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ECE453Fall2010HW5[1]

# ECE453Fall2010HW5[1] - (8x8 identity matrix Problem 2...

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Fall 2010 ECE 453 HW #5 Due 10/22 HW # 5 Due Friday, October 22, 2010 This homework is based on QTAT, Sections 5.1-5.2, 6.1-6.2. Sections3.1-3.3, 4.1-4.2 provide useful background material. QTAT: S.Datta, Quantum Transport: Atom to Transistor, Cambridge (2005), ISBN 0- 521-63145-9, on reserve in Engineering Library and HKN lounge Central result: [ h ( r k )] [ H nm ] exp( i r k .( r d m r d n )) m Bandstructure Problem 1: A molecule (NOT a solid) consists of eight carbon atoms arranged at the corners of a regular octagon of side ‘a’. Assume (1) one orbital per carbon atom as basis function and (2) the Hamiltonian matrix is given by H n , n (site energy) H n , m t if n, m are neighboring atoms H n , m 0 if n, m are NOT nearest neighbors What are the eight energy eig envalues in terms of ‘ ’ and ‘t’ ? Write down the corresponding eigenvectors. Assume the basis functions to be orthogonal so that the overlap matrix [S] is a

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Unformatted text preview: (8x8) identity matrix. Problem 2: Consider an infinitely long linear 1-D lattice (lattice constant: a) with one s-orbital per atom (assumed orthogonal) and having a site energy of ε , so that the Hamiltonian looks like <-- a --> Lessons from nanoelectronics: Copyright 2010 [email protected] All Rights Reserved 2 H te i te i te i te i te i Find the dispersion relation E(k). Problem 3: In class we have seen that an infinitely long linear 1-D lattice (lattice constant: a) with a Hamiltonian H t t t t t has a dispersion relation E ( k ) 2 t cos ka where ka (1) Do the same problem using a unit cell of two atoms (instead of one) and show that E ( k ) 2 t cos ka where /2 ka /2 (2) Are (1) and (2) equivalent? Explain....
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