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Simulation methods
for electronic structure calculations:
Which method for which material
science problem?
Virginie SERIN,
Lionel CALMELS, CEMES Toulouse
EELS - EFTEM tutorials
Lausanne, September 2004 Toulouse
Out line
• Introduction
• Band Structure
Approach
• Multiple
Scattering
• Multiplet
• Conclusion
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Electronic Structure
and
Theoretical approaches
Band
Structure
Multiple
Scattering
Periodic
Structure
Molecular
Orbitals
Cluster
radius 5 Å
(7 shells)
Multiplets
Cluster
radius 2 Å
1(2) shell(s)
Single
Atom
+ crystal
field
Increasingly local description
Correlated systems
Metal
Semi-Conductor
Influence
Insulator . L 2,3 for 3d et 4d TM
. M4,5 for Rare Earth
of Core Hole
To understand
• Geometrical aspect:
Structure, site defect: Grain boundary, dislocation
core, interface, surface, disorder,
• Electronic structure aspect:
Valency, charge transfer, bonding, …
Toulouse
• Physical properties
Band
Multiple Molecular Multiplets
Structure Scattering Orbitals
Periodic
Structure
Cluster
radius 5 Å
(7 shells)
Cluster
radius 2 Å
1(2) shell(s)
Single
Atom
+ crystal
field
Codes
DFT + LDA or
MS:
MO
Multiplets
GGA
FEFF
SCF LCAO
LMTO-FLAPW CONTINUUM SCF X
WIEN2k
ICXANES
Pseudo-Pot:
VASP,
CASTEP, Material Studio
PARATEC,
Ab-INIT,
LKKR
AIMPRO
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ELNES
EELS
CB
E
EF
VB
E
Core state
E - E
Interpretation
One electron Theory
I(E)  IM(E) I2. (E)
“Atomic-like”
transition matrix element
Site- & symmetry-projected DOS
For small momentum transfer (q a 0)
Dipole selection rules l= ± 1
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Summary
• Introduction
• Band Structure
Approach
• Multiplet
• Multiple
Scattering
• Conclusion
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Band Structure approaches:
Infinite crystal with periodic structure
Calculation in the reciprocal space
Experimental parameters +
qualitative
Semi-empirical methods
(Tight binding -Huckel-)
• DFT:
 Ab initio methods: DFT +
Kohn-Sham + LDA or GGA:
KS: many body pb
•
 Space is divided in atomic
spheres (AS) and interstitial space
AS
AS
 The AS are not defined,
and a plane wave basis set
is used,
pseudo-potential methods
E(n)  F(n(r))   vex t (r)n(r)d 3 r
AS
AS
single
particle pbs
• in spheres: RnlYlm
• in interstitial:
- PW (FLAPW)
- SW (LMTO, ASW)
FP: non constant
potential outside AS
Pseudo-potential method:
- only valence electrons,
- core electrons and ions
potentials replace by
Toulouse
functions that vary
slowly
Reciprocal
space
Band Structure
Approach L
H
M
A
Starting point
K :
Unit cell + Periodicity
G
+ Analysis in terms of:
MoC-Hcp
 Hybridization
Mo-d
C-p
C-s
1,6
Graphite from Wien 2k
DOS
1,2
0,8
30.0
30.0
 Charge
transfer
20.0
20.0
0,4
0
-15
-5
0
Energ y ( eV)
5
TOTDOS Vmin
10.0
10.0
 Relaxation,
MoC-hpr
TOTDOS V1
TOTDOS V2
DOS
compression
0.00.0
-10
EFEF
-10.0
-10.0
-20.0
-20.0
- 12
- 10
-8
-6
-4
-2
0
-30.0
 Bonding
or anti-bonding
character
(COOP)
-30.0
DA
GG
M M K K LGLG
DA
(COOP=Crystal Orbital overlap Population)
2
4
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 complementary info /
experimental measurements

Physical Properties
 Optimisation of the
geometry
 Total energy
- Mechanical
- Optical
- Magnetic
Structural
parameters
