The Inelastic Mean Free Path (IMFP): Theory, Experiment, and Applicationsbnl.gov/ps/nsls/workshops/2009/haxpes/files/Powell_Tanuma.pdf · The Inelastic Mean Free Path (IMFP): Theory,

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The Inelastic Mean Free Path (IMFP): Theory, Experiment, and The Inelastic Mean Free Path (IMFP): Theory, Experiment, and ApplicationsApplications

S. Tanuma, C. J. Powell*, D. R. Penn*, and K. Goto**S. Tanuma, C. J. Powell*, D. R. Penn*, and K. Goto**National Institute for Materials Science (NIMS)National Institute for Materials Science (NIMS)

*National Institute of Standards and Technology (NIST)*National Institute of Standards and Technology (NIST)**National Institute of Advanced Industrial Science and Technolo**National Institute of Advanced Industrial Science and Technology Chubugy Chubu

1. Introduction1. Introduction2. Calculation of IMFPs from optical data2. Calculation of IMFPs from optical data

-- Evaluation of energyEvaluation of energy--loss function dataloss function data-- Fano Plots Fano Plots -- Modified Bethe equation fitsModified Bethe equation fits-- Comparison with IMFPs from the TPPComparison with IMFPs from the TPP--2M equation2M equation

3. Experimental determination of IMFPs3. Experimental determination of IMFPs-- Results for 13 elemental solidsResults for 13 elemental solids

4. IMFP Applications4. IMFP Applications-- Effective attenuation lengthsEffective attenuation lengths-- Mean escape depthsMean escape depths-- Information depthsInformation depths-- Modeling XPS for ThinModeling XPS for Thin--Film StructuresFilm Structures

5. Summary5. Summary

I will be talking about Electron scattering in Surface electron spectroscopy. My talk consists of 4 part.The first is introduction.The second is calculations of IMFPs for wide variety of materials. I will talk about the evaluation of energy loss functions, Fano Plot,.The third is experimental determinations of IMFPs with elastic peak electron spectroscopy.The last is summary.

1. Introduction

The electron inelastic mean free path (IMFP) is a basic materialparameter for describing the surface sensitivity of XPS and other surface electron spectroscopies.

IMFP is needed for quantitative analyses by XPS (matrix correction), determination of film thicknesses (with effective attenuation lengths), and estimates of surface sensitivity (meanescape depths and information depths).

IMFPs have been determined for 75 materials from optical energy-loss functions with the Penn algorithm for energies from 50 eV to 30,000 eV (previously 50 eV to 2,000 eV)

The TPP-2M formula for predicting IMFPs has been evaluated for the 50 eV to 30,000 eV range (previously 50 eV to 2,000 eV)

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2. Calculation of IMFPs from optical data2. Calculation of IMFPs from optical data

Flow of calculationFlow of calculation

Experimental optical Experimental optical datadata: optical constants: optical constants: atomic scattering : atomic scattering factorsfactors

(ELF) (ELF) qq=0=0

-- check with Sum Rulescheck with Sum Rules

IMFPsIMFPs-- function of function of EE

-- energy dependence : energy dependence : optical energy loss function (optical energy loss function (q q = 0)= 0)

-- qq dependence : dependence : Lindhard model dielectric function (RPA)Lindhard model dielectric function (RPA)singlesingle--pole approximation (pole approximation (E E > 300 eV)> 300 eV)

Penn algorithmPenn algorithm

d 2dqd

me0

2

N hEIm 1

(q, )

1q

n 1

-5

0

5

10

15

20

25

30

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

10 100 1000 104 105

Co

f1

f2

n

k

atom

ic s

catte

ring

fact

or (f

1, f2

)

optical constants (n,k)

Electron Energy (eV)

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

10-2 10-1 100 101 102 103 104 105

cobalt

Ene

rgy

Loss

Fun

ctio

n

E(eV)

Im 1/ E

D. R. Penn, Phys. Rev. B 35, 482 (1987).

This shows the flow of IMFP calculation from energylossfunction.Electron inelastic mean free path in solid can becalculated from imaginary part of inverse dielectricfunctionorenergylossfunction.InPennalgorithm, theenergydependenceofELFcanbeobtained fromexperimentalopticalenergy loss function.As forqdependence, it is verydifficult tomeasurewithexperimentalmethod.ThentheLindhardmodeldielectricfunctionwasused.Over300eV,weusedthesinglepoleapproximationforqdependenceofELF.

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Conditions and materials for IMFP calculationsConditions and materials for IMFP calculations

- Energy range for IMFP calculation: 50 eV to 30,000 eV- calculated at equal intervals on a logarithmic energy

scale corresponding to increases of 10 %.

