abstract - baylor-ir.tdl.org · mark r. mastin, b.s. a thesis approved by the department of physics...

ABSTRACT

An SEGF Study Of The Rh(001) Surface

Mark R. Mastin, M.S.

Advisor: Gregory A. Benesh, Ph.D.

A self-consistent study of Rh(001) has been performed using the surface

embedded Green function method. Calculations have been performed by

embedding a three-layer slab onto a bulk rhodium substrate. The calculated

work function and surface core level shifts are in good agreement with other

theoretical work. Additionally, bulk band structures have been calculated, and

the projected bulk band structure has been constructed. Densities of states have

also been calculated in order to identify surface states and surface resonances. In

addition to a contour plot of the total charge density, charge densities have been

plotted for several of the surface states.

Page bearing signatures is kept on file in the Graduate School.

An SEGF Study of the Rh(001) Surface

by

Mark R. Mastin, B.S.

A Thesis

Approved by the Department of Physics

___________________________________

Gregory A. Benesh, Ph.D., Chairperson

Submitted to the Graduate Faculty of

Baylor University in Partial Fulfillment of the

Requirements for the Degree

of

Master of Science

Approved by the Thesis Committee

___________________________________

Gregory A. Benesh, Ph.D., Chairperson

___________________________________

Kenneth T. Park, Ph.D.

___________________________________

Joe C. Yelderman, Jr., Ph.D.

Accepted by the Graduate School

August 2007

___________________________________

J. Larry Lyon, Ph.D., Dean

iii

TABLE OF CONTENTS List of Figures ..................................................................................................... v

List of Tables ....................................................................................................... vii

Acknowledgments.............................................................................................. viii

Dedication ........................................................................................................... ix

Chapter

I. Introduction................................................................................................... 1

II. Surface Electronic Structure Calculations ........................................................ 4

A. The Surface Problem.................................................................................. 4

B. Surface Embedded Green Function Method ............................................... 8

C. Embedding Potential.................................................................................. 13

D. Basis Functions.......................................................................................... 15

1. Interstitial Region................................................................................. 16

2. Muffin-Tin Spheres.............................................................................. 16

3. Vacuum Region ................................................................................... 17

E. Self-Consistency ........................................................................................ 17

F. Mixing....................................................................................................... 18

III. Computational Details ..................................................................................... 21

A. Embedding Potential.................................................................................. 21

B. Self-Consistent Charge Density.................................................................. 24

C. Bulk Band Structure and Projected Bulk Band Structure ............................ 24

D. Density of States........................................................................................ 26

iv

IV. Results and Analysis of Clean Rhodium .......................................................... 27

A. Work Function........................................................................................... 27

B. Surface Core-Level Shifts .......................................................................... 30

C. Projected Bulk Band Structure ................................................................... 34

D. Densities of States...................................................................................... 42

E. Charge Density .......................................................................................... 46

V. Conclusion ...................................................................................................... 60

References ........................................................................................................... 62

v

LIST OF FIGURES

Figure 2.1 Depiction of the Surface Region I (including vacuum) and Substrate

(bulk) Region II utilized in the SEGF method. ..................................... 9 2.2 Surface and bulk are separated by the complicated surface S, but the effective embedding interface S0 is used............................................... 14 2.3 Sketch depicting the three distinct sub-regions of the Surface Region. 15 3.1 Top view of two layers of rhodium....................................................... 22 3.2 Top view of Rh(001) demonstrating basis vectors. .............................. 23 3.3 Atoms in the surface region. .................................................................. 23 3.4 k points used to determine the projected bulk band structure........... 25 4.1 Bulk band structure at ! [(0,0)] plotted as a function of k

!. ................. 35

4.2 Bulk band structure at (0.125, 0) plotted as a function ofk

!. .................. 36

4.3 Bulk band structure at X [(0.5,0)] plotted as a function ofk

!. ............... 36

4.4 Bulk band structure at M [(0.5,0.5)] plotted as a function ofk

!. ........... 37

4.5 PBBS of even symmetry for the ! X line in the SBZ. ......................... 38 4.6 PBBS of odd symmetry for the ! X line in the SBZ. ........................... 39 4.7 PBBS for the ! X line in the SBZ. ......................................................... 40 4.8 PBBS for the entire SBZ. ........................................................................ 41 4.9 DOS at !. ................................................................................................ 43 4.10 DOS at X. ................................................................................................ 43 4.11 DOS at M. ............................................................................................... 44

vi

4.12 PBBS with DOS points overlayed.......................................................... 45 4.13 Total charge density on Rh(001) surface sampled on ten special

k points. ................................................................................................... 47 4.14 Total charge density on Rh(001) surface sampled on ten special

k points. ................................................................................................... 48 4.15 Valence charge density of the surface resonance of ! (0, 0) at

EF – 5.14 eV. ............................................................................................. 49 4.16 Valence charge density of the (0.25, 0) surface state at EF – 4.80 eV. .. 50 4.17 Valence charge density of the (0.5, 0) surface state at EF – 4.12 eV. .... 51 4.18 Valence charge density of the (0.5, 0.125) surface state at EF – 2.56 eV. 52 4.19 Valence charge density of the (0.5, 0.25) surface state at EF – 0.20 eV 53 4.20 Valence charge density of the (0.5, 0.3125) surface state at EF – 3.98 eV. 54 4.21 Valence charge density of the (0.5, 0.5) surface state at EF + 0.98 eV. . 55 4.22 Valence charge density of the (0.3125, 0.3125) surface state at

EF – 0.58 eV............................................................................................... 56 4.23 Valence charge density of the (0.1875, 0.1875) surface state at

EF – 3.92 eV............................................................................................... 57 4.24 Valence charge density of the (0.0625, 0.0625) surface state at

EF – 3.03 eV............................................................................................... 58

vii

LIST OF TABLES

Table

4.1 Comparison of experimental and theoretical work functions. ..................... 28

4.2 Surface and Sub-surface Core Level Shifts. .......................................... 33

viii

ACKNOWLEDGMENTS

I am forever grateful to Dr. Greg Benesh for his seemingly infinite

patience through this long process. He never gave up on me and his

encouragement and persistence helped to carry me through. His wisdom and

guidance were invaluable and he was consistently generous with both. Thank

you, also, to Dr. Kenneth Park and Dr. Joe Yelderman for their willingness to

serve on my committee on short notice.

I would also like to thank the Graduate School and the Department of

Physics for allowing me the opportunity to study and for giving me “one last

chance.”

I must thank my colleagues in Information Technology Services that

helped me throughout the process: they encouraged me; they provided

necessary technical assistance; and they allowed me the time required to

complete my study.

Last, but certainly not least, thank you to my family for encouraging me

throughout this process. Thanks for always being there for me (even when I

could not return the favor). I hope now to give you back some of the time that

you have lent me.

ix

To

Regina, the love of my life. You never gave up on me!

Drew, Evan, and Lindsay. Don’t quit and never say never.

Mom and Dad. Thanks for teaching me to keep reaching.

Granddaddy. I’m sorry you didn’t see me finish, but you helped me get there.

1

CHAPTER ONE

Introduction

Solid surfaces come in two types: amorphous (in which there is no regular

array of atoms) and crystalline (in which a regular pattern of atoms is repeated

continually). However, all solids do form crystals under certain circumstances,

which results in a computational simplification. For this reason, crystal surfaces

are more commonly studied. A theoretical study that models a physical crystal

would require a solution to an N-body problem, with N on the order of 1023.

