A NEW RECURSIVE APPROACH TO PHOTOELECTRON DIFFRACTION SIMULATION.pdf

8/11/2019 A NEW RECURSIVE APPROACH TO PHOTOELECTRON DIFFRACTION SIMULATION.pdf

1/7

Anew recursive

approach

to

photoelectron

diffraction

simulation

1

'

2

F. J.

Garcfa

deAbajo ,

'

3

M.

A. VanHove, and

1

'

3

C.S.

Fadley

1

Materials Sciences

Division,

Lawrence Berkeley NationalLaboratory, Berkeley, CA 94720, USA

2

Departamento de CCIA

Facultad

deInformdtica) andDonostia International Physics Center

(DIPC), San Sebastian,

Spain

3

Department

of

Physics, University

of California,

Davis,

CA

95616,

US A

Abstract. A new

recurs ive

method for the simulationof photoelecton

diffraction

in

solids within the cluster approach is presented. No approximations are made beyond

the

mu ff in - t in

model, and in particular, an exact representation of the

free-electron

Green f unc t i on is used. The new method relies upon a convenient separation of the

f ree-e lectron

Green f unc t i on in vo lv in g rotation matrices to reduce the computation

time and storage demand. The multiple scattering expansion is iteratively evaluated

usinga

divergence-free

recursionmethod. The resulting computational demand scales

as

A

r2

(/

max

-f I)

3

with

the

number

of

atoms

in the

cluster

TV and the

maximum

of

the

relevant angular momentum quantum numbers

/

max

-

Actual examples

are given

where

TV

>

1000

is

needed

for

convergence within

5% in the

calculated photoelectron

intensity.

INTRODUCTION

Multiple elastic scattering (MS) plays a central role in the description of electron

transport

inside solids in

different

experimental

spectroscopies, and in particular in

core-level photoelectron

diffrac t ion

(PD) [1-3]. In this context, the relatively high

electron energies usually employed (> 50 eV) permit approximating the atomic po-

tentials by spherically-symmetric muffin-t in potentials [4]. Besides, inelastic scat-

tering can be treated in a phenomenological way via complex optical potentials

[ 4 ] -

The cluster model adopted provides a natural approach to simulate MSeffects in

PDthat issuggestedby thefact thatexcited electrons cannot travel large distances

in realistic solids withoutsuffer ing inelastic losses,

sothat the

region which actually

contributes

to the

emission

of

elastically scattered electrons defines

a

fini te cluster

surrounding the emitter [2,5-9].

Typically, the electron wave function is expressed in spherical harmonics and

spherical Bessel

functions

attached to each

atom

of the cluster in order to in-

corporate curved-wave

effects.

Unfortunately,

the

propagation

of

these functions

CP514,Theory and Computation for Sy nchrotronRadiation Spectroscopy,

edited by M .

B enfatto,

C. R. Natoli, and E.Pace

2000Am erican Institute of Physics l-56396-936-X/00/ 17.00

123

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2/7

between cluster

atoms is

computationally

very

demanding [10,11,4]. Therefore,

approximations have been

introduced

in the

past

[12-14,5,15,16,7,8,17,18], some

of them inspired in the high-energy limit,

where

the electronpropagation reduces

to

plane-

wave factors

(plane- wave approximation) and

each

term

in the MS se-

ries becomes a product of scattering amplitudes [ 1 9 ] . Expansions that take into

consideration the

fin i te

sizeof the atomshave also been developed [ 5 ] .

Beyond this, full curved- wave

formulations

of the problemrequire dealing with

^ m a x+ I)

2

partial wave components per atom,where /

m a x

is the maximum of the

significant

angular

momentum

quantum

numbers, which scales roughly

as

/

m a x

~

kr

mi

with

the

electron

momentum k and the atomic

muffin-tin

radius r

m t

, and

therefore, each

propagation

of the electron wave

function between each pair

of

atomsinvolves (/

max

+ I)

4

complex

products.

In order to overcome the rapidly-growing

computational

demandwith increasing

^ m a x ,

Rehr

and Albers [7] R - A ) providedaclever procedurebasedupona

separable

representation

of the

free-electron

Green

function that allows

one to generalize the

scattering

amplitudes, substituting

them

by

matricesthat produce

reliable

results

when keeping only

a few of

their

leading

elements

[ 7 ] .

This

is

particularly

suitable

tocalculate the contribution ofdiffe rent individualelectron paths.

In

this paper,

the MS expansion is

evaluated

using an exact

representation

of

the

Green

function propagator. An iterative

procedure

is

followed

that requires

(10/3)]V

2

(/

max

+ I)

3

multiplications

per iteration. Moreover, previously reported

divergences

in the MS series [ 2 0 ] are

prevented

by

using Haydock's

recursionmethod

[ 2 1 ] .

THEORY

Within

the

muffin-t in

model

adoptedhere,

eachatomin the

cluster

is

represented

by an

atomic potentialthat

vanishes outside a sphere of

radius r^

t

t h e

muffin- t in

radius) centered atR

a

. These arenon-overlapping

spheres

and the total

potential

is set to a

constant

t h e

muffin-tin

zero) in the interstitial

region.

