1. Ab-initio study of work functions of element metal surface
Xiang Ma
Materals Process Design and Control laboratory
MAE 715 final project, May 7th, 2007
Instructor: Professor Zabaras
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

2. Definition of Work function Slab model and super cell
Outline
Definition of Work function
Slab model and super cell
Computation Methods (Density functional theory)
Change of work function due to the orientation of clean surface
Change of work function due to absorption of H atom
Conclusion
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

3. A image of surface for fcc(111)
From Solid to Surface
A image of surface for fcc(111)
In this course, most of the problems we deal with are bulk properties.
In nature, crystals are not infinite but finite macroscopic three-dimensional objects terminated by surfaces.
Many phenomena and processes occur at the interface between a condensed phase and the environment.
Modeling surfaces is then of great theoretical and practical interest.
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

4. Atomic resolution on Pt(100)
From Solid to Surface
Atomic resolution on Pt(100)
The key-ingredient to surface science experiments is ultra-high vacuum (UHV).
To main a low pressure to assure that a surface stays clean for a time long enough to do some experiments.
With the development of density functional theory, we can also explore the surface properties through the ab-initio study.
A very good surface science tutorial.
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

5. From Solid to Surface
A lot of phenomenon associated with surface can be studied by first-principal calculation
--- surface reconstruction and surface relaxation
--- surface energy
--- adsorption on surfaces
--- interface
--- work function
With the adsorption of atoms or molecules, the surface electronic structure is modified and the work function can change by several eV.
The measurement of the work function changes can give valuable insight in to the condition of a given surface.
Nowadays, the work function can be calculated by ab-initio methods in the framework of density functional theory.
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

6. Work function definition
Work function is defined as the minimum energy necessary to extract an electron from the metal at 0K.
For a crystal with electrons, if is the initial energy of the metal and
that of the metal with one electron removed to a region of electrostatic potential
, we define
Note: The removed electron is assumed to be at rest, and therefore possesses only potential energy.
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

7. Work function definition
At 0K, the chemical potential is by definition
In the limit of large systems, all polarisation effect can be neglected after removing the electron. Then chemical potential is then shown to coincide with the Fermi energy
The work function, finally, is the difference between the Fermi level and the vacuum level.
Schematic energy diagram of a metal
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

8. Work function definition
The calculation of work function is then divided into two parts.
First to perform a self-consistent calculation to find the Fermi energy of the slab.
Second, we need to find the electrostatic potential in the vacuum level.
Macroscopic average
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

9. Macroscopic average
The electronic density is the basic variable calculated by DFT.
Introduce the plane-averaged electronic density:
where z axis is perpendicular to the slab surface
The macroscopic-average electronic density is then defined from the integration over the interplanar distance d of the slab:
The potential is related to the charge density via the Poisson equation. So we can get a similar relation between plane-averaged potential and macroscopic average
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

10. Macroscopic average
By plotting the macroscopic average over the z axis, the vacuum level is found.
Because the curve of the average is nearly flat in the vacuum provided the vacuum is large enough.
Subtracting this vacuum level from the Fermi level get the work function for the metal surface.
Work Function
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

11. Slab model and supercell approach
Slab model is the most popular way to model the surface.
The slab model consists of a film formed by a few atomic layers parallel to the crystalline plane of interest.
Using plane waves needs to force a 3-D periodicity. The thin slabs needs to repeat in one direction.
To perform a supercell calculation, one defines a unit cell oriented with one axis perpendicular to the surface of interest, containing the inequivalent atoms of a crystalline thin film and some vacuum layers.
Ideally, the thickness of the vacuum layer and of the slab must be large enough for two successive metal surfaces not to interact significantly.
supercell
thin slabs
vacuum layer
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

12. Slab model and supercell approach
It is not trivial to construct the slab model at first. You need to visualize them.
A nice web tool Surface Explorer is used for this purpose.
fcc(110)
fcc(100)
fcc(111)
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

13. Slab model and supercell approach
XCrysDen is an application for visualizing crystalline and molecular structures.
All of the slab models studied were viewed using XCrysDen to ensure that their
geometries were described correctly.
Surface primitive cell is two-dimensional, which is different from conventional bulk primitive cell.
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

14. DFT calculations
As a preliminary step towards the study of surface, we have to find the equilibrium lattice constant;
It is well known that the equilibrium atomic positions in a crystal surface are generally different from those in the ideal bulk-terminated surface. We need to perform a relaxation calculation to find the equilibrium geometry of the surface.
The relaxed coordinates are put into another input file to perform a self-consistent calculation to find the Fermi energy in the slab
Using post-process code to extract the electrostatic potential from the output file.
Calculate the macroscopic average potential to determine the vacuum level
Put the two values into the definition of the work function to determine the final solution.
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

15. DFT calculations Basis sets --- plane wave
Cut-off energy of 16 Ry for the plane wave expansion
ultrasoft pesudopotentials
The Fermi level is positioned using the Methfessel-Paxton (MP) scheme, with the smearing parameter set to 0.01 Ry.
8x8x1 special Monkhorst-Pack special k-points
slab models
A surface unit cell with a slab of 8 atom layers and 8 equivalent vacuum layers was chosen to model the metal surface. H atom coverage is a full monolayer.
Exchange-Correlation approximation
LDA(Perdew-Zunger form)
Software: Quantum Espresso (opEn-Source Package for Research in Electronic Structure, Simulation, and Optimization), version 3.2
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