Periodic boundary conditions and crystalline symmetry only reduce the problem

so far: N remains on the order of 109. Unfortunately, this problem is

unmanageable, even with today’s supercomputers. Certain assumptions and

approximations must be made to reduce the size of the problem in order to solve

accurately and within a reasonable time period.

Previously, the most common theoretical approach to the surface problem

has been to model the crystal as a slab, typically five to nine atomic layers thick,

and extending to infinity in the x-y plane. This method allows determination of

several aggregate surface electronic properties such as work functions, charge

densities, and spin densities. While this approximation reduces the problem to a

manageable size, it also contains limitations. A thicker slab would presumably

give more accurate results, but the amount of computational resources required

becomes too great. Another limitation of these slab-based methods is the

inability of the valence electrons to properly screen two surfaces that are in close

proximity to each other. As a result, states arising from the two surfaces interact

2

unphysically with each other, creating pairs of hybridized states at different

energies.

Benesh and Inglesfield1,2,3 have developed a technique which more

accurately describes the surface using a semi-infinite geometry. The Surface

Embedded Green Function (SEGF) method treats the surface as a semi-infinite

crystal using an embedding potential to exactly represent the influence on the

surface of an infinite bulk substrate. The SEGF method has been used to obtain

excellent results for Al(001)4, Al(111)5, Al(110)6, Fe(001)7, Pt(001)8, as well as

chemisorbed surfaces such as O/Pt(001)9, CO/Fe(001)10, and N/Fe(001)11. The

general nature of the surface problem and the specifics of the SEGF method will

be described in chapter two.

In chapter three, the computational details of the SEGF method as applied

to the surface of the 4d transition metal rhodium are discussed. Rhodium has an

important industrial application in its use as an exhaust catalyst for nitric oxide

reduction and carbon monoxide oxidation.12 The present characterization of the

clean Rh(001) surface is necessary for a later determination of the interactions

between rhodium and various overlayers, such as NO and CO, atop the surface.

The intent is to gain a fundamental understanding of such reactions in an effort

to achieve greater efficiency and increased gas reduction. As noted, the SEGF

method has been used very successfully for many metals, both clean and with

overlayers; however, this study of rhodium represents the first application of the

SEGF method to a 4d transition metal surface.

The fourth chapter discusses the results of the SEGF study of clean

rhodium. The work function is calculated and compared to other theoretical and

experimental results. The accuracy of the work function is commonly viewed as

3

a measure of the accuracy of the method. The surface core level shift (SCLS) is

also evaluated and compared with other results. The band structure of Rh(001)

has been investigated to determine the positioning of surface states and surface

resonances. These are compared to other theoretical models and to available

experimental results.

4

CHAPTER TWO

Surface Electronic Structure Calculations

A. The Surface Problem

There are no known closed-form solutions of the Schrödinger equation for

a system of N interacting particles when N is greater than two. Thus, in order to

find the electronic properties of the surface of some material, one must make

some approximations. To simplify the problem, it is desirable to recast the

many-body problem as a system of single-particle equations. The separated

equations can then be solved individually for each particle. The regular array of

atoms in a crystal allows the problem to be reduced to solving a Schrödinger-like

equation for each electron in a single unit cell:

!!2

2m"2

+Ui(r)

#$%

&'()

i(r) = *

i)

i(r) (2.1)

where

�

!!2

2m"2 is the kinetic energy operator, U

i(r) is the one-electron potential

(which is non-local due to many-body effects), εi is the associated energy

eigenvalue, and !i(r) is the electronic wavefunction. This equation is for an

independent electron, with the effect of the other electrons and ions in the system

taken into account in the potential energy term Ui(r) .

In the Hartree approximation, the wavefunction solution to the original N-

particle problem is the product of the one-electron wavefunctions. Labeling the

5

N-particle wavefunction as Ψ, the N-electron Schrödinger equation for the

crystal system is

H! = "!2

2m i

2# ! " Ze21

ri " R!

R

$%

&'(

)*+1

2

e2

ri " rj! = E!

i+ j$

i=1

N

$ . (2.2)

The second term, the negative potential energy term, represents the electrostatic

potential at ri due to the fixed nuclei at the points R of the Bravais lattice. The

positive potential energy term is the electron-electron repulsion; it is this

interaction term that makes equation (2.2) intractable. Some simplification must,

therefore, be made to approximate this electron-electron interaction.

Returning to the one-particle equation (2.1), an initial approximation is

made for Ui(r) . U

i(r) should contain the potential of the ions:

Uion(r) = !Ze

2 1

r ! RR

" (2.3)

But Ui(r) also needs to contain, at least approximately, the influence of each of

the remaining electrons. If these other electrons are treated as a distribution of

negative charge, then the potential energy due to electrostatic repulsion is

Uel(r) = !e dr '"(r ')

1

r ! r '# , (2.4)

where

!(r) = "e #i(r)

2

i

$ (2.5)

is the charge density and the sum is over all occupied one-electron levels. The

iterative process begins with an initial guess of !(r) . (!(r) is often approximated

from calculated atomic charge densities.) With the starting charge density, Uel is

6

then calculated and added to Uion to give the initial approximation of Ui(r) . The

Schrödinger equation is solved, and the output charge density constructed from

the one-electron wavefunctions. This process is repeated until the output charge

density of two successive iterations does not change significantly. This point is

known as self-consistency.

Although the Hartree approximation has been successful in obtaining

many properties of atomic system (at least qualitatively), it has its shortcomings

as well. First, it fails to account for the anti-symmetry of the electron

wavefunctions. Second, it does not adequately incorporate correlation effects

between pairs of electrons, since electrons interact directly with one another−not

just through the averaged density.

If Hartree’s product wavefunction is replaced by a Slater determinant of

one-electron wavefunctions in the form:

!(r1s1,r2s2,…,r

NsN) =

"1(r1s1) "

1(r2s2) … "

1(r

NsN)

"2(r1s1) "

2(r2s2) … "

2(r

NsN)

! ! " !

"N(r1s1) "

N(r2s2) … "

N(r

NsN)

, (2.6)

the proper anti-symmetrization of the total wavefunction is obtained. Inserting

(2.6) into the variational expression for the energy,

H!=

! H !

! !, (2.7)

then minimizing with respect to the !i's produces the Hartree-Fock equations,

!!2

2m

2

"# i (r) +Uion(r)# i (r) +U

el(r)# i (r) !

! d $re2

r ! $r# j

%( $r )# i ( $r )# j (r)& si s j'

j

( = )i# i (r)

(2.8)

7

with Uion and Uel defined previously. The last term on the left hand side is

called the exchange term and distinguishes the Hartree-Fock equations from that

of Hartree. (An additional slight variation is that the Uel(r) in Hartree does not

contain the contribution of the ith electron, while Hartree-Fock does.) Although

the Hartree-Fock equations are an improvement over the Hartree equations, they

are still not simply solvable.13

One approach, which includes exchange-correlation effects and justifies

the use of one-electron wavefunctions, is the Hohenberg-Kohn-Sham14,15 (HKS)

Density Functional (DF) theory. This method considers the ground-state energy

for the N-particle system as a functional of the ground state charge density, !(r) .

