basics of dft applications to solids and surfaces

7/29/2019 Basics of DFT applications to solids and surfaces

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Peter Kratzer

Physics Department, University Duisburg-Essen, Duisburg, Germany

E-mail: [email protected]

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 , each

perpendicular to a pair of lattice vectors

real space

Reciprocal (wave-vector)

space


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From molecules to solids

Electronic bands as limit of bonding and anti-bonding combinations of

atomic orbitals

R. Hoffmann,

Solids and Surfaces - A chemists view of bonding in extended structures, VCH Publishers, 1998

Blochs 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 uj,k(r).

The index kis 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.


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Brillouin zones

in two dimensions

for a square lattice

for a hexagonal lattice

in three dimensions

for a face-centered cubic (fcc)lattice

Band structure - schematic

The orbital energies are smoothfunctions ofk

Example: chain of Pt-R4 complexes

Adopted from: R. Hoffmann, Solids and Surfaces - A chemists view of bonding in extended

structures, VCH Publishers, 1998

k=0 k=/a

0 k /a

z2xy

xz,yz

x

2

-y

2

z

energy

z

xy

xz

z2


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Brillouin zone sampling

Charge densities (and other quantities) are

represented 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 bands

are only partially occupied by electrons.The Fermi energy Fdefines the highest

occupied 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/

Fermi

surface

of Cu

Fermi

surface ofFe (spin )

Fermi

surface ofFe (spin )


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Band structure of a magnetic material

A. Yamasaki & T. Fujiwara, J. Phys.

Soc. Japan 72, 607 (2003)

Spin-resolved band

structure for fcc iron

GW method (full)

LSDA (dashed)

Correct description of

magnet properties

requires dense k-spacesampling !

F

How to treat metals in DFT

Fermi distribution functionfFenters 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. Blchl et al., Phys. Rev. B 49, 16223 (1994)]


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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) surface

E. 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 lat atomI

Recovery of the chemical interpretation in terms of orbitals

Qualitative analysis tool; ambiguities must be resolved by truncating

the r-integral orby Lwdin orthogonalization of the


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energy

Ferromagnetic half-metal: Co2MnSi

DOS (states / eV)

SiMn

Cominority spin band structure

Surfaces: geometric andelectronic structure


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atomic structure: relaxation and reconstruction

change of interlayerdistances near the surface

response of the surface

atoms 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 theslabs

Sufficient slab thickness to mimic semi-infinite

crystal

Semiconductors: saturate dangling bonds on theback 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:


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LM

Surface Brillouin zone

L

U

XZW

K

Y'

Y

K 6

RQ

kz

kx

ky

example: fcc crystal, (111) surface

M

K

Surface band structure of Cu(111)

(111)

Shockley surface state

Tamm surface state


10/17

Surface states (I)

gap

2 |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 dangling

orbitals split off from the

band edge Tamm surface state


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CoCo

MnSi

CoCo

MnMn

Out-of-plane

dorb.In-plane dorb.Minority

bands

Spin-down components of d

orbital ofsurface Mn andsubsurface Co

Surface states derived

mainly from Co d3r2z2 in

subsurface layer

Orbital mixing of Co orbitals withMn dxz dyz orbitals pushes Co

states out of the gap.

Co2MnSi: spin-polarized surface state

P. Kratzeret al., J. Appl.

Phys. 101, 081725 (2007)PDOS (states/eV)

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

top view

side view

[001]

bulk-terminated atomic structure

reconstructionside view

[1 1 0]

[1 1 0]


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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 the

splitting 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. Krger & J. Pollmann, Phys. Rev. Lett. 74, 1155 (1995)


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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


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n

_a

[001]

a(O)

O

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


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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 one

chemical 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


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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 conditions

commonly used in

molecular beamepitaxy, the 2

surface recon-

struction is stable.

Only at lower

temperatures and

higher As2 pressures,the c(4 4) structure

prevails (terminating

As double layer).


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Summary & Acknowledgements

Bjrn Hlsen,FHI

Sung Sakong,

UDE

Matthias Timmer,

UDE

S. Javad Hashemifar,

Isfahan University of

Technology

The chemical physics of solid surfaces and interfaces is an active

and exciting field of current research !

basics of dft applications to solids and surfaces

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