For a certain device (that can be used as a detector of optical radiation) we fabricate a coupled well structure, where electrons sitting inside GaAs layers see the interface with AlGaAs layer as a potential barrier of height v= 60 eV. Assume that the width of the GaAs wells is d=1 nm, while the width of the barrier is w=0.7 nm , the mass of electrons inside GaAs is 0.092m, mass of AlGasAs= 0.092m, where m is the mass of the free electron.

1. Compute energies of the first three energy levels for a single well with given parameters. Do it, first assuming that the AlGaAs barrier has infinite height, then repeat these calculations for the actual barrier height. Compare and comment on the results.

2. Plot eigenfunctions for each of the three levels found in the case of the barrier of the finite height.

3. Still considering a single well, assume now that a constant electric field of strength is applied along the well. Using non-degenerate perturbation theory, find second order corrections to energies and first order corrections to the wave functions assuming potential barrier of the finite height . Plot the unperturbed and the perturbed wave functions on the same plot. Comment on the results. (Take into account all discrete energy levels, but do not include into consideration states corresponding to continuous spectrum).

4. Now consider the system of two wells as shown in the figure. Derive an equation for energies of the bound states in this system, applying boundary conditions at each interface (you will end up with 8 equations). But if you choose the origin of the coordinate system at the center of the AlGaAs barrier, you can use the symmetry arguments and consider odd and even solutions separately, which will reduce the number of equations to four for solutions of each parity. Find discrete energy levels using first graphing method

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