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following quadratic relationship:
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Table 3.1 displays the energies of the lowest ten levels within a Gai_ x Al x As parabolic quantum well. The layer thicknesses have been taken as a = b = 100 A, and the effective mass has been taken as a constant, i.e. 0.067 mo; however, the maximum alloy concentration, xmax, has been taken artificially as high as '10', in order to produce a large number of confined levels. As E\ = hui/2, then the third column of data displays the 'half-integer' (n + |). Clearly, the agreement with the analytical theory is exact for the lower states, although as the levels approach the top of the barrier, they experience the finiteness of the potential, and hence some discrepancy with the infinite parabola solution arises.
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Figure 3.11 The three confined eigenstates of a finite parabolic quantum well
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The energy step hw between the levels is a useful quantity for describing the properties of the well, even though w doesn't represent the angular frequency of oscillation. With this in mind, consider the conversion of the alloy concentration of equation (3.35) into a potential profile, (see Appendix A); the conduction band profile of Ga 1 _ x Al x As would then become:
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Recalling that dV/dz = Cz, then:
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Ignoring the origin shift, b a/2:
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Then, uj follows from equation (3.31) as:
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By using the values corresponding to the data in Table 3.1, i.e. VCB 0.67, EgA1As EgGaAs =1247 meV, Xmax = 10, Zmin = 0, a = 100 A and m = 0.067 m0 then hw 871.879 meV, which is in excellent agreement with the numerical result as displayed in the table.
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Figure 3.12 Eigenenergies for a GaAs/AlAs parabolic quantum well
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Figures 3.11 and 3.12 display the wave functions and energies, respectively, of the three confined states of the more realistic parabolic quantum well, with x max =l. The wave functions resemble those of the single quantum well and the energy levels are to all intents and purposes equally spaced. In summary, it has been demonstrated that the simple shooting-method solution of the Schrodinger equation, taken together with the stated starting conditions, can produce energy levels in exact agreement with the analytical values for the curved potential of the parabolic quantum well. In addition, equation (3.40), summarizes the behaviour of parabolic wells within semiconductor multilayers. The energy level spacing is inversely proportional to both the well width and the square root of the mass.
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Figure 3.13 Poschl-Teller potential hole; a = 0.05 and A = 2.0 The modified Poschl-Teller potential hole [60], an example of which is shown in Fig. 3.13, is important in that it resembles the profile of a diffused quantum well, but has the advantage of analytical solutions. Therefore as is currently the aim, it will serve well for validation of the shooting method and the choice of starting conditions described in Section 3.1. The potential is given by the following [61]:
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where a is known as the width parameter and A as the depth parameter (see Fig. 3.14). Clearly, values of 0 < A < 1 give potential barriers, A = 1 gives a flat band and the area of interest here, namely potential wells, are given by values of A > 1. The eigenvalues of the resulting Schrodinger equation are given by [60]:
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Table 3.2 compares the analytical solutions given by equation (3.42) for the two lowest energy confined states of the Poschl-Teller potential holes of Fig. 3.14, with that obtained by the numerical shooting method. The agreement is very good for all values of A and for both the ground state, of energy E\, and the first excited state, of energy E2. However, for the largest value of A used, which does represent a very deep well, of a magnitude not found in semiconductor heterostructures, the discrepancy
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