theoretical methods for surface science
TRANSCRIPT
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
International Max-Planck Research School Theoretical Methods for Surface Science Part II Slide 2
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
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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)
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-
+
Work functionsurface dipole d
Jellium model
Lang and Kohn, PRB 1,4555(1970)
Potential difference
=()-(-)=4d
d
Work function
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Work function
Lang and Kohn, PRB 1,4555(1970)
Potential difference
=()-(-)=4d
Work function
Chemical potential of the electrons
=E(N+1)-E(N)=EF
Work function
=()- =
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Nearly Free electron model (NFE)Periodic potentialV(z) = -Vo+2Vgcos(gz)
The energies and wave functions:
Band gap opens at the zone boundaries
V02Vg
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Surface states
The solution for imaginary values of is also possible at the surface:
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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)
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DFT bandstructure for Cu(111)
2x21x1
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Bandstructure of Cu(111)
Euceda et al., PRB 28,528 (1983)
6-layer slab 18-layer slab
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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.
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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.
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Bandstructure of Cu(111)
K
M
Surface BZ
Schockley surface state
Tamm state
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Adsorption
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Adsorption
Ediss
EadsPhysisorption well
Chemisorption well
Activation barrier
Ener
gy
z
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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}
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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
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Phase diagram
Reuter and Scheffler, PRB 68, 045407 (2003)
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Physisorption
-+
+-zr’ z
metal
r
The electrostatic energy:
Taylor expand in terms of 1/z:
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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).
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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)
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Chemisorption
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Adsorption sites
H
B
FT
T
B
F
H
Top site
Bridge site
Hollow FCC-site
Hollow HCP-site
Close packed (111)-surface
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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
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Potential energy surface
O2 on Pt(111), Gross et al., Surf. Sci., 539, L542 (2003).
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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)
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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.
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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.
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Newns-Anderson model continuedCalculate the Green’s function for the Hamiltonian
as
and identify the self-energy components:
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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
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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.
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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)
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Chemisorption on a metal surface Na/Cu(111)
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Adsorbate induced work function change
[e
V]
Tang et al., Surf. Sci. Lett. 255, L497 (1991).
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Charge transfer for Na/Cu(111)charge depletion
+
-
charge accumulation
adsorbate induced dipole
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Properties for Na/Cu(111)
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Quantum well state for Na/Cu(111)
Carlsson and Hellsing, PRB 61, 13973 (2000)
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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.
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Harding’s compensating surface charge Qs
(Surf. Sci. 422, 87 (1999))
r1 r2a0
Q1Q2QsQp
repeat unit
Qs=Q1, where
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Ex: Properties of ZnO
a) Ground state structure for ZnO: Wurtzite structure
b) High pressure structure: Rock salt
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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
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The polar ZnO{0001}-surfaceZn-terminated [0001]-surface
O-terminated [0001]-surface
[0001]
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The polar ZnO{0001}-surface
A
B
Carlsson, Comp. Mat. Sci. 22, 24 (2001)
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The polar ZnO{0001}-surface
Carlsson, Comp. Mat. Sci. 22, 24 (2001)
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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
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Surface Phase diagram of ZnO[0001]
Kresse et al., PRB 68, 245409 (2003)
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Summary•Surface energy
•Atomic structure relaxation
•Charge redistribution
•Work function
•Surface states
•Adsorption
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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