573lec38

573lec38 - 38 - 1 5.73 Lecture #38 Infinite 1-D Lattice II...

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38 - 1 5.73 Lecture #38 updated September 19, 20032:22 PM LAST TIME : Infinite 1-D Lattice II H 2 + localization tunneling: overlap bonding and antibonding orbitals R vs. distance below top of barrier an 0 2 1-D lattice: 1 state per ion tunneling only between nearest neighbors H matrix 0 = c q (E 0 – E) – A(c q–1 + c q+1 ) # of coupled equations Usually solve for {c q } by setting determinant of coefficients = 0 and solving for E. Can’t do this because determinant is . TRICK: expect equal probability of finding e on each lattice site by analogy to plane wave e ikx , where probability density is uniform at all sites along x, try c q = e ikq l x q = q l integer distance between lattice sites H H = = = =−∞ EA AE E c qq q 0 0 0 0 00 O O OO νϕ ν ϕ ϕν TIGHT-BINDING (Kronig-Penney) Model (see Baym pp. 116-122) Notice that this is similar to free particle e ikx , which seems rather strange because particle is never really free in “tight-binding” model. Variational wavefunction. Minimize E.
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38 - 2 5.73 Lecture #38 updated September 19, 20032:22 PM c q 2 1 = divide through by e EEA k ikq l l 02 0 =− () cos Ek E A k 0 2 cos l E(k) k E 0 −π l l 0 E varies continuously over an interval 4A, where A is the adjacent site interaction strength or the “tunneling integral” What happens when we look at k outside - π / l k < π / l “1st Brillouin Zone” ce kk e e e k ikq k ik q ikq i q ikq = =+ π == = + π π l l l ll l 2 2 2 (one additional wavelength per lattice spacing l ) wavefunction is unchanged! So if k goes outside 1st Brillouin Zone, get same ψ , so get same E nothing new!
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This note was uploaded on 11/28/2011 for the course CHEM 5.74 taught by Professor Robertfield during the Spring '04 term at MIT.

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573lec38 - 38 - 1 5.73 Lecture #38 Infinite 1-D Lattice II...

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