homework4 - 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 4 ` Due on Feb. 17, 2009 at 5:00 PM Suggested Readings: a) Lecture notes Problem 4.1 (1D lattice) Consider a 1D lattice of lattice constant a equal to 5 Angstroms. In this problem you will find an almost exact numerical solution to the problem of an electron in a periodic 1D lattice and compare this solution with approximate analytical methods discussed in the lecture. Suppose the lattice potential can be written as: () = x a V x V π 2 cos 2 1 Where 1 V equals 0.2 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 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. Now the solution looks more like a variational solution. a) Plug the assumed form of the solution in the Schrodinger equation and show that the resulting matrix equation looks like as shown below for the case when N =2.
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2 () = + + + + M M M M O O 2 1 1 2 2 1 1 2 2 1 1 1 1 1 1 1 1 1 1 2 0 0 G c G c c G c G c k E G c G c c G c G c G k e V V G k e V V k e V V G k e V V G k e The advantage of this approach is that now one can solve the matrix eigenvectors and eigenvalues numerically, and in the limit N becomes large the solution obtained is pretty much the exact solution. For numerical solution, you will need to use software like Matlab or Mathematica. a) Assume N =10. For each value of k in the FBZ from 0 to π / a , numerically solve for the smallest three eigenvalues of the matrix above. This will give you the energies, ( ) k E 1 , ( ) k E 2 , and k E 3 of the lowest three bands. Plot these energies vs the k-vector from 0 to to π / a .
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This note was uploaded on 04/04/2009 for the course ECE 4070 taught by Professor Rana during the Spring '08 term at Cornell.

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homework4 - Department of Electrical and Computer...

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