interpreting eels fine structure - cornell...

David Muller 2006 1

Interpreting EELS Fine Structure

Notes to accompany the lectures delivered by David A. Muller at the SummerSchool on Electron Microscopy: Fundamental Limits and New Science heldat Cornell University, July 13-15, 2006.

Reading and References:

Muller, Singh and Silcox, Phys Rev B57, 8181 (1998) Theory of EELS as a LDOS

Fingerprints of most materials

V. Keast et al,(review of EELS)

Electron Shells and Transitions

Electron State Notation

4 f 5/2

Principal quantum number

Angular momentum state (ℓ)

ℓ=0 sℓ=1 pℓ=2 dℓ=3 f

J=L+S quantum number (jj coupling)

EELS transitions are named by the initial state (e.g. 2p3/2→ [3s1/2 3d5/2,3d5/2] is just L3 )EDX transitions are named by the initial core-hole state, the emission line, and the line strength (1 is the strongest, ) (e.g. 3d5/2→ 1s1/2 is Kα1 )

AES transitions are named by initial ionization, filling shell, shell of ejected electron (e.g. KL2L3)

K K

SHELL

L1

L2

L3

L

M1

M2

M3M

1s1/2

2s1/2

2p1/2

2p3/2

3s1/23p1/23p3/23d3/23d5/2

M4

M5

N1

N2

N3

4s1/24p1/24p3/24d3/24d5/2

N4

N5N

N6

N7 4f5/24f7/2

O1

O2

O3

5s1/25p1/25p3/25d3/25d5/2

O4

O5OO6

O7 5f5/25f7/2

O8

Κα1Κα2

Κβ1Κβ3

Κβ2

Lα1Lα2 Lβ1

Lβ2

Lβ5Lγ3

K

L1

L2

L3

M1

M2,3

M4,5

N1

N2,3

N4,5

N6,7

EELS Edges

X-raylines

David Muller 2006 3

10 4

10 5

10 6

10 7

10 8

0 100 200 300 400 500 600 700

Inte

nsity

(arb

. uni

ts)

Energy Loss (eV)

O-K edge

Si L edge

Incident Beam

Valence Excitations

Electron Energy Loss Spectrum of SiO2

Each edge sits on the tails of the preceeding edges -> Backgrounds are large

David Muller 2006 4

Energy Loss Spectrum of a 100 keV Electron Beam in Si

1 0

1 0 0

1 0 0 0

1 0 4

1 0 5

0 2 0 4 0 6 0 8 0 1 0 0

1 7 n m2 1 0 n m

CC

D C

ount

s

E n e r g y L o s s ( e V )

1 st P la s m o n

2 n d P la s m o n3 r d P la s m o n

Z e r o L o s s P e a k

In the thinner film (17 nm thick), only single scattering has occurred, and there is a single peak at the plasma energy (~17 eV) – this is also called a plasmon.

In the thicker film (210 nm), a significant portion of the electron beam has undergone inelastic scattering many times. In each scattering event it loses ~ 17 eV – so those electrons that have scattered twice show up as a peak at 2x17 = 34 eV, those that scattered 3 times at 3x17=51 eVand so on.

Plasmon mean free path λ~ 120 nm

t/λ = 0.14t/λ = 1.75

David Muller 2006 5

Effect of Thickness on the Si L23 Edge at 100 kV

• Plasmon Mean Free Pathin Silicon λp≈ 120 nm.

• When the thickness t/λp> 1 multiple plasmon scattering dominates the EELS spectra.

• At 210 nm, a ratio map at the 100 eV will measure the 6th plasmon, not the Si L edge!

2000

4000

6000

8000104

3 104

5 104

50 100 150 200

CC

D C

ount

s

Energy Loss (eV)

4th Plasmon

5th Plasmon

6th Plasmon

Si L2,3

Edge 17 nm

38 nm

78 nm114 nm

148 nm174 nm210 nm

Incr

easi

ng T

hick

ness

Moral: EELS needs thin samples!

