Self-consistent localized-orbital study of chemisorbed oxygen on FE(001)
by Hong Huang
A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in
Physics
Montana State University
© Copyright by Hong Huang (1985)
Abstract:
The electronic structure and magnetism of a p(lxl) chemisorbed oxygen layer on the Fe(001) surface
was studied by the SCLO method. The interface geometry was suggested by a previous LEED analysis.
We found a good agreement between the calculated DOS and the UPS data. We found the oxygen
atoms have significant bonding to both the surface and subsurface Fe atoms, and atomic bonding
pictures were derived. We did not find a magnetically dead layer of Fe surface, consistent with the
spin-resolved photo emission experiments on Fe-based glass. The calculated work function change
disagrees with the experiments, and this may be due to oxygen incorporation. The surface-state bands
were predicted. Comparing these bands with ARPEs may clarify the chemisorption/oxidation model.
S ELF - CONS I S TE NT LOCAL I ZED- ORB ITAL
STUDY
OF CHEMISORBED OXYGEN ON F E ( O O l )
by
Ho n g H u a n g
A th esis submitted in p a r tia l fu lfillm en t
of th e r e q u i r e m e n t s f o r th e d eg ree
of
Doctor
of
Philosophy
in
Phy s i c s
MONTANA STATE UNI VE RS I T Y
Bozeman,Montana
August
1985
ArcM ^
%
ii
APPROVAL
of
a thesis
submitted
by
Ho n g H u a n g
T h i s t h e s i s h a s b e e n r e a d by e a c h m e m b e r of t h e
t h e s i s c o m m i t t e e a n d h a s b e e n f o u n d t o be s a t i s f a c t o r y
re g ard in g c o n te n t, English usage, form at, c i t a t i o n s ,
b i b l i o g r a p h i c s t y l e , and c o n s i s t e n c y , and i s re a d y f o r
s u b m i s s i o n t o t h e C o l l e g e of G r a d u a t e S t u d i e s .
Date
ChaiArperson,
Approved
for
Date
the
Major
for
the
College
of
Graduate
p a r t me n t
Graduate
-//
Date
Committee
Department
‘H e a d , . M a j d r
Approved
Graduate
Dean
Studies
iii
STATEMENT OF P ERMI S S I ON TO USE
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University,
available
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argee
only
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Requests
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for
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for
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the
and t h e
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U. S.
or
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degree
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copying
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partial
the Library
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and d i s t r i b u t e
abstract
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extensive
be
in
a doctoral
prescribed
should
national,
to
I
for
thesis
in
and from
and d i s t r i b u t e
by
iv
ACKNOWLEDGEMENTS
The
author
gladly
takes
P r o f e s s o r J o h n Herman son,
research
patient
reported
discussions
Helpful
General
here,
opportunity
wh o s u g g e s t e d
for
during
discussions
this
his
the
with
Research L ab o rato ries
advice
the
and
3 years.
J a c k Gay
are
thank
and s u p p o r te d
unlimited
past
to
and John S m it h
greatly
of
acknowledged.
V
TABLE OF CONTENTS
page
, -i '■
APPROVAL............................................................................................................................
ii
STATEMENT OF P ERMI S S I ON TO U S E . . . . . . . . . . . . . . . . . . . . . . .
iii
ACKNOWLEDGMENT...........................................................................................................
iv
TABLE OF CONTENTS....................
v
L I S T OF T A B L E S ..................................................................................................
v i
L I S T OF F I G URES......................................................................................................... v i i
ABSTRACT............................................................................................................................ v i i i
CHAPTER I .
I NTRODUCTI ON..................................................................................
I
CHAPTER 2 .
METHODOLOGY............ .........................................................................
11
S l a b M o d e l .................................................................................................................
L o c a l D e n s i t y A p p r o x i m a t i o n ............................
M a t r i x F o r m o f t h e S c h r o d i g e r E q u a t i o n ..................................
G a u s s i a n E x p a n s i o n o f A t o m i c Wave F u n c t i o n
a n d P o t e n t i a l ...............................................................................
S t a r t i n g - m a t r i x C a l c u l a t i o n .................................................................
S y m m e t r y .......................................................................................................................
S e l f - c o n s i s t e n t I t e r a t i o n .......................................................................
L o w d i n R e p r e s e n t a t i o n ..................................................................................
11
14
20
CHARPTER 3 .
23
27
35
51
57
RESULTS AND D I S C U S S I O N ...................................................
65
T w o - l e v e l B o n d i n g M o d e l .............. .............................................................
D ensity
o f S t a t e s ..................................................................................
A t o m i c - o r b i t a l O c c u p a n c i e s . . . . . ..........................................
M a g n e t i s m ....................................................................................................................
Charge
a n d S p i n D e n s i t i e s ..........................................
S u r f a c e - s t a t e B a n d s ........................................................................................
65
71
83
88
89
90
CHAPTER 4 .
SUMMARY AND FUTURE WORK...................................................
97
S u m m a r y ...........................................................
S u g g e s t i o n s f o r F u t u r e S t u d y ...............................................................
97
98
R E F R EN CES
CITED
100
vi
L I S T OF TABLES
1.
A t o m i c Wave F u n c t i o n
and P o i n t Group
2.
Representation
matrices
3.
Representation
matrix
4.
O c c u p a t i o n Numbers
5.
Magnetic
Moment s
for
for
for
for
R e p r e s e n t a t i o n ..........................
Fe,
Fe,
40
C^ v G r o u p ..................................
44
Z - R e f l e c t i o n G r o u p ...............
45
a n d 0 / F e * S l a b s ....................
86
and O/Fe $ s l a b s
87
vii
L I S T OF FI GURES
pa ge
1.
Slab
model
2.
Slab
Model
for
p(lxl)
for
V i e w ....................
12
O / F e ( 0 0 1 ) . T o p v i e w .........................
13
S y m m e t r y ...............................................................
46
p(lxl)
O/Fe(001).
Side
3.
Application
of
C4 y
4.
Application
of
Z—R e f l e c t i o n
5.
Two-level
Bonding.
We a k C o u p l i n g .................................................
69
6 . Two-level
Bonding.
Strong
C o u p l i n g . ........................................
70
0 / F e ( 0 0 1 ) ...........................................
76
7.
Layer-projected
8.
Surface
9.
Subsurface
DOS's
3d-orbi tal
of
S y m m e t r y . . . . ................
DOS's o f
3d-orbital
O / F e ( 0 0 1 ) ..................................
DOS’ s o f
O / F e ( 0 0 1 ) ..........................
3 d - o r b i t a l DOS's o f
77
78
10.
Central-layer
11.
P lanar Bonding
I . . . . ...................
80
12.
Planar
I I ........................................................................................
81
13.
Vertical
B o n d i n g ...........................................................................................
82
14.
Charge-density
15.
Spin-density
16.
Symmetric
17.
Antisymmetric
Bonding
O / F e ( 0 0 1 ) ..................
47
79
C o n t o u r s . . . . . . ......................................................
93
C o n t o u r s ......................................................... .. ................
94
Surface
B a n d s .......................................................................
Surface
Bands
95
96
viii
ABSTRACT
The e l e c t r o n i c s t r u c t u r e a n d m a g n e t i s m of a p ( l x l )
c h e m i s o r b e d o x y g e n l a y e r o n t h e F e ( 0 0 1 ) s u r f a c e wa s
s t u d i e d b y t h e S CL0 m e t h o d . T h e i n t e r f a c e g e o m e t r y w a s
s u g g e s t e d b y a p r e v i o u s LEED a n a l y s i s . Ve f o u n d a g o o d
a g r e e m e n t b e t w e e n t h e c a l c u l a t e d DOS a n d t h e UPS d a t a . Ve
found the oxygen atoms have s i g n i f i c a n t bonding to both
t h e s u r f a c e a n d s u b s u r f a c e Fe a t o m s , a n d a t o m i c b o n d i n g
p i c t u r e s w e r e d e r i v e d . Ve d i d n o t f i n d a m a g n e t i c a l l y d e a d
l a y e r o f Fe s u r f a c e , c o n s i s t e n t w i t h t h e s p i n - r e s o l v e d
p h o t o e m i s s i o n e x p e r i m e n t s o n F e - b a s e d g l a s s . The
c a l c u l a t e d work f u n c t i o n change d i s a g r e e s w i t h the
e x p e r i m e n t s , a n d t h i s ma y be d u e t o o x y g e n i n c o r p o r a t i o n .
T h e s u r f a c e —s t a t e b a n d s w e r e p r e d i c t e d . C o m p a r i n g t h e s e
b a n d s w i t h ARPEb ma y c l a r i f y t h e c h e m i s o r p t i o n / o x i d a t i o n
model.
I
CHAPTER I
INTRODUCTI ON
One
of
the major
goals
of
understand
the
microscopic
corrosion,
for
which
The
study
a few
to
of
this
decades
be a p p l i e d
brought
[1-15]
under
have
that
the
the
step
incorporation
mechanism
O/Fe
is
system,
so
that
control.
Since
performed
of
of
oxidation
0 into
the
Finally
of
the
bulk.
consensus
regarding
the
adsorbed
coverage,
the
oxidation
products,
sticking
surface.
the
thin
to
techniques
began
could
and
both
agreed
oxygen on i r o n
form a
two-
layer
there
the
is
by
is
the
begins
saturation
them b r i e f IyT
of
studies
oxidation
to
no
nature
experimental
and
be
studies
generally
structure/
chemisorption
H e r e we r e v i e w
until
oxide
However,
Several
possible
followed
probability,
etc.
is
of
selvedge
towards
on
It
Fe,
this
example.
processes
chemisorption
grow
Fe ( 0 0 1 )
not
system,
and t h e o r e t i c a l l y .
to
chemical
extensive
this
oxide.
been r e p o r t e d
was
then,
on
is
important
oxidation
dimensional
inward
of
a very
however,
the
dissociative
first
science
ago when u l t r a - h i g h - v a c u u m
been
experimentally
surface
of
the
have
the
2
S i mmo n s
a n d Dwy e r
diffraction
to
study
the
oxidation
clean
( LEED)
of
iron
[5]
used
I o w- e n e r g y - e l e c t r o n -
and A u g e r - e l e c t r o n
structural
changes
F e (001)
room
sample
at
showed a
and
sharp
deposited
on t h e
LEED s p o t s
gradually
appeared,
p(lxl)
surface,
and
sharpness
w e r e m a x i m i z e d w h e n AES
monolayer
of
further
exposure
h a l f —o r d e r
the
became
of
shift
p (Ixl)
at
integral-order
c (2x2)
the
spots
was
features
interpreted
as
authors
somewhat
was
alternate
to
is
due
shift
formed.
than well
not
The
to
substrate.
This
ordered.
this
The
Fe
the
oxide
but
experiment
is
the
of
peaks
is
the
showed
the
c a n be
interpretation
of
the
oxygen a t
surface
followed
to
the
corresponding
This
diffusing
by o x i d e
indicated
amorphous
LEED p a t t e r n ,
also
With
intensity
F e(001)
due
a
simultaneously
surface.
however,
The p ( I x l )
half
The
chemisorbed
AES i r o n
oxide,
and
T h e LEED p a t t e r n
from
original
surface,
of
away.
relative
different)
the
Wh e n
c (2x2)
that
surface.
peaks.
clean
on
the
intensity
occurred
different
sites
At f i r s t ,
indicated
went
the
the
the
LEED p a t t e r n .
their
spots
the
( though
beneath
nucleation.
oxide
of
four-fold
sites
but
( AES)
L=I O * T o r r - s e c o n d ) ,
AES i r o n
1 0 L,
diffraction
I
gradually
the
of
1 0 L,
of
h a l f —o r d e r
f or me d on th e
to
LEED s p o t s
di s a p p e a r e n c e
with
(up
kinetics
temperature.
o xy ge n was
o x y g e n was
spectroscopy
further
the
rather
therefore,
underlying
that
that
is
iron
oxygen
3
exposure
no
caused
gradual
diffraction
2 0 L.
This
oxide
is
is
seen.
this
features
than
heating
coverage,
AES i n d i c a t e d
the
during
no
the
of
epitaxial
The
and
the
the
Fe
fact
of
path
the
in
exposures
that
of
a
Fe
can
be
at
again.
concentration
o f LEED s p o t s
disorder-order
relationship
substrate
incident
a few m i n u t e s
The r e a p p e a r a n c e
result
the
showed up
oxygen
above
amorphous
of
for
and
the
underlying
LEED p a t t e r n
change
heating.
as
layer
to
reflections*
for
sample m i l d l y
p (Ixl)
that
all
t h e mean f r e e
the
interpreted
F e O.
due
of
observed
s o no LEED p a t t e r n
After
occurred
were
probably
thicker
electrons,
weakening
between
is
transition
the
FeO o x i d e
is
( O O l ) F e O Il ( 0 0 1 ) Fe .
AES a l s o
this
indicated
stage
was
that
about
4
c (2x2)
oxygen
adlayer.
oxygen
needed
to
structure
on t h e
substrate
and
that
the
oxide
form
Fe
the
has
for
epitaxial
structure
the
sample,
in
This
higher
is
substrate.
addition
The
is
change
the
oxygen a t
the value
the
amount
reflections
4.5%
expected
when
spots
the
for
of
of
the
smaller
that
oxide
iron
suggests
than
this
layer
and m ild h e a t i n g
of
the
Na Cl
coincidence
75 L e x p o s u r e
to
of
FeO w i t h
parameter
So i t
w ill
After
of
overlayer
a lattice
than
exactly
t wo l a y e r s
FeO b u l k .
thicker.
concentration
times
oxide
the v alu e
becomes
the
Fe(OOl),
there
of
4
appeared hexagonal
FeO(Ill)
layer
Brucker
LEED s p o t s
epitaxialy
and Rhodin
ultraviolet
which
grown on
[8]
studied
photoemission
suggest
the
the
work-function-change
measurements.
For
1.5
little
(a
L,
o f Fe
data
they
reported
d —b a n d —p e a k
from
about
emission
a clean
5.5
Fe
eV b e l o w
sample.
stage
Fe(OOl)
surface.
confirming
that
the
Fe
its
m a x i mu m ,
For
large
exposures,
substantially,
attenuated.
out
gradually
exposures
changes
to
further
those
the
at
at
exposure
7 L.
reported
at
which
suggest
the
and
2p b a n d
by
the
on
of
at
the
this
the
as
f o r m e d on
stage
clean
at
authors
of o x y g e n was
reached
surface value.
eV p e a k b r o a d e n e d
e m i s s i o n was
F e O.
caused
after
the
that
S i mmo n a n d
beginning
of
the
the
distribution
the
c ( 2x 2 )
LEED
spots
p(lxl)
changes
Dwyer,
t h e LEED p a t t e r n s
sharply
energy
Regarding
T h e LEED p a t t e r n
by
the
structure
photoelectron
of
than
attenuation)
T h e LEED p a t t e r n
interpreted
5.5
that
4 L,
to
oxygen.
d —b a n d
At 6 0 L t h e
disappeared
with
and
similar
patterns,
due
eV a b o v e
the
less
peak c e n t e r e d
monolayer
0.2
exposures
and
An a d d i t i o n a l
The w o r k f u n c t i o n
about
LEED,
corresponding
0 c ( 2x2 )
was
a half
surface.
was v a r y
This
(BPS),
the
Ep a p p e a r e d
showed a c l e a r
surface.
s y s t e m by u s i n g
small
compared to
di s s o c i a t i v e l y c h e m i s o r b e d
this
change
Fe(OOl)
s a me
spectroscopy
an o r d e r e d
but
to
spots
agreed
not
changed.
oxidation
fade
well
the
These
after
1.5
L
5
exposure,
and t h e
oxide,
concluded
as
began to
decrease
mi n i mu m v a l u e
clean
Fe
1.5
formed.
superstructure
analysis
in
Jepson
chemisorption
also
used
to
of
reported
to
that
histories
c (2x2 )
half-
of
to
study
the
concluded
possible
a few
the
layers
sample
in
reported
performed
by
below
FeO a r e
of
aspect
of
a
on
of
the
surface
were
Fe
e v e n when
surface
to
and
thermal
produce
the
oxygen
( S o me f a i n t
amounts
of
c a rb o n on
the
surface).
caused
no
change
the
geometry
large
Exposure
to
oxygen
only
observed
existed,
of
They
LEED s p o t s
spots
presence
the
AES w a s
impurities.
other workers
but
that
a LEED
surface.
procedures
order
the
bulk oxide.
concentration
clean
the
which
of
formation
existence
a
the
dissociative
F e (001)
oxygen
of
to
causing
geometrical
on t h e
o x y g e n —e x p o s u r e
the
surface,
dipole,
[9]
no f r a c t i o n a l - o r d e r
structure
order
that,
leads
oxygen ch em iso rb ed on
they v a r i e d
corresponds
iron
the
and Marcus
oxygen
the
in
a
work f u n c t i o n
authors
is
amorphous
and r e a c h e d
This
the
of
The w o r k f u n c t i o n
three-dimensional
calibrate
and t o m o n i t o r
the
surface
Beyond
the
order
than
The
exposure
and
J ona,
the
system
stage.