The ground state energy can be written as:

E[!(r)] = T [!(r)]" Ze dr!(r)R " r#

R

$ +1

2drd %r

!(r)!( %r )r " %r## + E

xc[!(r)] (2.9)

where T [!(r)] is the kinetic energy functional, the second term on the right-hand

side is the ion-electron interaction, and the third term is the average electrostatic

repulsion energy of the electrons. The final term, the exchange-correlation term,

is a collection of all the remaining many-body effects associated with the

problem.

The advantage gained is that the charge density to minimize (2.9) can now

be found using a set of single-particle equations (setting e=m= ! =1):

!1

2"2# i (r) +Veff (r)# i (r) = $i (r)# i (r) (2.10)

Veff (r) = !Z1

r ! RR

" + d #rn( #r )r ! #r$ +Vxc[%(r)] (2.11)

8

where n(r) = !i

2" , summed over the occupied orbitals, and Vxc[!(r)] contains

the exchange and correlation interactions.

A common approximation to Vxc[!(r)] is made in the Local Density

Approximation (LDA). When applying the LDA, the exchange-correlation

energy density of each infinitesimal volume of the inhomogeneous electron

distribution is assumed to be equivalent to the exchange-correlation energy

density of a homogeneous electron gas with the same density, i.e.

Vxc(r) = V

xc!(r)[ ] . It is worthwhile to point out that the only approximation made

in the DF method is the calculation of many-body effects which are included in

the Vxc

term.16

B. Surface Embedded Green Function Method

As mentioned earlier, the most common geometry used in surface

calculations is the slab, in which a crystal is modeled using five to nine layers of

atoms. Slab-based methods are successful in calculating aggregate properties,

but are not as successful in calculating properties which require individual

wavefunctions on the surface or in the bulk. Benesh and Inglesfield have pointed

out certain problems associated with slab-based methods.3 For instance, the bulk

is not properly represented by the few (three to seven) layers which separate the

two surfaces. There is also an unphysical interaction between the two surface

layers, often splitting surface states arising from the two surfaces into hybridized

pairs.

Desiring to improve upon these methods for surface calculations, Benesh

and Inglesfield1,2,3 developed the SEGF method. In this method, an embedding

9

potential for a semi-infinite substrate (Region II in Figure 2.1) is obtained from

the substrate Green function. The vacuum along with a few surface layers

(Region I) are then embedded onto this substrate, as shown in Figure 2.1. The

inclusion of the embedding potential in the Schrödinger equation for Region I

causes surface wavefunctions to match onto bulk solutions across the embedding

surface.

Figure 2.1. Depiction of the Surface Region I (including vacuum) and Substrate (bulk) Region II utilized in the SEGF method.

10

To obtain the solutions in Region I, an expression is first derived for the

expectation value of the energy of a trial function !(r) , defined as a trial

wavefunction !(r) in Region I, and in Region II as ! (r) . ! (r) is the exact

solution of the Schrödinger equation at some energy ε in the bulk. An additional

requirement is that !(r)and ! (r) must match amplitudes across the interface S.

In atomic units e=m= ! =1:

H! (r) = ("1

2

2# +V

bulk(r))! (r) = $! (r) , with r in Region II. (2.12)

The energy expectation value for ! is given by:

E =! H !

! !=

" H "I+ # $ $

II+ ! H !

S

" "I+ $ $

II

(2.13)

where the first term in the numerator is the expectation value in the surface-

vacuum region, the second term is the expectation in the bulk, and the third term

is:

! H !S=1

2d2rs" *(r

s)#"(r

s)

#ns

$#% (r

s)

#ns

&

'()

*+s, (2.14)

This term accounts for a discontinuity in derivative between ϕ and ψ across the

substrate-surface interface, leading to the integral over the embedding surface S.

Therefore, the total expectation energy in integral form is:

E =

d3r! *

(r)H!(r)I" + # d

3(r)$ *

(r)$ (r)II" +

1

2d2rs! *(r

s)%!(r

s)

%ns

&%$ (r

s)

%ns

'()

*+,s

"d3r! *

(r)!(r)I" + d

3(r)$ *

(r)$ (r)II"

.

(2.15)

The Green function G0 for Region II satisfies the differential equation

11

(!1

2r

2" +V

bulk(r) ! #)G

0(r, $r ) = % (r ! $r ) , r, !r in II. (2.16)

G0 is chosen to satisfy the boundary condition that its normal derivative

vanishes at the interface r = rS

,

!G0(r

s, "r )

!ns

= 0 . (2.17)

Using Green’s theorem, a relation can be obtained between the amplitude of the

wavefunction in II and its normal derivative at the interface:

! (r) = "1

2d2rSG0(r,r

S)#! (r

S)

#nS

S$ . (2.18)

With r on the interface, this becomes:

! (rS) = "

1

2d2 #r

SG0(r

S, #r

S)$! ( #r

S)

$ #nS

S% . (2.19)

The inverse of (2.19) gives the desired result, ! "#

"nS

in terms of ! (rS) , which at

the interface is equal to!(rS) :

!" (rS)

!nS

= #2 d2 $r

SG0

#1(r

S, $r

S)%( $r

S)

S& (2.20)

An expression is still needed to replace the volume integral over Region II.

Inglesfield1 has shown that:

d3r! *

(r)! (r)II" = # d

3rS

S" d

3 $rS% *(r)

&G0

#1(r

s, $r

S)

&'%( $r

S)

S" (2.21)

E can now be expressed purely in terms of the Region I trial function

!(r) and the bulk Green function G0, such that (2.15) becomes:

12

E =

d3r! *

(r)H!(r)I" +

1

2d2rS! *(r)

#!(rS)

#nS

S" +

$

%&

d2rS

S" d

2 'rS! *(r)G

0

(1(r

s, 'r

S)!( 'r

S)

S" ( ) d

2rS

S" d

2 'rS! *(r)

#G0

(1(r

s, 'r

S)

#)!( 'r

S)

S"

*

+,

d3r! *

(r)!(r)I" ( d

2rS

S" d

2 'rS! *(r)

#G0

(1(r

s, 'r

S)

#)!( 'r

S)

S"

.

(2.22)

The trial function, !(r) , is expanded in terms of a set of basis functions, !i(r) ,

[!(r) = ai"i(r)

i

# ] and E is then minimized with respect to small changes in

!(r) , resulting in:

(Hij + (G0

!1)ij + (E ! ")

#(G0

!1)ij

#E)

j

$ aj = E Oijajj

$ (2.23)

where

Hij = d3r !k

*(r)H!i (r)

I" +1

2d2rS !k

*(rS )

#!i (rS )

#nSS" , (2.24)

(G0

!1)ij = d

2rS

S

" d2rS#G

0

!1(rS ,rS

#;E)$i (rS# )

S

" , (2.25)

and Oij = d

3r !i

*(r)! j (r)" . (2.26)

Equation (2.23) now shows G0

!1 as a potential added to the surface

Hamiltonian. As such, G0

!1 contains all the information about the substrate

needed for the surface calculation and is the embedding potential.

The surface Green function may be calculated instead of the individual

wavefunctions. We have chosen to expand G(r, !r ;E) in terms of a set of basis

functions as:

13

G(r, !r ;E) = gij (E)"i (r)" j ( !r )ij

# . (2.27)

A matrix equation comparable to (2.23) is then computed, and, evaluating E at ! ,

the following is obtained:

(Hij + (G0

!1)ij !

j

" EOij )gij (E) = # ij (2.28)

from which the gij ’s are computed. At this point, with the surface Green

function having been obtained, other interesting properties of the surface may be

calculated.

