2 }

where E s f (rd) is the spin-flip energy of the donor at position rd. The finite linewidth of any spin-flip signal will arise from microscopic fluctuations of the material parameters. For example, the well width, alloy fluctuations in the barrier, and the random nature of the donor distribution itself, will all give rise to a broadening of the signal from each of the donor positions rd along the axis of the quantum well structure. The standard deviation a is related to the linewidth l (full width at half-maximum) by:

1 78

In order to save considerable computational effort, the donor calculations were performed only at the points marked by circles on Fig. 5.23. A spline of this data was produced, given by the dashed line, and then this more detailed curve was used to produce the intensity versus energy data of Fig. 5.24, which also shows the effect of different linewidths on the predicted Raman spectrum of this single quantum well. In the lowest curve (l 0.5 cm- 1 ) it is possible in principle to resolve the donors in the well from those in the barrier, as observed experimentally [120]. Comparing Figs 5.23 and 5.24, it can be seen that the central peak at around 18.5 cm- 1 corresponds to spin-flips of electrons bound to donors in the barrier. This peak in intensity is due entirely to the proportionately larger fraction of donors in the barrier compared to those in the well. In addition, spin-flips from electrons bound to donors in the well are clearly resolved as a small peak at about 16 cm 1 . As the linewidth l of the signals is allowed to increase, the resolution and information in the simulated spectrum decreases. At a linewidth of 1 cm 1 , spin-flips due to donors in the well appear as just a shoulder on the larger 'barrier' peak, while by l = 2 cm 2 this information is lost altogether, and a broad central peak ensues.

where \k is the largest eigenvalue of the covariance matrix, Cz, of the inputs to the unit [Oja and Karhuner 1985]. A good initial value is given as nk(0) = 1 / [ 2 Z T Z ] , where Z is the input matrix. Cichocki and Unbehauen provided an adaptive learning rate which utilizes a forgetting factor, 7, as follows [Cichocki and Unbehauen 1993]:

arithmetic, definition of F (see remarks above) } heading for the induction hypothesis, we note that G 2 - G - I = 0. So, G"-1 = G n ~ 3 xG 2 = Gn~2 + G"'3. Similarly, G n = Gn~l + Gn~2 . }

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