David Muller 2006 6

Effect of Increasing the Illumination Angle (Effect of Increasing the Illumination Angle (α)α)(by reciprocity: increasing the collector angle in STEM)(by reciprocity: increasing the collector angle in STEM)

1 2

3 430 nm

3 mr 10 mr

80 mr ADF

α>>θobj

•No Phasecontrast

•No Diffractioncontrast

α<<θobj•Phasecontrast

•Diffractioncontrast

α≈θobj

•No Phasecontrast

•Diffractioncontrast

The incoherent BF image is the complement of the ADF image

David Muller 2006 7

Effect of Increasing the Illumination AngleEffect of Increasing the Illumination Angle(by reciprocity: increasing the collector angle in STEM)(by reciprocity: increasing the collector angle in STEM)

Coherent Illumination:• Condenser < Objective Aperture• Thickness fringes• Fresnel Contrast

Incoherent Illumination:• Condenser > Objective Aperture• Diffraction contrast suppressed

1.5 mr 80 mr

17 eV(Si Plasmon)

23 eV(SiO2 Plasmon)

Si

Si

SiO2

David Muller 2006 8

-200

0

200

400

600

800

1000

1200

1400

530 540 550

Inte

nsity

(a.u

.)

Energy (eV)

Is this real?

-200

0

200

400

600

800

1000

1200

1400

520 530 540 550 560 570

Inte

nsity

(a.u

.)

Energy (eV)

Always show the pre-edge background. Gives noise level & confidence in background subtraction

Interpreting Experimental Data

David Muller 2006 9

CoreCore--Level Level EElectron lectron EEnergy nergy LLoss oss SSpectroscopypectroscopy

EELS measures a local density of statespartitioned by

•site - as the probe is localized,•element - the core level bindingenergy is unique

- probes the conduction band- provides local electronic information

O 1s

EFermi

Dipole transition: s p

David Muller 2006 10

EELS Theory

Some subtleties as to which density of states is measured see Muller, Singh and Silcox, Phys Rev B57, 8181 (1998)

This is very important if you want to measure charge transfers (you don’t – there is no unique definition).

Dipole selection rules: 1,0,1 ±=∆±=∆ jl

ps →1K-edge: ;,2:;2: 3,21 sdpLpsL →→L-edge:

David Muller 2006 11Muller, Singh and Silcox, Phys Rev B57, 8181 (1998)

EELS as a Local Density of States (LDOS)

If we project the total density of states on to a local set of states and examine the overlap of each eigenstate with the local state . The probability of finding an electron in the eigenstate at site is , so the local contribution to the density of statesfrom site is

The basis set chosen for is not unique, so the amount of charge at site i is also not Unique. (e.g. a sphere of arbitrary size).

For EELS, the oscillator strength is proportional to a LDOS with a basis set of

{ }i

Don’t compare EELS to calculations of charge transfer

•To caution against directly comparing EELS ‘‘whitelines’’ against calculated charges, we show the charge transfer from an atomic sphere surrounding a Ni atom in the B2 NiAl compound, calculated in the LMTO-ASA approximation.

•The choice of the relative sphere sizes for the Ni and Al sites are a matter of computational convenience, rather than being a physically measurable property of the system.

•By altering the ratio of the Ni/Al sphere sizes we can change not only the magnitude, but also the sign of the Ni-Al charge transfer.

The charge-transfer problem: Since there is no unique definition of a local density of states, there is also no unique definition for charge transfers between local states

Muller, Singh and Silcox, Phys Rev B57, 8181 (1998)

David Muller 2006 13P. Rez, Ultramicroscopy 28 1989 16-23

K Edge

L Edge

2p->d dipole2p->p monopole2p->s dipole

Dipole Approximation is good for Core Level EELS

1s->p dipole1s->s monopole1s->d quadrupole

(except when the probe < core orbital size – can happen during channeling)

David Muller 2006 14P. Rez, Ultramicroscopy 28 1989 16-23

Dipole Approximation is good for Core Level EELS

The solid line is the full calculation, the dashed line is the dipolecontribution, and the light dotted line is the nondipole 2p-3p term

