Theoretical Methods for Surface Science

International Max-Planck Research School Theoretical Methods for Surface Science Part II Slide 1

Theoretical Methods for Surface Science

part II

Johan M. CarlssonTheory Department

Fritz-Haber-Institut der Max-Planck-GesellschaftFaradayweg 4-6, 14195 Berlin


SummaryLast lecture:

The foundations of the DFT

How to calculate bulk properties and electronic structure

How to model surfaces

Surface structures

This lecture:

Electronic structure at surfaces

Adsorption


Charge distribution at Surfaceselectrons spill out from the surface

Jellium model

Lang and Kohn, PRB 1,4555(1970)

All-electron LCGO DFT-calculations for Cu(111)-surface.

Euceda et al., PRB 28,528 (1983)


-

+

Work functionsurface dipole d

Jellium model


Potential difference

=()-(-)=4d

d

Work function


Work function


Potential difference

=()-(-)=4d

Work function

Chemical potential of the electrons

=E(N+1)-E(N)=EF

Work function

=()- =


Nearly Free electron model (NFE)Periodic potentialV(z) = -Vo+2Vgcos(gz)

The energies and wave functions:

Band gap opens at the zone boundaries

V02Vg


Surface states

The solution for imaginary values of is also possible at the surface:


Surface statesMatching the two solutions at a/2 leads to a Schockley surface state. *This state has a large amplitude in the surface region, but decay rapidly into the bulk and into the vacuum region.*Its energy is located in the band gap.

Schockley, Phys. Rev. 56, 317, (1939)


DFT bandstructure for Cu(111)

2x21x1


Bandstructure of Cu(111)

Euceda et al., PRB 28,528 (1983)

6-layer slab 18-layer slab


Projected Bulk bandstructures

Bertel, Surf. Sci. 331, 1136 (1995)

kx

kz

k

kk

There is a range of k-vectors with a k-component along the perpendicular rod for each k-point in the surface plane.


Projected Bulk bandstructures

kx

kz

k

kk

Calculate the bands along the perpendicular rod.

The values between the lowest and highest values correspond to regions of bulk states.

Surface states can occur outside the bulk regions.


Bandstructure of Cu(111)

K

M

Surface BZ

Schockley surface state

Tamm state


Adsorption


Adsorption

Ediss

EadsPhysisorption well

Chemisorption well

Activation barrier

Ener

gy

z


Thermodynamics for adsorption

a

a

Host

Definition of adsorbate energy: Eads=G=G[host+ads]-{G[host]+Na a}

where G(T,p)= E-TS + pV=F+pVFtrans, Frot, pV negligible for solids, but not in the gas phaseThe adsorbates vibrate at the surface: Fvib(T,)=Evib (T,)-TSvib (T,)This gives the adsorption energy Eads={E[host+defect]+Fvib(T,)}-{E[host]+Na a}


Thermodynamics for adsorptionConvert the energy values of the chemical potential into T and p-dependence of the gas phase reservoir i(T,pi)=DFT+G(T,p0)+ kT ln(pi /p0)Interpolate G(T,p0) from tables.Reuter and Scheffler, PRB 65, 035406 (2002).

Eads(T,p)={E[host+defect]+Fvib(T)}-{E[host]+a(T,pa)} The adsorbate concentration can be estimated in the dilute limit

C=N exp(-Eads/kT)

where N is the number of adsorbtion sites


Phase diagram

Reuter and Scheffler, PRB 68, 045407 (2003)


Physisorption

-+

+-zr’ z

metal

r

The electrostatic energy:

Taylor expand in terms of 1/z:


van der Waals interaction

Cohesive energy for graphite as function of a- and c-lattice parameters. Calculated with GGA XC-functional

Rydberg et al., Surf. Sci. 532, 606 (2003).


Physisorption of O2 on graphite

h=3.4 ÅDFT-GGA: Eads=0.04 eV/O2

TPD-experiment: Eads=0.12 eV/O2

Ulbricht et al.,PRB 66, 075404 (2002)


Chemisorption


Adsorption sites

H

B

FT

T

B

F

H

Top site

Bridge site

Hollow FCC-site

Hollow HCP-site

Close packed (111)-surface


Finding the adsorption siteAdsorption without a barrier:

Non-activated adsorption:

can start the atomic relaxation anywhere

Calculation the Potential Energy Surface (PES)

Adsorption system with a barrier:

Locate the transition state at the barrier

Need to start the atomic relaxation inside the barrier

chemisorption sites

barrier


Potential energy surface

O2 on Pt(111), Gross et al., Surf. Sci., 539, L542 (2003).


