1 ab-initio study of work functions of element metal surface mae 715 – atomistic modeling of...

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Ab-initio study of work functions of element metal surface

CCOORRNNEELLLL U N I V E R S I T Y


MAE 715 – Atomistic Modeling of MaterialsN. Zabaras (5/7/2007)

MAE 715 final project, May 7th, 2007Instructor: Professor Zabaras

Xiang Ma

Materals Process Design and Control laboratory

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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

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From Solid to Surface

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.

A image of surface for fcc(111)

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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.

Atomic resolution on Pt(100)

http://www.uksaf.org/tutorials.html

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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.

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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

W

N NE 1NE

eV

1( )N e NW E V E

Note: The removed electron is assumed to be at rest, and therefore possesses only potential energy.

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Schematic energy diagram of a metal

At 0K, the chemical potential is by definition

1N NE E

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

FE

The work function, finally, is the difference between the Fermi level and the vacuum level.

e FW V E

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e FW V E 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

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Macroscopic average

The electronic density is the basic variable calculated by DFT.

Introduce the plane-averaged electronic density:

( )n r

1( ) ( )

Sn z n r dxdy

S

where z axis is perpendicular to the slab surface S

The macroscopic-average electronic density is then defined from the integration over the interplanar distance d of the slab:

( )n z

/ 2

/ 2

1( ) ( )

d

dn z n z z dz

d

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

( )eV z

( )eV z/ 2

/ 2

1( ) ( )

d

e edV z V z z dz

d

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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.

Lattice Units (A)

Ene

rgy(

eV)

0 10 20 30 40 50-10

-8

-6

-4

-2

0

2

4

6

fE

Work F

unction

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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

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o It is not trivial to construct the slab model at first. You need to visualize them.

o A nice web tool Surface Explorer is used for this purpose.

fcc(100) fcc(110)

fcc(111)

http://w3.rz-berlin.mpg.de/~rammer/surfexp_prod/SXinput.html

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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.

http://www.xcrysden.org/

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DFT calculations

1. As a preliminary step towards the study of surface, we have to find the

equilibrium lattice constant;

2. 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.

3. The relaxed coordinates are put into another input file to perform a self-

consistent calculation to find the Fermi energy in the slab

4. Using post-process code to extract the electrostatic potential from the output

file.

5. Calculate the macroscopic average potential to determine the vacuum level

6. Put the two values into the definition of the work function to determine the final

solution.

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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

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Summarize of the computation procedure

Fit E Vs V curve to find the theoretical lattice constant (pw.x)

Set up the appropriate thickness of slabs and vacuums

relax the geometry of the slab (pw.x)

self-consistent calculation to find the Fermi energy (pw.x)

Extract the electrostatic potential form the self-consistent calculation (pp.x)

Calculate the macroscopic average

(average.x)

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Numerical examples

1. Change of work function depends on the surface orientation

2. Change of work function due to the H atom adsorption

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Lattice constant

Energy Vs. Volume

-4.192

-4.19

-4.188

-4.186

-4.184

-4.182

-4.18

-4.178

-4.176

-4.174

7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8

Lattice constant (a0)

En

erg

y (R

yd)

Ecut = 20

Ecut = 40

Ecut = 60

Theoretic: 7.50 bohr

Experimental : 7.66 bohr

LDA underestimate

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Al and Al(100)

Al• fcc structure• a = 7.50 a.u

Al(100) side view Al(100) top view

1st layer

3rd layer

unit cell

(1 1)

0, 90a b c

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Al(100)

Lattice Units

Ele

ctro

nic

Den

sity

0 10 20 30 40 500

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Lattice Units (A)

Ene

rgy(

eV)

0 10 20 30 40 50-10

-8

-6

-4

-2

0

2

4

6

fE

Work F

unction

Plane-averaged electronic charge density (dashed line)

Macroscopic average (solid line)

Plane-averaged electrostatic potential (dashed line)


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Al(110)

Al(110) top viewAl(110) side view

1st layer

unit cell

0, 90a b c

(2 1)3rd layer

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Al(110)





Lattice Unit (A)

Ele

ctro

nic

Den

sity

0 10 20 300

0.01

0.02

0.03

Lattice Unit (A)

Ene

rgy(

eV)

0 10 20 30

-6

-4

-2

0

2

4

6

Work

function

fE

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Al(111)

Al(111) top viewAl(111) side view

A

unit cell

, 90 , 120o oa b c

(1 1)C

B

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Al(111)





Lattice Unit (A)

Ele

ctro

nic

Den

sity

0 10 20 30 40 500

0.005

0.01

0.015

0.02

0.025

0.03

Lattice Unit (A)

Ene

rgy

(eV

)

0 10 20 30 40 50

-12

-9

-6

-3

0

3

6

fE

Work Function

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Result

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

Calculations of Work function of Al

4.41 0.02

4.28 0.02

4.24 0.03

The results are in a good agreement with the experimental values.

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Result

Cu Fermi Level (eV)


Experimental (eV)

(100) 5.551 10.391 4.84

(110) 2.390 7.105 4.715

(111) 5.581 10.780 5.199

Calculations of Work function of Copper

4.59 0.03

4.48 0.03

4.94 0.03

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.

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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

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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

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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.

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H on the Al(111) surface (top view)

bridge

hcp hollow fcc hollow

ontop

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H on the Al(111) surface (side view)

ontop bridge hcp hollowfcc hollow

H/Al(111) Fermi Level (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

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H on the Al(111) surface

H coverage Fermi Level (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

Calculations of Work function of H/Al(111) ontop site

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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.

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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.

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Acknowledgement

Prof. Zabaras

MPDCC cluster for the computation

Software: Quantum Espressor

Thank you!