Fundamentals of DFT Bandstructure Calculations of

“Fundamentals of DFT Bandstructure Calculations of Defect-Free Materials” Kamil Walczak Department of Chemistry and Physical Sciences, Pace University, 1 Pace Plaza, New York, NY 10038 Formation of Bandstructures

INTRODUCTION We show how to use the density functional theory (DFT) as formulated within Kohn-Sham (KS) equations to perform banstructure calculations of defect-free materials by imposing periodic boundary conditions (which mimic Bloch waves). This self-consistent DFT method allows us to obtain the E-k diagrams in the whole range of available wavevectors, and the corresponding densities of states (DOS) as functions of electron energy. Since DFT calculations based on Local Density Approximation (LDA) for exchange-correlation (XC) potential underestimate bandgaps typically by 30-50%, it is also presented how to fix the bandgap problem within the GWA-DFT approach based on the theory of propagators (known also as a Green’s functions technique). The E-k diagrams and their corresponding densities of states (DOS) are shown for bulk silicon, bulk copper, and 2D graphene sheet, indicating contributions from individual atomic orbitals as incorporated into Slater determinants. Since we analyze defect-free materials, we neglect charge redistributions and band bending effects due to surface and dopant states.

Fixing Bandgap Problem: GWA-DFT

E-k Diagram & DOS: Bulk Silicon

Classification of Solids

E-k Diagram & DOS: Bulk Copper

Kohn-Sham Density Functional Theory A single-electron Kohn-Sham (KS) equation:

     KS     KS [n ] i ( r )  E i i ( r )  2m  2

The electron density expressed in terms of KS orbitals: N      2 3 3 2 n ( r )  N  d r2 ... d rN | ( r , r2 ,..., rN ) |   | i ( r ) | i 1

Electron density function must be normalized via the relation:

 3 n ( r ) d r  N 

Self-Consistent KS DFT Scheme

The KS potential as a functional of electron density:

 kq Z E XC [n ] kq n ( r ' ) 3 KS [n ]         d r '   | r  r '| n ( r )  1 | r  R  | M

2

 ext ( r )

Hartree [n ]

XC [n ]

   3 E XC [n]    X [n( r )]  C [n( r )]n( r )d r  1/ 3  X [n]   D [n( r )]

E-k Diagram & DOS: Graphene (2D)

The electronic structures of solids show bands of allowed energies, being separated by zones of forbidden energies as a consequence of periodic structure of defect-free materials!

2

Local Density Approximation (LDA) for exchange-correlation:

Dirac Exchange

CONCLUSIONS

Schilfgaarde et al. PRL 96, 226402 (2006).

Correlation Effects (Fit to Monte Carlo)

The bandstructure of a solid dictates its optical and transport properties, while GWA-DFT computational method provide reasonable results of E-k diagrams and densities of states! The unbound electrons in solids can be treated as nearly free particles which do not contribute to resistance, because there is no scattering on periodic potential created by crystal ions! Impurities and interfaces modify bulk bandstructures, while conventional bandstructure theory is not applicable to strongly correlated systems (Mott insulators, Hubbard model)!