-- 42 elemental solidsLi, Be, diamond, graphite, glassy carbon, Na, Mg, Al, Si, K, Sc, Ti, V, Cr, Fe, Co,

Ni, Cu, Ge, Y, Zr, Nb, Mo, Ru, Rh, Pd, Ag, In, Sn, Cs, Gd, Tb, Dy, Hf, Ta, W, Re, Os, Ir, Pt, Au, and Bi

- 12 organic compounds26-n-paraffin, adenine, beta-carotene, diphenyl-hexatriene, guanine, kapton,

polyacetylene, poly(butene-1-sulfone), polyethylene, polymethylmethacrylate, polystyrene, and poly(2-vinylpyridine)

- 21 inorganic compoundsAl2O3, GaAs, GaP, H2O, InAs, InP, InSb, KBr, KCl, LiF, MgO, NaCl, NbC0.712,

NbC0.844, NbC0.93, PbS, SiC, SiO2, VC0.758 , VC0.858 and ZnS.

TheIMFPswerecalculatedinthe50eVto30keVenergyrange.Theywerecalculatedatequalintervals

Theseshowthelistofthecalculatedmaterials.Wehavecalculatedthese42elementalsolids,12organiccompoundsand21inorganiccompoundsasshownhere.Becausetheyhaveopticalconstantssufficientenergyrange.

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Evaluations of EnergyEvaluations of Energy--Loss Functions with Sum Loss Functions with Sum RulesRules

f-sum rule (or oscillator-strength sum rule )

KK-sum rule (a limiting form of the Kramers-Kronig integral)

Zeff (2 /

2 p2 ) E Im[1 / (E )]d (E )

0

Emax

Peff (2 / ) E1

0

Emax

Im[1 / (E )]d (E ) n2 (0)

When

Zeff Z

Peff 1

Emax : total number of electrons for material

The accuracy of the energy loss function affects the reliability of IMFPs. Then it is very important to evaluated energy loss function with sum rules.We use two effective sum rules.Using F-sum rule, we can evaluate the accuracy of ELF at high energy region.With KK-sum rule, the accuracy of low energy region of ELF , especially under 100 eV,can be evaluated.

When DE max becomes infinity, f-sum value Z eff equals to the total number of electron for material and Peff equals to unity if ELF is correct.

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Energy loss function and sumEnergy loss function and sum--rule rule calculations for organic compoundscalculations for organic compounds(e.g., polyethylene)(e.g., polyethylene)

10-11

10-9

10-7

10-5

10-3

10-1

101

100 101 102 103 104 105

Polyethylene

Ener

gy L

oss

Func

tion

E(eV)

Valence-bandexcitations

Poly-C2H4

Nv = 12

0

5

10

15

20

100 101 102 103 104 105

Polyethylene

Z eff

Emax

(eV)

Nv(12.3)

C2H4 (Z=16)

0

0.2

0.4

0.6

0.8

1

100 101 102 103 104 105

Polyethylene

Pef

f

Emax

(eV)

KK-sum error: 0.9 %

f sum

Nv=12

f-sum error: 0.0 %

:saturates over 60 eV :evaluated low-energy region (and n(0) value gave large contribution)

C K-shell excitations

KK sum

This shows the results of sum rule calculations for polyethylene. This is the energy loss function of polyethylene. Since polyethylene consists of low atomic number elements Hydrogen and carbon, its ELF is very simple.This peak corresponds to the valence and conduction electron excitation. This is due to the carbon K-shell ionization.This figure shows the F-sum rule results. We see the contribution of valence electrons ; Nv =12,3 this value is almost the same as theoretical value.

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Al and AlAl and Al22OO33: sum rule results: sum rule results

0

10

20

30

40

50

10-3 10-2 10-1 100 101 102 103 104 105

Al2O

3

Z eff

Emax

(eV)

Error = - 5.3 %

0

2

4

6

8

10

12

14

10-2 10-1 100 101 102 103 104 105

aluminumZ e

ff

Emax

(eV)

Error = - 0.9 %

0

0.2

0.4

0.6

0.8

1

10-2 10-1 100 101 102 103 104 105

aluminum

Pef

f

Emax

(eV)

0

0.2

0.4

0.6

0.8

1

10-3 10-2 10-1 100 101 102 103 104 105

Al2O

3

Pef

f

Emax

(eV)

Error = 0.9 %

Error = 1.7 %KK-sumf-sum

These figures are the calculated results of sum rule for aluminum and aluminum oxide.

Theoretical values Al = 13 (total electrons ) Al2O3 =50. The error of f-sum rule for Al is -0.9%, -5.3 % for Al2O3.

-Valence electrons : 3 for Al not clear Al2O3-K-shell :about 2 for Al: -For Al2O3 : the shell structure is not clear compared to Al

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SumSum--rule results for elemental solids and inorganic rule results for elemental solids and inorganic compoundscompounds

-40

-20

0

20

40

-40 -20 0 20 40

Inorganic compounds

Err

or o

f KK

-sun

rule

(%)

Error of F-sum rule (%)

GaAsInAs

InSb

Results for 21 inorganic compoundsRMS KK-sum error: 14% (6.7%)

RMS f-sum error: 7.4 %

-40

-20

0

20

40

-40 -20 0 20 40

Eelements

Erro

r of K

K-s

um ru

le (%

)


Zr

Nb Sn

V

Results for 42 elemental solidsRMS KK-sum error: 9.0 %

RMS f-sum error: 7.6 %

Elements

These figures show the results of sum rule calculations for elements and inorganic compounds.For 42 elemental solids, the root mean square error of KK-sum rules is 9.0 %. F-sum rule error is 7.8% . These values indicate these ELF are sufficiently accurate for IMFP calculations.

