Solid State Theory: Band Structure Methods
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1 Solid State Theory: Band Structure Methods Lilia Boeri Wed., 11:15-12:45 HS P3 (PH02112)
2 Plan of the Lecture: DFT1+2: Hohenberg-Kohn Theorem and Kohn and Sham equations. DFT3+4: Solving K-S in practice; basis functions, augmented methods and psp theory. DFT5: Practical problems in DFT (k space integration, convergence etc) P1: EOS and band structure of silicon. ADV1+2: Linear Response theory (mostly for phonons). P2: Phonons of silicon ADV3: Wannier Functions and TB approximation. P3: Wannier Functions and BOM for silicon.
3 Introduction to Density Functional Theory(DFT 1-2): Quantum-Mechanical Many-Electron Problem. Main Concepts of Kohn-Sham (Spin)-Density Functional Theory. Wavefunction Theory. Density Functional Theory (Density is the Basic Variable). Uniform electron gas. For this and the following 2 lectures I will be following A primer in Density Functional Theory (Springer), chapters 1 and 6. A free (pdf) copy can be downloaded at this address:
4 Quantum-Mechanical Many-Electron Problem The material world of everyday experience is built up from electrons and a few (hundreds) kinds of nuclei. The interactions between them is electrostatic (Coulomb). In most cases, one assumes that the energy scales of electrons and nuclei are well separated in energy, so that the electrons maintain their fully quantum nature, while ions are treated semiclassically (Born-Oppenheimer approximation). The problem for the electrons is described by the following Hamiltonian: Ĥ = NX i=1 1 2 r2 i + X i X I I r i R I +1 2 X i X j6=i 1 r i r j +1 2 X I X J6=I I J R I R J Finding the solutions (eigenvalues, eigenfunctions) of this quantum-mechanical many-electron problem is the main problem of solid state physics, and is stil unsolved. Ĥ k(r 1 1,...,r N N )=E K k(r 1 1,...,r N N )
5 Main Concepts of Density Functional Theory: The quantum-mechanical many-electron problem can be greatly simplified if we are only interested in its ground-state properties. 0(r 1 1,...,r N N ) n 0 (r) = X 2... N d 3 r 2...d 3 r N 0(r,...,r N N ) 2 Hohenberg-Kohn Theorem: The ground-state energy of a system of interacting electrons is a function of its ground-state density only. The complications induced by the electron-electron interaction are dumped into an effective exchange and correlation energy, whose exact form is unknown (but good approximation exists). Kohn-Sham Equations: It is possible to find the ground-state density of the interacting system solving self-consistently a system of single-particle equations for the auxiliary (effective) Kohn- Sham quasi-particles.
6 Main Concepts of Density Functional Theory: In the (spin-)density functional method, we seek the ground-state total energy E and the spin densities for a collection of N electrons which interact with each other, and with a given external potential (typically a sum of nuclear potentials). The energy and densities can be found by self-consistent (scf) solution of a set of equations. 1 2 r2 + v(r)+u([n];r)+v xc ([n ",n # ];r) (r) = (r) Kinetic External Hartree Exchange- Energy Potential Potential Correl. Potential The Hartree and Exchange-Correlation energies are functionals of the (spin) electron density. The solutions of this set of one-particle equations are the auxiliary Kohn-Sham quasi-particles (Kohn, Sham, Phys. Rev. 1965).
7 Electron Density The ground state electron density is defined, in terms of the K-S quasi-particles, as: n (r) = X (µ " ) (r) 2 n(r) =n " (r)+n # (r) μ is the chemical potential, defined by: N = d 3 rn(r) FULL EMP Although the Kohn-Sham quasi-particles have no clear physical meaning, they are constructed in such a way that their ground state density is the same as that of the original, interacting μ
8 Main Concepts of K-S Density Functional Theory: The Hartree Potential is a functional of the K-S electron density: u([n];r) = d 3 r 0 n(r 0 ) r r 0, n(r) =n "(r)+n # (r)? functional is a rule which assigns a number F[f] to a function f. What is a functional? While a function is a rule which assigns a number f(x) to a number x, a EX: E [ ]=h Ĥ i The Hartree Potential represents the repulsive potential experienced by each K-S quasi-particle due to average effect of all other quasi-particles.
9 Main Concepts of Density Functional Theory: The Exchange and Correlation potential is an unknown function, which contains all the (unknown) manybody effects, not included in the first two terms of the Kohn-Sham equations (effective kinetic energy term, Hartree potential). This is usually expressed as the functional derivative of an (unknown) exchange and correlation energy: v xc ([n ",n # ];r) (r) Hohenberg and Kohn Theorem (Phys. Rev. 1964) states that if the exact form of the exchange and correlation energy was known, it would be possible to calculate the ground state energy exactly, knowing only the ground-state electronic density. This is not possible but good approximations exist: E LSD xc [n ",n # ]= d 3 rn(r)e xc (n " (r),n # (r)) Local (Spin) Density Approximation E GGA xc [n ",n # ]= d 3 rf(n ",n #, rn ", rn # ) Generalized Gradient Approximation
10 What is a self-consistent solution? The Schroedinger s equation for the Kohn-Sham quasi-particles depends on the ground-state density; on the other hand, the ground-state density is obtained summing over the occupied K-S orbitals. 1 2 r2 + v(r)+u([n];r)+v xc ([n ",n # ];r) (r) = (r) n (r) = X (µ " ) (r) 2 1. Start from a set of guess functions. 2. Calculate the corresponding charge density. 3. Compute the Hartree and xc potentials Iterate until the charge densities at step t and (t+1) differ by less than a given threshold 4. Insert into the Schoedinger equations.
11 DFT Total Energy The ground-state energy is a functional of the ground-state density: E = T s [n ",n # ]+ d 3 rn(r)v(r)+u [n]+e xc [n ",n # ] Kinetic Energy (of the K-S system), do not confuse with that of the interacting system!!! T s [n ",n # ]= X X (µ " )h 1 2 r2 i Hartree Energy (electrostatic): U [n] = 1 2 d 3 r d 3 r 0 n(r)n(r0 ) r r 0 Exchange-Correlation Energy (unkwown): v xc ([n ",n # ];r) (r)
12 DFT Total Energy DFT works because the x-c energy (the only approximate part) is a small fraction of the total energy of an atom, molecule or solid. However, this small energy is usually a large fraction (> 90%) of the energy needed to break up a solid or molecule (binding energy) or to take one electron away from an atom (atomization energy).
13 Introduction to Density Functional Theory(DFT 1-2): Quantum-Mechanical Many-Electron Problem. Main Concepts of Kohn-Sham (Spin)-Density Functional Theory. Wavefunction Theory. Density Functional Theory (Density is the Basic Variable). Uniform electron gas. For this and the following 2 lectures I will be following A primer in Density Functional Theory (Springer), chapters 1 and 6. A free (pdf) copy can be downloaded at this address:
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