Further
Legg,
of
the
layers
L exposure,
beneath
decease.
L exposure,
chemisorption
for
oxygen
to
a few
7 L exposure.
a reversal
ork function
1.5
eV h i g h e r
at
of
of
S i mmo n a n d D w y e r .
after
surface
to
by
0.05
incorporation
leads
formation
w h e n AES
in
indicated
of
the
the
6
LEED p a t t e r n
diffracted
p (lx I)
only
beams,
the
changes
remained
low
and c o n t r a s t
increased
broader
with
o f 25 L o r m o r e ,
revert
the
succeeded
full
oxygen
0 p(lxl)
in
exposure
p (l x I ) 0/Fe(001)
and
intensity
t h e O-Fe
the
substrate
Fe
apparently
the
hollow
The
first
Th e
up t o
found
After
but
that
proximity
interlayer
bulk metal.
c h e m i s o r p t i o n model
of
0/Fe
Fig.I
that
replaced
the
by
7-layer
slab
semi-infinite
Recently,
Sakisaka,
geometry
(SES),
AES,
Hiyano
a n d On c h i
and
initial
[10]
(EELS)
secondary-electron
and w o r k - f u n c t i o n - c h a n g e
chemisorption
is
This
and F i g . 2
should
be
geometry.
electron-energy-loss-spectroscopy
w i t h LEED,
there
to
spacing
8% c o m p a r e d w i t h
in
that
oxygen atoms
by
except
below
ordered
with
close
shown
they
6 L
expanded
is
high
to
determined
A,
in
L,
LEED c a l c u l a t i o n
they
0.48
0
grew"
it,
and well
Fe-Fe
6-10
concentration
They
sites
surface.
possible
By d y n a m i c a l
is
the
annealing
analysis,
the
of
exposure.
oxygen
it.
distance
atoms.
by
of
LEED s p o t s
always
a complete
voltage
4-fold
high
the
was
by h e a t i n g
interlayer
occupying
and
it
structure.
vs.
substrate
structure
provided
formation
increasing
reducing
monolayer
intensities
the
F e(001)
exposures
one
with
the
progressively
never
the
on
background
to
in
consistent
s tr u c t are
background
then
bat
in
emission
measurements
oxidation
of
used
conjunction
spectroscopy
to
study
Fe(OOl)/
Three
7
stages
of
oxidation
chemisorption
the
selvedge
2 0 L,
from
is
that
identified:
3 L,
(2)
between
leading
interest
to
the
the
formation
change
in
by
EELS.
ascribed
antibonding
about
until
but
the
and t h e
bond,
to
the
of
the
the metal
observed
clean
the
^^
are
in
former
as
the
peaks
play
characterized
Fe
phase
reaching
a ma x i mu m o f
oxide
of
3 L,
bonding
energy-loss
at
role
at
3 —2 0
in
process
due
L.
Fe
at
the
A 0 =+0.25
to
in
and
peaks
at
this
Th e a u t h o r s
3 d xy#IZ#
due
to
Fe
chemisorption
Fe
3dz » ^xa- y a
the
oxide
The
also
confirmed
a d- *- d
measurement
steeply
eV a t
the
the
570 C was
peak
increased
the
transition.
The w o r k - f u n c t i o n - c h a n g e
the work f u n c t i o n
the
involving
by p e a k s
Y - F e 2O j - ^ F e O
that
6 eV b e l o w
EELS s p e c t r u m
an e n e r g y - l o s s
the
3 d z » f x a- y 2 • Thus
3d c h a r g e - t r a n s f e r
transition.
at
range
O1 - 2 p - * - F e , +
by m o n i t o r i n g
of
peak remained unchanged
as
transition
properties
substantially
a major
The
above
special
that
between
incorporation
involved.
Of
into
characteristic
surface
the
involving
orbitals
and f o r
phase was
was
0 atoms
oxidation
to
peak
eV e n e r g y - l o s s
exposure
latter
orbitals
2
(3)
of
electronic
transition
The
dissociative
Y~Fe * 0 s .
An e n e r g y - l o s s
states.
interpreted
g^
of
5 a n d 8 eV w e r e w e a k e n e d
stage,
and
surface
c h e m i s o r b e d o x y g en was
being
(I)
incorporation
3 a n d 2 0 L,
characteristic
observed
the
below
were
up t o
3 L,
then
showed
3 L,
decreased
8
to
a mi n i mu m
increased
The
A<$> = + 0 . 0 2
again
initial
transfer
dipole
of
the
layer,
of
thus
of
oxidation.
agreed
by
observation
the
of
is
is
with
reviewing
at
of
p (Ixl)
s ome
the
layer.
beneath
The f i n a l
the
in-depth
measurement
process
at
thus
confirmed
the
p(lxl),
agreeing
with
[9].
However,
the
the
coverage
a coverage
of
I,
calibration
of
any
the
results
of
experimental
is
realized
stage
Fe(OOl)
that
of
system.
surface,
or
group.
example,
there
is
The q u e s t i o n
to
to
surface,
does
geometrical
For
oxidation
correspond
Fe(OOl)
other
the
controversial.
O / F e (001)
the
oxygen
to
seen
for
to
structure
oxygen on t h e
surface
subsequent
the
related
charge
a coverage
still
that
a
to
to
of
is
due
L.
corresponding
it
oxidation
to
oxidation
3 L exposure
the
is
350
barrier
The
due
T h e LEED p a t t e r n
investigations,
structure
energy
an o x i d e
a n d 20 L c o r r e s p o n d i n g
After
oxygen
is
function
L e g g j@_t
agree with
this
the
eV a t
creating
surface.
three-stage
stage
calibration,
agreed
to
function
oxygen,
the
and f i n a l l y
Ad> = 0 . 4
The w o r k - f u n c t i o n - c h a n g e
EELS a n a l y s i s .
not
work
work f u n c t i o n
the work
with
0.35
of
leading
chemisorption
of
the
2 0 L,
value
increasing
out
the
incorporation,
increase
of
chemisorbed
coming
decrease
a limiting
increase
to
electrons
to
eV a t
the
the
aspect
it
is
a p(lxl)
is:
does
chemisorption
incorporation
forming
amorphous
of
of
or
I
9
ordered
goes
F e O? Al t h o u g h
inside
geometry
task
of
of
we kn o w
the
Fe
surface,
the
oxygen
theoretical
the
the
experiments,
chose
chemisorption
A few
cluster
6 eV b e l o w
Fe-O
O-Fe,,
the
in
the
unrestricted
predicted
cluster
to
Fe-O bond
spacing
near
[13]
occupied
spacing
of
four-fold
0.48
spacing
level
site
and
bond
energy
range
5 to
states
( DOS) w a s
7
three
further
A.
for
o x y g e n 2p l e v e l s
an
equilibrium
o x y g e n was
i n an
[14] ,
In r e c e n t
used
of
For
0 2p levels
Ep d u e
been t
c omp u t a t i o n s
and
structure
near
have
Ri b a r s k y ,
0.38
E p • Th e
We
oxygen
calculation
length.
predict
the,model.
when t h e
site.
of
eV b e l o w
reduced
to
pick
O / F e (001)
A d a c h i ±_t j l .
[15],
the
A,
A,
to
The
—' :
obtained
cluster
study
0.5
the
level,
hollow
Eartree-Fock
a vertical
of
s p i n—p o l a r i z e d
Anderson
calculations
substrate
refine
calculations
models
In
highest
interlayer
placed
the
system.
is
and p o s s i b l y
or
and
unknown.
s u r f a c e . "■■■l
structure
cluster
extent
compare w i t h
justify
F e (001)
electronic
performed using
the
remains
p r o p o s e d kby ’ L e g g .ejt jil, f o r
on t h e
chemisorption
to
to
nature,
therefore,
study
system,
and t h e n
t h e model
to
oxygen e v e n t u a l l y
exact
investigation,
models,
of
the
the
incorporation
s o me e x p e r i m e n t a l
properties
that
Xa
a 9 - atom
0 / Fe(OOl)
an
were
versus
interlayer
obtained
in
d —b a n d
density
of
to
Fe-O b o n d i n g .
the
the
10
None
of
the
effects
of
adsorbate-adsorbate
extended
0/Fe(001)
2p levels,
bands
of
reflection
along
conjunction
( ARPES)
with
the
above
the
J
in
interaction
points
zone
included
interactions
This
general
detectable
reviewed
two-dimensional
Brillouin
parity
spectroscopy
energy
in
(IBZ)
the
but
broadens
the
bands.
These
[16] .
The
selection
use
O
wedge
definite
and A s y m m e t r y
angle-resoved
the
irreducible
have
using
the
lines.
They
photoemission
v
.
• ■ I
p o la ri z e d l i g h t in
of
rules
[17]
should
aid
in
identification.
Here
of
to
at
surface
s h o u l d be
system.
leading
hybridize
the
their
calculations
we r e p o r t
calculations
a p ( l x I ) OZFe( OOl )
slab
of
the
using
the
electronic
structure
self-consistent
' '
: '
m e t h o d d e v e l o p e d b y S m i t h , Gay
.
l o c a l iz e d - o r b i t a l
and A r l i n g h a u s
OZFe ( 0 0 1 )
( SCLO)
[18-20].
[21],
In
we a l s o
did
Ni ZCu(OOl )
[22],
CuZNi(OOl)
P dZF e(001)
[25],
and
incorporates
the
surface
results.
By t r e a t i n g
able
this
as
as
the
bonding
discussed
in
well
and e n erg y
detail
in
to
the
band
the
of
[26].
slab
a
from
study
calculations:
[24],
this
simple
the
of
method
picture
of
numerical
polarization
the
influence
magnetic
of
structure.
next
calculation
Because
basis,
electron-spin
we w e r e
the
HZNi ( OOl )
developed
explicitly,
system
[23],
orbital
be
to
a number
F e Z C n (001)
an a t o m i c
bond can
addition
the
behavior
magnetism
T h e SCLO m e t h o d
chapter.
of
on
is
11
CHAPTER 2
METHODOLOGY
SUb
Mi JdeI
Th e s e m i - i n f i n i t e
theoretically
the
by a n a t o m i c
XY ( s u r f a c e )
plane
the Z d i r e c t i o n
In our
slab
study,
adjustable
the
atomic
slab
are
center
considered
the
layer
are
bulk
The
to have
thick
like
number
See
charges
enough
electronic
layers
to
Z.
in
in
and F i g . 2.
properties
translation
[27,28].
of
physical
in
extension
Fig.I
equation
only
symmetry
calculated
infinite
electronic
m and n u c l e a r
assumed
simulated
surface).
Schrodinger
and r e f l e c t i o n
if
to
parameters.
positions
ma y be
surface
the
be
a finite
we c a l c u l a t e d
any
XY p l a n e
can
slab with
and
(normal
by a p p l y i n g
positions
surface
it
of
the
without
inputs
are
Atomic
symmetry
in t h e
the Z direction.
The
to
real
represent
properties
the
at
the
12
Fig.I
S l a b mo de l f o r p ( I x l ) O/ Fe ( 0 0 1 ) . S i d e Vi e w. C r o s s
s e c t i o n along ( H O ) plane. D i s t a n c e s in angstroms.
13
Fig.2
Slab
model
for
p(lxl)
0 / F e ( 0 0 1 ) . To p V i e w .
14
kocal
DenUt^
The
Aje^ r o x i m a t i o n
electronic
Schrodinger
wave
function
of
the
system
obeys
the
equation
H V ( T1 , S 1 ---------- r n , s n ) = E V Cr 1 , S 1 ---------- r n , s n ) ,
(I)
where
N
m is
H= J ( - 1 / 2
1=1
N N
j - ] y Zm/ I r 1- B i ) + 1 / 2 5
5 I / r ij »
m
T=Ij=I
the
position
atomic
in
the
system.
in
the
non-relativistic
elements
with
Atomic
atomic
and 4 d t r a n s i t i o n
In
the
exclusion
Slater
and N is
units
( a. u. ) a r e
approximation,
number
metals,
single-electron
principle,
the
the
number
(2)
of
used.
electrons
This
namely,
for
is
valid
all
Z< 5 O [ 2 7 ] ,
thus
including
w h i c h we a r e
most
interested
approximation,
wave
function
due
to
3d
in.
Pauli's
c a n be w r i t e s
as
a
determinant
^ 1 ( I ) ------0 n ( l )
V ( r I . sI
r n» s n
I)
(3)
*1
^ 1 ( n ) ----- ( n )
wh e r e
f i ( j ) - ^ i ( r j ) X 1( s j ) ,
that
of
is,
the
an o r b i t a l
(4)
single-electron
function
and a
wave
function
spin
function.
is
the
product
15
To m i n i m i z e
the
single-electron
e qua t i on
total
wave
en e rg y of
function
the
must
system
obey
the
<VlHlvp>, t h e
Hartree-Fock
[29,3 0,31] :
[ - l / 2V * - 5 2 Inz I r i - I n i + £ / i/r* , ( r ' J'A.-( r ' ) d * r ' / i r - r ' l ]
m
j
j
j
[ S s i S j / 4* * j ( r ' ) lAi ( T t ) J 1 r ' / i r - r ' l ]
The f i r s t
term
the
second one
the
third
one
electrons,
between
in
is
is
and
this
the
the
the
equation
potential
Coulomb
fourth
parallel-spin
( r )-E
one
is
the
energy
interaction
is
due
to
to
the
(5)
energy,
nucleus,
potential
exchange
electrons,
( r ).
kinetic
due
(r)
*
between
potential
Pauli's
exclusion
principle.
In
the
exchange
local
To
use
Density
potential
electron
i
derive
the
the
plane
the
can
density
V
consider
under
Local
be w r i t t e n
for
t
influence
to
each
of
a
of
Vfi ( r ) = e x p ( I k i ^ r )
,
of
the
the
[36],
first
a homogeneous
constant
represent
[32-35],
a function
r
expressions
case
as
( LDA)
spin:
o
above
extreme
waves
Approximation
wave
let
electron
potential
field.
<«>
us
gas
We
can
functions:
( 7 )
16
Then t h e
exchange
term
in
the
Hartree-Fock
equation
becomes
^
J
if
e xp ( - i ( k j - k j ) * r ' ) / I r - r ' i d * r ]
J
= I ^ / e x p ( i ( k j - k j ) *r ) / r
d*r]
e xp ( i k . • r )
J
exp( Iki • r )
= 4 n [ % I / ( k j - k j ) 1 ] IAi ( T )
kj<kF
— ► ! / ( 2 n a ) [ / I / ( k . - I t i ) 2 d * k . ] (Ai C r )
kj<kF
= ( k F / Jt) F ( k i / k F ) ( A- Cr )
(8)
where
F(x)=l +[C l-xa )/2x]In I(l+ x )/(l-x ) I .
Th e
exchange
several
inside
ways
the
to
sphere
and which
be
principle
charge
k i = kp*
of
density
kF= ( 3 n * p ) ^ / *
in
k.-dependence,
is
gives
[37] ,
is
<F > = 1 . 5
the
Fermi
the
b a s e d on
rigorous
There
for
the
are
example,
Slater
or
at
the
K o h n - Sh am a p p r o x i m a t i o n
[ 3 3 ] . We u s e
a more
the
k ^ —d e p e n d e n t .
ki < kF* which
which
< F >=1
is
this
alternatively,
thus
The v a l u e
sphere
and w hich
gives
derived,
term
average
Fermi
approximation
Fermi
energy
(9)
since
it
can
the v a r ia tio n a l
wa y
wavevactor
latter
[33],
is
related
to
the
by
(10)
17
for
each
spin
Eq. (5)
is
energy
for
over
Eq . ( 6 ) .
Now we
varying
that
since
summation
a particular
spin
....
i
■
consider
above
-
the
The
expression
system
is
for
inhomogeneous
essential
idea
the
expected
approximation
for
the
and t h e
potential
and where
solid
is
most
formed
T h e LDA i s
Fock exchange
the
only
potential,
correlated.
Following
electrons.
is
The
correlation
l/rjj
is
and t h e
a
it
depression
In
the
treated
as
distribution
the
of
the
be
do
is
a ,
case
is
of
the
atoms
a good
region,
change
where
slowly
o c c u r s when
of
the
the
the
as
between
it
Hartree-
many-body
electrons
is
moves
distribution
called
the
any
interaction
another
for
incorporates
Hartree-Fock
interaction
of
-'
atoms.
electron
in
the
"
method for
density
also
is
slowly
interstitial
The m o t i o n
each
a
This
approximation
depression
hole.
instantaneous
an
;;
rearrangement
isolated
effects.
system
the
charge
from
not
in
'
in
t h e LDA w i l l
electron
correlation
the
that
by
assumption
even
of
exchange
energy
system.
term
given
in
Thomas-Fermi
solids
of
is
moving
locally
varying
is
gas
exchange
a slowly
It
So t h e
fundamental
valid
behind
exchange
-
electrons
field.
the
only.
electron
homogeneous
[38,39] .
in
the homogeneous
potential
the
the
highly
through
of
other,
exchange-
equation,
pair
of
between
electron.
the
electrons
one
electron
In p a r t i c u l a r .
