MIT OpenCourseWare http ocw mit edu 8 512 Theory of Solids II Spring 2009 For information about citing these materials or our Terms of Use visit http ocw mit edu terms 1 8 512 Theory of Solids II Problem Set 4 Due March 10 2008 1 Consider a tight binding model on a lattice with hopping matrix element t Add an on site disorder potential Vi where Vi is a random variable distributed uniformly between W2 Consider the one dimensional case with N sites a For a given realization of Vi consider the eigenvalues for periodic boundary con ditions i e VN V1 and N 1 where 1 is the wavefunction on site i The eigenvalues are solved by diagonalizing an N 1 N 1 matrix Set up the form of the matrix b Now consider a twisted boundary condition i e VN V1 and N 1 ei How is the matrix modi ed from a c Show that the eigenvalues E in b are equivalent to a problem with complex hopping i e t is replaced by tei N 1 and with periodic boundary conditions This is the problem of a ring with N 1 sites with a magnetic ux through the ring What is the value of the ux in units of the ux quantum hc e d Diagonalize the matrix numerically for a given realization of disorder Choose W t 2 0 and N 20 Plot the energies of the 10 levels near E 0 as a function of Now increase N and observe how the picture changes e For the values of W t chosen in part d calculate the dimensionless conductance of the sample using the Thouless formula G ET where ET d2 E d 2 and is the average energy level spacing Calculate ET and by averaging over the 10 levels near E 0 and by averaging over a number of realizations of the random potential Check the dependence of G as a function of sample size N 2 f Optional If you are interested you may repeat the problem for a two dimensional square lattice and contrast the behavior Compare W t 4 and 9
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