Si L23 edge for a 100 keV incident electron

Collection angles of 100 mrad (q= 167 nm-1)

Collection angles of 12.5 mrad (q=20 nm-1),


Pearson, Fultz and Ahn, Phys Rev B47 (1993)

EELS Fine Structure of Transition Metals

L2 L3

L3

L2

d DOS

EELS edgeGround state interpretation of spectra as a LDOS (ignore core hole)


Leapman, Grunes, Fejes,Physical Review B26 614-635 (1982)Cu L2,3 EdgeTi L2,3 Edge

EELS Fingerprints of Oxidation States


Fingerprint EELS from our websitewww.weels.net : EELS spectra of common semiconductor

materials

Identify the local environment from the shape of the spectrum

(e.g. Cu vs. CuO vs. Cu2O)


What Happens when Al is added to NixAl1-x?

BA

Energy Loss (eV)

0.00

10.0

20.0

30.0

40.0

50.0

845 850 855 860 865 870 875 880 885

NiNi3AlNiAl

514±20508±20508±20

Area (barns)

Ni L2,3 Edge After Deconvolution with Low Loss

•The Total Areas under each curve are very similar (no charge-xfer).•Ni d is broadened, shifting states from the main band, to the tails

increased Ni-Al bonding (Ni p-d hybridization)


EELS: Final State Effects

metal

δErel

δEex

δErel

insulator

Initial State Electrons relax to screen core hole

Excited electron couples to core hole


Comparison of the Ni LComparison of the Ni L33 Edge measured by EELSEdge measured by EELSwith the calculated, unoccupied d DOS of Niwith the calculated, unoccupied d DOS of Ni

[D.A. Muller, D.J. Singh, J. Silcox, P.R. B57 (1998) 8181]

Mor

e N

i-Al b

onds

NiAl

Ni3Al

Ni

Calculated

0 5 10 15Energy (eV)

NiAl

Ni3Al

Ni

Measured

0 5 10-5 15Energy (eV)

G.O

.S. (

e- /eV

/Ato

m) 0.00

0.20

0.10

0.00

0.10

0.20

0.00

0.10

0.20

0.30


Comparison of the Al LComparison of the Al L33 Edge measured by EELSEdge measured by EELSwith the calculated, unoccupied GOS of Alwith the calculated, unoccupied GOS of Al

[D.A. Muller, D.J. Singh, J. Silcox, P.R. B57 (1998) 8181]


Ground State Theory: the Si L3 Edge

X. Weng, P. Rez, P. E. Batson, Sol. Stat. Comm. 74 1013 (1990).(PAO calculation with 0.3 eV broadening)

VASP

•Weng aligns theory at edge onset

•All major features present, but-peak a is too weak-peak b is too high-c,d,e are in good agreement

•PAO results reproduced by VASP (using s/d = 1.1)


Excited State Theory: the Si L3 Edge

(Adaptive coordinates calculation with 0.3 eV broadening, core hole in 64 atom cell)

•align theory at edge onset

•ch: All major features present, but-peak a is too high-peak b is split, too wide-peak d is too sharp

•Only difference from ground state is in the first 1eV

Core hole in Si does not add new features at 0.3eV, just sharpens old ones

Z

chp


Comparison of the Measured Comparison of the Measured SiSi--L Edge with L Edge with abab--initioinitio CalculationsCalculations

Strong core-hole effects on the silicon-L Edge(it does not reflect the ground state)

0

0.05

0.1

0.15

0.2

0 5 10 15 20

Inte

nsity

(arb

. uni

ts)

Energy (eV)

Experiment

Z+1 (core hole)

(ground state)

Neaton et al, Phys. Rev. Lett. 85 1298(2000)


Comparison of the Measured Oxygen-K Edge with ab-initio Calculations

Inclusion of a core-hole effectsoverestimates the influence of the exciton

0

0.5

1

1.5

2

2.5

3

0 10 20 30 40

O* p-DOSO-p DOSExpt

DO

S (e

/eV)

Energy (eV)


When are core holes important?

