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Theoretical Methods for
Surface Science
part I
Johan M. Carlsson
Theory Department
Fritz-Haber-Institut der Max-Planck-Gesellschaft
Faradayweg 4-6, 14195 Berlin
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Theoretical Methods for Surface Science Part I Slide 1
Bulk
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Theoretical Methods for Surface Science Part I Slide 2
Surfaces
A
I
II
The surface break the 3D-periodicity of the bulk crystal
Total energy of the system: GI+II=GI+GII+DGsurface
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Theoretical Methods for Surface Science Part I Slide 3
Surface effects
•Surface energy
•Atomic structure relaxation
•Charge redistribution
•Work function
•Surface states
•Adsorption
Lang and Kohn, PRB 1, 4555 (1970)
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Theoretical Methods for Surface Science Part I Slide 4
Real world problems are complex
L[m]
1
macroscopic
regime
10-3
10-6
EkaNobel, Bohus, Sweden
mesoscopic
regime
10-9 microscopic
regime
10-12 10-9 10-6 10-3 1
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t[s]
Theoretical Methods for Surface Science Part I Slide 5
Surface Science methods

AB+C=>AC+B
Experimental
surface science
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Computational
surface science
Construction
of models
Theoretical Methods for Surface Science Part I Slide 6
The Multi scale approach
L[m]
1
macroscopic
regime
10-3
10-6
mesoscopic
regime
Classical mechanics
kinetic Monte Carlo
simulations
10-9 electronic
structure
10-12 10-9 10-6 10-3
Thermodynamics
Electron structure methods
1
t[s]
It is necessary to combine different methods
in order to tackle realistic problems
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Theoretical Methods for Surface Science Part I Slide 7
Total energy methods
Intermediate methods
- Tight-binding,
- many-body potentials: EAM
Accuracy
Simple empirical potentials
- Force fields, pair potentials…
ab-initio techniques
- Hartree-Fock
- Density functional theory (DFT)
Beyond DFT:
- GW
- Quantum Monte Carlo
- Quantum Chemical: CI
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Theoretical Methods for Surface Science Part I Slide 8
Density Functional Theory (DFT)
DFT is nowadays an established method
DFT is capable of treating a few hundred atoms with very good
accuracy
DFT-Properties:
•Charge density
•Total energy
•Forces
•Structure determination
•Phonons
•Electronic structure
Walter Kohn received the Nobel prize in 1998
for the development of DFT.
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Theoretical Methods for Surface Science Part I Slide 9
Hohenberg-Kohn theorem
Hamiltonian for a many-electron system:
Variational principle:
< Y | H |Y >  < Y0 | H |Y0>= E0
Hohenberg-Kohn theorem: Phys. Rev. 136, B864 (1964)
The total energy of the electron system is a functional of the
electron density n(r):
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Theoretical Methods for Surface Science Part I Slide 10
Kohn-Sham equations
Phys. Rev. 140, A1133 (1965)
Minimize the total energy with the constraint to conserve the
number of electrons N:
N=  n(r) dr
where the electron density con be calculated from
This gives the Lagrange equation
This equation can be identified with a Schrödinger like equation
=Kohn-Sham equation for non-interacting electrons in an effective
potential:
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Theoretical Methods for Surface Science Part I Slide 11
Kohn-Sham equations
The effective potential contains three contributions:
The electron density appears in the effective potential which
means that the Kohn-Sham equations needs to be solved selfconsistently.
This means that the total energy of the electron system can
be obtained by solving the Kohn-Sham equations.
Add the ion-ion interaction EII to get the full total energy
E[n(r)]=EII+ EeI[n(r)]+{ Ekin[n(r)]+ EH[n(r)]+ Exc[n(r)]}
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Theoretical Methods for Surface Science Part I Slide 12
The self-consistent scheme
Payne et al., Rev. Mod. Phys. 64, 1045 (1992).
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Theoretical Methods for Surface Science Part I Slide 13
Exchange-Correlation functionals
Local density approximation (LDA):
Assume that the exchange-correlation is the same as the
value for a homogeneous electron gas with the same
density.
Generalized Gradient Approximation (GGA):
Take also the density variations into account by defining the
exchange-correlation as a function of both the density and its
gradients.
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Theoretical Methods for Surface Science Part I Slide 14
K-point sampling
Bloch’s theorem states that the wave function in a periodic
crystal can be described as:
where the wave vector k is located in the first Brillouin zone (BZ).
It is therefore necessary to sample the wave
function at multiple k-points in BZ to get a
correct description of the electron density
and effective potential.
ky
IBZ
kx
Using symmetry lowers the number of
necessary k-point to the ones in the
Irreducible Brillouin zone (IBZ).
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Theoretical Methods for Surface Science Part I Slide 15
Basis set
The wave functions are fourier expanded in a basis set.
Ex:
such that the Kohn-Sham equations are transformed from a
set of differential equations into a set of algebraic equations.
