Theoretical Methods for Surface Science part I

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Theoretical Methods for Surface Sciencepart I

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

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Bulk
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Surfaces
A
I
II
The surface break the 3D-periodicity of the bulk
crystal Total energy of the system
GIIIGIGIIDGsurface
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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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Real world problems are complex
EkaNobel, Bohus, Sweden
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Surface Science methods
ABCgtACB
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The Multi scale approach
Thermodynamics Classical mechanics
kinetic Monte Carlo simulations
Electron structure methods
It is necessary to combine different methods in
order to tackle realistic problems
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Total energy methods
Simple empirical potentials - Force fields, pair
potentials Intermediate methods -
Tight-binding, - many-body potentials
EAM ab-initio techniques - Hartree-Fock -
Density functional theory (DFT) Beyond DFT -
GW - Quantum Monte Carlo - Quantum Chemical CI
Accuracy
Accuracy
Number of atoms treated
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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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Hohenberg-Kohn theorem
Hamiltonian for a many-electron system
Variational principle lt Y H Y gt ? lt Y0 H
Y0gt E0
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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
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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 self-consistently.
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 En(r)EII
EeIn(r) Ekinn(r) EHn(r) Excn(r)
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The self-consistent scheme
Payne et al., Rev. Mod. Phys. 64, 1045 (1992).
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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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K-point sampling
Blochs 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).
IBZ
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. Using symmetry lowers the
number of necessary k-point to the ones in the
Irreducible Brillouin zone (IBZ).
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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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Basis sets
Two common basis sets are

• Plane waves
• Complete basis set
• Systematic way of improving the accuracy
• Many plane wave are needed to accurately describe
  localized wave functions
• Periodic boundary conditions necessary

Localized orbitals Only a few basis functions
needed per atom Hamiltonian matrix is sparse
Periodic boundary conditions not necessary - No
systematic way of improving accuracy
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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
x
valence electrons
core electrons
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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 ion-electron interaction, since all
corrugation is removed.
Lang and Kohn, PRB 1,4555(1970)
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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.
Hamann et al., PRL 43, 1494 (1979)
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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
• VpsVAE, rgtrc

II. eips eiAE
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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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Bulk properties
The bulk properties can be determined using the
Murnaghan equation of state E0total energy at
equilibrium lattice constant, B0Bulk modulus,
B1first derivative of B0 with respect to
pressure Murnhagan, Proc. Nat. Acad. Sci. USA 30,
244 (1944)
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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
Unit cell for Cu
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Band structure
The dispersion relation between the wave vector
and the energy eigenvalues
In general are the eigenvalues a complicated
function of k E(k)f(k)
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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
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Example Band structure of Cu
Bouckaert et al., Phys. Rev 50, 58 (1938).
Cu has FCC structure. High symmetry points in
the Brillouin zone Gcenter of the Brillouin
zone Lmid point on the zone boundary plane in
the 111-directions Wcorner point on the
hexagon of the kikj-plane Kmid 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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Band structure of Cu
Electronic configuration of Cu 3d94s2
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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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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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DOS for Graphene
Brillouin Zone
e-eF eV
DOS
K G M K
G
e-eF eV
kÅ-1
van Hove singularities
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PDOS for Graphene
Brillouin Zone
px,py
PDOS
s
DOS
pz
e-eF eV
e-eF eV
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Surfaces
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Surface energy
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
Solids (low T) G(T,p) G(0,0) Etot
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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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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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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 Cluster
geometries very cheap for small clusters
(local basis sets) ideal for local aspects
(defects etc.) ? slow convergence with cluster
size (embedding etc.)
Payne et al., Rev. Mod. Phys. 64, 1045 (1992).
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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. 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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Quantum size effects
The electronic states in the slab are quantized
perpendicular to the surface.
Boettger, PRB 53, 13133 (1996)
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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 Newtons second law
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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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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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Surface relaxation of Cu surfaces
the charge density is smoothened at the surface
Gross, Theoretical Surface Science
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Surface relaxation at semiconductor surfaces
Basic principle The observed surface structure
has the lowest free-energy 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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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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Surface reconstruction of GaAs(100)
Moll et al., PRB 54, 8844 (1996).
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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