In core-level photoemission, the direct

wave fu n c t io n

0 (i.e.,

b efo re

MS is c o n -

sidered) can be expressed as a combination of spherical outgoing waves

centered

around the

emitter atom

a

0

:

fo r

|r -

R

ao

|

> Ct,

where

h

(

+\kr)

=i

l

h\

+)

(kr)Y

L

(Sl

r

), h \

}

is a

spherical

Hankel

function

[ 2 2 ] ,

and

L

= ( / , r a )

labels

spherical harmonics

YL.

The

coefficients 0

oL

depend on the geometry, polarization, and energy of the

incident light

and on the

initial

core

state.

Single scattering of this direct wave leads to an

extra

contribution to the wave

function

than

can also beexpressed in termsofsphericaloutgoing

waves

centered

around eachof the clusteratoms.By iteratively

employing

this

argument,

onefinds

124



3/7

that the full self-consistent wave function f> has to be

made

of

spherical

waves as

well (even

after

MS has

been

carried out up to an infin i te order), so

that

+)

[ -Ra)]^,L

(2)

fo r

r outsidethe

mu ff in - t in . sphe res .

The coefficients

< / > , L must

be

determined

self-consistently by solving a secular

equation that can be

found

as follows: for each

atom

a,

< />

a?

L

is the sum of the

direct wave (only contributing fora

0

, that is, the emitter) plus the result of the

freepropagation

of the

electroncurved-wavecomponentsfrom everyother

atom/3

up to atom a, followed by scattering at the latter. More precisely,

a

=

< +*aG

3)

f3^a

where

G

a

ptakes care of the

abovementioned propagation,

t

a

is the

scattering

matrix

of

atom

a,

0

=

aa o

< / >

0

,

and

< j

a

is the vector ofcomponents

0

a

,L-

In the basis set

ofspherical harmonicsattachedto

each

clusteratom,one has

a

, z,L/ = sin5fe

l5

?SLU,

where 8? is the /

th

scattering phase

shift ofatom a

[22].

Besides, the matrix G

a

p

is

connected to the

translation

formula of spherical

harmonics

and its

detailed

expression

can be

written

Sl),

(4)

where

d

a/

3 = R

a

R#, for r R

a

|

oo limit. One finds

/(r)oc

wherek/ =kr/r and

a

,the

distance

fromatomato thesurfacealongthedirection

of

emission, is

introduced

to

account

for

inelastic attenuation

of the photoelectron

before

it leaves the

solid.

Three

different iterative

techniques have been used and

compared

in the present

work to evaluate Eq. 3): (a) direct Jacobi

iteration,

consisting in starting

with

0^0andusingEq. (3) as an iterationformula to improve

6 (this

resultsin the

intuitive MS

procedurewhere

every

iteration leads

to the

next

order of scattering);

(b ) simultaneous

relaxation

(SR)

[23],previously

usedinthis context

[20],

and

con-

sisting

in

both

using

the

latest

values

of the

coefficients

of

c j

as

soon

as

they

are

calculated and mixing the result of each iteration with the previous one to improve

125

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4/7

convergence (the fractional weight of the former will be denoted 7 7 ) ; and (c) Hay-

dock's recursion

method [21],

modified in a way

suitable

to

obtain

photoemission

intensities along

an arbitrary number of directions of

emission with

a

single

MS

calculation

for

each

electron

energy,

as

will

be discussed

elsewhere

[24].

Rather

than directly

using Eq.

(4), G

a

/3 will

be

constructed

in three

steps

as

follows

[10,7]: firs t , the

bond vector

d

a

/3 is

rotated

onto the z

axis

by using a

rotation matrixR

a

p

[22,7];

the resultingrotated

wave

functioncomponents arethen

propagated

a

distanced

a

p along

the

positive direction

of the

z axis

by multiplying

by G

z

a

p,

calculated

from Eq. (5) by

using

( 0 , 0 , d

a /

g )

instead of d

a /

g ; finally, the z

axis is

rotated

back onto the

d

a

p

direction, and one has

G

a

p = R~pG

z

a

pRap. (6)

The

rotation matrices

involved

here

can be in

turn

decomposed

into

azimuthal and

polar rotations.

Asignificant reductioninmemory demandcan be

accomplished

if the coefficients

of each polar rotation, each azimuthal rotation, and each propagationG^LI/ are

computed and

stored

once and for all the first time

that

theyare encountered during

the

full calculation.

Besides, all of the

matrices

that appearon the right

hand

side

of

Eq. (6) are sparse, and a

detailed

inspection leads to the

conclusion that

the

number of complex multiplications needed to evaluate each product

G

a

/ 3 < / > / 3 when

using this decompositionis cut

down

by a

factor

of3/

max

/10.

EXAMPLES AND DISCUSSION

The

performance

of the

various iteration methods discussed above

to

calculate

PD from a simple sample consisting of two carbon atoms is compared in Fig. 1,

where the inset illustrates the details of the geometry (the interatomic distance

corresponds to

nearest

neighbors in graphite).