Ex:
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Theoretical Methods for Surface Science Part I Slide 16
Basis sets
Two common basis sets are:
Plane waves:
Localized orbitals:
+ Complete basis set
+Only a few basis functions
needed per atom
+ Systematic way of
improving the accuracy
-Many plane wave are needed
to accurately describe
localized wave functions
-Periodic boundary conditions
necessary
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+ Hamiltonian matrix is sparse
+ Periodic boundary conditions
not necessary
- No systematic way of
improving accuracy
Theoretical Methods for Surface Science Part I Slide 17
Ion-electron interaction
All-electron, full potential method:
The true Coulomb potential from the ions is used and all
electrons are treated explicitly.
+All electrons are treated on the same footing
+ Very accurate
- Very expensive
E
Ri
Rj
valence electrons
x
core electrons
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Theoretical Methods for Surface Science Part I Slide 18
Jellium model
Smear out the potential from
the ions as a constant
positive background.
+Very easy to treat
mathematically
+Can anyway give
qualitative results
-Can at most give a crude
description of the ionelectron interaction, since all
corrugation is removed.
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Lang and Kohn, PRB 1,4555(1970)
Theoretical Methods for Surface Science Part I Slide 19
Pseudo potential method
Remove the core electrons and replace
the ion potential by a smooth pseudo
potential
+Much cheaper than the Full potential
method, since only the valence
electrons are treated explicitly, but
much more accurate the jellium model.
-The interaction between the core and
valence electrons is treated statically,
since the core electrons are frozen into
the pseudo potential.
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Hamann et al., PRL 43, 1494 (1979)
Theoretical Methods for Surface Science Part I Slide 20
Ab-initio Pseudo potentials
Start with an all-electron atom calculation
Immitate the effective potential felt by the
valence electrons by screening the potential
from the ion nucleus by the core electrons
Hamann et al. proposed four constraints:
I.
II.
Vps=VAE, r>rc
eips= eiAE
III. Norm conservation:
IV. Maintain scattering properties:
Hamann et al., PRL 43, 1494 (1979)
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Theoretical Methods for Surface Science Part I Slide 21
Modeling your system
Build your supercell
Check for convergence of basis set
and k-point sampling
Calculate the bulk properties using
the Murnaghan equation of state.
Calculate the electronic structure,
Density of states(DOS) and
bandstructure
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Theoretical Methods for Surface Science Part I Slide 22
Bulk properties
a0=equilibrium lattice parameter and V0 equilibrium volume
Ecoh=cohesive energy
B=Bulk modulus= V d2E
dV2
The bulk properties can be determined using the Murnaghan
equation of state:
E0=total energy at equilibrium lattice constant, B0=Bulk modulus,
B1=first derivative of B0 with respect to pressure
Murnhagan, Proc. Nat. Acad. Sci. USA 30, 244 (1944)
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Theoretical Methods for Surface Science Part I Slide 23
Calculating Bulk properties
Vary the lattice parameter and calculate the total energy. Make a
curve fit of the total energy values to the Murnaghan equation of
state:
x
x
x
x
x
x
Unit cell for Cu
x
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Theoretical Methods for Surface Science Part I Slide 24
Band structure
The dispersion relation between the wave vector and
the energy eigenvalues
Free electron model:
In general are the eigenvalues a
complicated function of k:
E(k)=f(k)
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Theoretical Methods for Surface Science Part I Slide 25
How to Calculate
DFT Band structure
I. Solve the Kohn-Sham equations
self-consistently to determine the
effective potential using an even
k-point sampling.
II. Use the effective potential while
solving the Kohn-Sham equations
non self-consistently along high
symmetry lines in the Brillouin zone
Bouckaert et al., Phys. Rev 50, 58 (1938).
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Theoretical Methods for Surface Science Part I Slide 26
Example: Band structure of Cu
Cu has FCC structure.
Bouckaert et al., Phys. Rev 50, 58 (1938).
High symmetry points in the Brillouin zone:
G=center of the Brillouin zone
L=mid point on the zone boundary plane in the {111}-directions
W=corner point on the hexagon of the {kikj}-plane
K=mid point on the edge between two hexagons {110}-direction
X= mid point on the zone boundary plane in the {100}-direction
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Theoretical Methods for Surface Science Part I Slide 27
Band structure of Cu
Electronic configuration of Cu: 3d94s2
Bouckaert et al., Phys. Rev 50, 58 (1938).
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Theoretical Methods for Surface Science Part I Slide 28
Density of states (DOS)
Method:
Calculate the Kohn-Sham eigenvalues with a very dense
k-point mesh.
Use a Gaussian or Lorentzian broadening function for
the delta function.
Perform the summation of the states over the Brillouin
zone.
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Theoretical Methods for Surface Science Part I Slide 29
Projected density of states (PDOS)
Method:
Calculate the Kohn-Sham eigenvalues ei and wave
functions yi.
Calculate the overlap between the Kohn-Sham wave
functions yi and atomic wave functions fal
Use a Gaussian or Lorentzian broadening function for
the delta function.
