Basics of DFT applicationsto solids and surfaces

Peter KratzerPhysics Department, University Duisburg-Essen, Duisburg, Germany

E-mail: Peter.Kratzer@uni-duisburg-essen.de

Periodicity in real space and reciprocal space

• Example: honeycomb lattice

Lattice vectors a1, a2, a3Unit cell volume �Crystallographic basis consisting of two atoms

Reciprocal-lattice vectors b1, b2, b3 , eachperpendicular to a pair of lattice vectors

real space

Reciprocal (wave-vector) space

From molecules to solids

• Electronic bands as limit of bonding and anti-bonding combinations ofatomic 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 mustbehave like

under translations by a lattice vector

• Consequently, we can write

with a lattice-periodic part uj,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.

Brillouin zones

• in two dimensionsfor a square lattice

for a hexagonal lattice

• in three dimensionsfor a face-centered cubic (fcc)

lattice

Band structure - schematic

The orbital energies are smoothfunctions of k

Example: chain of Pt-R4 complexes

Adopted from: R. Hoffmann, Solids and Surfaces - A chemist’s view of bonding in extendedstructures, VCH Publishers, 1998

k=0 k=�/a

0 k �/a

z2xy

xz,yz

x2-y2

z

energy

z

xy

xz

z2

Brillouin zone sampling

• Charge densities (and other quantities) arerepresented by Brillouin-zone integrals

• 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)

16 � 8 MP grid

In a metal, some (at least one) energy bandsare only partially occupied by electrons.The Fermi energy �F defines the highestoccupied state(s). Plotting the relation

in reciprocal space yields the Fermi surface(s).

The grid used ink-space must be suffficiently fine to accurately sample the Fermi surface.

Fermi surfaces

http://www.phys.ufl.edu/fermisurface/

Fermisurfaceof Cu

Fermisurface of Fe (spin� )

Fermisurface of Fe (spin� )

Band structure of a magnetic material

A. Yamasaki & T. Fujiwara, J. Phys.Soc. Japan 72, 607 (2003)

Spin-resolved bandstructure for fcc iron

GW method (full)LSDA (dashed)

Correct description ofmagnet propertiesrequires dense k-spacesampling !

�F

How to treat metals in DFT

• Fermi distribution function fF enters Brillouin zone integral:

• Simplest method: “smearing” of the Fermi function– artificially increased kBTel ~ 0.2 eV– Extrapolation of the total energy to Tel = 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, 16223 (1994)]

Total-energy correction for finite Tel

-60

-40

-20

0

20

40

60

80

0.01

0.03

0.05

0.07

0.09

electronic temperature [eV]

�E

[meV

]

extrapolated to 0

E(T)

substitutional adsorption of Na on Al(111)J. Neugebauer and M. Scheffler,Phys. Rev. B 46, 16067 (1992)

Indium adatom on GaAs(001)c(4x4) surfaceE. Penev, unpublished

Ezero = � (F(T)+E(T)) + O(T3) = F(T) – � Tel Sel(T) + O(T3)

E

Ezero

F

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 orbitalsQualitative analysis tool; ambiguities must be resolved by truncatingthe r-integral or by Löwdin orthogonalization of the

energy

Ferromagnetic half-metal: Co2MnSi

DOS (states / eV)

SiMn

Co minority spin band structure

Surfaces: geometric andelectronic structure

atomic structure: relaxation and reconstruction

• change of interlayerdistances near the surface

• response of the surfaceatoms to an adsorbate

• change of the symmetry• atomic rearrangement, formation of new bonds

relaxation reconstruction

oxygen on Ru(0001)

supercell

adatom(+superficial periodic images)

Supercell approach to surfaces

– 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

Points to consider:

�LM�

Surface Brillouin zone

L

U

XZW

K

�'

�

� �

� Q

kz

kx

ky

example: fcc crystal, (111) surface

M

� K

Surface band structure of Cu(111)

(111)

Shockley surface state

Tamm surface state

Surface states (I)

gap2 |VG|

�/a k�

i

(k�) matching condition

potential

exp[ i(k�+ i) z]~exp[�z]

� Shockley surface state

complex band structure

approximation of the ‘nearly-free-electron’ metal

Surface states (II)

In the tight-binding(localized orbital) picture,surface states mayappear due to ‘danglingorbitals’ split off from theband edge� Tamm surface state

CoCoMnSi

CoCoMnMn

Out-of-planed orb.