Scattering

fro m a cluster of carbon

atoms

is a severe

test

case, as multiple scattering between bonded carbon

atoms

is

particularly

strong. Within

the

resolution

of the figure , the recursion

method (solid

circles)

converges in

just seven iterations.

At single

scattering,

the direct Jacobi

iteration (open circles)

lies 4%

off

theexactresult, and

subsequent

scattering

orders

lead to divergence. The

latter

is not prevented by

using

the SR method (broken

curves) over a wide range of the

relaxation

parameter

77 .

The lower 77 , the slower

the increase in intensity withiteration step, but the divergent behavior

remains.

As is well known in LEED

[4],

divergences like this one are encountered in MS

when the

absolute value

of any of the

eigenvalues

of the

eigensystem

(3) is larger

than 1. The SR method provides a cure in

many

cases [20], but it is not

suf-

f ic ien tly general, as

illustrated

by Fig. 1.

Instead,

the

recursion method

has a

well-established

convergent

behavior

[21].

The

efficiency

of our new

method

allows us to perform

calculations

for

much

larger

clusters

than

with

other methods

(e.g., we

have calculated

full-hemisphere

distributions

for a

cluster

of

about 2500

atoms

[24]).

We can

thus investigate

the

126



5/7

question ofcluster sizeconvergence, as is done in Fig.2 forphotoemission from a

Cu2s

level situated on the third layer of a

Cu(lll)

surface. The geometry under

consideration is

illustrated schematically

on the lower left corner of the

figure.

Plotted here

is the reliability

factor defined

as

^ 7)

where the

average

is

taken

over

azimuthal

directions of emission,I

N

is the

intensity

calculated for an

Natom

cluster, and / is actually obtained forN = 1856. The

solid

curve and circles correspond to the result

obtained

from the recursionmethod,

where convergence is achieved in lessthan20 iterations. Asmoothconvergence can

be seenin the

T V >

oolimit. The insetshowsazimuthal scans

obtained

fo r diffe rent

cluster

sizes, in

order

to

facilitate

the understanding of the

actual meaning

of

R

in terms

of

curve comparisons.

For

T V

= 944

(dotted curve

in the

inset),

one has

H 3

recursion

850-eV

e-

photon

10 12 14 16 18 20 22 24

Iteration

step

FIGURE 1. Cls

photoemission intensity

in a cluster formed by two

carbon

atoms separated

by 1.4 A as a function of the

number

of

iteration

steps. The

incoming light

is

linearly polar-

i zed w i th the polarization ve cto r

parallel

to the interatomic

axis.

The emission

occurs

in the

forward-scattering

direction (see inset).

The electron energy is 850 eV.

Results

obtained

from

different iteration methods are compared: the recursion

method

(solid cu rve ) ; the direct Jacob

iteration (dotted and dashed curves); forwhich the number ofiteration steps

equals

thescatter-

in g

order;

and the

simultaneous relaxation

fo r

various

values

of the

relaxation

parameter

77

( th in

broken curves) . The

intensity

has

been

normalized to

that

of the

isolated

C

atom.

127



6/7

R 0.03 and convergence is

already

quite good as

compared

to the

T V

= 1856

case,

although some

small

discrepancies

can

still

be distinguished in the

height

of

the peaks around 30, 60 , and 90 , so

that

over 1000 atoms are needed to

obtain

convergence within

the

resolution

of the

figure

(deviations

above

5% in

intensity

are

observed

for some directionsof

emission

using

T V

= 944).

The

open circles

in Fig. 2

show

the reliability

factor

obtained from the

Jacobi

method

for

various scattering orders

(5, 9, 13, 17, 21, and 25) ,

where

the

spread

in

the position of the circles makes evident a

divergent

behavior. The

latter

is more

pronounced for larger clusters. In thissense, theJacobi methodhas to be regarded

as an asymptotic series

unable

to convergebelow a

certain

reliabilityfactor in the

present case.

3 6

Azimuthal angle (degrees)

25 500 750 1000

umber

o fatoms

125 15

FIGURE

2.

-Rfactor

[Eq.

(7)]

variation w i t h

the

number

of

atoms

N

for

Cu2sphotoemission

f rom the third layer of a Cu(ll l) surface. A z i m u t h a l scans have been considered

w i th

a polar

angle of emission of 35, a

photoelectron energy

of 100 eV, and p-polarized light

u n d e r

normal

incidence conditions, as shown

schematically

on the

lower left

corner of the figure . The inset

shows

the

intensity

as a

f unc t i on

of azimuthal angle for various cluster

sizes,

as

indicated

by

labels,

normalized

to that of the direct

emission

w i t h o u t inelasticattenuation.

128



7/7

ACKNOWLEDGMENTS

This work

was

supported

in part by the

University

of the Basque

Country

and

the Spanish Ministerio de Educacion y Cultura (Fulbright

grant

FU-98-22726216),

and in

part

by the

Director,

Office of Science,

Basic Energy Sciences, Materials

Sciences

Division,

of the

U.S.

Department of Energy under

Contract

No. DE-

AC03-76SF00098.

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