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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 fact that our electron is treated as a free particle Also one can take a particular solution of the Schrodinger equation to be a sine function n x A sin kx where A is a constant Solving for k the electronic wavefunction becomes n x n x A sin L where n is an integer number of half wavelengths between 0 and L The orbital energies are easily evaluated and take on the values n h 2 n h 2 k 2 2m 2m L 2 4 FIG 1 Energy levels in one dimension taken from Kittel B Fermi energy Let s concider now that we have N electrons confined in our one dimensional potential The Fermi energy F is defined as the energy of the topmost filled level in the ground state of the N electron system Since electrons have to obey the Pauli Principle each orbital can only be filled by 2 electrons at most The Fermi level corresponds then to the level with nF N 2 and the Fermi energy is F C h 2 N 2m 2L 2 Free electron gas in three dimensions The generalization of the problem from one to three dimensions is rather straightforward We now consider a free electron gas of N particles confined in a cubic box of edge L The Schrodinger equation now becomes h 2 2 k r k k r 2m for which the wavefunctions are required to be periodic in the three directions x y and z so that k x L y z k x y z and similarly for the y and z directions The wavefunctions satisfying those conditions are again the free particle planewaves thus 5 FIG 2 3D Fermi sphere within the k space taken from Britney Spears k r exp ik r with wave vector k 2 kx2 ky2 kz2 for which the components of the wave vector satisfy kx 0 2 L 4 L Similarly the energy k of the orbital with wave vector k is then 6 FIG 3 Density of states taken from Britney Spears k h 2 2 h 2 2 k k ky2 kz2 2m 2m x This has an important consequence and gives rise to the Fermi sphere In the ground state of the N electron system the orbitals may be represented as points within the 3dimensional k space The occupied orbitals are included in a sphere and the energy at the surface of this sphere is the Fermi energy The sphere is called the Fermi sphere and the Fermi energy is such that F D h 2 2 k 2m F Density of states The total number of electrons contained in the Fermi sphere within the volume element 2 L 3 is given by 7 2 4 kF3 3 V 3 k N 3 3 2 F 2 L which gives us an expression for the three dimensional Fermi vector kF 3 2 N V 1 3 and thus to the Fermi energy h 2 F 2m 3 2 N V 2 3 We define the density of states D as the number of electrons per unit energy range and is given by dN V D 2 d 2 II 2m h 2 3 2 1 2 BAND STRUCTURE I A CHEMICAL APPROACH We now go back to a one dimensional system In the previous section we have just been solving the Schrodinger equation for an electron in a finite system of arbitrary length L supposed small This cannot be a good approximation of an extended solid but it is reasonable to suppose that a solid could be modelled by an infinite collection of such systems in 3 dimensional space In this section we will take the fictitious linear chain of hydrogen atoms as an example to model our solid system and we will show that this gives rise to a band structure We will also introduce very briefly the ideas underlying Bloch s Theorem A Linear chain of hydrogen atoms A solid is nothing but a collection of atoms arranged within some symmetrical order In one dimension one can model such a system by i e a linear chain of N hydrogen atoms Each hydrogen atom is associated with an s state and our aim will be to find the expression for the molecular orbitals for the chain In the LCAO picture the total wavefunctions will be 8 FIG 4 Energy spectrum for the linear chain of hydrogen atoms taken from Sutton i N X cj ji j 1 where ji stands for the s wavefunction of the jth atom in the chain The task is then to find the molecular coefficients cj and the energy of the molecular orbital The Schrodinger equation to be solved is H i E i which in terms of the atomic orbitals becomes N X cj H ji E j 1 N X cj ji j 1 We have to solve for the Hamiltonian matrix elements using the secular equation N X cj hp H ji E j 1 N X cj hp ji j 1 where …