C. Embedding Potential

As shown in (2.23) and (2.28), the embedding potential

�

G0

!1 is included as

an extra (embedding) potential in the surface region Hamiltonian. Since

�

G0

!1 is

related to the reflection properties of a crystal, one technique for obtaining the

embedding potential is the layer-doubling method, commonly used in low

energy electron diffraction (LEED) analyses.18 In terms of the reflection matrix R,

the Fourier components of

�

G0

!1 are obtained from the expression:

G0,K

!1=

1

2" (1! R)(1+ R)!1 (2.29)

where K is the wave vector, R is the reflection matrix, and

! = K "m

2 # ! "m

2= K

m

2+ k

z

2 (2.30)

with Km=K + Gm, where Gm is a reciprocal lattice vector.

In figure 2.2, it is clear that the interface S may not be a simple surface

over which to evaluate the matrix-element integrals. Benesh and Inglesfield2,3

overcame this difficulty by transferring

�

G0

!1 from the surface S to S0, a more

14

convenient flat surface with a constant potential between S and S0. This

procedure is further justified by Crampin et al.17

Figure 2.2. Surface and bulk are separated by the complicated surface S, but the effective embedding interface S0 is used. (from Ref. 3, p. 6684)

15

D. Basis Functions

The surface region I is further divided into three sub-regions (shown in

Figure 2.3): muffin-tin (MT) spheres centered around the nuclei, the interstitial

region between the muffin-tin spheres, and the vacuum region. Each of these

sub-regions contains a different characteristic potential. Within the muffin-tin

Figure 2.3. Sketch depicting the three distinct sub-regions of the Surface Region. (Adapted from Ref. 10, p. 18)

16

spheres, the potential is largely spherical due to the atomic cores. In the

interstitial region, the potential is relatively flat; while in the vacuum, the

potential has a strong z -dependence, but relatively weak x- and y-dependence.

Due to the varying nature of the potential, solutions to the surface problem have

a different characteristic form in each of the three sub-regions. For this reason,

Benesh and Inglesfield use Linear Augmented Plane Waves (LAPWs) as the basis

functions for the SEGF method.

1. Interstitial Region

Since the potential is relatively flat, plane waves of the following form are

used:

!m,n

+(r)

!m,n

"(r)

#$%

&%=

2

'eiKm •R (

cos(knz), n even

sin(knz), n odd

)*+

(2.31)

kn=n!

D and ! = AD where D is the slab thickness, A is the surface area of the

unit cell, and D defines a plane slightly larger than D (see Figure 2.3), which

aids in the matching of the wavefunctions at the vacuum and bulk boundaries.

Note that n is even for the symmetric (+) case and odd for the antisymmetric (-)

case, as determined by reflection z! "z .

2. Muffin-Tin Spheres

Since the potential is highly spherical within the muffin-tin spheres, the

LAPW for this region is composed of radial u! and energy derivative

!u"

solutions of the scalar-relativistic Schrödinger equation:

17

!m,n

+(r)

!m,n

"(r)

#$%

&%=

AL ,'+(K)

AL ,'"(K)

()%

*%

#$%

&%+ u

!,' +

BL ,'+(K)

BL ,'"(K)

()%

*%

#$%

&%+ "u

!,'

,

-..

/

011YL(2) +

i!

i!"1

()%

*%L

3 (2.32)

Benesh and Inglesfield have found that ! -values up to !max

= 8 usually suffice to

reach convergence. A and B coefficients are determined by matching χ and its

radial derivative across the surface of the MT spheres, denoted by α.

3. Vacuum Region

In vacuum, the basis set should describe both the x-y periodicity and the

z -dependence of the potential. Thus, the basis functions are defined as two-

dimensional plane waves times a linear combination of z -dependent and

energy-derivative solutions of the Schrödinger equation for a planar-averaged

potential:

!m,n

±(r) = " ±

(m,n)vm(z) + # ±

(m,n) !vm(z)$% &'e

iKm •R (2.33)

where α and β coefficients are determined by making χ and !"!z

continuous at

the vacuum-slab boundary.

E. Self-Consistency

Once the surface Green function has been successfully obtained, a number

of physical properties can be determined. For instance, the local density of states

(DOS) can be found from the expression:

! (r,E) = "

i(r)

2# (E $ E

i)

i

%

=1

&Im G(r, 'r ;E + i().

(2.34)

The charge density may be obtained by integrating σ over the occupied states:

18

n(r) =1

!Im dE G(r, "r ;E)

#$

EF

% (2.35)

The integration is performed using a semi-circular contour in the upper half

complex plane.

Once the charge density is known, it may be used to solve Poisson’s

equation for the electrostatic potential

!2Ves(r) = "4#$(r) (2.36)

using a pseudo-charge method developed my Weinert.19 In Weinert’s method,

the MT charge density is (temporarily) replaced by a pseudo-charge density

which has the same multipole moments as the actual charge density in the MT

spheres. The pseudo-charge is then Fourier expanded and used to solve

Poisson’s equation. The resulting solution is correct throughout the interstitial

and vacuum sub-regions and on the MT boundaries. The MT boundary

conditions allow Poisson’s equation to be solved in the MT spheres using the

actual MT charge density. The exchange-correlation potential is then added to

the electrostatic potential to create the new surface potential. This process is

repeated until self-consistency is reached.

F. Mixing

As mentioned previously, the SEGF method is an iterative method.

Obviously, it is desirable to achieve convergence in a reasonable number of

cycles. One approach is to take the results from one iteration and insert it as the

input for the next iteration of the SEGF program. However, this approach is

unstable in atomic systems due to the sensitivity of the electronic energy levels to

changes in screening. This sensitivity causes instabilities in the system, resulting

19

in outputs from consecutive cycles that oscillate between different

configurations, with the amplitude growing larger for each subsequent iteration.

To avoid this problem, the output from a given cycle is mixed with the input for

that cycle to create the input for the next iteration. Let !in

i represent the input

charge density for ith iteration. Then for the i+1 iteration,

!in

i+1= (1"# )!

in

i+#!

out

i (2.37)

where α is the mixing factor, which is usually restricted to be less than one.

Equation (2.37) represents a simple mixing scheme, also known as an attenuated

feedback algorithm. Generally, α may range as high as five percent, but for

rhodium it was necessary to keep the mixing at three percent or lower in order to

approach convergence.

A more sophisticated scheme employed in the SEGF calculations is the

Broyden20 method, which resembles a multi-dimensional Newton-Raphson

method. In one dimension, Newton-Raphson attempts to find the solution, x* , to

the equation f(x) = 0 by iterating on

xi+1

= xi!f(x

i)

"f (xi)

(2.38)

where !f (xi) is the first derivative evaluated at x

i. In more than one dimension,

x is a vector and !f (xi) is replaced by the Jacobian. As pointed out by Johnson,21

this creates problems in that the storage and multiplication of N x N matrices are

required, where N may be on the order of 10,000. A modified technique, which

requires the storage of only m vectors of length N where m is the number of

iterations, has been adopted.

20

Vanderbilt and Louie22 have modified the Broyden method so that

information from all previous iterations is used in the updating procedure.