• When you have good energy resolution (<1 eV)• When screening is poor

– Metals (small), semiconductors(medium), ionic (huge)– The effect is larger on anions than cations– More noticeable in nanoparticles and clusters than bulk

• Batson’s Rule: core hole effects are more pronounced when the excited electron is confined near the core hole. (It shouldn’t work, but it does.) – Atoms surrounded by strong scatterers (often nodeless valence

wavefunctions 1s, 2p, 3d…) (e.g Si in SiOx, Al in NiAl, TiB2 out of plane)


Limits of Density Functional Theory (DFT)

A practical matter:•The theory provides total energies for the ground state and (with constraints)

also the energies for excited states.

•One total energy per calculation (minutes → days)

•Core level binding energy Eb = Eexcited - Eground (a difference of 2 total energies)

•A full EELS spectrum needs 1 total energy per excited state ( α E3) YEARS!

Need to understand errors in DFT in order to produce effective approximations

•which errors are large, which will cancel?

•Use EELS binding energies to study systematic errors

We use approximate methods instead (1 calc. per spectrum)

David Muller 2006 28More localized the core hole, the bigger the error

A DFT Test Case: Noble Gas Edge OnsetsTotal Energies from GGA: Eb = Eexcited - Eground

GGA (generalized gradient approximation) is more accurate than LDA, but still:

•LDA, no spin: 1% error

•LDA+spin : -1% error

•GGA + Spin : 0.1% error(~ 0.5 eV at O-K edge)

Why spin?Less self-interactionsin localized core hole

-5

0

5

10

0 200 400 600 800 1000

Erro

r (eV

)

Measured Edge Onset (eV)

He-K

Ne-K

GGA+Spin

LDA

LDA+Spin

GGAO-K


LDA vs. “Exact” EigenvaluesThe real excitations of the system obey the Dyson equation:

The local density approximation (LDA) replaces with

Self-energy

Effect of LDA on Eigenvalues

No lifetime broadening

Problems with changing densitiescomplexNon-local

Energy-dependent

real

localEnergy-inindependent Band gap & shape of DOS are wrong!

(LDA eigenstates are fictitious constructs to solve the Kohn-Sham equations)

LDA DOS (Z, Z+1, all-electron) cannot rigorously describe EELS

( ) ( ) ( ) ( ) ( ) ( )rEdrrErrrrrVc ψψψψ =Σ+⎭⎬⎫

⎩⎨⎧ +∇− ∫ '',',

21 2

( )Err ,',Σ ( )rΣ

( )Err ,',Σ ( )rΣ

A Physical Interpretation for LDA Eigenvalues(LDA eigenvalues do not reproduce the true quasiparticle excitation spectrum)

Taylor Series Expansion of the EELS excitation from state i to f

Instead:

ie

Self-interaction:(~2%)

(slowly varying)

Hartree Energy:(~10%)

(short ranged, willcompress DOS)

Core level shift = eigenvalue difference + “relaxation energy”

“relaxation energy”

ii

Tot en

E=

∂∂

L+Π++−=∆ → fifiiffi FeeE ,,

( ) [ ] ( ) [ ]( )∫ ∂∂+∂∂=Π ρερρρερρρ xcxcfifi dr 22, 2

( ) ( ) ''', drdrrrrrF fifi ∫∫ −= ρρ


In Metals the Core Level Shift tracks the eigenvalue shift

• A Valence band shift changed electrostatic potential

• Core levels experience a similar electrostatic potentiala similar shift in binding energy

• Most accurate for nodelessvalence states (1s,2p,3d,4f)

• (Only true for metals - no gap)0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3

Val

ence

Ban

d Sh

ift (e

V)

Core Level Shift (eV)

NiSi2

NiAlNi

3Si

Ni3Al

NiSi

Core level shifts can tell us about the occupied, valence bands!

