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similar to the infinitely deep wire case, with the same distribution of maxima and minima. However, the main difference is, of course, that the finite barrier height allows for a significant 'leakage' of the wave function into the surrounding material. It is this interaction with the 'potential pillars' located outside of the wire at each corner, which originate from the approximated potential in Fig. 8.8, which limits the applicability or accuracy of this simple model. For this relatively narrow (l00 Ax100 A) wire, the scale next to the ground-state charge density in Fig. 8.10 (top left), implies that, roughly speaking, probably around 80 90 % of the charge is confined within the wire. Thus the effect of the approximate potential outside of the wire will not be too substantial. However for the higherenergy states more of the 'lighter shade' is outside the wire, and thus it would be expected that this approximation would be worse. The same effect would be true when reducing the length of the wire side. This simple approach to the finite quantum wire is just a way of understanding the basic physics and being able to predict qualitatively how the electronic properties change with the system parameters. For a more complete description, a full twodimensional Schrodinger solution may be required (see [160,163]), or alternatively the empirical pseudopotential approach (see 13) may be employed. The energy eigenvalues Ey,z could be improved by considering a perturbation on the two-dimensional system which removed the '2V potential pillars. Using first-order perturbation theory, the change in energy of a level would be given by:

Qxd + r = Qxd'+r-n n = Qx(d'-d) (31: d = l+d:n = Qxl) . So, subtracting any multiple Qxl of Q from r, simultaneously adding I to d, maintains (15.32). Formally, for all I, (P = Qxd + r) [r,d:=r -Qxl, d+l] <= P = Qxd + r . By design, however, m is a multiple of Q; it is Qx2 k . Thus, subtracting m from r, at the same time adding 2 k to d, maintains property (15.32). In summary, the loop body takes the following form: { ( K r < m A P = Qxd + r A l ^ f c A m = Qx2k } ra.fc := m - ^ 2 , f c - l ; { Invariant: O^r A P = Qxd + r A CKk A m = Qx2 k Bound function: do r^m - r,d

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Figure 8.10 The charge densities of the four lowest energy confined states of a GaAs quantum wire of side 100 A with finite Ga0.8Al0.2As barriers, (top left) n y =l, nz=l; (top right) n y =l, nz=2; (bottom left) ny=2, nz=l; (bottom right) ny =2, n2=2; the edges of the wire are indicated by the boxes

The convergence speed of NNs is directly proportional to the learning rate 77. Considering stochastic GD, the momentum term added to the weight updates also has as its objective improving convergence time. Learning Rate The learning rate controls the size of each step toward the minimum of the objective function. If the learning rate is too small, the weight adjustments are correspondingly small. More learning iterations are then required to reach a local minimum. However, the search path will closely approximate the gradient path. Figure 7.5(a) illustrates the effect of small rj. On the other hand, large n will have large weight updates. Convergence will initially be fast, but the algorithm will eventually oscillate without reaching the minimum. It is also possible that too large a learning rate will cause "jumping" over a good local minimum proceeding toward a bad local minimum. Figure 7.5(b) illustrates the oscillating behavior, while Figure 7.5(c) illustrates how large learning rates may cause the network to overshoot a good minimum and get trapped in a bad local minimum. Small learning rates also have the disadvantage of being trapped in a bad local minimum as illustrated in Figure 7.5(d). The search path goes down the first local minimum, with no mechanism to move out of it toward the next, better minimum. Of course, all depends on the initial starting position. If the second initial point is used, the NN will converge to the better local minimum.

{ ( K r < m A P = Qxd + r A O^fe A m = Qx2 k } . So far, we have not made any use of the fact that the algorithm we are developing is based on binary (as opposed to, say, decimal) arithmetic. All our calculations are equally valid if '2' is replaced everywhere by ' ', where B ^ 2. Recall the discussion of long division. In decimal arithmetic, each step subtracts a multiple of Q of the form Qxnx 10k, for some k and n, where 0 < n < 10, from the current remainder r. In binary arithmetic, each step subtracts a multiple of

The perturbation to the potential V'(y, z} would be negative and of magnitude V, and employing the four fold symmetry of this square cross-sectional wire, would give:

Q of the form Qxnx2 k , for some k and n, where 0 ^ n < 2 , from the current remainder r. So, each step subtracts Qx2 k at most once. We can formally justify this within the development of our algorithm. Currently, the loop body we have developed subtracts m repeatedly from r until r < m. The following calculation investigates how many times this needs to take place. Specifically, noting that the value subtracted from r is m -r 2, where m satisfies 0 ^ r < m, we investigate when r -m + 2 ^ m~2. r-m-r2 < m+ 2 = <= { { arithmetic } m-1 s$ 2x(m + 2), transitivity of < }

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