Theoretical Methods for Surface Science part II

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Theoretical Methods for Surface Sciencepart II

• Johan M. Carlsson
• Theory Department
• Fritz-Haber-Institut der Max-Planck-Gesellschaft
• Faradayweg 4-6, 14195 Berlin

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Summary
Last 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 Surfaces
electrons spill out from the surface
Jellium model Lang and Kohn, PRB 1,4555(1970)
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Work function
Work function F
surface dipole d

d
-
Potential difference Dff (?)-f (-?)4pd
Jellium model Lang and Kohn, PRB 1,4555(1970)
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Work function
Work function F
Chemical potential of the electrons
mE(N1)-E(N)EF Work function Ff (?)-m
Df-m
Potential difference Dff (?)-f (-?)4pd
Lang and Kohn, PRB 1,4555(1970)
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Nearly Free electron model (NFE)
Periodic potential V(z) -Vo2Vgcos(gz)
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Surface states
The solution for imaginary values of k is also
possible at the surface
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Surface states
Matching 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)
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Bandstructure of Cu(111)
6-layer slab
18-layer slab
Euceda et al., PRB 28,528 (1983)
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Projected Bulk bandstructures
Bertel, Surf. Sci. 331, 1136 (1995)
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
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)
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Adsorption
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Adsorption
Energy
Activation barrier
Ediss
z
Physisorption well
Eads
Chemisorption well
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Thermodynamics for adsorption
a
ma
Host
Definition of adsorbate energy
EadsDGGhostads-GhostNa ma
where G(T,p) E-TS pVFpV Ftrans, Frot,
pV negligible for solids, but not in the gas
phase The adsorbates vibrate at the surface
Fvib(T,w)Evib (T,w)-TSvib (T,w) This gives the
adsorption energy EadsEhostdefectFvib(T,
w)-EhostNa ma
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Thermodynamics for adsorption
Convert the energy values of the chemical
potential into T and p-dependence of the gas
phase reservoir mi(T,pi)mDFTDG(T,p0) kT
ln(pi /p0) Interpolate DG(T,p0) from
tables. Reuter and Scheffler, PRB 65, 035406
(2002). Eads(T,p)EhostdefectFvib(T)-Eho
stma(T,pa) The adsorbate concentration can
be estimated in the dilute limit CN
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
The electrostatic energy
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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
h3.4 Å DFT-GGA Eads0.04 eV/O2 TPD-experiment
Eads0.12 eV/O2 Ulbricht et al.,PRB 66, 075404
(2002)
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Chemisorption
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Adsorption sites
Top site Bridge site Hollow FCC-site Hollow
HCP-site
T
B
B
F
H
H
F
T
Close packed (111)-surface
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Finding the adsorption site
Adsorption system with a barrier Locate the
transition state at the barrier Need to start the
atomic relaxation inside the barrier
Adsorption without a barrier Non-activated
adsorption can start the atomic relaxation
anywhere Calculation the Potential Energy Surface
(PES)
barrier
chemisorption sites
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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
Anderson, Phys. Rev. 124, 41 (1961) Newns, Phys.
Rev. 178, 1123 (1969)
Consider an adsorbate atom with a valence level
a gt interacting with a metal which has a
continuum of states k gt.
where
is the overlap interaction between the adsorbate
atom and the substrate levels k gt.
e
k
a gt
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Greens function techniques
The Greens function Gs(e) is the solution to
the equation
The Greens function describe the response of the
system to a perturbation and poles gives the
excitation energies.
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Greens function techniques
The imaginary part of the Greens function is
called the spectral function
it is equivalent to the projected density of
states.
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Newns-Anderson model continued
Calculate the Greens function for the Hamiltonian
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Weak chemisorption limit
If 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. D is then independent of energy which
means that L 0. The projected density of states
for the adsorbate atom is then a Lorentzian with
a width D, centered around ea
a gt
D
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Strong chemisorption limit
When the adsorbate interacts with a narrow
d-band, then the ek can be approximated by center
value ec such that the denominator in the Greens
function becomes
Solving this equation gives two roots
corresponding to bonding and anti bonding levels
of the absorbate system.
e
a gt
d-band
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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.
• Charge would be donated to the substrate if the
  atom has low ionization energy and
• 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
DFeV
Tang et al., Surf. Sci. Lett. 255, L497 (1991).
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Charge transfer for Na/Cu(111)
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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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Taskers rules(J. Phys. C 12, 4977 (1977))
Surface types in ionic crystals Type I Crystals
with neutral planes parallel to the surface
ex MgO100-surfaces Type II charged planes
where the repeat unit is neutral Layered
materials with stacking -1 2 -1 -1 2 ... Type
III charged planes leading to a net dipole
moment ex MgO111-surfaces Type III is
unstable unless surface charges set up an
opposing surface dipole which quench the internal
dipole moment.
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Hardings compensating surface charge Qs(Surf.
Sci. 422, 87 (1999))
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Ex Properties of ZnO

• Ground state structure for ZnO Wurtzite
  structure
• High pressure structure Rock salt

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Electronic structure of ZnO
EgapExp3.4 eV EgapDFT-GGA0.8eV
Under estimation of the bandgap in
semi-conductors is a common problem in
DFT-calculations with LDA or GGA
exchange-correlation functional.
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The polar ZnO0001-surface
Zn-terminated 0001-surface
0001
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The polar ZnO0001-surface
B
A
Carlsson, Comp. Mat. Sci. 22, 24 (2001)
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The polar ZnO0001-surface
Carlsson, Comp. Mat. Sci. 22, 24 (2001)
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STM of ZnO0001-surface
Dulub et al.,PRL 90, 016102 (2003)
Triangular islands Step height2.7 Åc/2 nO-edge
atoms b) Triangle of O-atoms n(n1)/2
of Zn-atoms n(n-1)/2 QZn / O 3/4 gt n7 L
(n-2)a 16.25 Å c) Triangle with internal
triangle of O-atoms 3n(n1)/2-3 of
Zn-atoms 3n(n-1)/2 QZn / O 3/4 gt n6 L
(2(n-1)-1)a 29.25 Å
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Surface Phase diagram of ZnO0001
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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Literature
Review 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