theoretical methods for surface science part ii

Theoretical Methods for Surface ScienceTheoretical Methods for
Summary
How to calculate bulk properties and electronic structure
How to model surfaces
Charge distribution at Surfaces
Jellium model
Euceda et al., PRB 28,528 (1983)
International Max-Planck Research School
Work function
Potential difference
Work function
Potential difference
m=E(N+1)-E(N)=EF
Nearly Free electron model (NFE)
Periodic potential
Band gap opens
Surface states
The solution for imaginary values of k is also possible at the surface:
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)
DFT bandstructure for Cu(111)
Bandstructure of Cu(111)
6-layer slab
18-layer slab
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.
k
k
kx
kz
k
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.
k
k
kx
kz
k
Bandstructure of Cu(111)
Adsorption
Adsorption
Ediss
Eads
Thermodynamics for adsorption
where G(T,p)= E-TS + pV=F+pV
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)
Eads={E[host+defect]+Fvib(T,w)}-{E[host]+Na ma}
Thermodynamics for adsorption
Convert the energy values of the chemical potential into T and
p-dependence of the gas phase reservoir
mi(T,pi)=mDFT+DG(T,p0)+ kT ln(pi /p0)
Interpolate DG(T,p0) from tables.
Eads(T,p)={E[host+defect]+Fvib(T)}-{E[host]+ma(T,pa)}
The adsorbate concentration can be estimated in the dilute limit
C=N exp(-Eads/kT)
Phase diagram
Physisorption
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 Å
Chemisorption
Adsorption sites
Finding the adsorption site
Adsorption without a barrier:
Adsorption system with a barrier:
Locate the transition state at the barrier
Need to start the atomic relaxation inside the barrier
chemisorption sites
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 >
e
Green’s function techniques
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 techniques
The imaginary part of the Green’s function is called the spectral function
it is equivalent to the projected density of states.
The self energy describes the interactions in the system
The real part L(e) leads to a shift of the energy eigenvalues, the imaginary part D(e) gives a broadening
Newns-Anderson model continued
as
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 >
D
sp-band
e
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 Green’s function becomes:
| a >
d-band
e
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.
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)
Chemisorption on a metal surface
Na/Cu(111)
Adsorbate induced
Charge transfer for Na/Cu(111)
Properties for Na/Cu(111)
Quantum well state for Na/Cu(111)
Carlsson and Hellsing, PRB 61, 13973 (2000)
Tasker’s rules
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 neutral
Layered materials with stacking -1 +2 -1 -1 +2 ...
Type III charged planes leading to a net dipole moment
ex 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
Qs=aQ1, where
Ex: Properties of ZnO
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
The polar ZnO{0001}-surface
Zn-terminated [0001]-surface
A
B
Carlsson, Comp. Mat. Sci. 22, 24 (2001)
STM of ZnO[0001]-surface
Dulub et al.,PRL 90, 016102 (2003)
Triangular islands
n=O-edge atoms
L = (n-2)*a = 16.25 Å
c) Triangle with internal triangle
# of O-atoms = 3n(n+1)/2-3
L = (2(n-1)-1)*a = 29.25 Å
Surface Phase diagram of ZnO[0001]
Kresse et al., PRB 68, 245409 (2003)
Summary
Literature
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
r
1
r
2
a
0
Q
1
Q
2
Q
s
Q
p

theoretical methods for surface science part ii

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