Stability
 Charge Density
Limits
 Time consuming for complex
structures
 Non periodical structures
 Non compact structures
 First 15 eV of the unoccupied band are
well reproduced
 Transition with atomic character - L or
M edges- mono-electronic calculation
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Ni/Graphite interface- Wien 2k
0.75
0.75
Bulk
+ 7%
2.122
2.011
0.65
0.60
- 22%
0.65
0.60
2.014
0.55
2.028
0.50
0.50
0.45
0.45
C
+ 2%
- 16%
0.70
Ni Magnetic moment (B)
d(Å)
Ni Magnetic moment (B)
0.70
C1 C2
+ 7%
Bulk
+ 2%
Ni(111)
0.55
B
Ni(111)
A
AC
E5
q
q
q
integrated
graphene layer
graphene layer
Intensity [arb. units]
Interface
(Model C)
graphene
E4
E3
E2
E1
C1
C2
graphite
-5
0
5
10
15
20
25
30
35
Energy Loss [eV]
G. Bertoni, L. Calmels V. Serin et al,
Phys . Rev. B 2004 (In press)
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OLCAO within LDA-DFT in AlN polytypes
Simulations revealed
that broadness of the
first peak in Al-L2,3
and Al-K in wurtzite
and zinc-blend AlN
is related to the
covalency of the
compounds
Mizoguchi et al Micron 34 (2003)
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• Core hole
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Core hole
For accurate and quantitative comparison
between theory and experiments
 By exciting a core electron the screening of the
nucleus potential by the core electrons is reduced,
resulting in an enhanced attraction by the nucleus
 One electron theory : not an exact description
 Core hole perturbs the final state
 ≈ effect of Core hole increases by + e the effective
charge seen by the outer electrons
Non negligible for semi-conductors and insulators
(in metal CH is screened within a short distance)
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Wien
Mg-K in MgO
Insulator
Metals
Core hole
screened
by valence
electrons
Cu-L in Cu
From Hébert, Schattschneider et al. Micron 34 (2003)
Z+1 Approach: the excited atom is replaced by the neighbouring atom
in the periodic table to simulate the absence of a core electron.
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(cf Elsasser et al, Ultramicroscopy 2001, 2003)
Band Structure
Core hole - MgO
O-K in MgO with CH correction versus supercell size
from Ching et al. PRB 62, 2000
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Z+1
lY(r)l2 of the lowest CB state
Ground
state
Band Structure
Core hole
e- from
Mg-2p
excited
e- from
Mg-1s
excited
FIG. 8. uC ( r)u2 of the lowest CB state in the supe rc ellc alc ulation2 for
MgOlowest
on a ~001!
of the
rocksa
lt structurecalculation
. The c enter
lY(r)l
of the
CBplane
state
in the
supercell
is a t the Mg ion. The dark ~white! shade s re pr esentre gion of low
for MgO
on a (001) plane of the rocksalt structure.
~high! density. The gray a rea s a re re gions of inte rm edia te density.
- The
is at the
Mgsa ion.
~a! Ground
statecenter
c alc ulation.~
b! The
m e re sult obta ined using the
- The h.dark
represent
region
Z 1 1 a pproac
Note(white)
in the Z 1shades
1 a pproac
h,no distinc
tion cof
an low
be
m ade
betwe en the Mg- 1s or Mg- 2p c ore e xc ita tion. ~c! A Mg- 1s
(high)
density.
e lecZ+1
tr onisapproach,
e xc ite dto the
A Mg2p c ore
e lec tr on
Notec ore
in the
nolowestCB.
distinction~d!can
be made
between
is e xc ite d to the lowest CB.
the Mg-1s or Mg-2p core excitation.