18
the
t wo a n t i p a r a l l e l - s p i n
the
exactly
s a me
space
antiparallel-spin
which w i l l
Co u l o mb
this
happen
for
up
both
term
this
The
local
in
term
density
the
it
is
i n LDA i s
both
is
t wo
Thus
energy
pt
of r s and
partly
is
needed
by
adding
is
the
average
and
p,
p 4,
of
Our
the
is
the
where
(11)
distance
between
electrons,
and
spin-polarized
Wilk and N u s a i r
interpolate
r s* to
low
was
homogeneous
[40].
( RPA)
parameter.
potential
accurate
approximation
small
(12)
polarization
correlation
the
or
P =(Pf-Pl)/(pf+p*),
which
a
Hamiltonian.
r s= [ 3 / ( 4 n x ( p t + p t ) ) ] 1 / J ,
which
in
although
a function
spins,
a function
for
region,
the
also
at
overestimated
electrons
to
be
system.
improvement
energy
to
when T1= T i )
physical
exchange
Further
of
vanish
density
and a n t i p a r a l l e l
correlation
equivalently,
high
correlation
electron
not
real
allowed
determinant
electrons
the
deficiency.
the
the
electrons
parallel
called
does
in
are
(Slater
between
especially
parallel-spin
makes
point
electrons
interaction
way,
for
never
electrons
They
[40-45]
density,
electron
used
results
adapted
for
a Fade
the
Mont e
studies
liquid
of
by Vo s k o ,
technique
to
random-phase-
correlation
where
from
energy,
Carlo
valid
results
for
19
[46,47]
the
are
available.
correlation
energy
per
To
energy,
electron
determine
the
[40]
a convenient
paramagnetic
was
first
form
of
correlation
fitted
as
[48-50]
e ( r s,0)=-A F(rs/R ),
(13)
where
F ( x ) = ( 1 + x 1 ) I n ( 1 + 1 / x ) + 2 / x - x 2- l / 3 ,
A is
48.6
a to
Eq.(13)
dependent
mRy,
and R is
with
15.
A*= 3 1 . I
correlation
We a l s o
.
fit
(14)
the
spin
mRy a n d R , = 1 6 . 4 .
energy
can
The
be w r i t t e n
as
e ( r s , p ) = e ( r $, 0 ) + ap * / 2 .
stiffness
spin[40]
(15)
Then
^ctorl
«< £-•.> _
3pt or4
= - A ln (1+R/r s )+ap+pp^,
(16)
where
P=I/2
To
obeys
A 'ln(l+ R '/rs)-a .
sum u p ,
i n LDA,
the
(17)
single-electron
wave
function
20
[ - I / 2V*
Z m/ I r - ml + V p ( r ' ) d 1 r ' / I r - r ' I + Vx c ( p ( r ) 1 ^ * )] ^ ( r )
m
= E « A( r ) .
(18)
Symbolic a l I y ,
H (p(r))^=E ^.
In
p(r)
order
to
first.
(19;
k now t h e
But
p(r)
wave
is
function,
given
we h a v e
to
know
by
p ( r ) = % ^ 1( T ) 2 .
o ccupied
So
This
in
order
to
indicates
IU lrix
know p ( r ) ,
we h a v e
the
Form of
The
(20)
to
we h a v e
solve
Schrodinger
Schrodinger
to
Eq.(18)
know VfI ( r )
first.
self-consistently.
Equation
equation
H ( r ) ( / f j ( r ) = Ei / f j ( r )
can
of
be w r i t t e n
basis.
lattice
in m atrix
The b a s i s
s u ms
(21)
of
form w ith
functions
atomic
wave
we
an
appropriate
choose
here
are
choice
the
functions:
^ i ( r , k ) = 5 e x p ( - i k* m) a t ( r - m ) ,
m
where
k is
a lattice
is
Th e
the
ith
in
the
site
atomic
subscript
irreducible
within
i
wave
one
layer
both
(22)
Surface
function
indicates
Bloch
of
Brillouin
the
located
the
slab,
at
atomic
Zone,
and
ajXr-m)
lattice
orbital
m is
site
and
m.
the
21
layer
at
which
transition
they
metals
behave
electrons.
be
are
as
are
There
are
19
2s,
functions
for
each
occupied
the
in
the
the
the
2p,
Fe 4 p ,
freedom
and
4s,
set
each
4p,
d-
lattice
atomic
basis
for
nearby,
atomic
the
the
the
3d,
of
of
wa v e
in
this
Fe a t o m
5s,
and 9
in
atomic
the
cell:
Is,
2 p,
3 s,
3p.
and
the
O 3s,
3p o r b it a ls
are
not
Ve a d d
in
orbitals
them t o
describing
surface
of
increase
charge
regions
the
2 s,
can
in
atoms.
[51]
interstitial
extended v i r t u a l
5s
Thus
for
3p,
0 atom
nucleus
corresponding
functions
3s,
d - electrons
the
introduction
atomic
isolated
variational
in
like
Since
hound by
good c a n d i d a t e s
Is,
them
located.
a p ur t e r b a t i o n .
cell:
Amo n g
is
tightly
Therefore,
functions
the
atom
somewhat
considered
case.
the
of
surface
the
atom
r e a r r augment
solids.
Th e
are
•
especially
describe
thus
important,
the
could
charge
not
get
calculation
shows
4d orbitals
is
The
k is
into
t h e m we
could not
i n t o vacuum
properly,
the
correct
work
that
whether
or
not
crucial
atomic
a good quantum
t/tj ( r , k ) = 5 Ci J
without
expansion
eigenfunction
expanded
for
of
the
Bloch
in
the
function.
not
we
'
and
Trial
include
the
Fe
calculation.
Schrodinger
functions
equation
with
a given
is
k
since
number:
r , k)
(23)
22
We p l u g
the
left,
J
(22)
and
and
(23)
perform
into
the
[Hl i ( k ) - E j 0 l i ( k ) ]
(21),
multiply
integration,
C i j =O,
a^tr)
from
obtaining
1=1,N
(24)
where
D l i ( k ) = ] j j e x p ( - i k * M X a j ( r ) I H( r ) I a £ ( r - M ) > ,
m
(25)
0 l i ( k ) = J e x p ( - i k * M) < a j ( r ) I a £ ( r - M ) > .
m
(26)
Symbolically,
the
equation
in m atrix
form
is
HC=EOC.
This
is
a
After
(27)
generalized
we
diagonalize
a n d Ej ( k ) .
We t h e n
high
the
up t o
of
will
explained
of
valence
k points.
charge
running
count
total
meaning
be
diagonalization
number
and
the
of
electron,
later)
thus
for
energy
as
of
S u mmi n g o v e r
density,
until
(2 4)
the
all
the
potential
problem.
every
levels
k,
we . ge t
Ej ( k )
the valence
opposed
slab
occupied
core
times
states,
new p o t e n t i a l .
is
’/'j ( k )
from low
electrons
to
cell
al I
(the
electron,
the
we
We k e e p
self-consistent.
to
number
get
the
23
Gap_s_siap Ex.ean_s_ion o f
In t h e
function
A jom lc
central-field
can
be w r i t t e n
Wave F g n c t l o n _and P o t e n t i a l
approximation,
as
the
atomic
wave
[29]
a ( r ) = Rn , i < r ) Y l f m( 0 <£) ,
(28)
where
Rn , l < r ) = |
(29)
Dn , j K i C a j . r ) ,
2 l+ '/% jl/'+ '/4
8 1 ( a j » r ) = ---- - r - ------------- --------------------- —------ r 1 e x p ( - a , r * ) ,
J
It1 Z4 I l x S x ------x ( 2 1 + l ) ] 1 / »
J
and
gi (a j , r )
[52].
best
is
a ,normalized
Wh a t we w a n t
represents
finite
set
of
wave
Oj
Gaussian
a n d Dj
function
primitive
( j = l , N)
with
which
a given,
N.
Solve
to
get
these
t h e LDA E q u a t i o n
spherically
2)
a
an a t o m ic
The p r o c e d u r e
1)
is
radial
(30)
symmetric
o's
for
a n d D' s i s
as
an atom w ith
potential
follows:
a
V( r ) ,
V ( r ) = - Z / r + ( 4 n / r ) / p ( r ' ) r , a d r ’ + Vx c ( r ) ,
(31)
obtaining
[53].
Choose
the
a ' s for
Oj=OqP^- 1
self-consistent
the
j-l.N
Gaussian
V (r) numerically
basis
set
as
(32)
24
3)
These
are
e v e n —t e m p e r e d G a n s s i a n s
cover
the
whole
Resolve
[54]
2
the
using
Vary
to
5)
atomic
the
Ad d
repeating
best
a's
N+2 d i m e n s i o n a l
to
ma k e
this!
E2 ,
best
2)
and D 's
to
equation
to minimize
matrix
form
set:
(33)
E i ( E i ( ------<En a n d t h e
the
almost
Gaussians.
with
basis
which
i=l,N
t wo m o r e G a u s s i a n s
a n +2
relevant
given Gaussian
and P ,
get
of
LDA e q u a t i o n
( H i j - E S i j ) Dj = O ,
obtaining
4)
spectrum
[52]
corresponding
3)
for
the
of
above
above
basis,
atom,
the
2 s function
to minimize
E^,
(Is).
the
and u s e
D's.
solve
vary
Schmidt
a n+i
the
and
procedure
orthogonal
to
the
Is
function.
6)
Repeat
the
functions
7)
Repeat
for
the
functions
8)
procedure
given
procedure
with
Add v i r t u a l
diffuse
a
Gaussian
rearrangment
Under
Fe
describes
the
ho w we
0),
for
for
when
assumption
and 2 s * 2 p 4 f o r
for
orbitals
of
we
all
occupied
angular
different
ma x i mu m v a r i a t i o n a l
This
for
each
all
obtain
atomic
can
occupied
atomic
orbital)
in
the
to
is
atomic
the
single
charge
constructed.
wave
configurations
determine
(a
provide
describing
slab
wave
momenta.
s and p f u n c ti o n s
atomic
wave
mome n t u m.
angular
freedom
the
atomic
functions.
( 3 d 74 s 1 f o r
atomic
charge
25
distribution,
atomic
potential
the
first
the
second
into
and
one
therefore
c a n be
finite
atomic
separated
contains
is
the
a
into
singularity
everywhere,
so
potential.
The
two p a r t s
[55]:
at
it
the
origin,
c a n be
and
expanded
Gaussians:
V(r)=V1 ( r ) + V ,(r),
(34)
Vi ( r ) = - ( Z / r ) e x p ( ~ r 2 ) ,
(35)
Va ( r ) = ( Z / r ) ( 1 - e x p ( - r * ) ) +VC o u l o m b + V x c .
(36)
Now we a r e
before
doing
n a me i t s
that,
expansion
V '(r)=J
j
where
ready
the
to
expand Va i n t o
we f i r s t
as
G a u s s i ans.
r e na m e Va ( r )
But
a s V( r ) ,
and
V' ( r ):
Di g ( a i , r ) ,
J
J
a ’ s obey
(37)
E q . ( 3 2) ,
that
is,
they
are
even-
tempered G au ssian s.
A least-squares-fitting
the
expansion.
That
is,
we
procedure
try
to
is
find
used
to
optimize
V' ( r ) s u c h
that
/ I V ( r ) - V ' ( r ) ] 2 W( r ) r 2d r = M i n ,
where
<o ( r )
V(r)
and
is
a weighting
V' ( r ) h av e
to
factor,
(38)
a n d we
set
<o( r ) = r .
obey
/ V ( r ) 2 w ( r ) r 2d r = / V ( r ) * w ( r ) r 2d r = C o n s t .
(39)
26
(38)
/V(
with
(39)
is
equivalent
to
r ) V ’ ( r )<i> ( r ) r * d r = Max.
When
(37)
K a,D )=|
is
Dj/
used
to
V (r)g
(40)
expand
V '(r)
in
(40),
( a j ( r ) ) a > ( r ) r ad r = |
we h a v e
Dj Pj = Max,
(41)
where
Pj = /
V ( r ) g ( <i j , r ) o » ( r ) r a d r .
Squaring
(40),
(42)
we h a v e
I 1 ( a , D ) = £ ^ Di Dj P i Pj = Ma x .
i j
(37)
^ ^
i j
is
Di D j /
also
used
to
expand
(43)
(39),
a n d we h a v e
g ( a i , r ) g ( a j , r ) u i ( r ) r ad r
= ^ ^ Di Dj S i j = C o n s t ,
(44)
wher e
S i j = / g ( a i , r ) g ( a j , r ) t » ( r ) r ad r .
Subtracting
tox ( 4 4 )
from
(43)
(45)
((U = C o n s t a n t ) ,
F ( D i , ----- »Dn ) = | J Di Dj ( O i j - CuSi j ) = M a x ,
I j
where
Oi j = P i Pj .
we h a v e
(46)
27
Differentiating
the
results
2
equal
F with
to
0,
respect
we h a v e
(O jj- USjj)Dj=O.
By
solving
(47),
to
the
all
D's,
secular
and
setting
equation
i=l,N
we f i n d
D's
(47)
corresponding
to
ma x i mu m w
and F.
As
in
the
wave
function
fitting,
a to
we v a r y
get
Max(<0m a x > We t h e n r e p e a t
Starting-matrix
The
step
the
procedure
the
whole
time.
calculation
calculation.
Hamiltonian m atrix,
computing
the
expansion
of Vx c .
Ca l c u l a t i o n
starting-m atrix
in
for
the
it
a very
Without
iteration
Furthermore,
is
the
would
could
right
take
be
important
starting
longer
trapped
a t wrong
points.
Since
together,
the
it
solid
is
is
f o r m e d by b r i n g i n g
natural
to
begin with
the
isolated
atomic
atoms
charge
distribution,
Po(r)=J Patom(r-m)•
m
N o t i ce
(48)
that
V xc(r)=V xc(P°(r)'/')
(49)
28
is
a non-linear
density
(48).
(49)
atomic-charge
sum o f
function
potential.
individual
I
with
atomic
the
l a t t i c e s um o f
(48)
is
called
So
(49)
w ill
the
not
the
charge
overlappingequal
the
lattice
potentials
i
' --,Al
Hl i - i
.
of
ii T ;
■>;
(50)
V % c ( r ) = % v I C t P t r - mV1/ * ^
m '
which
is
called
take
the
the
*
for
for
difference
between
and
^
the
that
by
the
potential
potential.
*
as
convenience,
a term
=
the
atomic
consists
:
i,-::
..'J
potential
of
four
we
-
starting
we h a v e
equal
If
to
to
the
overlapping-atomic-charge
matrix
"i ' . :
atomic
. . .
adding
overlapping
starting
.-I'.
atomic
mathematical
compensate
The
-
overlapping
potential
(50)
overlapping
potential
(49).
terms:
.
^ T"'
1)
< » , ( , ) I-IZZVa I a 1 ( r - n ) >
(51)
2)
<ai ( r ) | [ J v(r-m)]
m
exp ( - r y * ) | aj ( r - n ) >
(52)
3)
<a.(r)|[]k v(r-m)]
in
( 1-exp ( - ry * ) ) | aj ( r - n ) >
4)
where
j
< » ! ( r ) Vxc
y is
Al I w a v e
chosen
p ( r-m) ] - ^
as
functions
v xc [ p ( r - m ) ]
J aj
(r-n) >
(53)
(54)
0.15.
are
expanded
into
Kubic-Harmonic-
Gaussians
G( r ) = g ( r ) K ( x , y , z )
(55)
29
where
K is
Spherical
S,
Harmonics,
X,
Th e
a Kubic Harmonic
Y,
Z,
such
[39],
or
XZ,
YZ,
consists
of
t wo p a r t s
- ( Z / r ) e xp ( - r a ) , w h i c h
diverges
at
the
part,
Term I
is
the
overlapping
between
the
which
screened
potential
we
energy,
potential,
between
first
to
term 2
Gaussians
located
at
is
the
screened
the
difference
potentials,
and te r m 4
is
overlapping-atomic-charge
atomic
calculation
examine
and a non­
Gaussians.
term 3 is
overlapping
go i n t o
would l i k e
the
(34,35,36):
origin,
into
and u n s c r e e n e d
and t h e
Before
expanded
kinetic
atomic
difference
is
of
XY.
V(r)
singular
combination
as
3 Z 2 - R a , X1 - Y a ,
potential
linear
how
different
of
potential.
