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Introduction to Solid State Physics Band Structure Theory and Periodic Density Functional Theory Vincent Cocula Emily A Carter Contents I Free Electron Gas 3 A Energy levels in one dimension 3 B Fermi energy 4 C Free electron gas in three dimensions 4 D Density of states 6 II Band Structure I A Chemical Approach 7 A Linear chain of hydrogen atoms 7 B Translational invariance Bloch s theorem 10 III Band Structure II A Physical Approach 10 A Reciprocal lattice Brillouin zones 10 B Bloch s theorem Generalization 14 IV Planewave density functional theory 15 A Periodic Calculations 15 B The Hohenberg Kohn theorems 17 C The Kohn Sham equations 19 D The exchange correlation potential 20 E Electron ion Interaction 22 F Self Consistent Field 23 G Planewave basis convergence and k point sampling 25 H Successive improvement of the trial wavefunction 27 I Electronic Temperature and Fermi surface smearing 28 2 Reference material C Kittel Introduction to Solid State Physics Ed Wiley Sons 1986 Ashcroft Mermin Solid State Physics Saunders College Publishing 1976 A Sutton Electronic Structure of Materials Oxford Science Publications 1996 P D Haynes Linear scaling methods in ab initio quantum mechanical calculations PhD Thesis Cambridge 1998 Britney s Guide to Semiconductor Physics http britneyspears ac lasers htm 3 I A FREE ELECTRON GAS Energy levels in one dimension Let s consider a free electron i e no forces confined in an infinite square well potential In one dimension the potential can be defined as V x 0 V x 0 x L elsewhere The wavefunction n x of the electron is a solution to the one dimensional Schrodinger equation h 2 d2 n x n n x H n x 2m dx2 where n is the energy of the electron occupying the orbital n The wavefunction should vanish at the boundaries so that n 0 n L 0 and the general solution of the eigen equation should be an imaginary exponential like function n x A exp ikx One can notice that the solution is simply a planewave with wave vector k consistent with the