However, this method still requires the storage and multiplication of large

matrices. Further modifications by Johnson21 allow the advantages of Vanderbilt

and Louie’s method without the disadvantages associated with a large matrix, as

described in detail in Ref. 11. The result is the following construction for the

input of the m+1 cycle:

nm+1

= nm

+Gm+1

Fm , (2.39)

where nm is the input for the previous cycle, Gm+1 is the inverse Jacobian, and

Fm

= nm!1

! nm . This mixing scheme allows for the use of information from

more than one previous cycle through the Gm+1 term.

Dissimilar systems may respond differently to these types of mixing, but

some combination of mixing schemes is usually sufficient to achieve

convergence. It is often useful to alternate runs using the attenuated feedback

method with runs of Broyden cycles, sometimes varying the number of cycles for

each run. This will often produce more rapid convergence than using either

method exclusively.

21

CHAPTER THREE

Computational Details

For the purpose of this study, the Surface Embedded Green Function

(SEGF) method was applied specifically to rhodium through the use of a series of

FORTRAN programs. As previously described, the SEGF method uses an

embedding potential to include the effect of the infinite bulk substrate on the

surface being studied. The substrate is treated as a semi-infinite crystal and

attention is focused on the top two to three layers: a significant improvement

over slab-based methods, which use only a few layers to screen the two surfaces.

In this study, the SEGF method has been applied to a clean surface of Rh(001).

Three primary programs have been used for this process: SPIN.FOR,

EMBD.FOR, and SEGF.FOR, although most of the work has been done using the

EMBD and SEGF programs.

A. Embedding Potential

The initial surface charge density is calculated using the SPIN program, by

computing atomic charge densities, overlapping contributions from neighboring

atoms, and summing. To do this, basic information about the geometry of

rhodium is needed as input. Rhodium crystallizes in a face-centered cubic (fcc)

crystal. As the name implies, an fcc lattice has a unit cell consisting of a cube

with atoms at each of the corners, as well as in the middle of each face of the

cube. A lattice constant of 3.80 Å (7.18 a.u.) has been measured for the rhodium

crystal.

22

Figure 3.1. Top view of two layers of rhodium.

Figure 3.1 represents two layers of rhodium. The lattice constant, a, is

defined as the edge length between two corners of the cube in the same surface.

As shown in the figure, adjacent layers are shifted by half a lattice constant along

the edges, so that atoms in one layer (yellow) are centered over the four-fold

hollows in the next layer (green). The face diagonal has a length of a√2 and is

equal to 4 times the radius of a muffin tin (MT) sphere. Thus, the radius of each

muffin tin is a√2/4, or a/2√2.

This study focuses on the (001) surface of rhodium. This being the case,

the surface is shown as Figure 3.2, with the basis vectors being defined from the

center of a corner atom to the center of the closest face atoms.

This geometric information is used as input for the SPIN program, which

computes the initial charge density for the SEGF program. The EMBD program

uses the bulk potential and additional structural information to produce the

embedding potential for the SEGF program. For this work, the rhodium bulk

muffin tin potential of Moruzzi, Janak, and Williams (MJW)23 was used. The

Fermi energy for rhodium is 0.341 Hartrees (9.29 eV) with respect to the average

23

Figure 3.2. Top view of Rh(001) demonstrating basis vectors.

Figure 3.3. Atoms in the surface region.

24

interstitial potential. (Hartrees are the energy units used in the EMBD and SEGF

programs.) The embedding plane can be seen in Figure 3.3. The embedding

potential is the key ingredient for computing the self-consistent charge density

and densities of states.

In addition to producing the embedding potential, the EMBD program

can be used to calculate the bulk band structure. The bulk band structure was

calculated at several k points in order to compute the projected bulk band

structure (see section III.C). These will be discussed further in chapter 4.

B. Self-Consistent Charge Density

Once the embedding potential has been determined using the EMBD

program, it is used as an input to the SEGF program. The SEGF program then

iterates the charge density to self-consistency. This can be a lengthy process: for

Rh(001), over a thousand cycles were required for the charge density to

converge to a norm (root mean square of the charge difference vector) of 10-6

a.u. Once this was achieved, the charge density was considered converged. This

was accomplished by alternating between three cycles of simple mixing and ten

cycles of Broyden mixing. Other physical quantities were subsequently

computed from the converged results.

C. Bulk Band Structure and Projected Bulk Band Structure

Using the EMBD program, the bulk band structure can be calculated at

various k points within the Surface Brillouin Zone (SBZ). Since there is no

periodicity perpendicular to the surface, the three dimensional bulk Brillouin

zone must transform into the two-dimensional SBZ. Consequently, the bulk

bands (which all extend to the surface) must be labeled with only the two

25

components of the surface wavevector k. Where bands in the bulk could be

distinguished by their k

!component, at the surface they become band continua

plotted as a function of k (= k! ). For each k point, band energies were calculated

between 0 Ha and 0.42 Ha to ensure that the bottom of the valence band was

included and to observe any bands crossing the Fermi energy. Figure 3.4 shows

the k points along the lines of high symmetry used in the projection of the bulk

bandstructure. The points are given in terms of 2!a

, where a is the lattice

spacing of the surface unit cell.

Figure 3.4. k points used to determine the projected bulk band structure.

26

To construct the projected bulk band structure (PBBS), all bands were

analyzed for parity and to determine their energy minima and maxima. The

minima and maxima of different k points were then connected using a cubic

spline-fitting curve to illustrate the projected bulk band continua. The PBBS plot

can be used to identify surface states and surface resonances. In practice, bands

at additional k points were analyzed to better identify boundaries of various

band continua.

D. Density of States

The EMBD and SEGF programs can also be used in combination to

produce a density of states (DOS) plot. The EMBD program is used to produce

an embedding potential over the energy range of interest, but with energy points

located just above the real axis, instead of along the semi-circular contour used in

self-consistent cycles. The SEGF program is then run using the converged charge

density, to produce the density of states. At each k point the results are plotted,

and surface states and resonances are identified by sharp peaks in the DOS.

Typical DOS plots can be seen in Figures 4.9 – 4.11. Peak energies are next

plotted on the PBBS to identify surface states and surface resonances.

27

CHAPTER FOUR

Results and Analysis of Clean Rhodium

As discussed previously, crystalline rhodium’s structure is face-centered

cubic (fcc). For the Rh(001) surface, an embedding potential was produced from

an existing potential for the bulk atomic configuration. With this embedding

potential, three layers of surface atoms were effectively embedded onto an

infinite bulk substrate. The self-consistent surface charge density was computed,

and then used to study various physical properties. In this study, the work

function and core level shifts were computed and compared to experiment and

other calculated results. In addition, bulk band structures were calculated and

analyzed to produce the projected bulk band structure (PBBS) for the Surface

Brillouin Zone. Densities of states were also calculated and compared to the

PBBS to identify surface states and surface resonances.

A. Work Function

The work function is one of the more commonly calculated results, since it

is experimentally accessible and easy to compute. However, the accuracy of the

calculated work function is often viewed as a measure of the accuracy of the

calculation, since it depends sensitively on the arrangement of charge at the

surface. The work function for the (001) surface of rhodium was calculated to be

5.546 eV. As can be seen in Table 4.1, while this result does not correspond

closely to experimental values, it does compare favorably to other theoretical

values.