Newns-Anderson model

Consider an adsorbate atom with a valence level |a > interacting with a metal which has a continuum of states | k >.

where

is the overlap interaction between the adsorbate atom and the substrate levels | k >.

k| a >

Anderson, Phys. Rev. 124, 41 (1961)Newns, Phys. Rev. 178, 1123 (1969)


Green’s function techniquesThe Green’s function G()

is the solution to the equation

The Green’s function describe the response of the system to a perturbation and poles gives the excitation energies.


Green’s function techniquesThe imaginary part of the Green’s function is called the spectral function

The self energy describes the interactions in the system

The real part () leads to a shift of the energy eigenvalues, the imaginary part () gives a broadening

it is equivalent to the projected density of states.


Newns-Anderson model continuedCalculate the Green’s function for the Hamiltonian

as

and identify the self-energy components:


Weak chemisorption limitIf the interaction between the substrate and the adsorbate is weak, i.e. Vak is small compared to the bandwidth of the substrate band. Ex for a sp-band.

is then independent of energy which means that =0. The projected density of states for the adsorbate atom is then a Lorentzian with a width , centered around a

| a >sp-band


Strong chemisorption limitWhen the adsorbate interacts with a narrow d-band, then the k can be approximated by center value c such that the denominator in the Green’s function becomes:

| a >d-band

Solving this equation gives two roots

corresponding to bonding and anti bonding levels of the absorbate system.


Charge transfer Gurney suggested that the atomic levels of a adsorbate atom

would broaden and that there would be a charge transfer between the substrate and the adsorbate atom.

a) Charge would be donated to the substrate if the atom has low ionization energy and

b) charge would be attracted from the substrate if the atom has a high ionization energy.

Gurney, Phys Rev. 47, 479 (1933)


Chemisorption on a metal surface Na/Cu(111)


Adsorbate induced work function change

[e

V]

Tang et al., Surf. Sci. Lett. 255, L497 (1991).


Charge transfer for Na/Cu(111)charge depletion

+

-

charge accumulation

adsorbate induced dipole


Properties for Na/Cu(111)


Quantum well state for Na/Cu(111)

Carlsson and Hellsing, PRB 61, 13973 (2000)


Tasker’s rules(J. Phys. C 12, 4977 (1977))

Surface types in ionic crystals

Type I Crystals with neutral planes parallel to the surface ex MgO{100}-surfaces

Type II charged planes where the repeat unit is neutralLayered materials with stacking -1 +2 -1 -1 +2 ...

Type III charged planes leading to a net dipole momentex MgO{111}-surfaces

Type III is unstable unless surface charges set up an opposing surface dipole which quench the internal dipole moment.


Harding’s compensating surface charge Qs

(Surf. Sci. 422, 87 (1999))

r1 r2a0

Q1Q2QsQp

repeat unit

Qs=Q1, where


Ex: Properties of ZnO

a) Ground state structure for ZnO: Wurtzite structure

b) High pressure structure: Rock salt


Electronic structure of ZnO

Under estimation of the bandgap in semi-conductors is a common problem in DFT-calculations with LDA or GGA exchange-correlation functional.

EgapExp=3.4 eV

EgapDFT-GGA=0.8eV


The polar ZnO{0001}-surfaceZn-terminated [0001]-surface

O-terminated [0001]-surface

[0001]


The polar ZnO{0001}-surface

A

B

Carlsson, Comp. Mat. Sci. 22, 24 (2001)


The polar ZnO{0001}-surface

Carlsson, Comp. Mat. Sci. 22, 24 (2001)


STM of ZnO[0001]-surface Dulub et al.,PRL 90, 016102 (2003)

Triangular islandsStep height=2.7 Å=c/2n=O-edge atoms

b) Triangle# of O-atoms = n(n+1)/2# of Zn-atoms =n(n-1)/2Q=#Zn / #O =3/4 => n=7L = (n-2)*a = 16.25 Å

c) Triangle with internal triangle# of O-atoms = 3n(n+1)/2-3# of Zn-atoms = 3n(n-1)/2Q=#Zn / #O =3/4 => n=6L = (2(n-1)-1)*a = 29.25 Å


Surface Phase diagram of ZnO[0001]

Kresse et al., PRB 68, 245409 (2003)


Summary•Surface energy

•Atomic structure relaxation

•Charge redistribution

•Work function

•Surface states

•Adsorption


LiteratureReview article about DFT implementations:

Payne et al., Rev. Mod. Phys. 64, 1045 (1992).

A. Zangwill, Physics at Surfaces, Cambridge University Press

A. Gross, Theoretical Surface Science A microscopic perspective, Springer Verlag

F. Bechstedt, Principles of Surface Physics, Springer Verlag

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Theoretical Methods for Surface Science