Term I
and 2,
an i n t e g r a l ,
sites
(called
with
a
we
t wo
two-center
integr a l) ,
I = / e x p ( —a r a ) e x p ( —p ( r —n ) 1 ) d 1r ,
can
be
done
This
of
three
analytically.
integral
terms w ith
(56)
the
c a n be
s a me
two-center
l e l Z - V 1 Z-
decomposed
integrand
form,
as
the
that
product
is,
the
of
3
product
integrals:
(57)
30
where
(58)
I x= / e x p ( - a x * ) e x p ( - 0 ( x - x Q ) d x ,
and
similar
It
is
expressions
not
hold
difficult
to
for
show
Iy
and
I z•
that
I x= e x p [ - ( a 0 / ( a + 0 ) ) n x * ] / e x p ( - ( a + p ) ( x - p n x / a + p ) * )dx
(59)
= ( j t / a ) 1 ^ 1 e x p [ —( ap / ( a +P ) ) n X1 .
and
similar
Thus
results
a t w o —c e n t e r
integral,
which
three-center
reduced
The
to
can
for
I y and
integral
be
done
integral,
t w o —c e n t e r
two-
include
linear
hold
in
which
form,
combinations
reduced
closed
is
the
also
The
be
above
integrals
of
the
calculated
conclusion
with
centers
the
second
closed
is
also
part
Similarly,
in
term 2,
one-center
integrals
for
c a n be
form.
are
example,
)d r,
form
valid
of
a
which
(Kubic Harmonics
polynomials),
singular
(60)
[ 5 6 —5 8] .
for
the
three-center
- iii
p o t e n t i a l at one
[ 5 6 - 5 8]
I = < G j ( r ) j e xp ( —( r —m)
The
in
to
Gaussian
I = / x e xp ( - a r * ) ( y - y n ) e xp ( - p ( r - n )
can
a one-center
form.
then
polynomials
of
to
involved
and
and t h r e e - c e n t e r
m ultiplicative
is
I z-
) / I r —mi|G j ( r —n ) > .
derivative
p o l y n o m i a l —m u l t i p l i e d
of G i s
Gaussians:
a linear
(61)
combination
of
31
(62)
Then,
since
(63)
a i < r >=5 Di , m G i , m < r > ’
m
the
kinetic
energy
term
(SI)
may be w r i t t e n
as
(64)
O n c e we
are
in
fact
algebraic
kinetic
which,
the
and
term.
case
is
to
is
[56-58]
is
used
contributing
involved
in
the
Gaussi an s
true
that
to
a
[5 9 ] .
in
the
order
need
a very
wave
potential
to
wave
either.
which
only
calculate
is
needed,
function,
fine
the
would
m e s h w o u l d be
functions.
energy
sum o v e r
potential.
case
to
integration
localized
because
for
[ 5 6 - 5 8] ,
three-center-integral
the
atomic
needed
s u c h wave
a n d we
this
tabulated
of v e ry
describe
s a me
< G j ( r ) | g ( r - n ) P j ( r ~ n )>,
No n u m e r i c a l
time-consuming
except
expand
integrals
manipulation
in
required
(52)
the
worked out
energy
be v e r y
The
know
tabulation
all
No n u m e r i c a l
That
function
is
calculation
the
and atom ic
atomic
sites
inlegation
is
r e a s o n wh y we
potential
into
32
In
the
screened
calculation
the
exponential
contribute
the
effect
potential
term
(52),
by m u l t i p l y i n g
exp ( - y r 2 ) ,
overlapping
is
screened
the
potential
factor
to
screening
between
atomic
of
so
that
fewer
atomic
artif itial.
and u n s c r e e n e d
it
we
an
atomic
potential.
sites
But
To i n c l u d e
the
potentials
we u s e
this
difference
term 3
(53) .
Term 3
in
order
is
to
c o m p u t e d by F o u r i e r
Transformation.
That
calculate
(65)
S V j j ( n ) = <a j ( r ) | S V ( r ) | a j ( r - n ) > ,
we f i r s t
is,
calculate
66)
8 V (Q ) = 1 / V c e l l x / 6 V ( r ) e x p (-iCT r ) d * r
cell
(
S j j ( n , Q) = < a j ( r ) e x p ( - i d * r ) a j ( r - n ) > .
(67)
and
Then,
by
s ummi ng
over Q space,
we
get
SVi j ( n )
6 V ( Q ) x S J j ( n , Q) .
S j j ( n , Q)
can
be
also
integrated
Since
SV( Q)
is
8 V(r )
will
is
vanish
in
a
at
(68)
a kind
closed
of
two-center
form
integral,
and
it
[58].
slowly
varying
large
Q ( large
function
in
real
Q means r a p i d
space,
33
oscillation
in
the
cancellation).
makes
only
it
integrand,
It
is
this
possible
to
expand
a modest
number
of
leading
slowly
to
phase
varying
6V(r )
into
property
plane
which
waves w ith
Q's :
S V ( Q ) = I Z V c e J j x / [ J 5 v ( r - m ) ] e xp ( - iQ* r ) d * r
c e l l m=«°.
= 1 / Vee i j x / [ £ 8 v ( r - m ) ] e xp ( - iQ* r ) d * r
os m= c e 11
= 1 / V c e j j x [ ^ e x p ( - iQ * m) ] x / 5 v ( r - m ) e xp ( - iQ* ( r - m ) ) d * r
m= c e l l
®
= I Z Vce j j x e x p ( - iQ* m ) ] x / S v ( r ) e x p ( - i Q * r ) d * r
m= c e l l
os
Now / 5 v ( r ) e xp ( - iQ* r ) d * r
m,
and
can
be r e d u c e d
to
is
independent
of
a one-dimensional
lattice
(69)
site
numerical
integr ation:
/8v(r)exp(-iQ* r)d*r
CO
00
’
= ( 2Z Q ) / Vft^ o m . c ( r ) ( I - e x p ( - y r * ) s i n ( Q r ) r d r .
o
We a l s o
term 4
apply
(5 4 ) .
dimensional
In
the
this
numerical
Fourier
case,
transformation
we h a v e
integration
to
to
(70)
technique
perform
a three-
calculate
SV'(Q)=IZVcel j x / S V ' ( r ) e x p (-iQ* r ) d * r .
cell
Fortunately,
SV '(r),
the
overlapping-atomic-charge
difference
potential
to
(71)
between
(49)
and
the
the
34
overlapping
Al t h o u g h
atomic
near
potential
the
have
potential
nucleus,
their
(50),
is
also
both atom ic
slowly
charge
varying.
and a t o m i c
maxi mum v a l u e
and change most
atomic
overlaps
v.
rapidly,
very
-I
little
charge
m a k e s & V1 ( r ) s m a l l .
In f a c t ,
contribution
interstitial
slowly.
in
As t h e
only need to
in
real
the
result
of
include
space
and
5 V' ( r ) h a s
this
region
slowly
modest numbers
reciprocal
H e r e we w o u l d l i k e
its
where
out
that
it
varies
nature,
of mesh
for
which'
largest
varying
space
to p o in t
there,
we
points,
the
both
integrations.
in our m a trix
element c a lc u la tio n ,
?'
: ' -I
'
approxim ation, as is
calculate
the
i,j,n
as
order
IO + 2 ,
the
we d i d n o t u s e t h e n e a r e s t - n e i g h b o r
'
:
.
.
o f t e n u s e d b y o t h e r w o r k e r s . We
. -. .. r.
m a t r i x e l e m e n t < a j ( r ) | H( r ) ( a j ( r - n ) > f o r a n y
small
atoms
nearest
as 1 0” * (The
for
are.
largest
comparison),
In
neighbors
fact,
we
( | n | =27
no m a t t e r
calculate
a.u.)
atomic wavefunction s lo c ated
V
'
Finally,
number
of
. . .
we w a n t
points
simplicity,
in
we o n l y
generalization
A periodic
expanded
.
into
to
: '
to
at
the
element
how
space
the
the
most
same
in r e a l
a Fourier
series
diffuse
"
relation
. t
''
between
[6 0 ] .
case
is
5s
plane.
■ r .
one-dimensional
three-dimensional
function
of
apart
lattice
and Q space
the
far
is
up t o 1 5 th
' V
discuss
consider
the
for
.
real
matrix
V »
the
For
case.
The
obvious.
s p a c e w i t h p e r i o d X c a n be
as
35
f(x)=2
(72)
F (n)exp( - i2nnx/X),
wh e r e
X
F(n)=(l/X)/f(x)exp(i2nnx/X)dx,
o
for
f (x ) given
f(x)
is
given
in n um erical
from an
at
continuous
only
at
discrete
calculation
integral
to
x.
by
(73)
If,
on t h e
x values,
computer,
other
as
is
hand,
always
F(n) will
be
changed
a summation:
N
F(n)=(l/N)X f ( ( j-l/2 )A x )e x p ( i2 n (j-l/2)n/N )
j=l
where
X4=O . 5
It
f (x )
is
sampled
Ax ,
as
incur
is
easy
to
at
is,
number
there
of
show
in r e c i p r o c a l
is
limited,
points
in
only
in
space.
the
spaced p o i n t s ,
and
integration.
*
N F's
real
If
we c a n n o t
(74)
that
for
are
points
N equally
numerical
F ( n +N) = - F ( n ) ,
that
true
which
space
the
are
equals
space.
distinct.
the
information
gain more
reciprocal
(75)
n=l , N
insight
number
in
by
the
Thus
of
real
using
the
points
space
more
36
S x mme t r j r
H e r e we w o u l d
of
the
system.
like
Without
w o u l d be mu c h m o r e
First
of
potential
is
any
only
to
there
the
a
symmetry,
tran slaton
that
one
is,
properties
the
calculation
unit
in
symmetry
of
the
V(r+R)=V(r ) , where R
vector
behaves
function
to
a
in
cell.
exactly
the
plane.
Every
the
different
good q ua nt u m
represent
is
k dependent.
in
k.
Thus,
we
physically
same
cells,
of
k vectors,
system w ith
infinite
a few
of
them (6
wedge
of
the
whole
first
surface
are
necessary
45 k points
on which
for
introduction
special
surface
in
the
we
other
of
t h e wave
number,
in
any
other
however,
obeys
in
K points
Brillouin
to
I BZ
can
are
[61]
zone
to
is
is
in
the
the
energy
other
effective
hand,
only
to
in t h e
our
self-consistency,
interpolate
the
36
according
the
k makes
infinite
irreducible
or
for
On t h e
then
block-diagonal
needed
k's.
basis
Fortunately,
(IBZ) ,
zone),
iterate
Th e
principal,
extent!on.
Brillouin
the 2-d im en tional
function.
The H a m i l t o n i a n m a t r i x
The n u m b e r
function
symmetry
c o n d i t i o n : vP k ( r + R ) = e xp ( - i k* R ) vVkX r ) • T h u s we
k vector,
based
is
XY p l a n e ,
introduce
the
the
time-consuming.
quantity
Bloch
tests,
using
consider
The wave
need t o
for
discuss
lattice-displacem ent
observable
the
all,
in the
need
cell.
to
last
and
iteration,
and wave
the
Hamiltonion
[62]
37
lose
its
makes
symmetry
the
include
existence
all
layers
need atomic
above.
both
three-dimensional
charge
density
in
large
the
slabs.
and
is
no
or
s o we o n l y
need
point
F.C.C.
The
They
for
every
our
need
group
The
no
to
the
or B . C . C .
C4y g r o u p
has
order
space
one
to
to
mentioned
with
perform
the
calculate
we m u s t r e p e a t
slabs
between
at
is
the
so
nearest
all
between
a k vector
is
has
slab
to
interaction
there
cell
as
between
a. u .
in
the Z
no E ( k z ) d i s p e r s i o n ,
k 2= 0 .
H am ilton!an of
crystalline
8
is
symmetry,
r e a s o n wh y we
layer,
introduce
consider
of
the
potential,
equivalently,
to
is
model
in
unit
transformation
therefore
is
Th e
That
However,
there
and t h i s
translation
slab.
compared w ith -5
that
direction,
no
surface.
for
Fourier
a. u.
There
The
now
Z direction.
(-100
neighbors)
the
sides.
is
the
function
Wh a t we h a v e
vacuum a t
for
of
k point,
difficult.
there
of
basis
a general
mo r e
the Z direction,
to
slab
for
diagonalization
In
due
C4y
symmetry
the
structures
operations
(100)
is
surface
C4 y XR=D4 h .
and 5 c l a s s e s .
are:
E
Ca
C4
Cl
*h
<t v
«d
a 'i
X
X
-X
Y
-Y
X
-X
Y
-Y
Y
Y
-Y
-X
X
-Y
Y
X
-X
(76)
38
Th e R g r o u p h a s
Because
would
be
in
z
Z
-Z
of
operation
physical
first
to
place
which
it
the
is
any
is
in
I BZ w i t h o u t
the
only
indeed
things
and 4
for
f
Vftff
is
Ca y ,
X
X
-X
X
-X
Y
Y
-Y
-Y
Y
A)
or
within
ffV
( for
the
E
ah
X
X
X
Y
Y
-Y
<jd
C4y
the
the
XY
a 90-degree
convention.
[ 6 2] ,
the
a n d M.
which
has
4
av
symmetry
Y) or
in Ve f f
BZ b u t
changing
are
I BZ
are :
<Th
]X a n d Y,
the
the
H= E^ + V( r ) + K* p
at
They
in
changed
only
classes.
in
s o me k i n d o f
arbitrary
Hamiltonian
retained
symmetry
are
involved
is
perform
Ca
A,
k's
k which
E
At
all
k's
always
The
effective
At X t h e
opera tio n s
For
V ( r ),
a n d r i g h t —l e f t —h a n d - c o o r d i n a t e
symmetry
(for
can
of
represent
BZ.
They
(77)
symmetry
to
situation.
rotation,
C4y
C4 y
I BZ we
coordinate
For
the
sufficient
the
symmetry o p e r a t i o n s .
E
in t h e w h o le
not
2
( for
(78)
is
^
reduced
):
(79)
39
At
E
CTv
X
X
-X
Y
Y
Y
E
ad
X
X
Y
Y
Y
X
a
general
o p e r a t i o n which
invariant.
So
symmetry,
which
w ill
the
sides
slab.
and odd
The
a factor
atomic
thus
to
of
s h o wn
I.
the
is
there
effective
left
ma k e
even
functions
no
is
the
no p o i n t
group
Hamiltonian
the Z r e f l e c t i o n
and odd l i n e a r
located
coupling
reducing
is
at
between
dimension
opposite
these
of
the
2.
orbitals
representation
in Table
ns
There
of
the
symmetry
a t o m i c wave
functions,
m a t r i x by
however,
leave
only
leads
of
the
(81)
k point,
combinations
of
(80)
belonging
group
of
the
to
the
different
wave v e c t o r
[62]
is
even
40
Table
I
A t o m i c Wa v e
Point
Group
Function
A t o m i c Wave
S i t e ( O 1O)
and P o i n t
Group R e p r e s e n t a t i o n
Function
Site(c/2,c/2)
Ti
S1 Z 1SZ2 - R 2
S, Z 1 S Z 2 - R a
r*
NULL
NOLL
r.
X 2- Y 2
X2 - Y 2
[4
XY
XY
Ti
( X 1 Y)
( X Z 1 YZ)
( X 1 Y)
( XZ 1 YZ)
M1
S1 Z 1S Z 1 - R 2
XY
M1
NULL
X1 - Y 2
M1
X 2- Y 2
NULL
M4
XY
S 1 Z 1SZ2- R 2
M1
( X1 Y)
( X Z 1 YZ)
( X1 Y)
( X Z 1 YZ)
X1
S 1 Z 1X2 - Y 2 1S Z 2 - R 2
X1 XZ
X1
XY
Y1 YZ
X,
X 1 XZ
S1 Z 1 X2- Y 2 1S Z 2 - R 2
X4
Y1 YZ
XY
Ai
S 1 X1 Z 1 XZ1 X1- Y 2 , S Z 2 -] 2
S 1 X1 Z 1 XZ1 X2- Y 2 1S Z 2 - R 2
A1
Y1 XY1 YZ
Y1 XY1 YZ
Y1
S 1 Y1 YZ1 X1- Y 2 1S Z 2 - R 2
X1 XY1 XZ
Y1
X1 XY1 XZ
S1 Y1 YZ1 X2 - Y 2 1S Z 2 - R 2
h
S 1 X+Y, Z 1 XY1 XZ + YZ, SZ - R 2
S 1 X+Y, Z 1 XY1 XZ + YZ, S Z 2 - R 2
X - Y 1 X Z - Y Z 1 X1- Y 2
X - Y 1 X Z - Y Z 1X2 - Y 2
41
In
the
the
irreducible
knowledge
identify
of
the
between
the
band
compared
These
directly
structure
the
I
in
be
any
interpolated
energy
bands
of
experimental
the
point
us
,
to
energy
level.
s a me
symmetry
can
data
then
be
obtained
from
s p e c t r e s copy.
system
group
star ting-matrix-element
A X Y M ^
interpolated
the
the
f
helps
having
with
the
can
BZ,
eigenvalues
The H a m i l t o n i a n
of
first
involved
angle-resolve d-photo emission
operations
the
and Table
orbitals
calculated
properties.
of
[ 7 6 ] , [ 7 7 —8 1 ]
atomic
Furthermore,
wedge
is
D4 y .
invariant
This
fact
calculation.
under
is
all
applied
We o n l y
need
in
to
calculate
j ( n ) = < a £ ( r ) I H( r ) I a j ( r - n )>
for
n in
1/8
of
symmetry
of
V(r),
parts
the
of
the
plane.
we
plane.
can
T h e n by
obtain
(82)
applying
H|j(n)
Furthermore,
for
s ome
n - m ( H e r e we
origin
of
from
the
to
atom
the
the
in
coordinate
the
corresponding
mz = - m z a n d
symmetry
of
central
matrix
( n ' - m ' ) „ = ( U- I n ) ll
H(r).