WHY?LMTO Valence shift vs. EELS CLS

[D. A. Muller, Ultramicroscopy 78 (1999) 163]


How Big is the Hartree Correction?Localization of the Ejected Electron alters the Spectrum

0

0.05

0.1

0.15

0.2

0.25

5 10 15 20

O+

O*

DO

S

Energy (eV)

O+: Excited e is delocalizedthroughout conduction band•Small overlap with core•Small Hartree correction

O*: Excited e- is localizedat edge onset.•Largest overlap with core•Largest Hartree energy

Cannot describe excitation quantitatively with 1 self-consistent calculation

Spectrum is stretched~ 2 eV

O-K Edge in α-Quartz


Core Level Pseudopotentials: Si L3

•Spin polarization is important (~ 0.9 eV correction )

•pseudopotential error increases with ionicity difference from the free atom.

( include the core hole in pseudopotential: easy to model large systems )

•(Z+1) total energy differences are as accurate

100

102

104

106

100 102 104 106

No spinSpin PolarizedZ+1 CLS

Cal

cula

ted

E(Si

L3)(e

V)

Measured E(Si L3)(eV)

c-Si

SiH

4

SiB

r 4 SiC

l 4

SiF4

α-Quartz

Si core hole has the same energetics as a P impurity


Core Level Binding Energies: O-K

•All published EB LDA calcs to date neglect spin.

•No spin: 8 eV error !

525

530

535

540

525 530 535 540

No spinSpin-polarizedCLS wrt O

2

Cal

cula

ted

E(O

-K) (

eV)

Measured E(O-K) (eV)

atom

ic O H

2 OO2

NO C

O α-Q

uart

z

( Total Energy GGA calculations - more accurate than LDA )

Absolute Error in DFT ~ 0.5 eV

Core level shifts are stillmore accurate thanabsolute values


Core Level Shifts in Insulators

•Excited-state eigenvaluedifferences track the

core-level shifts well

(0.6 eV rms error)

-3

-2

-1

0

1

2

3

4

5

-3 -2 -1 0 1 2 3 4 5

Excited StateGround State

Eige

nval

ue C

LS (e

V)

Total Energy CLS (eV)

H2O

α-Quartz

CONO

O2 •Scatter in ground state is

bad, but is there a trend?

(Bandgap will introduce errors ...)

O-K Edge CLS vs Eigenvalue differences wrt O2( )2sCBe eD -


Eigenvalue Errors vs Local Environment

•Errors tracks ionicity•Why?

Relaxation energy

depends on screening

from nearest neighbors.

-1

0

1

2

3

4

5

6

7

0 0.5 1 1.5

Ground StateExcited State

Eige

nval

ue C

LS E

rror

∆E

(eV)

Electronegativity Difference ∆χ

H2O

α-Quartz

CO

NO

O2

Eigenvalues work well when comparing atoms in similar local environments


Band edge onset 3 eV lower at interface - fewer O neighbors

O K-edge of Si-SiO2 Interface

a

b

0

50

100

150

200

520 530 540 550 560 570

Interfacial OxideBulk a-SiO

2

O-K

Edg

e C

ount

s (a

rb. u

nits

)

Energy Loss (eV)

David Muller 2006 38David Muller 2001

Oxygen K-edge at the Si/SiO2 Interface

GroundGround--StateStateTheoryTheory

ExperimentExperiment

0

5 0

1 0 0

1 5 0

2 0 0

5 3 0 5 3 5 5 4 0

In te r fa c ia l O x id e" B u lk -l ik e " S iO

2

O-K

Edg

e (a

.u.)

E n e rg y L o s s (e V )

00 .0 5

0 .10 .1 5

0 .20 .2 5

0 .30 .3 5

0 .4

5 3 5 5 4 0

In te r fa c e" B u lk -lik e " S iO

2

DO

S (e

V -1)

E n e r g y (e V )

00 .0 5

0 .10 .1 5

0 .20 .2 5

0 .30 .3 5

2 2 2 4 2 6 2 8 3 0 3 2

In te r fa c e" b u lk - lik e " S iO

2

DO

S (e

V -1)

E n e rg y (e V )

ExcitedExcited--StateState“Z+1 Approx”“Z+1 Approx”


EELS is very sensitive to Structural Changes

CLS ≅ C (Nd-5)/10 z1/2 exp(-κ r)

