Basics of DFT applications to solids and surfaces

Transcription

1 Basics of DFT applications to solids and surfaces Peter Kratzer Physics Department, University Duisburg-Essen, Duisburg, Germany Periodicity in real space and reciprocal space Example: honeycomb lattice real space Reciprocal (wave-vector) space Lattice vectors a 1, a 2, a 3 Unit cell volume Crystallographic basis consisting of two atoms Reciprocal-lattice vectors b 1, b 2, b 3, each perpendicular to a pair of lattice vectors

2 From molecules to solids Electronic bands as limit of bonding and anti-bonding combinations of atomic orbitals R. Hoffmann, Solids and Surfaces - A chemist s view of bonding in extended structures, VCH Publishers, 1998 Bloch s theorem In an infinite periodic solid, the solutions of the Kohn-Sham equations must behave like under translations by a lattice vector Consequently, we can write with a lattice-periodic part u j,k (r). The index k is a vector in reciprocal space taken from the first Brillouin zone. The number of different k-indices equals the number of unit cells in a crystal.

3 Brillouin zones in two dimensions for a square lattice in three dimensions for a face-centered cubic (fcc) lattice for a hexagonal lattice Band structure - schematic The orbital energies are smooth functions of k Example: chain of Pt-R 4 complexes x 2 -y 2 z k=0 k=/a z energy xz,yz xy xz xy z 2 z 2 0 k /a Adopted from: R. Hoffmann, Solids and Surfaces - A chemist s view of bonding in extended structures, VCH Publishers, 1998

4 Brillouin zone sampling Charge densities (and other quantities) are represented by Brillouin-zone integrals 16 8 MP grid Integrand smooth (for non-metals): replace integral by sum over sampling points H.J. Monkhorst and J.D. Pack, Phys. Rev.B 13, 5188 (1976); Phys. Rev. B 16, 1748 (1977) Fermi surfaces In a metal, some (at least one) energy bands are only partially occupied by electrons. The Fermi energy F defines the highest occupied state(s). Plotting the relation Fermi surface of Cu in reciprocal space yields the Fermi surface(s). Fermi surface of Fe (spin ) Fermi surface of Fe (spin ) The grid used in k-space must be suffficiently fine to accurately sample the Fermi surface.

5 Band structure of a magnetic material Spin-resolved band structure for fcc iron F GW method (full) LSDA (dashed) Correct description of magnet properties requires dense k-space sampling! A. Yamasaki & T. Fujiwara, J. Phys. Soc. Japan 72, 607 (2003) How to treat metals in DFT Fermi distribution function f F enters Brillouin zone integral: Simplest method: smearing of the Fermi function artificially increased k B T el ~ 0.2 ev Extrapolation of the total energy to T el = 0 Alternatives: Methfessel-Paxton distribution [ Phys. Rev. B 40, 3616 (1989) ] Marzari-Vanderbilt ensemble-dft method [ Phys. Rev. Lett. 79, 1337 (1997) ] Tetrahedron method Fermi surface is approximated by a polyhedron consisting of small tetrahedra in each plaquette [P. E. Blöchl et al., Phys. Rev. B 49, (1994)]

6 Total-energy correction for finite T el E zero = (F(T)+E(T)) + O(T 3 ) = F(T) T el S el (T) + O(T 3 ) E E zero F E [mev] E(T) extrapolated to electronic temperature [ev] substitutional adsorption of Na on Al(111) J. Neugebauer and M. Scheffler, Phys. Rev. B 46, (1992) Indium adatom on GaAs(001) c(4x4) surface E. Penev, unpublished Density of states (DOS) Information about the single-particle contribution to the total energy Projected density of states (PDOS) is the atomic orbital with angular momentum l at atom I Recovery of the chemical interpretation in terms of orbitals Qualitative analysis tool; ambiguities must be resolved by truncating the r-integral or by Löwdin orthogonalization of the

7 Ferromagnetic half-metal: Co 2 MnSi DOS (states / ev) energy Co Mn Si minority spin band structure Surfaces: geometric and electronic structure