The
we
for
can
all
other
changed
is
a.
is
easily
the
located
obtain
( n m ' ) z= - ( n - m ) z w i t h
by a p p l y i n g
procedure
n in
atom w h er e
layer),
element
C4y
o n c e we know
< a ^ ( r - m ) H( r ) a j ( r - n ) > f o r
the
the
the
the
Z-reflection
following:
Define
Hi J ( n ) = < a i ( r ) | H( r ) | a j ( r - n ) > .
(83 )
42
Performing
a C4y
operation
on r ' s ,
we h a v e
Hi j ( a n ) = < a j ( a r ) j H ( a r ) | a j ( a ( r - n ) ) > .
By C4 y
symmetry
of
the
(84)
Hamiltonian,
we h a v e
H(or)=H(r).
a^(a(r-n))
(85)
can
be
expanded
as
a i ( o ( r - n ) ) = J Dp j ( a ) a p ( r - n ) .
P
(86)
So
H.j ( a n )
D p i i p ( r ) | H ( r ) | 5 Dq j a q ( r - n ) >
q.
p
- I I D; i Dq j Hp q ( - >
p q
- I 5 O i p l + Hp q U )
Dj .
(87)
p q
In m a trix
form,
Hf = D+ HD.
Thus,
( 8 8)
o n c e we
wave f u n c t i o n
(86),
transformation
or
reflecting
element,
the
know
the
we
can e a s i l y
properties
the
lattice
calculated
transformation
obtain
of m a t r i x
site
results
of
the
the
elements.
n involved
for
properties
in
lattice
By r o t a t i n g
the
site
matrix
n in
the
43
1/8
of
plane
the
XY p l a n e
with
only
An e x a m p l e
The
easily
of
the
of
this
worked out
Table
operations,
D's
be
generalized
s o me a l g e b r a i c
coefficients
functions.
can
the
for
all
by a p p l y i n g
2
is
C4 y
(76)
given
to
all
D's
for
C4 a n d
D matrix
can
be
obtained
known.
whole
in
elements
gives
already
the
manipulation.
transformation
D(a)
into
a c a n be
atomic
av.
F i g . 3.
For
wa v e
all
other
by m u l t i p l i c a t i o n
Thus,
Ca=CjxC4 ,
(89)
Cjs CjxCjxCj,
(90)
E= C 2 x C 2 ,
(91)
Oh= C 2 X a y ,
(92)
a d= C 4 x a h ,
(93)
0 A=Caxad .
(94)
A similar
symmetry
argument
H(-z)=H(z)
to
allows
get
one
to
<a ^ ( r - m ) | H( r ) | a j ( r - n ) > w i t h
m' z= - m z
and
( n m ' ) „ ■( n - m ) „ .
of
this
reflection
<a . ( r - m ’ ) | H( r ) j a j ( r - n ' ) > o n c e
we k n o w
An e x a m p l e
apply
D(<rz )
( n ' - m ’ ) z = - ( n - m) z ,
is
transformation
s hown
is
in Table
given
in
3.
F i g . 4.
44
Table
2.
Representation
matrices
D j j ( a ) of
C4 v
D(C4 ) =
S
X
Y
Z
Z1
S
I
0
0
0
0
X
O
0
0
I
O
I
0
Z
O
0
Z1
O
Xa - Y 1 O
X1- Y 1
XZ
YZ
XY
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
I
0
0
0
0
0
0
0
0
I
0
0
0
0
0
0
0
0
0
0
0
-I
-I
XZ
O
0
0
0
0
0
0
YZ
O
0
0
0
0
0
I
0
0
XY
O
0
0
0
0
0
0
0
I
S
X
Y
Z
Z1
XZ
YZ
XY
S
I
0
0
0
0
0
0
0
0
X
0
I
0
0
0
0
0
0
0
Y
0
0
0
0
0
0
0
0
Z
0
0
0
I
0
0
0
0
0
Z1
0
0
0
0
I
0
0
0
0
X 1- Y 1 0
0
0
0
0
I
0
0
0
0
0
-I
0
D ((Ty.) =
-I
X2- Y 1
XZ
0
0
0
0
0
0
I
YZ
0
0
0
0
0
0
0
XY
0
0
0
0
0
0
0
-I
0
0
-I
45
Table
3 • Representation
matrix
Di j ( * z )
S
X
Y
Z
Z1
S
I
0
0
0
0
X
O
I
0
0
Y
O
0
I
Z
O
0
0
Z1
O
0
0
X1 - Y a O
0
Xa - Y 1
XZ
YZ
XY
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-I
0
0
0
0
0
0
I
0
0
0
0
0
0
0
I
0
0
0
0
0
XZ
O
0
0
0
0
0
YZ
O
0
0
0
0
0
0
XY
O
0
0
0
0
0
0
-I
-I
0
0
I
46
'
'
>
/
I
I
F i g . 3 A p p l i c a t i o n o f C4 v s y m m e t r y o f t h e c r y s t a l
potential
in the m a t r i x - e l e m e n t c a l c u l a t i o n .
<s(r)IH(r)|p (r-n)> =<s(r )l H( r)l px(r-n')>,
wn e r e n= ( 0 , a ) a n d n ' = ( a , 0 ) . D a s h e d l i n e s
r e p r e s e n t a C4 v " s y m m e t r i c p o t e n t i a l .
\
F i g . 4 A p p l i c a t i o n o f Z —r e f l e c t i o n s y m m e t r y o f t i e c r y ­
s t a l p o t e n t i a l in the m a t r i x - element- c a l c u l a t i o n .
<s ( r - m ) l H ( r ) I P 2 ( r - n ) > = - < s ( r - m' ) I H ( r ) I p z ( r- n ' ) > ,
w h e r e mz= a / 2 , n z = a , mz = - a / 2 , a n d n z = ~ a . Da s h e d
l i n e s r e p r e s e n t a n R- s y m m e t r i c p o t e n t i a l .
48
There
useful.
or
the
is
one
mo r e
For
t wo
atomic
opposite
s y mme t r y
layer
in
real
space
o r b ita ls located
from
the
center
which
in
of
the
the
is
s a me
layer
slab,
we h a v e
H j j ( n ) = < a j ( r ) j H( r ) | a j ( r - n ) >
= <a | ( r + n ) | H ( r + n ) Jaj. ( r ) >
= < a j ( r ) | H( r ) I a j ( r + n ) > *
= Hi j ( - „ ) *
provided
case
mentioned
Th e
of
H( r + n ) = H( r )
is
true.
This
is
indeed v a l i d
in
of
lattice
element.
That
site
n might
change
the
sign
is
H i j ( - n ) = + Hi j ( n ) .
Notice
I = C1 XOz .
sign w i l l
The
That
So b y
change
complex
(96)
or
(87)
we
can
determine
whether
the
not.
conjugate
might
also
lead
to
a
sign
change.
is,
Hij(n)*=±Hi j (n).
In
the
above.
inversion
the m a t r i x
(95)
fact,
we
(97)
changed
the
multiplying
each
t h e m by
Hi j ( k ) r e a l
and
of
basis
symmetric.
i
orbitals
t o ma k e
Depending
P x »Py*d x z , dyZ hy
the
on
lattice
the
n u mb e r
sum
of
49
imaginary
element
basis
will
orbitals
(will
complex-conjugate
signs,
not)
it.
need to
atomic
wave
plane s
situated
go
functions
(I
its
obtained
calculate
on
to
Combining
Hj £ ( n ) may be
we o n l y
involved
or
2),
negative
the
the
matrix
valne
w h e n we
t wo p o s s i b l e
from
(n ) .
minus
In o ther
the
matrix
element with
located
on th e
s a me p l a n e
the
opposite
sides
of
the
words,
i>j
or
for
on
central
plane.
When t h e
lattice
sum o f
the
Hamiltonian
H . j ( k ) = ^ e x p ( - i k * m) < a 4 ( r ) | H( r ) | ' a j ( r - m ) >
m
=5 [co s(k * m )- i s i n(k*m)]
m
is
performed,
sin
term
done
if
planar
above
the
involved
basis
by
sin
t wo
it
summation
s hown t h a t
the
H ^j ( m)
px,
is
py ,
is
H^j ( m)
easy
be
over
real,
m.
functions
sin
term,
summation over
to
and
be
show
that
while
m since
due
This
the
other
is
are
to
the
can be
under
the
each
function
imaginary,
involved
which
dy z
imaginary
will
are
E ^ j(k)
odd symmetry
dx z ,
On t h e
(98)
imaginary.
with
Then when o ne
term w i l l
imaginary
after
Ca ,
i.
imaginary
functions
in H .j(m ),
furthermore,
after
unless
inversion
multiplied
the
we w o u l d h a v e
H . . ( m)
J
is
and
H j j ( - m) = - H j j ( m) .
cos
term w i l l
hand,
in
imaginary,
H ^ ( - m ) = H i j ( m)
vanish
when none
H . j (m),
it
will
Thus
or
c a n be
vanish
in t h i s
case.
50
As
the
true
result,
for
the
to
symmetry
a D4J1 s y m m e t r y
space
sum
(k )
is
real.
The
s ame
there
is
is
OjjCk).
In a d d i t i o n
also
lattice
of
Fourier
in r e a l
in Q space,
that
space,
is,
the
reciprocal
transformation.
V(Q)=IZVc e l l x / V ( r ) e x p ( - i Q ' r ) d ' r .
Performing
a
symmetry
(99)
o p e r a t i o n R of
D4J1 on V ( Q ) ,
we
have
V(RQ)=I/Vc e l j x/ V ( r ) e x p ( - i R Q ' r ) d ' r
(100)
Noticing
RQ 1 T=R- 1 , RQ • R~ 1 r = Q* R- 1 r ,
(101)
we h a v e
V(RQ)=l/Vc e l l x /V (r)e x p (-iQ * R " 1r ) d , r
Due
to
the
D4J1 s y m m e t r y
in r e a l
(102)
space:
V( R- 1 r ) = V ( r ) ,
(103)
we h a v e
V ( R Q ) = I / V c e l l x / V ( r ‘ ) e x p ( - i Q * r ' ) d , r ' = V( Q)
w h e r e T 1=R- l T.
(104)
51
T h u s we n o t
also
only
in Q space.
t h e Q Xy
plane
have
Only
and w i t h
Self-consistent
D4^
those
symmetry
in r e a l
V ( Q ) ' s w i t h QXy
Qz >0 a r e
needed
in
space,
*n ^ e
the
but
1/8
of
calculation.
Iteration
D e f ine
V(r)=F(r)+Vxc(r)
where
(105)
F ( r ) = / p ( r ' ) / I r - r ' i d*r
potential
Also
between
is
electrons,
the
and
it
electrostatic
is
non-local.
define
6V(r)=V(r)-V0 (r) ,
(106)
8F (r)= F (r)-F 0( r ) ,
(107)
8 V x C ( r ) = V x c ( r ) - V x Co ( r ) ,
(108)
6p(r)=p(r)-p0(r),
(109)
w h e r e V0 ( r )
charge
total
potential,
density.
The
( F 0 ( r ) , Vxc o ( r ) )
(electrostatic,
and
They
Fourier
p0 ( r )
do n o t
is
the
change
transforms
of
is
the
overlapping-atomic-
exchange-correlation)
overlapping
during
the
8 p ( r ) and
atomic
charge
iteration.
8 Vx c ( r ) a r e :
8 p ( Q) =1 / Vce j j x / 6 p ( r ) e xp ( - iQ* * ) d * r ,
8 Vxc ( Q ) = I / V c e l J x Z S V xc ( r ) e x p ( - i Q * r ) d , r .
(HO)
(Ill)
52
Al t h o u g h
functions
atomic
is
over
nearby
the
that
in
can f r e e z e
Is,
Is-Iike
as
the
corresponding
is
called
necessary,
treat
the
large
number
using
now,
space,
so
potential
that
other
words,
the
other
from
at
it
their
is
in
rearrangement
up
as
the
derived
break
to
region).
from
values,
as w e l l
potential.
It
nor
even
is
as
to
unless
ma n y
a very
a s we a r e
and r e c i p r o c a l
That
in
the
vary
rapidly.
systems.
core
region,
altered
is,
is
the r e g i o n where
homogeneous
are
This
not
possible,
density
and s u r f a c e
the
core
atomic
down i n
a n d Vx c ( r )
due
are
t h e LDA f o r m u l a t i o n
for
of
and oxygen
10*
charge
the
states
space
applicable
atomic values.
occurs
to
real
unless
region
p(r)
above,
by
contributed
approximation.
t h e LDA w i l l
free
the
the
potential
existence
s e l f - c o n s i s t e n t Iy
both
t h e LDA i s
both
free
Near
the
BVx c ( r )
(called
density
their
and
the
varying
are.
affected
and
Sp-Iike
points,
as w e l l
interstitial
hand,
6p(r)
demonstrated
used
dense
slowly
exchange-correlation
me s h
that
Recall
In
3s,
and f u r t h e r m o r e ,
improved
the
are
BVx c ( r )
feel
region
electrons
of
and
is
hardly
f r o z e n —c o r e
core
quite
strongly
charge
2p,
states
a s we
so
words,
the
atomic
the
are
this
2s,
5p(r)
not
charge
they
In o t h e r
zero
atomic
the
electrons
atoms.
T h u s we
space,
where
nucleus
practically
iron
all
nucleus
deep,
other
p ( r ) a n d Vxc ( r ) a r e
on
region.
the
appreciably
charge
formation
of
the
solid
In
53
from
i s o l a t e d , atoms.
difference
This
is
between
also
true
transformations
number
of
Th i s ; c h a r g e
p(r)
for
the
p0(r)i
5 V xc' ( r )
(HO)
and
real-space
Although
and
rearrangment
in
(111),
varies
(109).
s l ow Iy
Thus
we o n l y
in
need
and r e c i p r o c a l - s p a c e
electrostatic
5p(r),
in-space.
Fourier
a modest
f
non-local
\
in r e a l
potential
*
»■
(112)
-'
space,
the
. ''
!
Fourier
\
transform
of
F(Q)=p(Q)/(4nQ*)
is
local
we
know
both
numerator
lim iting
Qy=O. ; a n d Qz = I O
On c e
(105).
now
it
except
is
easy
for
the
denominator
procedure
Q1 = O , Q = O
and
Gp( Q) ,
and
is
used.
to
are
to
the
we
know
We g i v e
of
GV(Q)*+i ,
t ^i e
GF( Q)
and
GV( Q) t a
as
potential
zero.
of
In
this
Q=O,
is
where
case,
we u s e
next-sm allest
subscript
GV( Q) JJu t ,
the
nth
*n Put
GVx c ( Q) ,
to
We o b t a i n
a
Qx =O,
Q vector:
and
indicate
iteration.
potential
superscript,
this
Th e n we
for
the
GV(Q) b y
is
the
a
convergence
factor
next
ranging
so
it
is
output
construct
iteration:
G V ( Q ) n^ nI = G V( Q) * n + o ( 6 V( Q J u u t - G V ( Q ) J n ) ,
a
GF( Q)
Q=O c o m p o n e n t ,
Instead
(compared
calculate
and Qz=O.I).
written
where
it
(113)
in Q space,
once
-
points.
F ( r ) = / p ( r ') / i r - r ' i d 1r
is
the
from
(114)
0.01
t o 0.1
54
(instead
of I ,
w o u l d be
the
in
which
input
of
over co m p e n s a tio n
5 V(Q) i n ,
iteration,
After
to
the
is
by
the
the
input
zero
case
the
next
output
iteration)
iterating
potential
by
from
to
an
iteration
prevent
correction.
difference
for
the
first
definition.
calculation
in Q space,
we s u m o v e r a l l Q p o i n t s
get
SHi j (n) = ^
(115)
a ( 5 V ^ u t ( Q ) - 6 V ^ n ( Q) I x S i j ( Ot Q) ,
where
(116)
S i j ( n , Q ) = < a i ( r ) [ e x p ( - iQ* r ) | a j ( r - n ) > ,
w h i c h was
the
calculated
starting
matrix
T h e n we u s e
Hamiltonian,
and
s t o r e d w h e n we w e r e
calculating
element.
this
correction
to
construct
a new
obtaining
(117)
H i j ( n ) n + l ==Hi j <n >n + 8 H i j ( ° ) n This
is
equivalent
to
the
convergence
technique
(114)
in
space.
Ve
then
begin
S H i j ( O) Za
for
all
criterion
<
another
until
(118)
0 . 1 eV
i , J = ( F e ) S d , 3d
for
iteration
and
convergence.
(0)2p,2p
This
orbitals.
on-site
This
is
the
matrix-element
Q
55
correction
is
Sd-Iike .and
a kind
0 2p - like
uncertainty,
is
experimental
In
the
diverge.
average
levels.
about
the
energy
0 . 1 eV,
s a me
correction
the
order
for
theoretical
as
the
corresponding
calculation,
S p ( Q)
That
has
to
we m e n t i o n e d
vanish,
that
otherwise
for
SF( Q)
t h e Q=O
would
is,
-
''
,
S p ( Q=O) = 1 / Vc e j j x / ( p ( r ) - p o ( r ) ) d r = 0 ,
or
the
total
charge
to
the
total
overlapping
to
the
finite
equal.
the
of v a l e n c e
In
judging
fact,
the
total
both
size
0. 4 5
a.u.
e x p l a i n how
the
to
be
equal
Unfortunately,
are
their
ideal
value:
unit
cell.
deviates
from
I b y a s mu c h
the
is
real
the
The
criterion
space
due
never
per
deviation
of
to
integrals
them
electrons
0.02
has
charge.
two
of
never
(119)
iteration
these
the mesh s iz e
constant
mesh.
