Magnetic Research Department

How do we work?

"Aright, nobody move! I've got a dragon and I'm not afraid to use it!" ["Shrek", the movie]

Applications of Density Functional Theory (DFT)
in band-structure calculations
by Wojciech Miiller

Since physicists realized, what is responsible for thermodynamic and transport properties of materials, they tried to find a hint - How to predict material's physical properties, using only composition and structure as an initial condition?

Material = Atoms + Bonds

Atoms, of course elements, are bricks of all matter around us and bands between them are a kind of clay, which glues material and is responsible for a fact, that one compound is hard and fragile and another one soft and rigid. For character of bonds are responsible electrons.

All bonds in a solid state body may be divided on four general categories:

ionic bond, where compound consists of separated ions (charged atoms) and there is no exchange of electrons between them (for example, rock salt - NaCl). Coulomb force between contrary charged ions is responsible for solidification of such material.
covalent bond, where some electrons between neighboring atoms are jointed. This bound is strictly directed and very strong. Examples: diamond, silicon.
metallic bond, where all atoms contribute some electrons to communite bond, called a valence band. This bond is not directed, that's why metals are ductile. Almost free electrons make metal a good heat and electricity conductor. Also electrons in such valence band are responsible for a light scattering, what makes metal's surface shiny.
molecular bound, but it is not such ordinary as previous-who's interested, please look on the Wikipedia ;-)

Of course, nothing is perfect. These four categories involve only few compounds. All other, thousands of materials contain a complex character of bonds - partially electrons are localized, partially itinerant, sometimes character of the electrons changes in function of temperature or other external parameters as pressure or magnetic field.

How to predict a character of such compound as, for example, FeSi₃? Iron is metallic and ferromagnetic, silicon is well known as a semiconductor, with covalent bounds. Is FeSi₃ metallic or semiconducting? Does it order magnetically or not? What is responsible for experimental behaviour? We can't find all answers, but clues leading to the solution: why this material behaves like that?

We can say: let us calculate all physical properties: electrical resistivity, specific heat, magnetic susceptibility, Young modulus etc. But how? We may assume some interaction between electrons and atomic lattice and between electrons. But there are about 10²³ electrons in bulk material! It would look like this:

1^st electron interacts with other 9 999 999 999 999 999 999 999 electrons,
2^nd electron feels influence of 9 999 999 999 999 999 999 998 (we have already calculated an energy of interplay between 1^st and 2^nd electron).
3^rd electron...

Number of steps in one iteration is easy to calculate: 10²³!. And we have calculated only interactions between electrons, but we should consider also lattice... It is unimaginable for common computers.

In 1965 two guys, Kahn and Sham, proposed: lets treat electrons not as separated balls (or wave functions) but as a cloud of charge delineated by a density of electrons. They assumed that energy of system may look like that:

E_tot(ρ_u,ρ_d) = T(ρ_u,ρ_d) + E_ee(ρ_u,ρ_d) + E_ne(ρ_u,ρ_d) + E_xc(ρ_u,ρ_d) + E_NN,

where:

E_tot(ρ_u,ρ_d) - total energy of the system
T(ρ_u,ρ_d) - kinetic energy of electrons (treated as non-interacting particles)
E_ee(ρ_u,ρ_d) - Coulomb repulsion between electrons
E_ne(ρ_u,ρ_d) - nuclei-electron attraction
E_xc(ρ_u,ρ_d) - electron-electron exchange correlations
E_NN - Repulsive energy of Coulomb interaction between ions in crystal lattice
(ρ_u,ρ_d) - density of electrons in two spin channels (we treat electrons like little magnets with magnetic moment up (u-index) or down (d))

The adaptation of this method in calculations is easy: valence electrons (from outer shells of the atom) may be treated as itinerant and extended states. They are described as plane waves, which contribute to the density of the valence band. Electrons with lower energies, close to the nuclei, are freezed as core states and treated as spherical waves.

In every iteration is calculated energy, program finds trend - which part of electrons (bands) should have lower or higher energy, to minimalise all systems energy. It is realization of 1^st principle of the thermodynamics. As input we describe sort of crystal structure and occupancy of electrons of each element of compound. After convergence (mineralization of energy) as output we receive, for example, density of states (number of electrons in function of their energy), magnetic moment on each kind of nuclei, character of the band structure (metal, semiconductor or insulator). We can't describe all forces between particles, but using some theoretical assumptions, it is possible to find probable scenario of phenomenon inside a bulk. Also all of that dependence on the assumptions (boundary conditions) made on the beginning of calculations. Every decision on start induces an avalanche of implications. We decide, which electrons are treated as valence states or core-levels. So, just like in everything in real life, band structure calculations need experience and patience.

In our department we use such packages as WIEN2K (www.wien2k.at) and FPLO (www.fplo.de).