As for these compounds ad elements, we plan to measure optical constants by EELS.We also plan to calculate optical constants from band structure calculations. yesterday my colleage Shinotuka talked abut the calculation of optical constants for several metals.

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SumSum--rule results for organic compoundsrule results for organic compounds

-20

-10

0

10

20

-20 -10 0 10 20

KK-sum

Err

or o

f KK

-sun

rule

(%)


26-n-paraffin

Polyacetylene

RMS sum-rule error (fi%)

Average f-sum error: 7.0%

Average KK-sum error: 2.5 %(all compounds are within 6% error)

rms(%) f 2i

n

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Calculated IMFPs for 42 elemental solidsCalculated IMFPs for 42 elemental solids

Minima: 10 eV to 150 eV

IMFP range- factor of 2.2 at 70eV- factor of 5.9 at 30,000 eV

100

101

102

103

101 102 103 104 105

42 elemental solidsIn

elas

tic M

ean

Free

Pat

ha (

)


This shows the calculated IMFPs for 42 elemental solids as functions of electron energy in the 10 eV to 30 000 eV energy range. We see the minimum values are in the vicinity of 70 eV. There are large variation at 30 000 eV and very low energy region. The factor at 30 000 eV is 5.9. This means very large material dependence.

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Calculated IMFPs for compoundsCalculated IMFPs for compounds

100

101

102

103

101 102 103 104 105

12 Organic compounds

Inel

astic

Mea

n Fr

ee P

aths

()


1

10

100

103

101 102 103 104 105

Inel

astic

Mea

n Fr

ee P

ath(

)


21 inorganic compounds

Minima at 70 eV to 80 eVIMFP range: < factor of 1.3 (at 70eV, 30 keV)

Minima at 50 eV to 100 eVIMFP range: factor > 2.0 (at 70eV, 30 keV)

These figures show the calculated IMFPs as functions of electron energy for 12 organic compounds and 21 elemental solids.For organic compounds ,there are minimum values around 70 - 80 eV. The factor is small, 1.3 at 70 eV and 30 000 eV. This is the reason why their variation of valence electron density are small for organic compounds.In 21 organic compounds, the minimum points are in the 50 to 100 eV energy range. In the 70 to 30 000, the material dependence is almost constant. It is about 2.0.

Modified Bethe equation for IMFPModified Bethe equation for IMFP

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Bethe equation for total inelastic-scattering cross section in matter at electron energy E

EE p

2 ln E C / E D / E 2

IMFP equation: Modified Bethe equation for 50 eV to 30,000 eV range

Ep = bulk plasmon energy, , C, and D are parameters

M. Inokuti, Rev. Mod. Phys. 43, 297 (1971).

The Bethe.. can be described as this equation.Mtot2 square of dipole matrix elements for all possible scattering process.Based on this equation, we derived this equation for describing IMFPs in the 50 30,000 eV, We call it modified Bethe equation for IMFP.

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Analysis of energy dependence of IMFPs by Fano Plots: Analysis of energy dependence of IMFPs by Fano Plots: 50 eV to 30 000 eV50 eV to 30 000 eV

EE p

2 ln E C / E D / E 2

E E p2 ln E C E D E 2

(())

(eV/(eV/))

Fano plot:Fano plot:vsvs lnElnE

C, DC, D --> parameters> parametersFor high energiesFor high energies: Fano Plots : Fano Plots --> straight line> straight line

E E p2 ln E C / E D / E 2 E p2 ln E

E /

In the analysis of energy dependence of IMFPs, it is convenience to use Fan plots.Fano plot is expressed as the energy over lambda versus log E.

Using Modified Bethe equation, Fano plots can be descried this equation. This has 4 parameters.

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Fano plots for Si and RuFano plots for Si and Ru E E p2 ln E C E D E 2

0

10

20

30

40

50

60

70

80

101 102 103 104 105

silicon

calculated IMFPs

Mod. Bethe Eq. Fit

Ene

rgy/

IMFP

(eV

/)


RMS = 0.64%

0

20

40

60

80

100

120

101 102 103 104 105

ruthenium

calculated IMFPs

Mod. Bethe Eq. Fit

Ene

rgy/

IMFP

(eV

/)


RMS = 0.46%

C D0.0311 0.1186 1.104 30.3

C D0.0186 0.0763 1.484 40.5

These figures show the Fano Plots for Si and Ru and results of curve fits with M. Bethe equation,The open circles show the calculated IMFPs from optical data. The solid line is the curve fit with Modified. Bethe equation. The curve fits gave excellent results on Si and Ru.