also s hown that the Z1 1 approxi mat ionfor the core-hole
Ching et al, PRB 62, 7901 (2000)
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Summary
• Introduction
• Band Structure
Approach
• Multiplet
• Multiple
Scattering
• Conclusion
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MS
Multiple Scattering
Number of most significant paths
SS
MS
- Cluster, - direct space calculations
- Infinite Crystal with a local defect,
which suppresses the translational
Symmetry (Dyson equation)
• Electronic Green Functions,
Full MS calculation or restricted to some
scattering paths
• Spherical Potential (Muffin Tin)
• Core hole can
be taken into
account
C-K in graphite
EELS
FEFF 8 (with core hole)
FEFF 8 (without core hole)
ELNES (arb. units)
3
2
1
From L. Laffont Toulouse,
Toulouse
FEFF8
0
280
300
320
E loss (eV)
340
360
Multiple
Scattering
approach
Starting point
cluster
DOS N-K in h-AlN
 Shell by shell
 fingerprint:
identification of
the structure
Number of shell
information
8
7
4
3
2
1
Limits
eV
0
10
20
30
40
 Covalent bonds
Transition with atomic character - L or M edgesmono-electronic calculation
First eV of the spectrum (e.g. for L edges )
Toulouse
Mn-doped GB in SrTiO3
Mn-L2,3
O-K
MS:
FEFF 7
Moltaji H. O.
Micron
31 (2000) 381
Toulouse
MS applied to amorphous materials:
Example of a-Si
From Hayakawa K. Chem Phys Let 37 (2003) 498
Calculated ELNES for bond angle fluctuation models
Only Si atoms from
the 1st shell are rotated
c-Si
Si atoms from the 1st
and 2d shells are rotated
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Å
N
3.0
7Å
4.98Å
1.89
Å
3 .0 7
1.92Å
Localization of O in O-doped wurtzite AlN
MS approach
Al
Calculated spectra
N-K in AlN( O)
O In octaedral sites
N-K in AlN( O)
O In tetraedral sites
11. N K e dge inV.O-doped
Serin AlN
PRBc alc
58ulated
1998using m
Toulouse
• MS and DFT comparison
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N-K edge in cubic GaN
FEFF8
From Arslan et al. Micron 34 (2003) 255-260
VASP
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Summary
• Introduction
• Band Structure
Approach
• Multiplet
• Multiple
Scattering
• Conclusion
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Multiplet Approach
Intra-atomic effects
multi-electronic approach
Are taken into account
 N electron
Schrödinger equation
• Total kinetic energy of
the electrons,
• Coulomb interactions enucleus,
• Electrostatic repulsions
between nucleus,
• Spin -orbit coupling
• Possibly Zeeman term
 Valid for localised
final states
• L2,3 Transition Metals
• M4,5 Rare Earth
• Electronic repulsion,
 Fitting
parameters
• Spin-orbit coupling
Toulouse
• Crystal field
Multiplet
To interpret transition like
2p63dn
2p53d n+1
+ Analysis in terms of:
 Valency
 Magnetic
properties
 Structure
 Hybridization
 Charge transfer
 Dichroïsm
From F. De Groot
Coordination Chemistry Reviews
In Press 2004
 Limits
 Systems with weakly localised electrons
 Numerous fitting parameters
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EXAMPLE
Multiplets
Calculation
V L3
EELS
0.7 eV
(a)
Intensity (arb.units)
d2 Oh
V L2
10Dq = 1 eV
10Dq = 1.5 eV
10Dq = 2 eV
510
515
520
525
d2 D4h
Intensity (arb.units)
535
530
(b)
Ds = 0.1 eV
Ds = 0.2 eV
510
515
520
525
Energy Loss (eV)
d1 Oh
Intensity (arb.units)
535
530
(c)
rpd = 0.7
rdd = 0.7
rpd = 0.40
rdd = 0.35
510
515
520
525
Energy Loss (eV)
530
535
V valency in vanadium oxide
EELS experiments and d2 simulations
A. Gloter, V. Serin al. EPJ B 2000
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• Chemical shift
Toulouse
EXAMPLE:
Experiment:
Cu L3 EXAFS of Cu compounds
Grioni et al
PRB 39 1989
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Ferrous/Ferric in minerals
P. A. Van Aken et al. Phys Chem Minerals (2002) 29:188
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Energy shift calculation:
Comparison LDA (+ spin), GGA (+ spin)
10
O-K
5
He-K
GGA
Ne- K
LDA
0
GGA+Spin
-5
0
LDA+Spi n
800
600
400
200
Measured Edge Onset (eV)
1000
From Muller D. et al, Phys Rev Let 2002
respect to the experimental measu
G. 1. Errors (with
( ! np ) in free at
,22]) for EELS K -edge onsets 1s
culated from total-energy differences within the LD
GA approximations, both with and without spin polariz
Toulouse
• Gap
Toulouse
Gap
---- Im(e)
___ Im(-1/e)
Schamm S., Zanchi G.