In
as
for
the
u sed i s one t w e l f t h
of th e
to
integrated
guarantee
proper
charge.
We now
done.
this
proper
calculation,
lattice
factor
the
atomic
size,
So we n o r m a l i z e
number
0.02.
during
mesh
normalization
the
Fe
error.
S p ( Q)
component,
of
At
the
start
of
the
spin-polarized
iteration
system
is
paramagnetic,
dependence
of
the
continue
to
electronic
iterate
as
we
that
is,
procedure,
there
structure
described
iteration
of
above
is
the
is
we a s s u m e
no
spin-
system.
until
close
We
to
56
convergence
under
paramagnetism.
For
the
artificial
T h e n we
majority-spin
restriction
introduce
energy
of
a perturbation:
levels,
let
E f = E-SE.
For
(120)
minority-spin
energy
levels,
let
E 4 = E+ 6 E ,
(121)
w h e r e SE i s
of
the
final
introduce
symmetry
levels
extra
self-consistent
of
the
extra
distribution.
Hamiltonian
for
diagonalize
the H am iltonian
separately.
This
yet.
So we
reached,
spin
keep on
that
is,
spins.
states
we
From
for
splitting
iterating
until
to
(118)
to
creating
paramagnetic
while
form
get
energy
others
the
are
minority-
a correction
next
case)
iteration
to
on,
S p i n t and spin^
is
until
is
the
lowered
the
the
T h u s we
to break the
distribution,
empty
estimated value
Since
some a r e
As a r e s u l t ,
both
the
splitting.
state.
(compared
the m a jo r it y - s p i n
creating
is
artificially,
shifted,
states
and
spin
paramagnetic
rigidly
occupied
raised
number
ferromagnetism
are
to form
spin
a positive
not
self-consistent
self-consistency
valid
for
each
is
spin.
the
we
57
Lj)w_d_i_n R_e^_r_e_s_e
In
have
order
to
to
interpret
go b a c k t o
orthonormality
quantum
on
the
examine
condition,
mechanics,
is
calculated
our
basis
(22).
though v a l i d
not
valid
eigenfunctions,
we
The
i n ma n y
cases
in
here:
<F . ( r , k ) | F j ( r , k ) >
—^ e xp ( - i k* m)
m
<a j ( r ) J a j ( r - m ) >
(
because
the
the
off-site
orthonormality
atomic
wave
functions
do n o t
122)
obey
condition
(123)
u n l e s s t h e r e i s a s y m m e tr y r e a s o n or t h e
far
apart
wave
that
essential
overlap,
be
is
no o v e r l a p
between th e ir
so
atomic
functions.
The o v e r l a p
the
there
two a to m s are
role
there
of
the
atomic
in
solid
w o u l d be
wavefunctions
state
no
physics.
interaction
s o l i d w ould be a f r e e a t o m i c
exactly
the
orthonormality
same
is
as
for
intrinsic
there
between
an
were
atoms.
no
Then
gas. All p r o p e r t i e s would
isolated
in
If
plays
solid
atoms.
state
Non­
physics.
58
But,
with
difficult
namely,
to
But
C's in
the
(23).
probability
Bloch
the
non-orthonormal
interpret
the
occupation
atomic
this
orbital
meaning
of
components
The
for
in
square
the
set,
of
it
is
an e i g e n f u n c t i o n ,
o f C i w o u l d be
a given
eigenstate
of
not
clear
in
the
the
the
orthonorm al-basis-set
C s is
the
basis
ith
case.
general
case.
Mulliken
[63]
norm alized to
proposed
unity
that
the
C' s
s h o u l d be
by
(124)
i
J
and t h a t
Ci
be
I
interpreted
atomic
Bloch
During
Cs,
(125)
° i j cJ = p I
as
orbital
the
occupation
for
a
iteration,
b u t we do u s e
Another
the
we
are
the
usual
occupation
orthonormal,
the
this
old
basis
way,
the
do n o t
actually
basis
to
to
the
C s to
proposed
functions
the
In
ith
new
are
unity.
[64],
and then to
the
of
the
basis
linear
basis
the
by Lo w d i n
squares
making
taking
form
the
interpret
set orthonorm al,
probabilities.
functions
need
norm alize
interpretation:
we a r e
of
eigenstate.
technique,
i s t o f i r s t make t h e b a s i s
follow
given
(12 4) t o
projection
probability
C s
set
combinations of
functions.
no l o n g e r
pure
In
atomic
Bloch
orbitals
mixed up
as
long
T h e Low d i n
matrix
form
centered
as
at
certain
symmetry
does not
representation
of
layers.
is
the Schrodinger
They
prevent
derived
as
equation
is
are
all
mixing.
follows.
The
HC=EOC.
(126)
A unitary
conjugate
matrix
M are
M which
applied
to
d i a g o n a l i z e s O and i t s
the
equation
to
complex
obtain
M+ H MM+ C= E M+ O MM+ C,
(127)
where
M+ OM=O'
is
a diagonal,
( 1 2 8)
p o s i t i v e —d e f i n i t e
matrix.
Def i n e
H' = M+ HM
(129)
C = M + C.
( 1 3 0)
and
Then
the
Schrodinger
H'C=EO'C'.
equation
(126)
becomes
,
(131)
60
The
corresponding
but
not
normalized
Since
matrices
0'
is
from
basis
functions
are
definite,
we
already
orthogonal,
yet.
positive
can
define
two
O’ :
O’ V / i - f O ' i . O 1
(132)
and
0iV Z i1 M ultiply
I / (O'V i ) (131)
by
(132)
(133)
and
(133)
as
follows:
O*- 1 / 1 H1 O’ ™1 ^ 2 O’ 2 / 2 C
= E O ' - 2 / 2 O’ O’ - 2 / 2 O’ 2 / 2 C'
= EO' 2 / 2
C' .
(134)
Define
H11= O ' - 1 / 2 H'
O' - 2 / 2
(135)
and
C " = 0' 2 / 2 C' .
The S c h r o d i n g e r
H''C''=EC''
(136)
equation
(126)
becomes
(137)
61
The b a s i s
the
set
now
eigenfunction
functions
even and odd
will
not
are
expanded.
In
much w i t h
other
talk
new
the
about
basis
the
order
basis
showed
used.
is
not
last
physical
case.
Wh e n
(with
respect
to
each
3d a t o m i c
the
basis
O-w-C'-*- C " ,
k points,
into
if
meaning.
general
the
the
except
that
Z reflection)
other.
wavef u n c tio n s
functions.
in
there
numbers
the
too
I BZ
is
Perhaps
large
iteration,
interpolate
eigenfunctions
randomly
states
also
interpret
they
is
are
not
do n o t
So
it
still
of
3d l e v e l s
too
mix
meaningful
using our
for
iteration.
essentially
this
at
we u s e
is
the
no
45
c h a n g e w h e n 15
because
Fermi
A trial
the
Fe
energy.
k points
in
density
However,
the
I BZ
in
to
1)
2)
the
components of
at
occupation
are
states
for
to
Low d i n t r a n s f o r m a t i o n
6 k points
calculation
of
so t h e
set.
We u s e
k points
mixed
the
easy
definite
not
transformed
N e v e r t h e r l e s s,
to
is
functions
be
be
have
this
functions
orthonormal,
should
themselves
Unfortunately,
basis
is
of
generated
the
k dependence
the
Schrodinger
k points,
to
of
eigenenergies
equation
get
for
a smooth
and
about
density
10000
of
( DOS) .
draw
energy
a kind
directly
to
of
the
bands
along
interpolation)
ARPES.
high-symmetry
which
can
be
lines
(this
compared
is
62
Ve
Here
discussed
we
focus
2)
on
in
the
detail
DOS
in the
k point
energy
are
randomly
chosen,
the
by
the
'atomic'
self-consistent
After
obtain
the
the
N(E) = ^
j
uniformly
number
'atomic*
wave
of
function
interpolation,
density
of
all
of
we
the
first
interpolate
wave
calculations
to s e l e c t 10000 k p o i n t s
in
we q u a d r a t i c a I I y
and o ccu p a tio n
labeled
which
and
on S y m m e t r y .
calculation.
We u s e a M o n t e C a r l o t e c h n i q u e
scattered
section
BZ.
For
both
the
Lowdin o r b i t a l s ,
function
and th e
belongs,
from
the
begin
nearest
to
each
which
layer
to
previous
k points.
count
levels
to
states
% S ( E - E, ( k ) )
k
J
(138)
wher e
S ( E - Ej ( k ) ) = 1 ,
for
E<Ej(k)<E+0.OOl(Hartree)
(139)
S ( E - Ej ( k ) ) = 0 ,
for
E otherwise
(140)
and E i t s e l f
is
generated
by
counting
specified
intervals.
A five-point-weighting-average
technique
is
remove
Th e
Ith
used
to
Hartree
all
spaced.
energy
is
defined
„<e , 1 . 1 ) - % ^
Thus
values
statistical
o r b i t a l —l a y e r —p r o j e c t e d
layer
n
0.001
DOS f o r
a histogram
falling
noise
the
ith
within
of
is
the
DOS.
orbital
and
as
Ci , l ( j , k ) ' 5 ( E - E j ( k ) )
(141)
63
The l a y e r - p r o j e c t e d
N1 U , 1 ) = J
I
^
integration)
of
i, I
Ith
layer
is
defined
Ci # 1 ( j , k ) a S t E - E j C k ) )
occupation
indices
the
or
number
the
I
is
defined
DOS u p t o
are
Fermi
(142)
as th e
summation
energy.
The
omitted.
(143)
definitions
above,
im plied for a p a r tic u la r
indicate
Th e
a
spin
total
(138),(141) — (143),
fact,
explicitly.
number
is
defined
as:
(144)
the m agnetic
moment
is
defined
as:
m=nt ~ n t »
in
both
cases
These
DOS,
(145)
the
possible
especially
and l a y e r - p r o j e c t e d
useful
in
charge
transfer
variations
discuss
are
s p i n o n l y . Now w e h a v e t o
dependence
occupation
in
n = n f +n v
and
(or
possible
n = 5 N( E)
E<E f
The
as
N0 ( E, i , I )
=5
The
DOS f o r
developing
from
these
in
indices
DOS N1 * w i l l
insight
surface
detail
or
I
are
orbital-layer-projected
in
turn
into
between layers,
the
i, I
out
bonding
to
Chapter
center
3.
DOS N q
be v e r y
mechanisms,
and m a g n e tic
to the
omitted.
moment
layer.
We w i l l
64
Finally,
probability
we w o u l d l i k e
on a l a y e r
p(l,j,k)=|
i
s ums o v e r
layer
I. T h i s p ( l , j , k )
states
states.
Chapter
on a
with
all
certain
p(surface)
We w i l l
3.
for
define
layer
a given
occupation
eigenstate
as:
Ci f j ( j . k ) *
where
localized
I
to
(146)
atomic Block o r b i t a l s
tells
us how much
layer
I.
> 60% a r e
discuss
In
located
j(k)
particular,
defined
surf ace-state
as
bands
at
the
is
those
the
in
surface
detail
in
65
CHAPTER 3
RESULTS AND DI S CUS S I ONS
Two-Level
Bonding
Before
going
calculated
simple
the
Yet
are
for
the
two-level
use
this
detailed
we w o u l d
calculation
t h e model
slab
the
results,
sophisticated,
did not
to
calculation
slab
simple
Mo d e l
discussion
like
a two-level
basic
model
physics
is
model
be
used
calculation,
to
see what
to
see w hether
on,
calculation
intuitive
The
and
agrees
thinking
two-level
is
calculation.
two-level
to
Schrodinger
the
the
examine
or
of
s a me a s
slab
the
Although
in
our
of
the
processes
slab
simple,
problem.
equation
as
the
t h a t we
results
the
two-level
f u n c t i o n s iP x a n d tP 2 c a n b e w r i t t e n
a
calculation.
physical
not
with
matrix
model.
We e m p h a s i z e
in our
kind
the
much mor e
interpret
qualitatively
b a s e d on
first
bonding
we p e r f o r m e d
can
going
to
of
with
basis
follows:
(147)
w h e r e H j 1 =Hi a
by
the
Hermitian
property
of
the Hamiltonian.
66
By
solving
(147),
we
get
E= ( 1 / 2 ) x [ ( H 11 + Ha a ) + ( ( H1 1 - H a i ) * + 4 H i a a ) 1 / a ]
and
the
If
corresponding
there
is
no
(148)
t wo e i g e n f u n c t i o n s .
coupling
between
the
t wo l e v e l s ,
H1 1 =O,
(149)
then
Ha = H11
The
t wo
In
and
E 1= Ha a .
eigenfunctions
the
case
of weak
are
simply
coupling
( H1 1 - H a a ) 2 >> 4 H1 a 3 ,
assuming
(150)
pure
between
^ 1 and
the
<f>a .
t wo l e v e l s ,
(151)
H1 1 >Ha a , we h a v e
E 1 = H1 1 + Hi a 2 Z ( H 1 1 - H a a )
(152)
E 1- H 1 a -
(153)
and
for
H1 a / ( H 1 1 - H a a )
energies,
and
Ca ZC1 = H1 a / ( H1 1 - H 11 ) ,
(154)
wh e r e
Ca > > Ca ,
(155)
67
for
t h e wave
function
,
and
(156)
Ci / C a = Hi I / ( Hi i - H i a ) ,
where
(157)
Ca >>Ci ,
for
t h e wave
We s e e
these
function
that
after
t wo l e v e l s ,
function
is
no
of
0 a mixed
into
corresponding
has
a
In
small
the
we
level
wave
the
Va.
amount
strong
turn
I
is
longer
it.
on t h e
raised,
pure
Also,
the
and
but
level
wave
function
is
of
fPx m i x e d
into
bonding
interaction
no
particular,
in
the
the
it
bonding
ener gies,
Ca = + Ca
corresponding
a
small
amount
lowered,
pure
and
$ a but
(Fig.5).
case.
(158)
degenerate
case
(159)
E=Hl i : t H i a
f or
is
longer
H1 1 - H , ,
where
has
2
( H1 I - H a a ) 1 << 4 H1 a 2 .
In
the
between
is
strongest,
we h a v e
(1*0)
and
(161)
for
wave
functions.
(degenerate)
case,
50% o c c u p a t i o n
With
ready
to
this
calculation.
other
on
It
are
words,
generalized
the
each
simple
discuss
mechanisms
That
is,
two
of
model
the
turns
out
into
two-level
the
levels
the
strong
are
that
the
basis
functions
in mind,
sophisticated
the
same
picture
a many-level
bonding
perfectly
calculation
much more
essentially
the
in
basic
in
mixed
( F i g . 6).
we a r e
slab
physical
these
t wo
c a n be e a s i l y
picture.
with
bonding
cases.
In
69
F i g . 5 T w o - l e v e l b o n d i n g . We a k c o u p l i n g .
(a) Energy s p l i t t i n g .
( b ) O c c u p a t i o n p r o b a b i l i t y on T 1 and
iP a .
70
Ex
F i g . 6 Two-level bonding. Strong cou p lin g (degenerate
(a) Energy s p l i t t i n g .
( b ) O c c u p a t i o n p r o b a b i l i t y on
and 0 * .
case).
71
Densi tv
of
States
•;
The n a t u r e
clarified
in
Fig.7
by
of
for
In
for
Fe
bond
layer-projected
F i g . 7,
a clean
we a l s o
slab
a clean,
for
is
DOS' s
DOS's
show
the
(142)
(141)
a 4-eV-wide
agreement
given
addition
F i g . 7,
the
levels
that
excited
from
is
spectroscopy
(an
five
independent
layer
Fe
slab
Fe
DOS i n
effects
from
the
as
near
6
oxygen-like
eV.
[65]
was
states
and a l s o
These
two-level
for
a prominent
potential
also
I
electrons
identified
eV a b o v e
the
The m a g n i t u d e s
of
to
the
the
bonding model.
strength
surface
Ep .
surface
Fe
of
produces
Fe
the
correlated
features
in
unoccupied
[10]:
reduces
good
antibonding
states
about
in
In
3d b a n d s
Th e
have
in
[7,8].
Appearance
0 p band r e g i o n .
directly
Ep.
b y EELS
substantially
region,
the
band of
final
2 p band
measurements
Fe-O h y b r i d i z a t i o n .
expected
2
serve
bonding
the
below
eV a b o v e
d band
are
levels
features
eV b e l o w E p ,
UPS r e s u l t s
about
adsorption
DOS i n t h e
5.5
a broad
bands
( APS)
pronounced
DOS h a s
observed
band of
Oxygen
surface
the
has
centered
bonding
extend
these
layer
experimental
adlayer
of
an empty
the
these
portion
transition
oxygen
distribution
to
in
*
the
with
given
corresponding
comparison
unrelaxed,
easily
performed).
