141A_PS7_sp07 - PHYSICS 141A Problem Set #7 S. G. LOUIE...

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PHYSICS 141A S. G. LOUIE SPRING 2007 Problem Set #7 Due: Friday, 03/16/07 Reading: Finish Chapter 6 and begin Chapter 7 of ISSP (25) 1. ISSP, Ch. 6, Problem 9: Static magnetoconductivity tensor. (25) 2. Density of states—nanometric wire . a) Consider a nanometric wire in the form of a rectangular parallelepiped, with two sides L x " L y " 1 nm and the long axis L z " 1 cm. The single particle eigenstates of the system may be written as " = sin(n x # x / L x )sin(n y y / L y ) exp(i 2 Nz). Show that the energy of the eigenstate with quantum numbers n = (n x , n y ) and N satisfies: E( n ,N) " n = ( 2 $ h N) 2 / 2 m = AN 2 = 1 2 mv n (E) 2 , where n # E( n ,N = 0 ) and v is the electron velocity along the z-axis. Here A = ( 2 h ) 2 / 2 m and N = [(E " n )/ A ]. Then given that δ E = 2AN N , show that the density of states per unit length in the z direction D n at fixed n , with account of the two spin orientations and the two ± values of N , is D n (E) = 4 N /
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141A_PS7_sp07 - PHYSICS 141A Problem Set #7 S. G. LOUIE...

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