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MIT 8 512 - LECTURE NOTES

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MIT OpenCourseWarehttp://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 Due March 3, 2008 8.512 Theory of Solids II Problem Set 3 1. (a) Consider a one-dimensional chain of hydrogen atoms with lattice spacing a. Using a single 1s orbital per atom, construct the tight binding band. You may keep only the nearest-neighbor matrix element V (a) = �φ(r)|H|φ(r + a)� , and ignore the overlap �φ(r)|φ(r + a)� . Assume V < 0. Where is the Fermi energy? (b) Now assume that the nth atom is displaced by a small amount (−1)nδ along the chain direction. For small displacement show that the matrix elements are alternating V + Δ and V − Δ, where Δ = 2δ dV . What is the new band structure? Is the system a da metal or an insulator? (c) Calculate the change in the electronic energy upon distortion. Show that it is of the form Δ2 ln |Δ/V | in the limit |Δ| � |V |. Compute the coefficient of this term. Hint: Make use of the fact that |Δ| � |V |. Then the contributions to the energy change come mainly from momentum states near k = ±π/2a, where cos ka and sin ka can be expanded to leading order. (d) The displacement costs lattice energy which is of the form bδ2 in the harmonic approxi-mation. Show that the uniform chain is unstable to the distortion assumed in part (b). Similar arguments were put forward by Peierls in 1950 to show that a one-dimensional metal is unstable to distortions which turn it into an insulator. (e) Evaluate the polarization function Π0(q, ω = 0) for a one-dimensional free Fermion gas. Show that a logarithmic singularity appears at q = 2kF . 2. Consider a two-dimensional electron gas (electron motion is confined to the x-y plane). What is the plasmon dispersion ωpl(q) for small q in the plane? Show that ωpl is proportional to |q|1/2 . Hint: Note that while the electrons are confined to the plane, the electromagnetic field is not. March 2,


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