homework5

# homework5 - Department of Electrical and Computer...

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1 Department of Electrical and Computer Engineering, Cornell University ECE 407: Physics of Semiconductor and Nanostructures Spring 2009 Homework 5 ` Due on Feb. 24, 2009 at 5:00 PM Suggested Readings: a) Lecture notes Problem 5.1 (1D lattice energy bands outside the FBZ) In the last homework, in problem 4.1, you found the exact solution for an electron in a periodic 1D lattice. You will consider the same problem again here. Consider a 1D lattice of lattice constant a equal to 5 Angstroms. Suppose the potential has the form: () + = x a V x a V x V π 4 cos 2 2 cos 2 2 1 Where 1 V equals 0.3 eV and 2 V also equals 0.3 eV. The exact solution for any wavevector k in the FBZ can be written as a superposition of plane waves in the form: = = −∞ = + −∞ = + m x G k i m m G k m k k m m e L G c G c 1 φ ψ Where: a m G m 2 = A good approximation to the exact solution can be obtained by terminating the series above at both ends, as follows: = = + N N m G k m k k m G c Where N is some large number, say 10. The way you hopefully solved the problem in homework 4 was to first choose a value of the wavevector k in the FBZ, then setup a matrix, and then find its three smallest eigenvalues. The question is what if one chooses a value of the wavevector k that is not in the FBZ? Would one end up with some new energy eigenvalues and new energy eigenfunctions? The goal of the problem is to explore this point.

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homework5 - Department of Electrical and Computer...

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