16. Summarize of the computation procedure
Fit E Vs V curve to find the theoretical lattice constant (pw.x)
self-consistent calculation to find the Fermi energy (pw.x)
Set up the appropriate thickness of slabs and vacuums
Extract the electrostatic potential form the self-consistent calculation (pp.x)
Calculate the macroscopic average (average.x)
relax the geometry of the slab (pw.x)
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

17. Change of work function depends on the surface orientation
Numerical examples
Change of work function depends on the surface orientation
Change of work function due to the H atom adsorption
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

18. Lattice constant Theoretic: 7.50 bohr Experimental : 7.66 bohr
LDA underestimate
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

19. Al and Al(100) Al fcc structure a = 7.50 a.u unit cell
3rd layer
1st layer
Al(100) side view
Al(100) top view
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

20. Macroscopic average (solid line) Macroscopic average (solid line)
Al(100)
Work Function
Plane-averaged electronic charge density (dashed line)
Macroscopic average (solid line)
Plane-averaged electrostatic potential (dashed line)
Macroscopic average (solid line)
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

21. Al(110) Al(110) side view Al(110) top view unit cell 3rd layer
1st layer
Al(110) side view
Al(110) top view
unit cell
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

22. Macroscopic average (solid line) Macroscopic average (solid line)
Al(110)
Work function
Plane-averaged electronic charge density (dashed line)
Macroscopic average (solid line)
Plane-averaged electrostatic potential (dashed line)
Macroscopic average (solid line)
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

23. Al(111) Al(111) top view Al(111) side view unit cell C B A
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

24. Macroscopic average (solid line) Macroscopic average (solid line)
Al(111)
Work Function
Plane-averaged electronic charge density (dashed line)
Macroscopic average (solid line)
Plane-averaged electrostatic potential (dashed line)
Macroscopic average (solid line)
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

25. Calculations of Work function of Al
Result
Calculations of Work function of Al Al
Fermi Level (eV)
Vacuum (eV)
Work Function (eV)
Experimental (eV)
(100)
2.364
6.782
4.418
(110)
2.488
6.768
4.28
(111)
2.634
6.869
4.235
The results are in a good agreement with the experimental values.
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

26. Calculations of Work function of Copper
Result
Calculations of Work function of Copper Cu
Fermi Level (eV)
Vacuum (eV)
Work Function (eV)
Experimental (eV)
(100)
5.551
10.391
4.84
(110)
2.390
7.105
4.715
(111)
5.581
10.780
5.199
The results shows a little deviation from the experimental values. It may be due to the experiment is performed at room temperature, while the calculation is at 0K. Overall, it shows good accuracy using this method since the error is within the computational range.
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

27. Anisotropy of the work function
From the Cu, we see that it shows the trend (110), (100),(111) of increasing work function.
This is best explained by the Smoluchowski[1] smoothing.
This smoothing leads to a dipole moment which opposes the dipole created by the spreading of electron and thus reducing the work function
Surface orientations of high density experience small smoothing, inducing a small reverse dipole, and thus a high work function.
[1] R Smoluchowski, Phy. Rev. 60, 1941
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

28. Anomaly anisotropy of Al work function
However, from the calculation, it is seen that the Al doesn’t obey this increasing ordering.
In the paper [1], the author investigated this phenomenon and concluded that the trend of the work function Al can be explained by a charge transfer the atomic-like p orbitals of the surface ions perpendicular to the surface plane to those parallel to the surface, when compared to the bulk charge density.
Thus it results from a dominant p-atomic-like character of the density of states near the Fermi energy.
Overall, our methods recovered both the normal and abnormal anisotropy of then work function of the fcc metals.
[1] C.J.Fall, N.Binggeli and A. Baldereschi, Phy. Rev. B, 58,1998
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

29. Adsorption of H on the Al(111) surface
There are four inequivalent adsorption sites on an fcc (111) surface.
We consider a monolayer of H atom adsorpted on one Al (111) surface.
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

30. H on the Al(111) surface (top view)
ontop
bridge
hcp hollow
fcc hollow
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

31. H on the Al(111) surface (side view)
ontop
bridge
fcc hollow
hcp hollow
H/Al(111)
Fermi Level (eV)
Vacuum (eV)
Work Function (eV)
ontop
1.050
6.106
5.056
bridge
0.4527
4.791
4.338
fcc hollow
0.5922
4.807
4.215
hcp hollow
0.6391
4.803
4.164
Clean surface:
4.235 eV
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

32. Calculations of Work function of H/Al(111) ontop site
H on the Al(111) surface
Calculations of Work function of H/Al(111) ontop site
H coverage
Fermi Level (eV)
Vacuum (eV)
Work Function (eV)
0.25
1.299
5.772
4.473
0.50
1.174
5.887
4.713
1.00
1.050
6.106
5.056
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

33. H on the Al(111) surface
Adsorption at the ontop and bridge site increase the work function while at the hollow sites decrease the work function.
This is due to the dipole induced by H-adsorption: when the H atom at the ontop and bridge site, it pulls away the electron from the surface. However, when the induced dipole opposes the spill-out of the electrons, it reduces the work function.
The work function increases with the increase of the H coverage. This is because at the low coverage, the dipole-dipole interaction will keep the atoms apart , while at high coverage, the same interaction will cause a depolarization of the dipoles and increase the work function.
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

34. Conclusion
The method can be used to calculate the accurate work function.
The change of work function depends on the surface orientation, adsorption sites and the adsorption coverage.
work function is the fundamental properties of the electronic structure of the surface. Its measurement can give valuable insight into the condition of a given surface.
This method can also be extended to semiconductor.
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)

35. Thank you! Acknowledgement Prof. Zabaras
MPDCC cluster for the computation
Software: Quantum Espressor
Thank you!
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (5/7/2007)