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Electronic Structure Theory for Periodic Systems: the Concepts Electronic Structure Theory for Periodic Systems: The Concepts Christian Ratsch Institute for Pure and Applied Mathematics and Department of Mathematics, UCLA Motivation • There are 1020 atoms in 1 mm3 of a solid. • It is impossible to track all of them. • But for most solids, atoms are arranged periodically Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014 Outline Part 1 Part 2 • Periodicity in real space • The supercell approach • Primitive cell and Wigner Seitz cell • Application: Adatom on a surface (to calculate for example a diffusion barrier) • 14 Bravais lattices • Surfaces and Surface Energies • Periodicity in reciprocal space • Surface energies for systems with multiple • Brillouin zone, and irreducible Brillouin zone species, and variations of stoichiometry • Bandstructures • Ab-Initio thermodynamics • Bloch theorem • K-points Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014 Primitive Cell and Wigner Seitz Cell a2 a1 a2 a1 primitive cell Non-primitive cell Wigner-Seitz cell = + 푹 푁1풂1 푁2풂2 Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014 Primitive Cell and Wigner Seitz Cell Face-center cubic (fcc) in 3D: Primitive cell Wigner-Seitz cell = + + 1 1 2 2 3 3 Christian푹 Ratsch, UCLA푁 풂 푁 풂 푁 풂 IPAM, DFT Hands-On Workshop, July 22, 2014 How Do Atoms arrange in Solid Atoms with no valence electrons: Atoms with s valence electrons: Atoms with s and p valence electrons: (noble elements) (alkalis) (group IV elements) • all electronic shells are filled • s orbitals are spherically • Linear combination between s and p • weak interactions symmetric orbitals form directional orbitals, • arrangement is closest packing • no particular preference for which lead to directional, covalent of hard shells, i.e., fcc or hcp orientation bonds. structure • electrons form a sea of negative • Structure is more open than close- charge, cores are ionic packed structures (diamond, which is • Resulting structure optimizes really a combination of 2 fcc lattices) electrostatic repulsion (bcc, fcc, hcp) Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014 14 Bravais Lattices Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014 Reciprocal Lattice • The reciprocal lattice is a mathematically convenient construct; it is the space of the Fourier transform of the (real space) wave function. • It is also known as k-space or momentum space Example: Honeycomb lattice The reciprocal lattice is defined by Real space × Reciprocal space = 2 × 풂2 풂3 풃1 휋 풂1 ∙ 풂×2 풂3 = 2 3 ×1 2 풂 풂 풃 휋 2 ×1 3 = 2 풂 ∙ 풂 풂 × 풂1 풂2 풃3 휋 풂3 ∙ 풂1 풂2 Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014 Brillouin Zone The Brillouin Zone is the Wigner- The Irreducible Brillouin Zone is 3D Seitz cell of the reciprocal space the Brillouin Zone reduced by all symmetries Cubic structure Square lattice (2D) square lattice Body-centered cubic Hexagonal lattice (2D) hexagonal lattice Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014 Outline Part 1 Part 2 • Periodicity in real space • The supercell approach • Primitive cell and Wigner Seitz cell • Application: Adatom on a surface (to calculate for example a diffusion barrier) • 14 Bravais lattices • Surfaces and Surface Energies • Periodicity in reciprocal space • Surface energies for systems with multiple • Brillouin zone, and irreducible Brillouin zone species, and variations of stoichiometry • Bandstructures • Ab-Initio thermodynamics • Bloch theorem • K-points Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014 From Atoms to Molecules to Solids: The Origin of Electronic Bands Discrete states Localized discrete Many localized discrete states, states effectively a continuous band Silicon EC EC EV EV Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014 Bloch Theorem Suppose we have a periodic potential where is the lattice vector = 푹 + + 푹 푁1풂1 푁2풂2 푁3풂3 Bloch theorem for wave function (Felix Bloch, 1928): = 푖풌∙풓 ѱ풌 풓 푒 푢풌 풓 phase factor + = 푢풌 풓 푹 푢풌 풓 Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014 The meaning of k Free electron wavefunction = 푖풌∙풓 ѱ풌 풓 푒 푢풌 풓 = 2 / = 0 푘 휋 푎 푘 0 푘 ≠ = / Christian Ratsch, UCLA푘 휋 푎 IPAM, DFT Hands-On Workshop, July 22, 2014 Band structure in 1D Energy of free periodic potential weak interactions Only first parabola electron in 1D 휀 = 휀 휀 2 2 휀 ℏ 푘 휀 2푚 0 2 0 2 0 휋 휋 휋 휋 Extended zone scheme of 휋 푎 푎 푎 푎 푎 similar split at = , = Reduced zone scheme Repeated zone scheme 휋 2휋 푘 푎 푘 푎 We have 2 quantum 휀 휀 휀 • numbers k and n Energy • Eigenvalues are E (k) bands n 0 2 3 0 0 2 3 휋 휋 휋 휋 휋 휋 휋 Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014 푎 푎 푎 푎 푎 푎 푎 Band Structure in 3D • In principle, we need a 3+1 dimensional plot. • In practise, instead of plotting “all” k-points, one typically plots band structure along high symmetry lines, the vertices of the Irreducible Brillouin Zone. Example: Silicon • The band gap is the difference between highest occupied band and lowest unoccupied band. • Distinguish direct and indirect bandgap. Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014 Insulator, Semiconductors, and Metals Large gap gap is order kT No gap The Fermi Surface EF separates the highest occupied states from the lowest un-occuppied states Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014 Fermi Surface In a metal, some (at least one) energy bands are only partially occupied by electrons. The Fermi energy εF defines the highest occupied state(s). Plotting the relation = in reciprocal space yields the Fermi surface(s). 휖푗 풌 휖퐹 Example: Copper Fermi Surface Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014 How to Treat Metals Fermi distribution function fF enters Brillouin zone integral: = ( ) ( ) , 3 2 푗 푗 풌 푑 풌 푛 풓 � � 푓 휖 휓 풓 퐵퐵 • A dense k-point mesh is necessary.푗 Ω퐵퐵 Ω • Numerical limitations can lead to errors and bad convergence. • Possible solutions: el • Smearing of the Fermi function: artificial increase kBT ~ 0.2 eV; then extrapolate to Tel = 0. 1 = 2 1 + ( )/ 푗 푒푒 • Tetrahedron method푓 휖 [P. E. Blöchl휖푗−휖 퐹et 푘al.,퐵푇 PRB 49, 16223 (1994)] 푒 • Methfessel-Paxton distribution [ PRB 40, 3616 (1989) ] Occupation • Marzari-Vanderbilt ensemble-DFT method [ PRL 79, 1337 (1997) ] 0 EF energy Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014 k-Point Sampling Charge densities (and other quantities) are • For smooth, periodic function, use (few) special k- represented by Brillouin-zone integrals points (justified by mean value theorem). Brillouin zone replace integral by discrete sum: Irreducible Brillouin zone • Most widely used scheme proposed by Monkhorst and Pack. • Monkhorst and Pack with an even number omits high symmetry points, which is more efficient (especially for cubic systems) Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014 Outline Part 1 Part 2 • Periodicity in real space • The supercell approach • Primitive cell and Wigner Seitz cell • Application: Adatom on a surface (to calculate for example a diffusion barrier) • 14 Bravais lattices • Surfaces and Surface Energies • Periodicity in reciprocal space • Surface energies for systems with multiple • Brillouin zone, and irreducible Brillouin zone species, and variations of stoichiometry • Bandstructures • Ab-Initio thermodynamics • Bloch theorem • K-points Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014 Supercell Approach to Describe Non-Periodic Systems A infinite lattice A defect removes Define supercell to get periodicity back a periodic system the supercell the unit cell Size of supercell has to be tested: does the defect “see itself”? Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014 Adatom Diffusion Using the Supercell Approach Diffusion on (100) Surface Supercell Need to test Lateral supercell size Vacuum separation Adsorption Transition site Transition state theory (Vineyard, 1957): slab thickness Vacuum separation D = Γ0 exp(−∆Ed / kT) ∆Ed = Etrans − Ead 3N ν ∏ j=1 j Lateral Attempt frequency Γ0 = − 3N 1ν * ∏ j=1 j supercell size slab thickness Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014 Surface Energies Example: Si(111) Surface energy is the energy cost to create a surface (compared to 1 (E (DFT ) − Nµ) a bulk configuration) γ = Surface 2 area (DFT ) ∞ E : Energy of DFT slab calculation ∞ N : Number of atoms in slab Bulk system µ : Chemical potential of material Surface 1.38 1.37 1.36 ∞ 1.35 1.34 Surfaceenergy per (eV)atom 0 5 10 15 ∞ Number of layers Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014 Ab-Initio Thermodynamics to Calculate Surface Energies For systems that have multiple species, things are more complicated, because of varying stoichiometries Example: Possible surface structures for InAs(001) α(2x4) α2(2x4) α3(2x4) In atom As atom β(2x4) β2(2x4) β3(2x4) Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014 Surface Energy for InAs(100) (DFT ) γ InAs−Surface = EInAs−slab − N InµIn − N As µ As (bulk ) with µi ≤ µi Equilibrium condition: (bulk ) µIn + µ As = µInAs = µInAs α2 (2x4) This can be re-written as: (DFT ) γ InAs−Surface = EInAs−slab − N InµInAs − (N As − N In)µ As (bulk ) (bulk ) (bulk ) with µInAs − µIn ≤ µ As ≤ µ As Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014 Conclusions Thank you for your attention!! Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014 .
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