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The complete crystal structure is obtained by placing the atomic basis at each Bravais lattice point. For materials such as Si, Ge, GaAs, AlAs, InP, etc., this consists of two atoms, one at (1/8,1/8,1/8) and the second at ( 1/8, 1/8, 1/8), in units of A0. For the group IV materials, such as Si and Ge, as the atoms within the basis are the same then the crystal structure is equivalent to diamond (see Fig. 1.3 (left)). For III-V and II-VI compound semiconductors such as GaAs, AlAs, InP, HgTe and CdTe, the cation sits on the (18- / ,18 site and the anion on (+1/8,+1/8,+1/8);this type of crystal - / ,18- / ) is called the zinc blende structure, after ZnS, see Fig 1.3 (right). The only exception to this rule is GaN, and its important InxGa1-xN alloys, which have risen to prominence in recent years due to their use in green and blue light emitting diodes and lasers (see for example [13]) these materials have the wurtzite structure (see reference [2] p. 47).

Let ABC be a triangle with BAC = 90 . Let the lengths of sides BC, AC, AB be, respectively, a, b and c. We wish to prove that a2 = b2+c2. Construct a square IJKL, of side b+c, and a square BCDE, of side a, as shown in the figure. Clearly, area(IJKL) = (b+c)2 .

Figure 2.10 Effect of electron density, N=l, 2, 5, 10 (x 1010) cm- 2 , on the distribution function of the lowest subband of the infinite quantum well of Fig. 2.7

But,

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Product unit neural networks (PUNN) have neurons which compute the weighted product of input signals, instead of a weighted sum [Durbin and Rumelhart 1989, Janson and Frenzel 1993, Leerink et al. 1995]. For product units, the net input is computed as given in equation (2.2). Different PUNNs have been suggested. In one type each input unit is connected to SUs, and to a dedicated group of PUs. Another PUNN type has alternating layers of product and summation units. Due to the mathematical complexity of having PUs in more than one hidden layer, this section only illustrates the case for which just the hidden layer has PUs, and no SUs. The output layer has only SUs, and linear activation functions are assumed for all neurons in the network. Then, for each hidden unit yj, the net input to that hidden unit is (note that no bias is included)

Figure 2.11 Effect of electron density on the quasi Fermi energy describing the distribution of the ground state E\

area(IJKL) That is,

whose only physical meaning is to describe the population within a subband it is not the true Fermi energy of the complete system. Fig. 2.10 displays the distribution functions for a range of carrier densities, for this same ground state and at a lattice temperature of 77 K. Although not obvious

eEivji(zt,p)

Empty Range: (0k: false :T) = 1 . One-Point: (0k : k = e : T) = T [ k : = e ] . Splitting: (0k:P:r> e( k:Q:r> = < k:P v Q : T > < k:P A Q : T > . Splitting (idempotent e): < k : (Bj:R:S) : T) = ( j:R:( k:S:T}}.

from the mathematics, f FD (E) at any particular energy E appears to scale with N. The corresponding Fermi energy is illustrated in Fig. 2.11. Clearly, the Fermi energy starts below the subband minima at this mid-range temperature, as discussed above, and as expected increasing numbers of carriers in the subband increases the Fermi energy, i.e. the energy of the state whose probability of occupation is 1/2. 2.5 FINITE WELL WITH CONSTANT MASS

Trading. Type and Range: Range and Term: <0fceK : P A Q : T) = ( k<E{k<=K\P}:Q:T) . ( f e : P A Q : T ) = ( fc : Q : if P T n ^P - 1 fi> -

While the infinitely deep confining potential has served well as a platform for developing the physics of two-dimensional systems, more relevant to alternating layers of dissimilar semiconductors is the finite quantum well model, which under both the effective mass and envelope function approximations looks like Fig. 2.12. In particular, a layer of GaAs 'sandwiched' between two thick layers of Gai-xA1xAs would form a type-I finite quantum well, where the conduction band has the appearance of Fig. 2.12, with the potential energy V representing the discontinuity in the conduction band edge between the materials. Again taking the simplest starting case of a constant electron mass m* throughout the dissimilar layers, and neglecting movement within the plane of the layers, then the standard Schrodinger equation can be written for each of the semiconductor layers as follows:

(3.3)

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