28

Xie and Scheffler25 and Cho and Sheffler31 have performed two density

functional theory (DFT) calculations using different approximations for the

exchange-correlation. The generalized gradient approximation (GGA) produced

a work function very close to experiment, while the local density approximation

(LDA) produced a work function in good agreement with the present results. It

is also worth noting that Gay et al.24 calculated a work function close to, but lower

than, experiment using a seven-layer self-consistent local-orbital method—

generally considered the least accurate method. Other calculated values are

generally higher than the experimental values, but in fairly good agreement with

Table 4.1. Comparison of experimental and theoretical work functions.

Method Result

Present SEGF result 5.55 eV

expt.- photoelectric effect28 4.92 eV

expt.- photoelectric effect29 4.98 eV

LAPW with LDA30 5.49 eV

7-layer LAPW with LDA25 5.30 eV

7-layer LAPW with GGA25 4.91 eV

9-layer LAPW with LDA31 5.26 eV

9-layer LAPW with GGA31 4.92 eV

7-layer SCLO24 4.8 eV

7-layer LMTO32 5.25 eV

9-layer pseudopotential33 5.57 eV

LAPW, Wigner interp.36 5.5 eV

29

the present SEGF result. The SEGF-calculated work function is in excellent

agreement with that produced by Morrison et al.33 using a nine-layer

pseudopotential. It also agrees very well with results by Feibelman and Hamann

using both the LAPW total energy30 method and the surface LAPW36 method.

There are competing views as to why most calculated work functions

deviate from experiment. One explanation pertains to the relaxation of the

outermost layer spacing. When Cho and Sheffler31 performed the two

calculations mentioned, they also considered the relaxation of the first surface

layer of Rh(001). Their results predicted a contraction of the outermost layer

spacing of about 3% using the LDA, and 2.8% using the GGA. The

pseudopotential method by Morrison et al.33 calculated a 3.22% contraction of the

top layer spacing.

For this reason, a second SEGF calculation of the electronic structure of

Rh(001) was performed. The parameters for this calculation were the same as in

the previous, except for a 3% contraction in the top surface layer spacing. The

resulting work function was calculated to be 5.563 eV, very close to the value

found by Morrison et al.,33 and not significantly different from the original SEGF

result.

There have also been experimental results that purport to measure a first

layer lattice expansion of about 1%34,39. Feibelman and Hamann30 argue that the

sample used in the low energy electron diffraction (LEED) measurements may

have been contaminated by a layer of hydrogen. However, there has been no

experimental evidence to support this view.

30

To consider the possible lattice expansion, the SEGF program was also run

for a Rh(001) surface with a 1% expansion of the surface spacing. The charge

density was converged, and a work function of 5.548 eV was determined. Thus,

contraction and expansion had no significant effect on the calculated work

function, so it is apparent that the SEGF method is robust enough not to be

affected by small perturbations of the geometry.

Hence, the discrepancy remains between experimental and computational

work functions. Among the possible explanations is some other type of surface

imperfection or impurity that alters the experimental value. There is also the

possibility that the top layer of the surface may be disordered, rendering

assumptions made in the surface calculations inaccurate.

Since the MT radius was reduced so that the expansion and contraction

would fit within the boundaries of the surface, the surface charge density was

also converged to self-consistency with the reduced MT radius, but using the

original geometry. The total energies were then compared, and the total energy

for the expanded surface was found to be 0.003 eV per unit cell higher than that

of the unrelaxed lattice. The contracted surface was found to have a total energy

that is 2.21 eV per unit cell lower than the unrelaxed surface. Thus, the current

results are consistent with the other theoretical predictions of a contracted layer

spacing.31,33

B. Surface Core-Level Shifts

There is often a difference in binding energy of the core level electrons in

the surface atoms as opposed to those in the bulk. This difference in energy is

called the surface core-level shift (SCLS). In general, there are three contributions

31

to the surface core-level shift (

�

!SCLS): a shift due to environmental effects (

�

!env ), a

shift due to a change in atomic configuration (

�

!conf ), and a relaxation shift (

�

!relax )

due to the screening of the core hole.

! = !env + !conf + !relax (4.1)

The environmental and configuration shifts together make up the initial

state effect (

�

!init ):

!init = !env + !conf (4.2)

The initial state effect is calculated using data from the SEGF program.

Once the charge density is converged, the potential at the top surface layer is

compared to that in the third layer, which essentially becomes part of the bulk

through embedding. The difference in the 3d5/2 eigenvalues is the initial state

contribution to the SCLS.

The environmental effect is calculated by running the SEGF program, but

calculating only the core levels and stopping short of running a full cycle. An

input potential is first created by overlapping charge densities and then solving

Poisson’s equation for a specific configuration (in this case 4d85s1). In this way,

no re-configuration of electrons at the surface is allowed, since the electron

configuration remains fixed. Then, comparing the first and third layers, the

environmental effect is found.

The configuration effect, also known as the chemical shift, is a result of the

rearrangement of surface electrons between atomic subshells. It can be

calculated by taking the (converged) initial state shift and subtracting the

environmental shift:

!conf = !init " !env . (4.3)

32

The relaxation shift, or final state effect, is a result of the different

screening properties of bulk and surface atoms. The final state effect is

calculated by placing a “hole” in the electron configuration at the first layer and

iterating to self-consistency. The same is then done for the third layer, and the

surface energies of the two systems are compared.

The environmental effect was calculated by creating a starting charge

density using an electron configuration of 4d85s1. The 3d5/2 energies were

compared at the first and third layers for the surface shift and the second and

third layers for the subsurface shift. For the environmental shift, a value of 215

meV towards decreased binding was calculated at the surface and 8 meV at the

subsurface.

The initial state shift was calculated using the converged SEGF surface

charge density. The one-electron potential in the top layer was somewhat less

attractive than the bulk, producing an initial shift towards decreased binding in

the amount of 667 meV for the 3d5/2 electron. In contrast, the subsurface layer

shifts towards increased binding by 95 meV.

With the initial state and environmental effects calculated, the shift due to

configuration changes can be found by taking the difference. A value of 452 meV

towards decreased binding is found at the surface, along with 103 meV towards

increased binding at the subsurface.

According to Ganduglia-Pirovano et al.,40 much of the surface core level

shift is a result of a surface d-band shift (SDBS). The SDBS to reduced binding

causes d to s charge transfer, resulting in a d orbital core shift that accounts for

much of the SCLS. There is a corresponding narrowing of the d-band which is

necessary for the surface to remain electrically neutral.

33

Among the experimental evidence, Zacchigna et al.27 have used

synchrotron radiation to measure a value of 660 ± 5 meV for the total SCLS at the

surface, in excellent agreement with the present SEGF results, and 40 ± 0.01 meV

at the subsurface. Borg et al.26 used X-ray photoemission spectroscopy (XPS) to

produce a slightly different result of 620 ± 10 meV. Erbudak et al.43 have also

used XPS and produced a measurement of 580 ± 5 meV.

If, as suggested by Feibelman and Hamann,36 the relaxation shift is small,

then our initial shift is approximately equal to the total shift, and this value can

be compared to other calculations and experiment. In order to obtain an

approximate value for the relaxation shift, a “hole” has been placed in the 3d5/2

Table 4.2. Surface and Sub-surface Core Level Shifts (all units are meV).

