MIT OpenCourseWare http 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 8 512 Theory of Solids II 1 Problem Set 3 Due March 2008 March 2 3 2009 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 r H r a and ignore the overlap r r a Assume V 0 Where is the Fermi energy b Now assume that the nth atom is displaced by a small amount 1 n along the chain direction For small displacement show that the matrix elements are alternating V and V where 2 dV da What is the new band structure Is the system a metal or an insulator c Calculate the change in the electronic energy upon distortion Show that it is of the form 2 ln V in the limit V Compute the coe cient of this term Hint Make use of the fact that V Then the contributions to the energy change come mainly from momentum states near k 2a where cos ka and sin ka can be expanded to leading order d The displacement costs lattice energy which is of the form b 2 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 0 q 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 con ned to the x y plane What is the plasmon dispersion pl q for small q in the plane Show that pl is proportional to q 1 2 Hint Note that while the electrons are con ned to the plane the electromagnetic eld is not
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