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Fano plots for Fano plots for --carotene and polyethylenecarotene and polyethylene

E E p2 ln E C E D E 2

0

10

20

30

40

50

60

101 102 103 104 105

-carotene

calculated IMFPs

Mod. Bethe Eq. Fit

Ene

rgy/

IMFP

(eV

/)


0

10

20

30

40

50

60

101 102 103 104 105

Polyethylene

calculated IMFPs

Mod. Bethe Eq. Fit

Ene

rgy/

IMFP

(eV

/)


RMS = 0.1% RMS = 0.2%

C D0.0183 0.183 1.16 15.5

C D0.0188 0.168 1.28 18.5

These are the examples of organic compounds. Beta-carloten and polyethleye.The Modified Bethe equation gives also excellent fits for both compounds over 50 to 30 000 eV energy range.

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Fano plots for AlFano plots for Al22OO33 and GaAsand GaAs E E p2 ln E C E D E 2

0

20

40

60

80

100

101 102 103 104 105

Al2O

3calculated IMFPs

Mod. Bethe Eq. Fit

Ene

rgy/

IMFP

(eV

/)


RMS = 0.5%

0

10

20

30

40

50

60

101 102 103 104 105

GaAs

calculated IMFPsMod. Bethe Eq. Fit

Ener

gy/IM

FP(e

V/)


RMS = 1.1%

C D0.0146 0.0815 1.16 31.1

C D0.0306 0.0469 1.05 53.4

These are the examples of inorganic compounds. Al2O3 and GaAs. For Al2O3, M. Bethe fit show a good agreement for with energy range. For GaAs, the agreement is not good at higher energy region. This compounds is the worst case in 75 materials in this study. Fortunately, the difference is not so large,

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Results of Modified Bethe equation fits to calculated Results of Modified Bethe equation fits to calculated IMFPs over the IMFPs over the 50 eV to 30,000 eV 50 eV to 30,000 eV rangerange

Average RMS deviationAverage RMS deviation

Modified Bethe equation could be usedto fit IMFPs over the 50 eV to 30,000 eV energy range with low RMS deviations(less than 1.1%)GaAs and Co were the worst case!

42 elemental solids 42 elemental solids :: 0.47 %0.47 % 12 organic compounds12 organic compounds :: 0.12 %0.12 % 21 inorganic compounds :21 inorganic compounds : 0.56 %0.56 %

0

5

10

15

20

0 0.2 0.4 0.6 0.8 1 1.2

M. Bethe fit to IMFPS of 42 elemental soldis, 12 organic and 21 inorganic cpmpounds

elementsorganic compoundsinorganic compounds

Freq

uenc

y

RMS deviation (%)

I will show the results of modified Bethe equation fits to calculated IMFPs over the 50 to 30, 000 eV energy range.This show the histograms of the curve fit results using % of rms errors. The average of rms error of elements is 0.5%,These values are very nice results. Then, Modified Bethe equation could be used to fit IMFPs over the 50 to 30,000 eV with low rms deviations.

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TPPTPP--2M equation for IMFPs2M equation for IMFPs

EE p

2 ln E CE 1 DE 2 0.10 0.944 E p

2 E g2 0.5 0.069 0.1

0.1910.5

C 1.97 0.91UD 53.4 20.8U

U N vM

E p

2

829 .4

M: atomic or molecular weight: density

Nv: number of valence electronsEg: band-gap energy

- original energy range: 50 eV to 2000 eV (now extended to 30 keV)- based on IMFP data for 27 elemental solids and 14 organic compounds

S. Tanuma, C. J. Powell, and D. R. Penn, Surf. Interface Anal. 21, 165 (1994).

Then, We have developed the general formula TPP-2M empirically from the curve fit results with M. Bethe equation. Using this equation, we can estimate the IMFPs in the 50 30000eV at any material. IMFP can be calculated with these 4 physical parameters.

From now, we will compare the present calculated IMFPs with those of TPP-2M equation.

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Comparison of calculated IMFPs Comparison of calculated IMFPs for for Si and Ru Si and Ru with TPPwith TPP--2M 2M IMFPsIMFPs

1

10

100

103

101 102 103 104 105

silicon

calculated IMFPsMod. Bethe Eq. FitTPP-2M

Inel

astic

Mea

n Fr

ee P

ath(

)


RMS = 3.0 % ( 50 eV to 30,000 eV)

1

10

100

103

101 102 103 104 105

ruthenium

calculated IMFPs

Mod. Bethe Eq. Fit

TPP-2M

Inel

astic

Mea

n Fr

ee P

ath(

)


RMS = 5.0 % ( 50 eV to 30,000 eV)

These figures shows the comparison of calculated IMFPs from optical ELF and those of TPP-2M. This is silicon. This is ruthenium. Red circles show the calculated IMFPs, Blue solid line is the results of the curve fit with M. Bethe equation.Green doted line show the IMFPs of TPP-2M.

For both elements, the TPP-2M IMFP values are in good agreement with those of calculated IMFPs over 50 - 3000 eV energy range,The rms deviations are 3.0 % and 5 %.

