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

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

• 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

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

• 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