HOMEWORK ASSIGNMENT #3, PHYSICS 5335, …



BONUS HOMEWORK #1, PHYSICS 5335, FALL, 2020NOTE: The solution to the following problem requires the use of Quantum Mechanical Perturbation theory. Since some in this class might not know about this, this problem is optional and can be turned in for EXTRA CREDIT any time before the last class day, Tuesday, December 1!Deep Levels: A simple one dimensional model #1. See Figures. Consider a one-dimensionalFigure alattice with electronic bandstructuresdescribed by The Kr?nig-Penney Model,which was introduced & discussed near thebeginning of our band structure discussion.Recall that this model consists of a onedimensional periodic array of alternatingpotential wells & barriers, as shown inFigure a. The barriers all have height Vo & width c. The wells all have width b, & the periodic repeat distance is a = b + c. The resulting band structure wasthoroughly discussed during the band structure lectures. See Ch. 2, pages 33-35Defect Siteof the book by Balkanski & Wallis. Vd Figure bNow, consider a substitutional defect atone of the lattice sites, as in Figure b.bUse the approximation made by theHjalmarson theory of deep levels discussed in some detail in class thatthe defect potential consists of ashort-ranged, central cell potentialonly so that there is no long-ranged, screened Coulomb potential contribution to the defect potential. In particular, let the defect potential be confined to only one lattice site, as shown. Thisapproximation also assumes that there is no lattice relaxation at the defect site.For definiteness, let the defect barrier height be Vd, where Vd > Vo & assume thatthe defect well & barrier widths retain their host lattice values b & c, respectively.Use quantum mechanical perturbation theory & go as far as you can toapproximately calculate the “deep level” energy E associated with this localizeddefect. The energy in the first-order perturbation theory approximation issufficient. Note: In approaching this problem with perturbation theory, theperturbation that should be used is V = Vd – Vo. Of course, perturbation theoryalso requires that you use the wave functions for the unperturbed system. Theelectronic wave functions for The Kr?nig-Penney Model are discussed on page34 of the book by Balkanski & Wallis. ................
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