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Comparison of calculated IMFPs for Comparison of calculated IMFPs for --carotene and carotene and polyethylenepolyethylene with TPPwith TPP--2M IMFPs2M IMFPs

RMS = 13.1 % (50 eV to 30,000 eV) RMS = 7.9 % (50 eV to 30,000 eV)

1

10

100

101 102 103 104

-carotene


Inel

astic

Mea

n Fr

ee P

ath(

)


1

10

100

101 102 103 104

Polyethylene

calculated IMFPs

Mod. Bethe Eq. Fit

TPP-2M

Inel

astic

Mea

n Fr

ee P

ath(

)


These are the results of organic compounds. For beta carotene, TPP-2M shows the lower IMFP values over 50 - 30 000 eV compared to optical IMFPs. For polyethlene, the TPP-2M IMFPs are in good agreement with calculated IMFPs. Its rms difference is 8%.

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Comparison of calculated IMFPs Comparison of calculated IMFPs for for AlAl22OO33 and GaAsand GaAs with with TPPTPP--2M IMFPs2M IMFPs

RMS = 37.3 % (50 eV to 30,000 eV)RMS = 14.5 % (50 eV to 30,000 eV)

1

10

100

101 102 103 104

Al2O

3

calculated IMFPs

Mod. Bethe Eq. Fit

TPP-2M

Inel

astic

Mea

n Fr

ee P

ath(

)


1

10

100

101 102 103 104

GaAs


Inel

astic

Mea

n Fr

ee P

ath(

)


These are the results of inorganic compounds. For Al2O3, the IMFP values calculated from TPP-2M are larger than the optical IMFPs over 50 to 30,000 eV.On the other hand TPP-2M gives lower values for GaAs like this. The difference is rather large.

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RMS differences between calculated IMFPs (from optical energyRMS differences between calculated IMFPs (from optical energy--loss loss functions) and TPPfunctions) and TPP--2M IMFPs over the 50 eV to 30,000 eV energy 2M IMFPs over the 50 eV to 30,000 eV energy

range: range:

0

10

20

30

40

50

60

70

80

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

RMS differences between calculated IMFPs and TPP-2M IMFPs

50 - 2,000eV

50 - 30,000eV

RM

S d

iffer

ence

(%)

U (mol/cm 3)

diamond

graphite

Cs

Average RMS: 13.3 % (10.3 %)

elemental solids elemental solids in organic and organic compoundsin organic and organic compounds

0

10

20

30

40

50

60

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

50 - 30 000 eV50 - 2000 eV

RM

S d

iffer

ence

(%)

U= Nv /Aw (mol/cm 3)

LiF

GaAs

InAs, InSb, InP

Average RMS: 8.1 % for organic14.3 % for inorganic

This shows the RMS differences between calculated IMFPs and TPP-2M IMFPs for elemental solids as a function of valence electron density . The average rms is 13 % for 42 elemental solids over 50 to 30,000 eV energy range.This figure show the rms differences as function of the valence electron density U for element.Cs, graphite diamond show the large difference. Except these elements, the rms decrees to 10 %.

This is the results for compounds.The average rms is 8 % organic compounds and 14 % for inorganic compounds. However, Lithium fluoride and gallium arsenide gave large rms differences. These are mainly d t th i f th i l

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Summary of IMFP calculationsSummary of IMFP calculations

We have calculated IMFPs for 50 eV to 30,000 eV electrons in 42 elemental solids, 12 organic compounds, and 21 inorganic compounds using their energy-loss functions and the Penn algorithm. The IMFPs were calculated at equal energy intervals on a logarithmic scale corresponding to increments of 10 %.

These IMFPs could be fitted to the modified Bethe equation forinelastic scattering of electrons in matter for energies from 50 eV to 30,000 eV. The average RMS deviations in these fits were 0.1 % (organic) and 0.5 % (elements and inorganic compounds).

This RMS values are almost the This RMS values are almost the same as those found in our fits of same as those found in our fits of IMFPs calculated previously for the IMFPs calculated previously for the 50 50 -- 2000 eV energy range.2000 eV energy range.

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Summary (continued)Summary (continued)

The optical IMFPs were also compared with IMFPs from the TPP-2M equation; the average RMS deviations were 13 % (elements), 8 % (organic compounds) and 14 % (inorganic compounds) for energies from 50 eV to 30,000 eV.

Relatively large RMS deviations were found for diamond, graphite, and cesium in the elements group. If these elements were excluded, the average RMS deviation was 10.3 %. Large deviations were also found for GaAs and LiF.

We conclude that the TPP-2M equation is useful for IMFP estimation in other materials for energies up to 30,000 eV with an average uncertainty of about 11 %.