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Theoretical Methods for Surface Science Part I Slide 30
DOS for Graphene
Brillouin Zone
K
DOS
e-eF [eV]
G
M
K
G
M
k[Å-1]
K
G
van Hove singularities
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e-eF [eV]
Theoretical Methods for Surface Science Part I Slide 31
PDOS for Graphene
Brillouin Zone
G
M
K
DOS
PDOS
px,py
s
pz
e-eF [eV]
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e-eF [eV]
Theoretical Methods for Surface Science Part I Slide 32
Surfaces
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Theoretical Methods for Surface Science Part I Slide 33
Surface energy
A
I
II
Gibbs free energy:
G(T,p) = E-TS + pV= SjNjmj
where the chemical potential is defined
Surface energy g = Energy cost to create a surfaces
g= 1 (GI+II(T,p)-SiNimi])
A
Solids (low T):
G(T,p) ~ G(0,0) ~ Etot
g= 1 (Esurf -Ebulk)
A
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Theoretical Methods for Surface Science Part I Slide 34
Modeling Surfaces
The jellium model
Fridell oscillations
in the electron
density near the
surface
electrons spill out
from the surface
Lang and Kohn, PRB 1,4555(1970)
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Theoretical Methods for Surface Science Part I Slide 35
Modeling Surfaces
Lang and Kohn, PRB 1,4555(1970)
The surface energy diverges for metals with high electron
density when the Jellium model is used!
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Theoretical Methods for Surface Science Part I Slide 36
Modeling Surfaces
Supercell geometries:
+ proper surface electronic structure
+ good convergence with slab thickness
+ suitable for plane wave basis sets
 artificial lateral periodicity: “ordered arrays”
 inherently expensive
Payne et al., Rev. Mod.
Phys. 64, 1045 (1992).
Cluster geometries:
+ very cheap for small clusters (local basis sets)
+ ideal for local aspects (defects etc.)
 slow convergence with cluster size (embedding etc.)
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Healy et al,
PRL 87,
016105 (2001)
Theoretical Methods for Surface Science Part I Slide 37
Convergence of slab models
The slab should be thick enough that the
middle layers obtain bulk properties and that
the two surfaces do not interact with each
other through the slab.
vacuum
ts
slab
The vacuum region should be thick enough
that the two surfaces do not interact with
each other through the vacuum region.
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Theoretical Methods for Surface Science Part I Slide 38
Quantum size effects
The electronic states in the slab are quantized
perpendicular to the surface.
Boettger, PRB 53, 13133 (1996)
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Theoretical Methods for Surface Science Part I Slide 39
Atomic Relaxation
It is necessary to relax the forces on the atoms in order to find
the lowest energy ground state of the crystal.
Calculate the forces on the atoms:
The ions are so heavy that they can be considered classical
Move the atoms according to the discretized version of
Newton’s second law:
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Theoretical Methods for Surface Science Part I Slide 40
Atomic Relaxation
To get a rapid convergence it is necessary to have a good choice
of the step length.
Local minima
Global minima
However, the system might get trapped in a local minima, so it is
sometimes necessary check different reconstructions and compare
the surface energies!
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Theoretical Methods for Surface Science Part I Slide 41
Surface relaxations
at metal surfaces
Smoluchowski smoothing at metal surfaces,
Finnis and Heine, J. Phys. Chem. B 105, L37 (1973)
The charge density will be redistributed at the surface such the
charge is moved from the regions directly above the atom cores
to the regions between the atoms.
The atoms in the surface layer experience a charge imbalance.
This give rise to an inward electrostatic force which leads to a
compression of the separation between the surface layers.
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Theoretical Methods for Surface Science Part I Slide 42
Surface relaxation of Cu surfaces
the charge density is smoothened at the surface
All-electron LCGO DFT-calculations
for Cu(111)-surface.
Euceda et al., PRB 28,528 (1983)
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Gross, Theoretical Surface Science
Theoretical Methods for Surface Science Part I Slide 43
Surface relaxation at
semiconductor surfaces
Basic principle: The observed surface structure has the lowest freeenergy among the kinetically accessible structures under the
paricular preparation conditions.
Principle 1: A surface tend to minimize the number of dangling
bonds by the formation of new bonds. The remaining dangling
bonds tend to be saturated.
Principle 2: A surface tend to compensate charges.
Principle 3: A semiconductor surface tend to be insulating.
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Theoretical Methods for Surface Science Part I Slide 44
GaAs
GaAs is a compound material:
Remove the chemical potential for Ga and express the surface
energy as function of As and GaAs
The limits for chemical potential of
As is given by
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Theoretical Methods for Surface Science Part I Slide 45
Surface reconstruction of GaAs(100)
Moll et al., PRB 54, 8844 (1996).
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Theoretical Methods for Surface Science Part I Slide 46
Summary
The foundations of the DFT
How to calculate bulk properties and electronic structure
How to model surfaces
Surface structures
Next lecture:
Electronic structure at surfaces
Adsorption
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Theoretical Methods for Surface Science Part I Slide 47