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