S. Tanuma, C. J. Powell, and D. R. Penn, Surf. Interface Anal. 37, 1 (2005).

3. Experimental Determination of Electron 3. Experimental Determination of Electron Inelastic Mean Free Paths by ElasticInelastic Mean Free Paths by Elastic--Peak Peak Electron Spectroscopy (EPES)Electron Spectroscopy (EPES)

25

assess the reliability of IMFPs calculated from energy-loss functions with the Penn algorithm (optical IMFPs) and from the TPP-2M equation.

determine IMFPs for Ag, Au, Cr, Cu, Fe, Ga, C (graphite) , Mo, Pt, Si, Ta, W, and Zn over the 50 eV to 5,000 eV energy range from backscattered elastic-peak intensities (EPIs) using a Ni reference material.

C. J. Powell and A. Jablonski, J. Phys. Chem. Ref. Data 28, 19 (1999).S. Tanuma, T. Shiratori, T. Kimura, K. Goto, S. Ichimura, and C. J. Powell, Surf.

Interface Anal. 37, 833 (2005).

From now, I will talk about the experimental determination ofIt is very important to know.We have determine

We will compare the resulting experimental IMFPs with the

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Elastic-peak electron spectroscopy (EPES)

Theoretical Modelfor ElectronTransport

Electron EnergyIMFPs for theStandard Material

Experimental Ratio ofElastic BackscatteringProbabilities

Elastic ScatteringCross Sections

ExperimentalConfiguration

Material Parameters(composition, density)

IMFP

Elastically scattered electrons (E)Incident electron beam (E)

Sample (or reference material)

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Measurements of EPIs by absolute CMAMeasurements of EPIs by absolute CMA

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Measurements of EPIs by absolute CMAMeasurements of EPIs by absolute CMAMeasurements of EPIs by absolute CMA

Measurement of Elastic Peak IntensityMeasurement of Elastic Peak IntensityEnergy range : 1 eV to 5,000 eV Energy range : 1 eV to 5,000 eV (50 eV to 5,000 eV)(50 eV to 5,000 eV)Instrument: Absolute Auger SpectrometerInstrument: Absolute Auger Spectrometer--detection angle (42.3 detection angle (42.3 66))

Primary beam: 1Primary beam: 1AADetector: Faraday cup Detector: Faraday cup

This slide shows the instrument used for the measurement of elastic peak intensities. This system was made by Professor Goto. This system have a CMA and the detector is Faraday cup to measure the electron current directly. The measurements of EPIs were done in 1 to 5000 eV energy range.

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Measured elastic-peak intensity ratios (with respect to Ni) as a function of electron energyMeasured elasticMeasured elastic--peak intensity ratios (with peak intensity ratios (with

respect to Ni) as a function of electron energyrespect to Ni) as a function of electron energy

0

1

2

3

4

100 1000

Au/Ni

Ag/Ni

Cu/Ni

Si/Ni

Ela

tic P

eak

Inte

nsity

ratio


0

1

2

3

4

5

100 1000

Cr/NiFe/NiGa/Nigraphite/Ni

Ela

stic

pea

k in

tens

ity ra

tio (N

i-std

)

Electron energy (eV)

0

1

2

3

4

5

100 1000

Mo/NiPt/NiTa/NiW/Ni Zn/Ni

Ela

stic

pea

k in

tens

ity ra

tio (N

i-std

)


low Z: graphite, Si, similar Z: Fe, Cr, Zn, Cu, Gamedium Z: Ag, Mo high Z: Au, Pt, Ta, W

This viewgraph show the measured elastic peak intensity ratios as functions of electron energy for 13 elemental solids using Ni reference.

classify things into three types 4

Low atomic number : graphite, siliconMiddle atomic number: Fe, Cr, Zn, Cu Ga 28

29

Calculation of Elastic-Peak Intensities from Monte Carlo SimulationsCalculation of ElasticCalculation of Elastic--Peak Intensities from Monte Peak Intensities from Monte Carlo SimulationsCarlo Simulations

S: total path length = s1 + S2 + S3+ S4 ....

S1

S2S3

S4

e-

I Gt fs ddS

/N 0 exp

S

0

dS

Surface excitation factor

path-length distribution of electrons detected by CMA (42.3 6)

IMFP

0 10 0

2 10 -4

4 10 -4

6 10 -4

8 10 -4

1 10 -3

1.2 10 -3

1.4 10 -3

0 20 40 60 80 100 120

(d /dS)/N0

(d /dS)/N0x exp(-S/ )

(d

/dS)

/N0

Total pathe length ()

Ni at 1000 eV

:ElasticElastic--scattering cross sections fromscattering cross sections fromthe the ThomasThomas--FermiFermi--Dirac potentialDirac potential

:Pseudo random number generator:Pseudo random number generatorMersenne TwisterMersenne Twister

Instrumental function

This one show the calculation of , From Monte Carlo method, we can calculate the peak intensity from this equation.The Gt is a instrumental factor; mainly due to the transmission efficiency of CMA mesh. Fs is surface-excitation factor.S means the total path length. This graph is an example of the histograms of path length distribution.