T h e DOS o f
these
chemical
and o r b i t a l - l a y e r - p r o j e c t e d
calculation
also
surface
examining
F i g . 8 - F i g . 10.
DOS' s
the
■
of
DOS a r e
The r e d u c t i o n
72
of
the
d band
reduction
of
emission,
the
which
corresponding
experiments
[7,8].
a bulk-like
appearance,
negligible
are
inclose
length
A,
for
[9]
than
peaks
for
are
DOS o f
central
in
the
layer
DOS o f
well
calculated
Th e
the
planar
DOS's
the
in
orbital
components
surface
bond.
Fig.10.
We b e g i n w i t h
Among t h e
narrow
about
for
3 z a- r 1 o r b i t a l ,
atoms
(equivalent
by
as
oxygen,
suggested
layers
DOS
be
as
presumably
its
adatoms
Fe-O bond
plane,
A.
( F i g . 7),
resolved
DOS's
and x*-y*
have
but
earlier
of
The
somewhat
certainly
[10].
xz,
yz
The
the
and a g r e e s
too.
into
in
in
atomicthe
Fig.8-
F i g . 8.
have
strong,
Ep f o r m a j o r i t y
because
2.02
The O 2 p
concerning
interaction
sites.
not
with
0 2 p DOS's
eV b e l o w
have
expected.
[27,28],
insight
is
weak O 2 p h y b r i d i z a t i o n
hollow
symmetry)
in te ra c tio n with
bonding
in
film
3 z * —r *
which minimizes
positioned
2.08
agrees well
Fe
Fe
the
second
is
t o p - Iayer-Fe
1.9
very
it
i n UPS
oxygen
that
the
3d a n d
both
F ig.8.
orientation,
Fe
deeper
center,
O/Fe
bulk
the
compare
the
film
further
3d o r b i t a l s ,
Only
for
clean
We p r e s e n t
DOS p e a k s
Note
F i g . 7 can
for
of
since
where
of
the
hybridization
them.
plane,
corresponding
with
DOS's
smaller
to
was o b s e r v e d
Fe a t o m s ,
to
first
weak
DOS,
the
subsurface
the
correlated
however,
actually
very
the
Though
proximity
is
is
its
with
spin:
occurs
vertical
oxygen
orbitals
stronger
do n o t
Instead
it
dominate
is
the
the
73
planar
o r b i t a l s,
of
x a- y 1 a n d
strongest
hybridization
DOS p e a k s
below
with
the
nearly
calculation.
directed
3 d xy
features
are
because
the
is
similar
to
and b e c a u s e
is
as
high
the
as
of
primary
bonding
of
that
of
of
the
energy-loss
peaks,
affected
by
oxygen a t
orbitals
involved were
(xy,xz,yz),
and
calculation
shows,
xy o r b i t a l
the
oxygen,
bonding
at
further
of
is
x y DOS
( 2 p y )»
bonding,
is
it
0 2p r e g i o n
of
which
assigned
is
mainly
chemisorption
to
that
xz
it
is
not
responsible
the
t w o Fe
strongly
The a t o m i c
3de
and yz.
the
strongly
In
that
were
be
as
stage.
the
t h * EELS
stage.
to
most
is
observed
chemisorption
adlayer,
0 2p region
conclude
eV r e s p e c t i v e l y ,
that
the
p x ,Py«
was
referred
these
bonding.
in the
we
Fe d x y / 0
however,
and t h u s
the
2 p x , 2 py b a n d
first
Fe w h i c h
the
intense
repulsion
of
in
[10],
the
Both
xy DOS
O 2px
5 and 8
strong
the
the
lobes '
the
two-level
two-level
e_t a_l
Note
strong
of
mechanism
Sakisaka
energies.
largest
the
of
the
amplitude
strong
the
of
with
the
the
the
has
levels.
as
the
in
sites,
have
consistent
orbital,
well
higher
shape
that
feature
study
as
characteristic
Also,
so
eV,
towards
xy
0 2p
is
assumed
hollow
with
below -4
levels
the
that
and hence
result
geometry
four-fold
hybridization
DOS s t r u c t u r e
This
particular,
the
character,
oxygen,
d bands.
co-planar
In
into
strongest
of
the
with
xy
xz,
Onr
yz b u t
affected
for
fact,
by
chemical
i f we
chose
74
oxygen 2 p p l a n a r
(rotating
F i g . 12
of
the
become more
exp(-ik'm)
of
this
ma y
the
wave
after
correspond
in
weak
F i g . 11
considerations
, :
r :■
case,
sizeable
the
Fe
f
? <
1
of
:
in
-
:
; ’
however,
lattice
Thus
is
site
this
wave
changing
Fig.8
the
this
is
sign
sign
function
the
k-integrated,
k points :
while
k-dependent
s ome
of
others
bonding
is
geometrical
the
■
kind
factor
bonding,
Also,
this
mechanism
and
possible
This
that
the
and
p,
^
'
orbital
,
t
should
play
f
a
I. ;
role.
this
there
bonding.
at
thus
strong
and F i g . 12.
Now c o n s i d e r
In
to
suggests
s
minor
all
Figlll
bonding,
change,
function,
over
correspond
to
ma y
T h e DOS g i v e n
summing
in
summation.
superposition
wave
"h\
px
ma k e
exponential
function
of
bonding
function
function
the
the
planar
by a n
Bloch
affect
k points
shown
the
situation.
is,
the
a t o m i c wave
neighboring
bonding
that
in
did
automatically
This
modulated
atomic
change
and
is
a , we
needed),
obvious.
The
a j ( r —m) ,
if
I
instead
p x± p y
degrees),
combination
k-dependent.
m,
l o b e s 45
as
(diagonalization will
linear
will
orbitals
as
adlayer.
mechanism
in
DOS's
F ig .9,
below -4
eV.
and n o t
We ma y
infer
that
the
s e c o n d Fe
layer.
3 Z1- T 1 DOS h a s
the
oxygen
s c h e m e Fe
understandable
the
Moreover,
0 2 pz
to
of
only
the
according
is
orbital
seen
amplitude
resembles
atoms
the
the
DOS
a
shape
0 2 p x —2 p y b a n d o n
bonds
to
subsurface
dg z 1 - r 1 / 0 ^ p z .
considering
This
the v e r t i c a l
75
relationship
planar
of
bonding
virtually
t h e more
is
because
the
not
p bands
thick
enough
structure
the
DOS o f
at
to
xz
is
no
slab
is
In
thick
the
fewer
shows
that
at
enough
bond
this
length
and
d 3 z a_ r 1/ 0 2 p z
d xy. / 0 p x + y b o n d i n g
neigbors
(1:4)
and
oxygen h y b r i d i z e s very
the
would
at
slab
be
center,
for
that
that
the
as
as
that,
the
in
DOS o f
the
center
our
later
slab
is
electronic
it
is
required
the
a
completely
discussed
In F i g . 10
same,
s a me a s
to
bulk-like
plane.
to
Although
and m agnetism ,
nearly
the
center.
required
the
addition
reason
is
a t wo
of
Fe ( S - I )
nearest
occupancies
the
spite
the
(1:2).
central
almost
symmetry
is
has
and y z are
considerations.
orbital
Fe(S)
as
to
bonding
k is
shorter
strong
provide
the
the
contrast
that
In
this
slab
of
us
XY p l a n e .
In
vertical
dz 1 fu n ctio n ,
hybridization
sections
this
reminding
nature,
orbitals
thicker
eliminate
the
Fe
(Fig.13).
Fe/0 ,
the
involved
F i g . 10
with
slightly
as
former
Finally,
in
bonding
extended
bonding
in
atoms
surface
vector
k - i n d e pe n d e n t
little
of
t wo
k-independent,
dimensional
fewer
the
by
DOS o f
xz
slab
plane
s hown t h a t
symmetry
the
and yz.
case
is
xy
There
unless
the
bulk-like.
76
O Z F e(O O I) D O S
ADLAYER
Fe SURFACE PLANE
SUBSURFACE PLANE
CENTRAL PLANE
E N E R G Y (eV )
F i g . 7 L a y e r - p r o j e c t e d DOS' s o f O / F e ( 0 0 1 ) . Ep i s t h e
e n e r g y z e r o . The v e r t i c a l s c a l e i s a r b i t r a r y .
M a j o r i t y - ( M i n o r i t y - ) s p i n DOS ' s a r e i n d i c a t e d b y
t ( I ) . R e s u l t s o b t a i n e d f o r t h e c l e a n Fe(OOl ) f i l m
( s h i f t e d t o a l i g n E p ) a r e shown w i t h d a s h e d l i n e s .
77
(a) S U R F A C E D O S
i
-8
I
-6
I
I
- 4
I
- 2
I
I
I
O
I
I
2
E N E R G Y (eV )
F i g . 8 F e 3 d - o r b i t a l DOS’ s o f O / F e ( 0 0 1 ) f o r s u r f a c e p l a n e .
For c om parison p u r p o s e s ,
t h e O 2 p x (2p ) - o r b i t a l
DOS' s a r e a l s o
shown ( d a s h e d l i n e s ) .
78
'8
~6
—4
- 2
O
2
E N E R G Y (eV )
F i g . 9 Fe 3 d - o r b i t a l D O S ' s o f O / F e ( 0 0 1 ) f o r s u b s u r f a c e
p l a n e . F o r c o m p a r i s o n p u r p o s e s , t h e O 2 p z- o r b i t a l
DOS' s a r e a l s o s h o w n ( d a s h e d l i n e s ) .
(c) C E N T R A L - P L A N E D O S
D E N S IT Y O F S T A T E S
I
vVx2",y2 . S
r K
J
V-
I
. . .
V
--- ------------
JWI
xy
_
K
I
i/
R
___ ,1_____1I
-------6
I
/V
I —A,
-8
7—
Tl
, I 1 I , i 1 I,
- 4 - 2
i
I
0
I
J y -
I,
I
2
E N E R G Y (eV )
F i g . 10
F e 3 d - o r b i t a l DOS' s o f
for the c e n tra l-p la n e .
OZ F e ( OOl )
80
r bonding.
Fe(S)
d i y / O P jJ+ Py .
k=n / ( 2 a ) ( I , I )
81
F i g . 12
Planar
bonding.
Fe(S)
d xy / 0 p x - p y .
k=n/(2a)(-1,1).
82
F i g . 13
V ertical
bonding.
F e(S-I)
dz */ 0
pz
k - i n d e pe nd e n t .
83
A t o m i c- o r b i t a l
0_c_cu_E_an_cj.e_s
Occupation
numbers
orthogonalized
atomic
d—o r b i t a l
occupancies
potential
and
clean
film,
larger
the
all
than
is
due
higher
to
the
dehybridization
at
the
is
nearly
On t h e
Fig.IS
anisotropic,
1-
is
T 1 ,
adsorption
occupancies,
bonding.
atom,
from 1.31
are
due
to
two-level
into
the
number
while
to
the
the
reflecting
(metal)
model,
levels
xz,
increase
levels
have
the v ario u s
occupancies,
the
the
d shell
density
s h o wn i n
dominance
involved
of
e^
plane,
1.40
3z -r
is
the
electron
present.
The
the
to 1.09
as
a transfer
Ep.
d-orbital
in
repulsion,
above
the
5).
from
and
the
slightly
for
d-band
the metal
subsurface
level
the
surface
reduced
3d
spin
of
s h o u l d be
similar
in
The
surface
are
except
of
4.
surface
Because
isotropic
those
the
numbers
nominal
surface.
(Table
Table
surface-induced
[49]:
when o x y g e n
0 2 p/Fe
bonding
Th e
in
the
results
perturbs
is
1.21
antibonding
At
occupation
hand,
especially
Within
occupation
per
other
x 4- y 2 ) o r b i t a l s
Oxygen
adlayer.
atom h av e
(Fig.14).
Z
by b o t h
affected
bulk.
d - o r b i t a l s on a g i v e n
(3
are
central-plane
of
density
L owdin b a s i s
c e n t r a l —p l a n e v a l u e s
d character
charge
the
given
d-orbital
The
representative
in
o r b i t a l s are
oxygen
their
yz o rb ita ls.
( 1 4 1 —1 4 4 )
of
Both
Fe-O
xy
electrons
count
falls
decreases
predicted
by
d character
• 'i :a
non-bonding x -y
the
84
orbital
experiences
occupation.
increases
is
Finally,
from 1.38
adsorbed.
reduced
almost
the
to
no o x y g e n - i n d u c e d
surface
1.47
total
number
by 0 . 2 6 ,
0.02,
and
and c e n t r a l
of
0.12
planes
per
atomic
on
of
of
( y z) occupancy
electrons
The
subsurface,
xz
changes
the
the
a t o m when o x y g e n
d electrons
n^ is
surface,
oxygenated
film,
respectively.
S u r f a c e — a n d o x y g e n —i n d u c e d
studied
using
changes
in
valence
occupation
metal
per
attracts
surface
results
n^ n o t e d
film
atom)
the
above,
to w ith in
0.03
In
of
per
addition
While
(8
the
to
the
clean
valence
atom,
electrons
In
c a n be
n $p and t o t a l
given.
electron
layers.
transfer
for
neutrality
number
subsurface
4.
results
charge
significant
and
Table
n = n ^ +n $ p a r e
exhibits
a
in
charge
electrons
oxygen
from
particular,
the
0.63
■
electrons
atom,
0.11
behind
on p l a n e s
see
meaning
due
transferred
leaving
addition
we
are
to
to
the
that
of
the
S,
these
fact
atomic
Block
at
the
central
is
needed
the
metal
to
a
charge
deficit
S-I,
a n d C,
respectively.
covalent
ionic
from
bonding
bonding
also
occupation
the
exists.
Lowdin o r b i t a l s
orbitals,
layer
to maintain
the
is
an
0.11
is
are
‘
0.05,
Tnus
and
in
iron,
Although
the
somewhat
ambiguous,
no l o n g e r
that
neutrality
_
oxygen
oxygen and
electrons
indication
charge
of 0 . 5 2 ,
between
numbers
each
-
of
deficit
a
pure
per
thicker
the
atom
slab
central
85
layer
of
charge
slab
the
is
not
model,
which
is
and
given
in
the
for
was
function
shift
the
also
difference
eV [ 6 9 ] .
are
brought
surface
the
at
stage
full
a test
may h e l p
to
One
the
are
Fes
of
coverage
of
oxidation
clarify
the
than
Fe
model
a
the
slab
lattice
work-
dipole
eV.
Fe
layer
The w o r k
in e x c e l l e n t
4.4
eV [ 6 8 ]
O/ Fe ( 0 0 1 )
0.25
eV,
the
Our
and 4.3
in th e
perhaps
ideal
Fe
calculation
we u s e d
process
Fe
s l a b when t h e
eV,
below
given
a large
A(I) f o r
(1x1).
oxidation
3
Fe
A<t> =1 . 4
values
oxygen
or
eV ( a n a u x i l i a r y
creating
larger
oxygen
the
clean
of
oxygen,
clean
expects
is 4.4
[8,10]
no
8.0
the value
experimental
incorporation
provides
surface.
for
the
oxygen w ith
together,
We f o u n d
calculated
chemisorption
of
of 2
of
equivalently
eV v s .
performed).
Observed v a l u e s
the
charge
or
of
finite-thickness
between
4.4
compared w i t h
agreement with
to
a formal
a monolayer
interface.
f u n c t i o n we
our
transfer
electronegativity
oxygen m onolayer:
and O s l a b s
a large
within
huge
assigned
work f u n c t i o n
constant
due
the
such
compound f o r m a t i o n ,
calculation
at
However,
unreasonable
often
electrons
huge
slab.
of
and th u s
the
Fe
86
Table
4 ^ Occupation
numbers
plane
Fe ( 0 0 1 )
F e5,
atoms
positioned
film,
0 / F e 5. The
planes
are
iron
labeled
layer
is
Film
Layer
U
I
N
CO
denoted
the
Fe,
S
1.31
S-I
1.28
C
1.27
S,
atom
for
a clean
five-
and a f i v e - p l a n e ' f i l m w i t h
f o u r —f o l d h o l l o w
surface,
S—I ,
and
sites
subsurface
C.
Th e
on
oxygen
both
and c e n t e r
oxygen
adsorbate
a s A,
H
W
I
%
*»
faces,'
in
per
xz(yz)
iy
nd
1.31
1.38
1 .41
6.80
1 .23
8.03
I .22
1.42
1.40
6.74
I .25
7.99
1.2 9
1 .41
1.35
6.74
1.23
7.97
6.63
6.63
O/ Fe 5 A
n sp
n
S
1.21
1.31
1.47
I .09
6.54
0.94
7.48
S-I
1.21
1.27
1.43
1.43
6.72
1.23
7.95
C
1.21
I .28
1.35
1 .42
6.62
1.27
7.89
87
T a b l e 5.
Fe( OOl )
Magnetic
film,
moments
the
f o u r —f o l d h o l l o w
sites
The
iron
surface,
labeled
S,
S -I,
and
denoted
a s A.
S
a clean
film
O Z F e 1.