Method !

Present SEGF initial state results -667

+95

Synchrotron Radiation27 -660 ± 5

-40 ± .01

Photoemission Sprectroscopy26 -620 ± 10

Angular-resolved XPS43 -580 ± 5

LMTO with LDA40 -830

LMTO with LDA42 -620

LAPW, Wigner interpolation36 -750

-50

34

level of the surface layer and the calculation was re-converged. The results were

compared with similarly treated holes in the second and third layers. However,

since the unit cell is repeated continually, the hole is also repeated for all atoms

in the layer, resulting in an overestimate of the effect. When the core level

eigenvalues are compared, the top layer has shifted to 599 meV reduced binding

compared with the third layer eigenvalue. This value is a combination of all

contributions and is not just the relaxation shift. If this result is taken to be a

calculated total shift, it also is in fair to good agreement with the two

experimental measurements of the total core level shift (620 meV26 and 660

meV27).

Among other calculated results, Methfessel et al.42 report a total SCLS of

620 meV, in good agreement with the present findings and experiment.

Additionally, an approximate initial state shift of 580 meV and a relaxation shift

at 40 meV were calculated. Feibelman and Hamann36 calculated the total SCLS to

be 750 meV toward decreased binding at the surface and 50 meV at the

subsurface, while Ganduglia-Pirovano et al.40 calculated an SCLS of 830 meV

towards decreased binding.

C. Projected Bulk Band Structure

The projected bulk band structure (PBBS) is the projection of the bulk

energy spectrum onto the Surface Brillouin Zone (SBZ). It is constructed from

the bulk band structure for k points along high symmetry lines in the SBZ, as

shown in Figure 3.4. A bulk band structure is plotted for each k point and then

each band is evaluated for even and odd reflection symmetry across the high-

symmetry lines. The minimum and maximum energies for each band are then

35

determined. Figure 4.1 shows the bulk band structure at ! , with the energy set

to zero at the Fermi level. Note: The zone boundary occurs when

k!= 2"

a(# 0.875 a0

$1), where a is the bulk edge length.

Figure 4.1. Bulk band structure at ! [(0,0)] plotted as a function of k

!.

When the band structure is evaluated for adjacent k points, the bands

shift, as seen in Figure 4.2, the bulk band structure at (0.125, 0).

The bottom valence band is completely symmetric. Along the symmetry

lines, bands of the same parity cannot cross due to the Pauli exclusion principle.

At points of high symmetry, such as ! , there may be degeneracies and band

crossings due to the higher-order symmetry. Away from the high-symmetry

points, only the reflection symmetry across the symmetry lines is retained.

Figures 4.3 and 4.4 show the bulk band structure for the other two high-

symmetry points, X and M .

36

Figure 4.2. Bulk band structure at (0.125, 0) plotted as a function of k

!.

Figure 4.3. Bulk band structure at X [(0.5,0)] plotted as a function of k

!.

37

Figure 4.4. Bulk band structure at M [(0.5,0.5)] plotted as a function of k

!.

After the band structures have been calculated and the minimum and

maximum band energies determined, this information is combined to form the

PBBS. The maxima and minima are plotted in the SBZ and then the points are

connected using a cubic spline fit to produce the bulk energy band continua.

When necessary, additional k points have been evaluated to further clarify some

areas of band formation. For purposes of clarity, the PBBS for the ! X section of

the SBZ is shown in three different figures. Figure 4.5 displays the even-

symmetry bands and band gaps, and Figure 4.6 displays the odd-symmetry

bands and band gaps. The vertical lines represent continua of bulk bands of

even symmetry, while the horizontal lines represent bands of odd symmetry. In

these figures four symmetry gaps may be observed: three of even symmetry and

one odd. Figure 4.7 combines the previous figures to show the total PBBS for the

38

! X high-symmetry line. There are two absolute gaps displayed in this region.

Figure 4.5. PBBS of even symmetry for the ! X line in the SBZ. The vertical lines represent bands of even symmetry. There are three relative band gaps.

39

Figure 4.6. PBBS of odd symmetry for the ! X line in the SBZ. The horizontal lines represent bands of odd symmetry. There is one relative band gap.

40

Figure 4.7. PBBS for the ! X line in the SBZ. The horizontal lines represent bands of odd symmetry and vertical lines represent bands of even symmetry. There are two absolute band gaps.

41

Figu

re 4

.8.

PBBS

for t

he e

ntire

SBZ

. Ev

en a

nd o

dd sy

mm

etry

are

mar

ked

as p

revi

ousl

y no

ted.

42

Continuing with the rest of the SBZ, the PBBS for the X M and M ! high-

symmetry lines are produced. These are combined to produce the total PBBS, as

seen in Figure 4.8.

The PBBS in this study compares favorably to that produced by Eichler, et

al.41 for rhodium using a calculation incorporating pseudopotentials and an

iterative solution of the Kohn-Sham equations in the local density approximation

(LDA). The PBBS in Figure 4.8 also matches rather nicely with PBBS for iridium35

and platinum.8 Like rhodium, these are both fcc transition metals. Although

rhodium is a 4d metal and the others are 5d, the similarities in crystal structures

and numbers of valence electrons produce similar PBBS.

D. Densities of States

The densities of states (DOS) have been evaluated at each of the k points

shown in Figure 3.4. The peak position energies and wavevectors have been

plotted on the PBBS to identify the various surface states (SSs) and surface

resonances (SRs). Figures 4.9 through 4.11 are the DOS plots for the ! , X , and

M points of high symmetry. Figure 4.12 shows the position of all SS and SR

peaks with respect to the projected bulk bands.

Again with Ir(001)36 and Pt(001)8 as reference, the Rh(001) surface bands

are very similar to those computed for the other two surfaces. Platinum has one

more valence electron, and thus a higher Fermi energy; however, iridium is

isoelectronic to rhodium.

Very little experimental identification of rhodium surface states and

surface resonances has been performed. Morra et al.37 performed angle-resolved

photoemission experiments and found evidence for a strong surface state at

43

Figure 4.9. DOS at !.

Figure 4.10. DOS at X.

44

Figure 4.11. DOS at M.

about 0.15 eV below the Fermi energy at M . Looking more closely at the current

Rh(001) SSs and SRs (Figure 4.12) and the DOS at M (Figure 4.11), it is clear that

there are no computed SSs or SRs near this energy at M . In fact, it is not clear

why a state would be seen in photoemission since there is no gap in the PBBS at

this energy and wavevector.

Although the present results differ from experiment, they are in good

agreement with Feibelman and Hamann36 and Eichler et al.38,41 At M , Feibelman

and Hamann have used a surface-linearized augmented plane wave method

with Wigner interpolation to obtain a surface state at approximately 1.5 eV above

the Fermi energy. Using a pseudopotential method with LDA, Eichler et al.38,41

found a surface state at about EF + 1.6 eV. The current study finds a surface state

at EF + 1.52 eV, in agreement with the two previous studies.

45

Figu

re 4

.12.

PBB

S w

ith D

OS

poin

ts o

verla

yed.

Sur

face

stat

es a

re re

pres

ente

d by

solid

circ

les,

surf

ace

reso

nanc

es b

y op

en ci

rcle

s.