-4

-3

-2

-1

0

1

2

1.5 2 2.5 3 3.5

Cor

e L

evel

Shi

ft (e

V)

Bond Length (Å)

V

β-Cr

α-Cr

fcc Ni

bcc Ni

Fe

Equilibrium Bond Length

We expect

CLS is ~ 4 eV/A:

Moral: EELS calculations of defects must be done on relaxed structures


Core Hole LifetimesCore Hole Lifetimes

Resolution Limits:

0.1 eV Z~14 (Si L3 Γi=0.15 eV)

0.2 eV Z~21 (Sc L3 Γi=0.19 eV)

0.3 eV Z~25,14 (Na K Γi=0.30 eV)

Better than 0.1 eV is still useful for valence EELS-image electrically active defects, --Doesn’t require sub nm probe

Krause and Oliver, J. Phys. Chem. Ref. Dat. 8 (1979) 329.


20 eV abovethe edge, no feature can be sharper than 3 eV


Photon Energy

Photon Energy

50 meV

XAS of the Si L2,3 Edge at 15 meV Energy Resolution

R. Puttner et al, Phys. Rev. A57 297(1998).

•Vibrational modes are important at 100 meV resolution•Core hole lifetimes are measured at 50-80 meV (~ 50 – 100% larger than theory)


XAS of N-K in N2

F. Esaka et al, J. Elec. Spectr. and Rel. Phen. 88-91 (1998) 817-820

Instrument resolution is 70 meVVibrational states are resolved, but core-hole lifetime depends on the environment

N2 gas

N2 in Oxidized TiAlN


Inte

nsity

(arb

. uni

ts)

450 455 460 465 470 475Energy (eV)

Many-Body Corrections for the Ti-L Edge in SrTiO3

Single Particle Theory: (Ogasawara, PRB 2001)•includes core-hole self-consistently• L3: 2p3/2→ Ti 3d (t2g, eg)L2: 2p1/2→ Ti 3d (t2g, eg )

•Wrong oscillator strengths and positions

Configuration-Interaction Theory: (Ogasawara 2001)•4 main peaks analogous to SP theory•Peak a is multiplet of the 2p3/2→ t2g •No lifetime broadening (gaussian used)•2p spin-orbit splitting is too small

Experiment:XAS: van der Laan 1990

(2 eV too high)EELS: Muller 2002

100 meV resolution50 meV absolute accuracy

L3

L2

t2g eg

t2g

eg

L3

L2

a

a


The Muffin-Tin Approximation

This is a shortcut that makes it easier to solve Schrödinger's equation by approximating the potential

A good approximation for close-packed structures like metals(DOS inside spheres looks like an EELS final states)

From Joly


Calculation of the Final States•Cluster Methods: good for defects & clusters, often easier to run

•Muffin-Tin Potential (OK for Metals, bad for semiconductors)•FEFF7 – no self-consistency: must guess charge transfers•FEFF8 – self –consistent: good for metals

•Full Potential•FDMNES – no self-consistency, but it can input potentials from WIen2k

•Bandstructure methods: (3D periodic structures or supercells)•Almost all bandstructure codes are self-consistent now•Muffin-Tin Potential

•LMTO – good for close-packed structures, esp. metals•Full Potential

•FP-LAPW •Wien2k – easy to calculate matrix elements & core hole effects

•Plane-wave codes (faster and less prone to artifacts than APW codes•ABINIT (free, open-source and downloadable from abinit.org)•VASP (commercial)•CASTEP (commercial, fancy user interface)


Many-Body Calculations

•GW : Correct bandgap, but only a few atoms/supercell

•Configuration-Interaction (CI): very accurate for 1-6 atoms-good for transition metal oxide clusters

•Multiplet: (de Groot, van der Laan) single-atom in a crystal field – good for transition metal oxide crystals. - like CI, except it has adjustable parameters

Strongly-correlated systems, and materials with large core-hole effectscannot be calculated using DFT codes. The options are limited


Wien2k vs FEFF8TiC0.8N0.2

expt

expt

Wien2k

Wien2k

FEFF8

FEFF8

interpreting eels fine structure - cornell...

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