Fe,
for
and a f i v e - p l a n e
in
Layer
atom
F e 1,
positioned
Film
per
3z*-ra
0.63
C.
subsurface
T he
oxygen
X 1- J 2
xz(yz)
0. 5 8
0.54
with
on
oxygen atoms
both
and c e n t e r
adsorbate
f i v e —p l a n e
faces,
planes
layer
are
is
xy
0. 4 9
2.79
0 .1 0
2 .8 9
S-I
0.58
0.62
0.38
0.42
2.37
-0.06
2.31
C
0.65
0.5 9
0.42
0.46
2.53
-0.04
2.49
0.2 4
0.24
O/Fe , A
S
0.73
0.67
0.44
0.64
2.92
0.010
2.93
S-I
0.56
0.63
0.46
0.48
2 . 5 8 —0 . 0 06
2.57
C
0.69
0.59
0.50
0.42
2.72
-0.004
2.72
88
Magnetism
Recent
suggest
spin-resolved
that
oxygen
Ni(IlO) , while
magnetism
enough,
stage,
of
even
it
it
has
case,
peak
Fe
the
to
the
mo me n t
compared w i t h
of
surface
atom,
corresponding value
subsurface
central
of
oxygen
dead
the
layer
the
clean
adsorption
and c e n t r a l
that
of O
kind of
Fe
are
to 2.68
slab
bulk-like
is
not
pg,
is.)
are
(16%
This
Fe
slab
does
also
not
magnetically.
of
in
the
increase
enhancement
from
is
mo me n t
the
(from 2.89
to
produce
moments
of
increased
respectively.
thick
Ou r
given
The m a g n e t i c
slightly
Fe
behavior
enhancement
The m a g n e t i c
layer
it
magnetic
film
oxygen.
increases
layer.
and from 2 . 4 9
indicates
feature
surface
of
surface
and c l e a n Fe(OOl)
adsorbed
induced
a reduction
on Fe(OOl)
clean metal
m,
(Surprisingly
chemisorption
and
what
of
on t h e
an oxygen
c e n t e r —p l a n e v a l u e ) .
per
Thus
Ep,
a universal
the
the
magnetically
be
is
oxygen
substantial
r e m o v e d by
pg).
with
no m a t t e r
not
to 2 .5 2
eV b e l o w
OZFe(OOl)
the
Fe
there
at
orbital-layer-resolved
Table
Note
that
5.5
on Fe,
of
magnetic
influence
glass
this
(141-143,145)
5.
slight
[68]
magnetism
in
seems
for
surface
( F e e i Bi a S i * ) .
in ag ree m en t
This
the
glass
chemisorption
2.93
only
was o b s e r v e d
3d p e a k ,
results
removes
experiments
a Fe-based
photoemission
[7,8].
photoemission
enough
to
from 2 .3 1
This
lead
Apparently,
the
89
interaction
between
oxygen
adsorbate
very
and has
longer
range
strong
introduced
magnetic
by
Fe
et
m's,
a_l
slab,
2 . 5 2 jiB ,
layer,
surface.
moments,
by O h n i s h i
clean
the
surface
clean
Fe
2 .35 , 2 . 3 9 ,
and 2.25
[70].
calculation,
is
0.24
jig p e r
atom.
This
and i s
that
*^g
magnetic
for
well
subsurface
for
also
oxygen
spin-polarized,
about
not
8% o f
surprising,
the
ground
2 .94,
with
an o x y g e n m o l e c u l e .
and
and c e n t e r
a
( C)
seven
2.98,
and C l a y e r ,
itself,
in our
a magnetic
however,
is
layer
2.32,
for
S-2,
the value
state
slab
moments a r e
adlayer
is
calculated
a five
found
S-I ,
Fe
m's
(S-I) ,
magnetic
S,
clean
the
are
substrate
perturbation
for
moments
the
j i g,
the
with
found
They
that
Th e
atom,
is
ourselves
[69].
slab,
respectively
(S) ,
than
Fe
calculated
They
magnetic
respectively
layer
agree
[69,70].
the
for
Th e
and
on a
if
mo m e n t
surface
of
Fe
we r e m i n d
*Pa f o r
an o x y g e n a t o m
Both
these
of
have
moments.
C h a r g e a n d jSj)_in D e n s i t i e s
The
spin
self-consistent
density,
C h a r g e —d e n s i t y
drawn
in
contours
that
a
pf (r)-p t ( r ) ,
contours
(HO)
plane
differing
these
charge
figures
by
are
s hown
differing
normal
a factor
have
de n s i t y ,
to
by
the
of 2
a b u l k —l i k e
p f ( r ) +p j ( r ) ,
in
the
F i g . 14
ratio
surface.
are
also
and
and F i g . 15.
1.414
are
Spin-density
given.
appearance
below
Note
the
90
F e - O i n t e r f a c e . : At
o x y g e n a n d Fe
are
evident
atoms
in
i n t e r n u c l ear
deep w i t h i n
oxygen,
which
Because
the
metal
density.
itself
substrate,
though
surface
not
[ 7 0]
probability
divided
latter
bands
surface
that
is,
bands
simple
t wo
just
ma y be
of
magnetism
in
F i g . 16
found
below
the
described
p bands
strong
and
in
the
as
spin
the
on t h e
60% o c c u p a t i o n
These
surface
4
states
These
were
states,
the
These
expected,
to 8
levels,
on
The
selected
As
bands
the
atoms.
F i g . 17.
bonding
each
Fe
t h a n 80 %.
range
by
negative
greater
F e 3d b a n d s .
energy
for
than
Fe-O b i l a y e r .
weak and
of * t h e
of
vacuum n o t e d
o n O / F e ( 0 0 1 ) , we
Fe-O o c c u p a tio n
tight-binding
sets
bands
on t h e
into
are
smoothed
neighboring
I BZ w h i c h h a d m o r e
s hown
bands
into
located
magnetization.
surface
eruptions
surface
further
are
the
also
small
is
t h e vacuum
between
(146)
having
a
is
in
found
Bands
the
obtained
So me w e a k ,
Surface-state
the
smoothing
absent.
is
in
quench
along
definite
Fig.IS,
between
second metal' la y e r s
are
magnetization
To d e t e r m i n e
is
bonds
oxygen la y e r
a
acquires
the
and
charge
surface,
corrugation
spin
covalent
first
axes.
does
states
the
of
adsorption
clean
in
buildup
charge-density
Th e
interface,
the
the
region.
the
strong
eV b e l o w
Ep,
oxygen-1ike
and d i s p e r s e
p orbitals.
direction
There
as
are
corresponding
91
to
symmetric
( F i g . 16)
com bination of
of
the
slab.
bonding
Th e
antisymmetric
interaction
splitting
in
this
surface
vanishs,
respect.
is
figures,
pz band
with
are
p x+y
from 0.5
The
to
the
of
[17]
bands
using
the
planes.
In
them
M1 a n d
fs,
light
excited
by l i g h t
planes,
while
If
this
enough
In
these
it
f
with
in
to
(I).
T h e p Z' P y
px being
with
the
the
lower
even band
0 2p b an d s
and t h e i r
energetic
relatively
candidates
together
lines,
with
the
vary
in
odd b a n d s ,
the
be
separation
flat
for
observation.
the
selection
Th e
rules
of
contain mirror
labled
A1 a n d 2 i
respective
A1 a n d J
potential
Ep s h o u l d
identification
which
even bands
polarized
the v ecto r
8 eV b e l o w
Their
facilitate
A and ^
the
4
[16] .
good
particular,
of
along
the
in our
and 17.
band
slab.
thick
true
follows:
bands
levels
ARPES w o u l d
component
be
as
splittings
ARPES
polarized
along
to
the
of
and
eV.
b u l k Fe
for
said
pz band h y b r i d i z e s
.Exchange
d i s p e r s i o n ma k e
use
at
of
faces
symmetric
a measure
essentially
p x (py)
oxygen-induced
resolvable
from
The
1.0
is
extends
degenerate
along ^
is
s h o w n b y F i g . 16
the
along T and f .
is
t wo e q u i v a l e n t
surfaces
slab
( F i g . 17)
between
bands
t wo
the
as
the
state
This
on t h e
splitting
the
calculation,
bands
levels
energy
between
hybridizes
and a n t i s y m m e t r i c
%,
are
mirror
require
perpendicular
a .
to
these
92
planes.
A broad
eV below
Ep i n
A second
angle-integrated
set
and F i g . 17.
and have
o xy ge n - i n du c e d r e s o n a n c e
of
These
higher
surface
are
Fe
Because
have
exchange
large
on t h e
degree
of
In
oxygenated
surface
below
Th e
Ep.
band
is
affected
is
One
by
this
band to
near
DOS o f
M4 ( x y
along
atoms
ma y
agreement
X1 ( 3 z —r
or
to
See
the
the
It
the
in
to
is
Fe
the
2
A
quite
3z*-r*
be o n l y w e a k l y
in
four-fold
Fig.8
of
also
hollow
strong.
surface
It
orbital-
orbitals.
We a k b a n d s
are
f*
Cxy) n e a r
the
edge,
,
x —y ) •
states
between
bands)
3d
eV
t wo
disagreement
identify
1.7
mainly
b a n d Y1 M1^ » i s
bulk
o x y g e n —l i k e
and h a s
located
eV.
these
convert
(especially
to
-1.8
character),
A1 n e a r
states
help
expects
An x 1- y a s u r f a c e
projected
near
eV,
bands
bands).
prominent.
-1.9
as
bands
depend
[70],
found
m a j o r i t y —s p i n
near
oxygen
centered
surface-state
particularly
centered
character.
sites.
is
only
as w ell
is
the
values
Fe ( 0 0 1 )
concentration
(We d i s c u s s
X1Y i M i ^ i
flat,
greatest
fewer
3d b u l k c o n t i n u u m ,
actual
clean
[8].
F i g . 16
amplitudes,
localization
comparison with
in
5.5
bands j u s t
Fe
The
data
seen
Fe
the
large
splittings.
has
also
the
than
their
surface
character.
region.
of
is
in
character
discussed.
photoemission
bands
located
was o b s e r v e d
Some o f
in
d-band
and
t h e weak s u r f a c e
thicker
calculated
a n d ARPES
found
slabs.
surface
should
chemisorption/oxidation
Th e
bands
greatly
model.
CHARGE DENSITY
F i g . 14
C h a rg e -d e n si ty c o n to u rs in the (HO) plane normal
t o t h e s u r f a c e o f t h e O Z F e j ( OOl ) s l a b . C o n t o u r
v a l u e s i n c r e a s e by t h e f a c t o r 1.414 a s t h e c o r e
region is approached.
94
SPIN DENSITY
OZFe(OOI)
F i g . 15
S p i n - d e n s i t y c o n t o u r s f o r O / F e ( OOl ) , p l o t t e d i n t h e
(HO) plane.
S u c c e s s i v e c o n t o u r s h a v e t h e r a t i o 2,
i n c r e a s i n g t o w a r d s t h e n u c l e i . We a k n e g a t i v e m a g n e ­
t i z a t i o n i s i n d i c a t e d by t h e s h a d i n g .
95
(a) S Y M M E T R IC S U R F A C E B A N D S
E N E R G Y (eV )
O— - ( I ) - O - O - O - - O - O - '
F i g . 16
S p i n - s p l i t s y m m e t r i c s u r f a c e b a n d s of 0 / F e ( 0 0 1 ) .
S t r o n g s u r f a c e s t a t e s ( f i l l e d c i r c l e s ) have more
t h a n 80% F e - O i n t e r f a c e c h a r a c t e r , w e a k s t a t e s
( o p e n c i r c l e s ) m o r e t h a n 60% b u t l e s s t h a n 80%.
t(*) bands are in d i c a t e d w ith fu ll(d a sh e d ) lin e s .
Ep i s t h e e n e r g y z e r o . A l o n g t h e s y m m e t r y l i n e s
Y, Z a n d A, b a n d s l a b l e d I a r e e v e n w i t h r e s p e c t
t o t h e a p p r o p r i a t e v e r t i c a l m i r r o r pla n e . Those
l a b l e d 2 a r e odd.
96
POUJ
(b) A N T IS Y M M E T R IC S U R F A C E B A N D S
F i g . 17
S p i n - s p l i t a n t i sy m m e tr ic s u r f a c e bands of 0/F e(0 0 1 ).
S t r o n g s u r f a c e s t a t e s ( f i l l e d c i r c l e s ) have more
t h a n 80% F e - O i n t e r f a c e c h a r a c t e r , w e a k s t a t e s
( o p e n c i r c l e s ) m o r e t h a n 6 0% b u t l e s s t h a n 80%.
t O ) bands are in d ic a te d w ith full(d ash ed ) lines.
Ep i s t h e e n e r g y z e r o . A l o n g t h e s y m m e t r y l i n e s
Y. % a n d A, b a n d s l a b l e d I a r e e v e n w i t h r e s p e c t
t o t h e a p p r o p r i a t e v e r t i c a l m i r r o r p l a n e . Those
l a b l e d 2 a r e odd.
97
CHAPTER 4
SUMMARY AND FUTURE STUDY
S u mma r y
The
electronic
chemisorbed
studied
by
structure
oxygen
the
layer
and m a gnetism
on
SCLO m e t h o d .
the
Fe(OOl)
The
interface
suggested
b y L e g g _ejb _al. b a s e d
analysis
[9] .
in
eV a b o v e
the
Ep,
3d-band
a n d EELS
[8]
a n d APS
both
the
bonding
is
and
adlayer
the
subsurface
[65]
the
surface
is
of
and
good a g r e e m e n t
O 2pI+y
was
5. 5
1
eV b e l o w
Fe
surface
with
UPS
Ep
DOS
[7,8,68]
experiments.
subsurface
due
was
geometry
the
oxygen atoms have
primarily
Fe
in
surface
centered
a reduction
region,
We f o u n d t h a t
to
and
a p (Ixl)
o n a p r e v i o u s LEED
We f o u n d o x y g e n —i n d u c e d p e a k s
and I
of
to
the
Fe l a y e r s .
surface
orbitals.
accomplished
significant
through
Planar
Fe 3 d
Vertical
bonding
orbital
bonding
of O t o
Fe 3 d z 1 a n d 0 2 p z
orbitals.
The
atom,
oxygen a d l a y e r
while
essentially
surface
Fe
unchanged
has
a magnetic
atoms
from
have
the
We d i d n o t f i n d a m a g n e t i c a l l y
moment
a moment
of
of
c l e a n —s u r f a c e
dead la y e r
0 . 2 4pg p e r
2.9 3pg,
value
o n t h e Fe
2 . 8 9 py.
98
surface,
consistent
experiment
on F e - b a s e d
Oxygen a t o m s
from
the
have
surface
substrate,
in
due
of
to
was
found
bands
help
energy
Fe,
clarify
to
order
we h a v e
1)
To
the
the
Fe(OOl)
2)
oxygen
To
of
into
0. 5
the
using
energy
whether
the
clean
the
to
discrepancy
metal.
4 to
1.0
8 eV b e l o w
eV.
Another
surface
Fe
spin).
These
ARPES,
and
character,
surface
this
should
model.
forms
oxidation
suggestions
bands
data.
the
the
It
for
is
for
future
c(2x2)
still
process
O /F e(001),
or
p(lxl)
layer(s)
energy
beneath
bands
the
for
Fe(OOl)
and
an e x p e r i m e n t a l
chemisorbed oxygen la y e r
c(2x2)
of
study.
structure
at
its
on
surface.
calculate
set
Study
understand
ARPES
drawn
As a r e s u l t
The
obtained
(majority
following
coverage
shifts.
oxygen
with
planes.
chem isorption/oxidation
fully
calculate
controversy
the
to
Ep
Future
compare w i t h
maxi mum
mainly
electron
m e a s u r e m e n t s on O / F e (001)
smaller
detectable
Sjng^g e_sti _on s f o r
In
with
of
0.6
eV c o m p a r e d w i t h
bands were
2 eV b e l o w
be
of
iron
1.4
splittings
bands,
should
to
much
absorption
exchange
surface
charge
increases
photoemission
[68].
subsurface
obtain
Oxygen-like
Ep w i t h
glass
disagreement
which
may be
spin-resolved
a net
and
the work f u n c ti o n
[8,10],
with
c(2x2)
surface,
and p ( l x l )
and
to
99
compare w i t h
process,
about
oxygen-like
of
oxygen
3)
ARPES t o
on t h e
possible
f.c.c
[72],
go
in
rather
Fe
the
In
incorporation
this
different
case,
from
the
the
bands
surface.
the
total
to
energy
evolves
bulk
this
respect.
than
b . c . c.
is
oxygen
known.
be q u i t e
system
stage
and t h i s
density
would
is
the
chemisorption/oxidation
chemisorption
to
little
calculate
u n d e r s t a n d how
way
which
bands
To
understand
from
[73].
the
oxide.
to
the
the
all
and to
oxygen
There
T h e LDA c a l c u l a t i o n
due
of
configurations,
crystalline
probably
approximation
iron
[27,71]
structure
failure
is
a long
predicts
for
of
bulk
the
Fe
local
100
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MONTANA STATI UNIVERSITY UIRARIES
Sith D378.H86
^
^
Self-consistent kxaHzed-orbital study
I_ _
3 1762 00190135 2