In MC calculation, pseudo random number generator is very important. We use Mersenne twisterm which was developed by Mtaumoto.It is proved that the period is 2^19937-1, and the 623-dimensional equidistribution property is assured

29

30

Calculation of Elastic-Peak Intensity Ratio to Ni ReferenceCalculation of ElasticCalculation of Elastic--Peak Intensity Ratio to Ni ReferencePeak Intensity Ratio to Ni Reference

I x

I Ni

cal

d dS x / N 0x

0

exp S x dSd dS Ni / N 0Ni

0

exp S Ni dS: Determination of IMFPs using Ni standard

Then,

Solve above equation for parameter x (IMFP) for the target material using optical IMFP for Ni (with solver in Excel)

f sx / f s

Ni 1 - Assume initially that surface-excitation effects are negligible

I x

I Ni

mesaured

I x

I Ni

cal

2

min

; remove ; remove GGttDetermination of IMFPs from EPIs using Ni standardDetermination of IMFPs from EPIs using Ni standard

Using path length distributions of Ni-reference and the target material, the peak intensity ratio can be described by this equation.Then, optimizing this condition, we can get lambda x for target material. This calculations can be done easily with solver command in excel.

30

31

IMFPs determined from EPI ratios for Au, Ag, Si, and CuIMFPs determined from EPI ratios for Au, Ag, Si, and Cu

0

10

20

30

40

50

60

100 1000


Ag

IMFP

()

0

10

20

30

40

50

60

100 1000


Au

0

10

20

30

40

50

60

100 1000


Cu

0

20

40

60

80

100

100 1000


Si

IMFP

()

Solid circles: present work (EPES) solid lines: optical IMFPs dotted lines: TPP-2M

This viewgraph shows the determined IMFPs of Ag, Au, SI and Cu from their elastic peak in the 50 to 5000 eV,.The solid circles.In Au, Ag, Cu, the present IMFPs from EPES are in excellent agreement with theoretical values over 200 eV region like this.However, in Si the EPES-IMFPs coincide well with optical and TPP-2M IMFPs in the 100 to 1000 eV. Over 1000 and under 100 eV region, the determined IMFPs are larger than the theoretical values.

31

32

IMFPs determined from EPI ratios for Cr, Fe, Ga, and GraphiteIMFPs determined from EPI ratios for Cr, Fe, Ga, and Graphite

010203040506070

100 1000


Cr

0

10

20

30

40

50

60

100 1000


Fe

0

20

40

60

80

100

100 1000


graphite

IMFP

()

0

20

40

60

80

100

100 1000


Ga

32

33

IMFPs determined from EPI ratios for Mo, Pt, Zn, Ta, and WIMFPs determined from EPI ratios for Mo, Pt, Zn, Ta, and W

0

1

2

3

4

5

100 1000

Mo/NiPt/NiTa/NiW/Ni Zn/Ni

Elas

tic p

eak

inte

nsity

ratio

(Ni-s

td)


0

10

20

30

40

50

60

100 1000


Mo

IMFP

()

0

10

20

30

40

50

100 1000


Pt

0

10

20

30

40

50

60

100 1000


Ta

0

10

20

30

40

50

100 1000


W

01020304050607080

100 1000


Zn

Solid circles: present work (EPES) , solid lines: optical IMFPs,Solid circles: present work (EPES) , solid lines: optical IMFPs, dotted lines: TPPdotted lines: TPP--2M2M

33

34

Analysis of IMFPs with Fano PlotAnalysis of IMFPs with Fano Plot-- We have analyzed experimentally determined IMFPs using Fano PloWe have analyzed experimentally determined IMFPs using Fano Plots which ts which were constructed by plotting values of were constructed by plotting values of E/E/ versuversus ln s ln EE..

100 1000 104

Ener

gy/

(eV

/)


Si

Cu

Ag

Au

0

0

0

0

20

60

100

140

100 1000 104

Ener

gy/

(eV

/)


Zn

Mo

TaPt

0

0

0

0

40

80

120

160

0

W

solid symbols :experimentally determined IMFPssolid symbols :experimentally determined IMFPssolid lines: fit to the energy/IMFP values with the Bethe equatisolid lines: fit to the energy/IMFP values with the Bethe equation on

E E p2 ln E Simple Bethe equation

100 1000 104E

nerg

y/ (

eV/

)Electron energy (eV)

graphite

Ga

Fe

Cr

0

0

0

0

20

60

100

140

E / b a ln( E )

Sincethedatapointsliesufficientlyclosetostraightlines,FanoplotcanbefitbysimpleBetheequation.Thenthevaluesoftheparametersbetaandgammawerefoundfromalinearleastsquaresanalysis.

34

35

RMS difference of IMFPs determined by EPES from optical IMFPs (PRMS difference of IMFPs determined by EPES from optical IMFPs (Penn enn algorithm) and IMFPs from the TPPalgorithm) and IMFPs from the TPP--2M equation2M equation

Average Average RMSRMS (100 eV to 5000 eV): (100 eV to 5000 eV): 11.0 % (optical)11.0 % (optical)10.7 % (TPP10.7 % (TPP--2M)2M)

finally I will show you the results of IMFP comparison.This is the results of the comparison with optical IMFPs and, this is the results with TPP-2M.The average rms is ..