46

Feibelman and Hamann have argued30 that the experimental samples may

have been contaminated by an overlayer of hydrogen, which may account for the

difference between the present work and experiment. However, this suggestion

does not explain the theoretical results of Gay et al.,24 who find a surface state at

about 0.15 eV below the Fermi energy, exactly where the state was found

experimentally. As Feibelman and Hamann have stated, this is a point of concern

that should be further investigated.

E. Charge Density

The following figures are charge density plots of the Rh(001) surface

calculated using the SEGF program. They all represent the top three layers of the

sample. The vacuum is at the top in all figures. Figures 4.13 and 4.14 are total

charge density plots sampled over ten special k points. Figure 4.13 is taken

across the face of the cube; so it displays the expected five atom face

configuration. Figure 4.14 is taken across the cube diagonal and represents the

(110) plane; it makes clear the charge density between the “next nearest

neighbors.” As expected, the two plots show a sizable interstitial charge

everywhere, as would be expected for a metal. In each figure, the uppermost

charge density contour represents 0.009 e a0

3 . The average valence charge

density of 9 electrons per atom (0.049 e a0

3 ) is represented by the ninth charge

density contour.

47

Figure 4.13. Total charge density of the Rh(001) surface sampled over ten special k points. The minimum charge density is 0.009 electrons per cubic bohr radius ( e a

0

3 ), the maximum is 0.089 e a0

3 , and there are 17 contour lines.

48

Figure 4.14. Total charge density of the Rh(001) surface sampled over ten special k points. The minimum charge density is 0.009 e a

0

3 , the maximum is 0.089 e a0


49

Figure 4.15 is a surface resonance at ! which lies 5.14 eV below the Fermi

level. The surface and subsurface appear to display dz2 orbital behavior with

some bonding between the two. Most of the valence charge lies in the top layer,

while very little resides in the third layer.

Figure 4.15. Valence charge density of the surface resonance of ! (0, 0) at EF – 5.14 eV. The minimum charge density is 0.002 radius e a

0

3 , the maximum is 0.042 e a

0


50

Figure 4.16 is the charge density of the surface state at the (0.25,0) k point,

4.80 eV below the Fermi energy. It is in the lower absolute gap along the ! X

high-symmetry line. The bulk and subsurface exhibit dz2 orbital character.

There is less covalency at this surface state, perhaps representing some

antibonding behavior.

Figure 4.16. Valence charge density of the (0.25, 0) surface state at EF – 4.80 eV. The minimum charge density is 0.005 e a

0



51

Figure 4.17 is the charge density of the X surface state, which lies 4.12 eV

below the Fermi level. It is in the bottom absolute gap that occurs along both the

! X and X M high-symmetry lines. All three layers contribute to the surface

state, with the top two layers sharing some charge, and the subsurface displays

dxz,yz orbital character.

Figure 4.17 Valence charge density of the (0.5, 0) surface state at EF – 4.12 eV. The minimum charge density is 0.001 e a

0



52

The charge density of the surface state at 2.62 eV below the Fermi energy

at the (0.5, 0.0625) k point is displayed in Figure 4.18. It lies in a narrow absolute

gap in the middle of the bulk band continuum that exists along the X M high-

symmetry line. The subsurface and bulk layers exhibit dx2! y

2 orbital character

while the surface layer is dz2 . All three layers contribute to the surface valence

charge.

Figure 4.18. Valence charge density of the (0.5, 0.0625) surface state at EF – 2.62 eV. The minimum charge density is 0.005 e a

0


0


53

Figure 4.19 shows the surface state at the (0.5, 0.25) k point which lies just

below the Fermi level (0.20 eV). All three layers contribute to the surface state

charge and exhibit dz2 orbital character.


0



54

The charge density contours in Figure 4.20 represent the SS which lies in a

large absolute gap in the middle of the PBBS, about 4 eV below the Fermi level.

The surface and third layers exhibit dx2! y

2 orbital character. The valence charge

comes primarily from the first and third layers, and there appears to be some

charge sharing between the top two layers.


0



55

The SS at M , which lies 0.98 eV above the Fermi level, is very interesting,

as several band continua meet at this point. The charge density of this SS is

shown in Figure 4.21 and displays dyz,xz orbital character at all three layers.

Although all three layers are involved, the third layer appears to contribute the

most. As this state is above the Fermi level, it is unoccupied and should be

detectable by inverse photoemission.

Figure 4.21. Valence charge density of the (0.5, 0.5) surface state at EF + 0.98 eV. The minimum charge density is 0.0001 e a

0



56

Figure 4.22 displays the charge density of the (0.3125, 0.3125) SS in the

absolute gap which lies 0.58 eV below the Fermi level. All three surface layers

exhibit dz2 orbital character and contribute to the valence charge.


0


0


57

Figure 4.23 is the charge density of the (0.1875, 0.1875) surface state at 3.92

eV below the Fermi level. The third layer appears to display dxz,yz orbital

character and the surface appears to display dz2 orbital character, while the

subsurface may be an admixture of orbitals. There is some covalency between

the layers and they all contribute to the charge.


0


0


58

Finally, Figure 4.24 is the charge density of the (0.0625, 0.0625) surface

state, which lies 3.03 eV below the Fermi level. All three layers exhibit dxz,yz

orbital character and can be seen to contribute charge, though the top layer

contributes very little, while the bottom two layers seem to show covalency.


0


0


The calculations presented are an important first step to understanding

the catalysis that occurs when NO and CO are placed atop the surface. It is

59

hoped that future experiments will examine the involvement of these SSs and

SRs in the catalytic activity of the surface.

60

CHAPTER FIVE

Conclusion The surface charge density for three layers of Rh(001) embedded on bulk

rhodium has been converged to self-consistency. Using the calculated charge

density, various electronic structural properties have been obtained. The

calculated work function compared favorably with other theoretical models, but

is in only fair agreement with experiment. Additional calculations were

performed to account for both lattice contraction and expansion, but the work

function was essentially unchanged. This model can be extended to other

surface geometries in order to resolve the discrepancy between theory and

experiment.

Components of the surface core-level shift (SCLS) were also calculated,

including the environmental and initial shifts, from which the configuration shift

was determined. These were in good agreement with both theory and

experiment. Our calculated initial state shift of 667 meV was in excellent

agreement with an experimental value of 660 ± 5 meV measured by Zacchigna et

al.27 Our attempt to calculate the relaxation shift was not as successful.

However, the current model could be used in conjunction with a 2 x 2 (or larger)

unit cell to better approximate the screening effect of a single atom, rather than a

plane of surface atoms.

The projected bulk band structure (PBBS) was constructed and densities of

states calculated and overlayed on the PBBS to identify surface states and surface

61

resonances. The structure of the PBBS is in good agreement with other models,

as well as those of iridium and platinum, which are closely related. While there

are few experimental studies with which to compare, the current calculations are

not in agreement with the experimental finding of an occupied surface state at

M .37 However, our results are in agreement with a theoretical study that

suggests that the experimental sample may have been contaminated with

hydrogen.30 The SEGF model has the capability of performing calculations with

impurities atop the metal surface, so it can be used to model a contaminated

surface if the geometry is known. Since rhodium is used as an industrial catalyst

for carbon monoxide and nitrogen oxide reduction, future studies could focus on

those reactions in hopes of improving both yield and efficiency.

62

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abstract - baylor-ir.tdl.org · mark r. mastin, b.s. a thesis approved by the department of physics...

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