Ultramicoscopy 2003
Simulations:
GGA-LDA: Gap underestimated
Toulouse
• Momentum resolved EELS
Toulouse
Anisotropy and angle dependence
of the ELNES

specimen

d

q
q //
q
2
2
2
E
q  q  q// avec q  ko et q//  ko 2E


2 1 /2
   
2
o
• or experiments are done at the Magic angle to
average over anisotropic effect,
• or angle resolved experiments
Simulations have to take into acount
the ELNES angle dependence
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Exemple of V2O5
Wien
O-K
Incident beam parallel
to the crystal c axis,
At different
collection angles,
And after removal
of the V-L edge
From Su D.S. , Hébert C. et al. Micron 2003
Toulouse
• Low Loss
Toulouse
Modelling EE low Loss spectra of dislocation
in Diamond, using AIMPRO
Atomic structure along the dislocation core in the gap region,
part of the dielectric function and predicted EELS spectrum for
A. a 90° Glide dislocation with a double period reconstruction,
B. a 60 ° shuffle dislocation,
C. and a 30 ° Partial dislocation with an open core
Toulouse
From C.J. Fall et al. Physica B 308-310 (2001) 577
ZrO2: Low Loss EELS and simulation:
using a LDA-DFT pseudo potential code
Integrated loss function for pure Zirconia: a) monoclinic,b)
tetragonal, c) cubic phases.
Solid line: experiment,
Dashed line Theory with local fields.
Insert: Theory without local fields
From N. Vast, M.C. Cheynet, et al. PRB (2004) In press
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• Method refinements
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PARATEC:
Parallel total energy code,
Pseudo potential, GW
TiO2 Rutile
Ti-K
E. Gaudry Thesis: http://www.lmcp.jussieu.fr/~antoine
PRB 67 094108 (2003)
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LSDA +U: Proposed by Anisimov in 1991 to bridge
the gap between LDA DFT and many body approaches
Useful for strongly correlated electronic systems
: partly filled d of f bands
LSDA +U: attempt to develop an effective one-particle
approach to the electronic structure of materials with
Strongly correlated electrons
DFT-LDA + UHF
Full Potential
on-site repulsion between electrons
See Dudarev , Botton et al, Micron 31 (2000)363
on NiO and UO2
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Continuum: developed by Natoli in 1980 in the FMS
formalism
 Absorbing atom and surrounding atoms are modelled by
a monoelectronic Muffin tin potential.
 Relaxed and screened potential calculated in the Z+1
Approximation.
 Combination of Molecular Dynamic and Multiple
Scattering.
The use of both MD and MS formalisms provides an
opportunity to validate the structural model by comparison
with XANES experimental data.
 Large cluster calculations, calculation averaged on
several clusters
 Allow site disorder from the absorbing atom to be taken
into account
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Summary
• Introduction
• Band Structure
Approach
• Multiplet
• Multiple
Scattering
• Conclusion
Toulouse
Rough Comparison
Calculation WIEN2k FEFF8
specificities
Full
potential
X
Material.
Studio,
VASP, ..
PARATEC
X
Pseudo
potential
X
X
X
Multi
electronic
Excited
states
Physical
properties:
Electronic
Density
Time
consuming
Convivial
Multiplet
X
X
X
X
XX
X
X
X
X
X
X
X
X
X
X
X
X
Localised
Defects
X
X
X
X
Strongly
correlated
system
X
Covalent
system
X
Materials
Disordered
materials
X
X
X
X
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To be thought :
 User friendly codes : point of view of
experimentalist/theoretician
 Manual adjustment, long and expensive
studies
 Quantitative and accurate investigations
 Too much defects : disordered materials,
pseudo-continuum of states
 Valence states often determine
physical properties
Toulouse
WIEN: http://www.wien2k.at/order/
FEFF: http://leonardo.phys.washington.edu/feff/
Paratec: http://www.nersc.gov/projects/paratec/
Material Studio: http://www.accelrys.com/mstudio/
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