Transcript Theoretical Methods for Surface Science

Theoretical Methods for
Surface Science
part II
Johan M. Carlsson
Theory Department
Fritz-Haber-Institut der Max-Planck-Gesellschaft
Faradayweg 4-6, 14195 Berlin
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 1
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
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 2
Charge distribution at Surfaces
electrons spill out from the surface
Jellium model
All-electron LCGO DFT-calculations
for Cu(111)-surface.
Lang and Kohn, PRB 1,4555(1970)
Euceda et al., PRB 28,528 (1983)
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 3
Work function
surface dipole d
Work function F
+
d
Jellium model
Potential difference
Df=f ()-f (-)=4pd
Lang and Kohn, PRB 1,4555(1970)
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 4
Work function
Work function F
Chemical potential of the electrons
m=E(N+1)-E(N)=EF
Work function
F=f ()-m = Df-m
Potential difference
Df=f ()-f (-)=4pd
Lang and Kohn, PRB 1,4555(1970)
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 5
Nearly Free electron model (NFE)
Periodic potential
V(z) = -Vo+2Vgcos(gz)
V0
Band gap
opens
at the zone
boundaries
2Vg
The energies and wave functions:
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 6
Surface states
The solution for imaginary values of
k is also possible at the surface:
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 7
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)
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 8
DFT bandstructure for Cu(111)
2x2
1x1
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 9
Bandstructure of Cu(111)
6-layer slab
18-layer slab
Euceda et al., PRB 28,528 (1983)
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 10
Projected Bulk bandstructures
kz
k
k
k
kx
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.
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 11
Projected Bulk bandstructures
kz
k
k
k
kx
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.
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 12
Bandstructure of Cu(111)
Surface BZ
G
M
K
Schockley
surface state
Tamm state
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 13
Adsorption
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 14
Energy
Adsorption
Activation barrier
Ediss
Eads
z
Physisorption well
Chemisorption well
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 15
Thermodynamics for adsorption
a
ma
Host
Definition of adsorbate energy:
Eads=DG=G[host+ads]-{G[host]+Na ma}
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)
This gives the adsorption energy
Eads={E[host+defect]+Fvib(T,w)}-{E[host]+Na ma}
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 16
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.
Reuter and Scheffler, PRB 65, 035406 (2002).
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)
where N is the number of adsorbtion sites
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 17
Phase diagram
Reuter and Scheffler, PRB 68, 045407 (2003)
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 18
Physisorption
metal
r’
-
+
z
z
+
r -
The electrostatic energy:
Taylor expand in terms of 1/z:
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 19
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).
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 20
Physisorption of O2 on graphite
h=3.4 Å
DFT-GGA: Eads=0.04 eV/O2
TPD-experiment: Eads=0.12 eV/O2
Ulbricht et al.,PRB 66, 075404 (2002)
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 21
Chemisorption
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 22
Adsorption sites
B
T
Top site
B
Bridge site
F
Hollow FCC-site
H
Hollow HCP-site
H
F
T
Close packed (111)-surface
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 23
Finding the adsorption site
Adsorption 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
barrier
chemisorption sites
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 24
Potential energy surface
O2 on Pt(111), Gross et al., Surf. Sci., 539, L542 (2003).
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 25
Newns-Anderson model
Anderson, Phys. Rev. 124, 41 (1961)
Newns, Phys. Rev. 178, 1123 (1969)
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 >.
e
k
International Max-Planck Research School
|a>
Theoretical Methods for Surface Science Part II Slide 26
Green’s function techniques
The Green’s function Gs(e)
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.
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 27
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
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 28
Newns-Anderson model continued
Calculate the Green’s function for the Hamiltonian
as
and identify the self-energy components:
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 29
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
e
|a>
sp-band
International Max-Planck Research School
D
Theoretical Methods for Surface Science Part II Slide 30
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:
Solving this equation gives two roots
corresponding to bonding and anti bonding levels of the
absorbate system.
e
|a>
d-band
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 31
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.
Gurney, Phys Rev. 47, 479 (1933)
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.
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 32
Chemisorption on a metal surface
Na/Cu(111)
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 33
DF[eV]
Adsorbate induced
work function change
Tang et al., Surf. Sci. Lett. 255, L497 (1991).
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 34
Charge transfer for Na/Cu(111)
charge depletion
charge
accumulation
+
adsorbate induced dipole
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 35
Properties for Na/Cu(111)
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 36
Quantum well state for Na/Cu(111)
Carlsson and Hellsing, PRB 61, 13973 (2000)
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 37
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 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.
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 38
Harding’s compensating
surface charge Qs
(Surf. Sci. 422, 87 (1999))
Qs
Q1Q2 Qp
repeat unit
a0
r1 r2
Qs=aQ1, where
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 39
Ex: Properties of ZnO
a) Ground state structure for ZnO: Wurtzite structure
b) High pressure structure: Rock salt
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 40
Electronic structure of ZnO
EgapExp=3.4 eV
EgapDFT-GGA=0.8eV
Under estimation of the bandgap in semi-conductors is a common
problem in DFT-calculations with LDA or GGA exchangecorrelation functional.
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 41
The polar ZnO{0001}-surface
Zn-terminated [0001]-surface
[0001]
O-terminated [0001]-surface
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 42
The polar ZnO{0001}-surface
B
A
Carlsson, Comp. Mat. Sci. 22, 24 (2001)
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 43
The polar ZnO{0001}-surface
Carlsson, Comp. Mat. Sci. 22, 24 (2001)
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 44
STM of ZnO[0001]-surface
Dulub et al.,PRL 90, 016102 (2003)
Triangular islands
Step height=2.7 Å=c/2
n=O-edge atoms
International Max-Planck Research School
b)
Triangle
# of O-atoms = n(n+1)/2
# of Zn-atoms =n(n-1)/2
Q=#Zn / #O =3/4 => n=7
L = (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)/2
Q=#Zn / #O =3/4 => n=6
L = (2(n-1)-1)*a = 29.25 Å
Theoretical Methods for Surface Science Part II Slide 45
Surface Phase diagram of ZnO[0001]
Kresse et al., PRB 68, 245409 (2003)
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 46
Summary
•Surface energy
•Atomic structure relaxation
•Charge redistribution
•Work function
•Surface states
•Adsorption
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 47
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
International Max-Planck Research School
Theoretical Methods for Surface Science Part II Slide 48