496_HW4sol

# 496_HW4sol - ECE 453 Fall’05 EE 117 MWF 330P — 420p...

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Unformatted text preview: 10/07/05 ECE 453, Fall’05 EE 117, MWF 330P — 420p Introduction to N anoelectronics All exercises, page numbers refer to S.Datta, Quantum Transport: Atom to Transistor, Cambridge (2005) HW#5: Solution Problem 1: Exercise E.5.4, Page 128. E = a+2\$0m4éa+2¢06011£6b M £30 160 ﬂ , 2/6 + 2/0 Problem 2: Consider an infinitely long linear l—D lattice (lattice constant: a) with one s— orbital per atom (assumed orthogonal) and having a site energy of , so that the Hamiltonian looks like 10/07/05 Impose periodic boundary conditions and assume a solution of the form (15” = ([50 eikna to find the dispersion relation E(k). Problem 6: The E(kx,ky) relation for some solids is often written in the form E = E0 — 2V (coskxa + coskya + 20!. coskxa cos kya) Where 0t is a dimensionless number. How would you choose the nearest neighbor and next nearest neighbor overlap matrix elements in a square lattice of side 'a' so as to correspond to this dispersion relation ? 00‘ —t .0. 0 Next nearest neighbor overlap : — t' —t .0. Nearest neighbor overlap : — t ._ t' (ei(kx+ky)a + ei(kX—ky)a + e—i(kx +ky)a + e—i(kx —ky )a) = E0 — 2t (cos kxa + cos kya) — 4t' (cos kxacos kya) Hence, V =t and Va = t' ...
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496_HW4sol - ECE 453 Fall’05 EE 117 MWF 330P — 420p...

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