Recalling the centred finite difference expansion for the first derivative, i.e. in Java

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Recalling the centred finite difference expansion for the first derivative, i.e.
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EXTENSION TO VARIABLE EFFECTIVE MASS
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Table 3.4 Comparison of the numerical solution with the analytical solution for a single quantum well with differing effective masses in the well and barrier
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Well width (A) 20 40 60 80 100 120 160 200
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Analytical solution (meV) El (nieV) E2 (meV) 126.227914 80.111376 53.276432 37.619825 27.884814 21.463972 13.820474 9.629394 166.522007 137.330295 106.557890 83.606781 54.647134 38.282383
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Numerical solution E E1 I (meV) E2 (meV) 126.204335 80.087722 53.260451 37.609351 27.877769 21.459068 13.817861 9.627854 167.766634 137.308742 106.535858 83.589149 54.636614 38.275914
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By gathering terms in u(z) on the right hand side, then:
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Making the transformation, 2Sz > Sz, then gives:
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which is the variable effective-mass shooting equation, and is solved according to the boundary conditions (as in Section 3.1). The effective mass m* can be found at the intermediate points, z 8z/2, by taking the mean of the two neighbouring points at z and z 6z. Clearly, equation 3.53 collapses back to the original form in equation (3.11) when m* is constant. Table 3.4 compares the ground-state and first excited-state energy levels, EI and E-2, respectively, calculated with this extended shooting equation, with the analytical solution from Section 2.6, for a GaAs quantum well surrounded by Gao.8Alo.2As barriers. In this series of calculations, the step length Sz was taken as 1 A and it can be seen from the data in the table that the agreement is very good for both the ground-state energy EI and the first excited-state energy E2 across the range of well widths. The discrepancy between the solutions of the two methods is largest for the excited state of the narrower wells, at which point it is of the order of 1 meV. For
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NUMERICAL SOLUTIONS
Table 3.5 Comparison of the numerical solution with the analytical solution for a single GaAs quantum well surrounded by Gao.2sAlo.75As barriers, with differing effective masses in the well and barrier
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(A) E Well width (A) E11 20 20 40 60 80 100 120 160 200
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Analytical solution solution (meV) (meV) E2 (meV) E1 1
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Numerical solution Numerical solution (meV) (meV) (meV) E2 (meV) 512.047324 306.736857 200.986873 141.724429 105.323933 64.766509 43.842871
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20746 7.699 270.357809 270.764969 129.450671 129.450671 512.393113 129.247261 75.774672 307.023049 75.669905 307.023049 49.774624 201.176843 49.715086 201.176843 35.204788 141.851288 35.168092 35.168092 26.217761 105.411306 26.193670 26.217761 105.411306 26.193670 64.812071 16.146909 16.158835 64.812071 16.146909 10.943102 10.949827 43.869233 10.943102
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the wider wells, the discrepancy reduces to less than 0.1 meV. Such accuracies are entirely acceptable when modelling, e.g. experimental spectroscopic data. As this shooting equation for a variable effective mass will be used widely, it is worthwhile performing a few more convergence tests in order to increase confidence in its applicability. In particular, Table 3.5 repeats the calculations of the previous table but with a much higher barrier Al concentration. The effect of this is twofold, i.e. there is an increased difference in the potential between the well and barrier, but more importantly for this present section, there is an increased difference in the effective masses. The discrepancies between the analytical solution and the numerical solution are of a similar order as before, for this step length 6z of 1 A.
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Table 3.6 Comparison of the numerical solution with the analytical solution for a 20 A single GaAs quantum well surrounded by Gao.25Alo.75Asbarriers, as a function of the number (N) of points per A in the mesh note the step length 5z 1/N (A)
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N (A TV ( A 11) Analytical solution Numerical solution Analytical solution Numerical solution E1 (meV) E1 (meV) 2 4 6 8 10 12 270.764969 270.764969 270.764969 270.764969 270.764969 270.764969 270.663106 270.739499 270.753649 270.758601 270.760894 270.762139
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THE DOUBLE QUANTUM WELL
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However, as before, the agreement between the numerical solution and that obtained from the analytical form can be improved by increasing the computational accuracy of the shooting method, i.e. increasing the number of points per A (decreasing 8z). This is highlighted in Table 3.6, which compares the solutions from the two methods for a decreasing step length 6z. When using the form for the variable effective mass shooting equation in equation (3.53), the BenDaniel-Duke boundary conditions are 'hard-wired' in. Thus it is unnecessary to repeat the analysis described in Section 3.4 in order to recover them.