In-plane d orb.Minoritybands

Spin-down components of dorbital of surface Mn andsubsurface Co

Surface states derivedmainly from Co d3r2–z2 insubsurface layer

Orbital mixing of Co orbitals withMn dxz dyz orbitals pushes Costates out of the gap.

Co2MnSi: spin-polarized surface state

P. Kratzer et al., J. Appl. Phys. 101, 081725 (2007)

PDOS (states/eV)

Dimerization at (001)-surface of group IV-elements

top viewside view

[001]

bulk-terminated atomic structure

reconstructionside view

[1 1 0]

[1 1 0]

Stabilization of dimerized Si(100) surface

sp3 sp3 sp3 sp3

�

��

pz

s-like

�-bond re-hybridisation and charge transfer

sp2-backbondsp-backbondsJahn-Teller-like effect enhances thesplitting of the surface state. [see, e.g., J. Dabrowski and M. Scheffler,Appl. Surf. Sci. 56-58, 15 (1992)]

Different reconstructions of group-IV (001)-surfaces

�-bonding

Jahn-Teller effect (buckling)

Jahn-Teller effekt (buckling)

C

Si

Ge

P. Krüger & J. Pollmann, Phys. Rev. Lett. 74, 1155 (1995)

Surface reconstruction of Si(001)

top view of Si(001)

A. Ramstad, G. Brocks, and P. J. Kelly, Phys. Rev. B 51, 14504 (1995).

Thermodynamics ofsurfaces

n

�a

[001]

(�)

�

surface free energy

definition:

�(n) = free energy per unit arearequired to create a surface

Depends on orientation of the planein a crystal, and on temperature andpressure

�(n)

�

Simple metal: potassium

M. Timmer, unpublished

DFT calculations at T=0 K: � A = Eslab - Natom Ebulk

Quantum size effects: Al(110)

A. Kiejna, J. Peisert and P. Scharoch, Surf. Sci. 432 (1999) 54

LDA, 20Ry plane wave cut-off,16 k-points in IBZ

EF

binary material: GaAs

• Gibbs free enthalpy G(p, T, NGa, NAs)• Surface (free) energy

• equilibrium with bulk GaAs fixes onechemical potential: �Ga + �As = �GaAs

• remaining variable ( for instance, �As )determined by gas pressure

• coexistence with elemental bulk phases setsbounds for �As(p,T)

�=[G(p,T,NGa, NAs) �NGa�Ga � NAs�As]/A

As crystal face

(001) Ga crystal face

Surface relaxation: GaAs(001) �2 (2x4)

LDA, 10 Ry plane-wave cut-off,2 x 4 k-points in full BZ

Slabs with n layersEbulk(n) := [Eslab(n) � Eslab(n � 2)]/Natsurface energy�(n)A := Eslab(n) � (NAs+NGa) Ebulk(n) � (NAs-NGa)�As(p,T)

As2 pressure determines the stable structure

Under the conditionscommonly used inmolecular beamepitaxy, the �2surface recon-struction is stable.

Only at lowertemperatures andhigher As2 pressures,the c(4�4) structureprevails (terminatingAs double layer).

Summary & Acknowledgements

Björn Hülsen,FHI

Sung Sakong,UDE

Matthias Timmer,UDE

S. Javad Hashemifar,Isfahan University ofTechnology

The chemical physics of solid surfaces and interfaces is an activeand exciting field of current research !