8 atomic structure: relaxation and reconstruction relaxation change of interlayer distances near the surface response of the surface atoms to an adsorbate reconstruction change of the symmetry atomic rearrangement, formation of new bonds oxygen on Ru(0001) Supercell approach to surfaces Points to consider: adatom(+superficial periodic images) Approach accounts for the lateral periodicity Sufficiently broad vacuum region to decouple the slabs Sufficient slab thickness to mimic semi-infinite crystal Semiconductors: saturate dangling bonds on the back surface Inequivalent surfaces: use dipole correction With some codes, slabs with inversion symmetry (for metals) are computationally more efficient Alternative: cluster model supercell

9 Surface Brillouin zone example: fcc crystal, (111) surface k z L Q K U ' X Z W k y k x M M K L Surface band structure of Cu(111) Shockley surface state Tamm surface state (111)

10 Surface states (I) approximation of the nearly-free-electron metal (k ) gap 2 V G complex band structure i ~exp[z] matching condition exp[ i(k + i) z] potential /a k Shockley surface state Surface states (II) In the tight-binding (localized orbital) picture, surface states may appear due to dangling orbitals split off from the band edge Tamm surface state

11 Co 2 MnSi: spin-polarized surface state MnSi CoCo Minority bands Out-of-plane d orb. In-plane d orb. Spin-down components of d orbital of surface Mn and subsurface Co Surface states derived mainly from Co d3r 2 z 2 in subsurface layer MnMn CoCo PDOS (states/ev) Orbital mixing of Co orbitals with Mn dxz dyz orbitals pushes Co states out of the gap. P. Kratzer et al., J. Appl. Phys. 101, (2007) Dimerization at (001)-surface of group IV-elements [001] side view top view [1 1 0] bulk-terminated atomic structure reconstruction [1 1 0] side view

12 Stabilization of dimerized Si(100) surface p z sp 3 sp 3 sp 3 sp 3 s-like -bond Jahn-Teller-like effect enhances the splitting of the surface state. p-backbonds sp 2 -backbonds [see, e.g., J. Dabrowski and M. Scheffler, Appl. Surf. Sci , 15 (1992)] re-hybridisation and charge transfer Different reconstructions of group-iv (001)-surfaces C -bonding Si Jahn-Teller effect (buckling) Ge Jahn-Teller effekt (buckling) P. Krüger & J. Pollmann, Phys. Rev. Lett. 74, 1155 (1995)

13 Surface reconstruction of Si(001) A. Ramstad, G. Brocks, and P. J. Kelly, Phys. Rev. B 51, (1995). top view of Si(001) Thermodynamics of surfaces

14 surface free energy definition: (n) = free energy per unit area required to create a surface Depends on orientation of the plane in a crystal, and on temperature and pressure [001] n (n) () a Simple metal: potassium DFT calculations at T=0 K: A = E slab - N atom E bulk M. Timmer, unpublished

15 Quantum size effects: Al(110) E F LDA, 20Ry plane wave cut-off, 16 k-points in IBZ A. Kiejna, J. Peisert and P. Scharoch, Surf. Sci. 432 (1999) 54 binary material: GaAs Gibbs free enthalpy G(p, T, N Ga, N As ) Surface (free) energy =[G(p,T,N Ga, N As ) N Ga Ga N As As ]/A (001) Ga crystal face equilibrium with bulk GaAs fixes one chemical potential: Ga + As = GaAs remaining variable ( for instance, As ) determined by gas pressure coexistence with elemental bulk phases sets bounds for As (p,t) As crystal face

16 Surface relaxation: GaAs(001) 2 (2x4) LDA, 10 Ry plane-wave cut-off, 2 x 4 k-points in full BZ Slabs with n layers E bulk (n) := [E slab (n) E slab (n 2)]/N at surface energy (n)a := E slab (n) (N As +N Ga ) E bulk (n) (N As -N Ga ) As (p,t) As 2 pressure determines the stable structure Under the conditions commonly used in molecular beam epitaxy, the 2 surface reconstruction is stable. Only at lower temperatures and higher As 2 pressures, the c(44) structure prevails (terminating As double layer).

17 Summary & Acknowledgements The chemical physics of solid surfaces and interfaces is an active and exciting field of current research! S. Javad Hashemifar, Isfahan University of Technology Sung Sakong, UDE Björn Hülsen, FHI Matthias Timmer, UDE