35

SummarySummary We have performed experimental determinations of IMFPs for 13 We have performed experimental determinations of IMFPs for 13

elemental solids over the 50 eV to 5000 eV energy range from elemental solids over the 50 eV to 5000 eV energy range from backscattered elasticbackscattered elastic--peak intensities using a Ni reference peak intensities using a Ni reference together with Monte Carlo simulations.together with Monte Carlo simulations.

These IMFPs determined could be fitted with the simple Bethe These IMFPs determined could be fitted with the simple Bethe formula over the 100 eV to 5000 eV energy range using Fano plotsformula over the 100 eV to 5000 eV energy range using Fano plots((average RMS deviation: 9 %average RMS deviation: 9 %))

The IMFPs of Ag, Au, Cr, Cu, Fe, Pt, Si, Ta, and W were in The IMFPs of Ag, Au, Cr, Cu, Fe, Pt, Si, Ta, and W were in excellent agreement (excellent agreement (RMS deviations less than 11%RMS deviations less than 11%) with those ) with those calculated from the Penn algorithm (optical IMFPs) over the 100 calculated from the Penn algorithm (optical IMFPs) over the 100 eV to 5000 eV energy range. eV to 5000 eV energy range.

36

36

37

We conclude that the accuracy of IMFPs for elemental We conclude that the accuracy of IMFPs for elemental solids solids calculated from measured energycalculated from measured energy--loss functions is loss functions is about 10 % over the 100 eV to 5000 eV energy range.about 10 % over the 100 eV to 5000 eV energy range.

37

4. IMFP Applications4. IMFP Applications- Quantitative XPS [1]

- Other important parameters [2]:- Effective attenuation lengths (EALs)- Mean escape depths (MEDs)- Information depths (IDs)

- Modeling XPS for Thin-Film Structures [3]

1. C. J. Powell and A. Jablonski, J. Electron Spectrosc. Relat. Phenom. (in press).2. C. J. Powell and A. Jablonski, Nucl. Instr. Methods Phys. Res. A 601, 54 (2009).3. W. S. M. Werner, W. Smekal, and C. J. Powell, Surf. Interface Anal. 37, 1059 (2005).

38

Effective Attenuation LengthsThe effective attenuation length (EAL) is the parameter to be used in an equation in place of the IMFP to account for the effects of elastic scattering.

Jablonski and Powell [1] proposed a simple empirical formula for the average EAL for film-thickness measurements, L, for emission angles between 0 and 50:

where in is the IMFP and tr is the transport mean free path (TMFP) which is derived from the differential cross section for elastic scattering. This formula was developed for electron energies between 100 eV and 2 keV,but should be valid for higher energies.

The plot shows as a function of electron energy for six illustrative elemental solids [2].

1. A. Jablonski and C. J. Powell, J. Vac. Sci. Technol. A 27, 253 (2009).2. C. J. Powell and A. Jablonski, J. Electron Spectrosc. Relat. Phenom. (in press).

)735.01( inLtrin

in

39

Mean Escape DepthsThe mean escape depth (MED) is the average depth normal to the surface from which the detected electrons escape.

In the absence of elastic scattering, the MED, D, iselectron emission angle (with respect to the surface normal).

When elastic scattering is considered, an empirical formula for D proposed by Jablonski and Powell [1] can be used for emission angles between 0 and 50:

This formula was developed for electron energies between 100 eV and 2 keV,but should be valid for higher energies.

1. A. Jablonski and C. J. Powell, J. Vac. Sci. Technol. A 27, 253 (2009).

trin

in

cosinD

)736.01(cos inD

where is the

40

Information DepthsThe information depth (ID) is the depth normal to the surface from which useful signal information is obtained.

In the absence of elastic scattering, the ID, S, iswhere P is a specified percentage of the detectedsignal.

When elastic scattering is considered, an empirical formula for S proposed by Jablonski and Powell [1] can be used for emission angles between 0 and 50:

This formula was developed for electron energies between 100 eV and 2 keV,but should be valid for higher energies.

1. A. Jablonski and C. J. Powell, J. Vac. Sci. Technol. A 27, 253 (2009).

trin

in

)100/(1

1lncosP

S in

)100/(1

1ln)787.01(cosP

S in

41

Modeling XPS for Thin-Film Structures

The NIST Database for Simulation of Electron Spectra for Surface Analysis (SESSA) [1,2] provides data for quantitative XPS for X-ray energies up to 20 keV. SESSA can also perform efficient Monte Carlo simulations of XPS spectra for multi-layered thin-film samples.

Version 1.2 of SESSA (expected to bereleased in fall, 2009) will include thecapability to perform XPS simulationswith varying amounts of X-raypolarization (both linear and circular).This capability will make SESSAuseful for XPS applications withsynchrotron radiation.

1. http://www.nist.gov/srd/nist100.htm.2. W. Smekal, W. S. M. Werner, and C. J. Powell, Surf. Interface Anal. 37, 1059 (2005).

42

5. Summary1. IMFPs have been calculated for 42 elemental

solids, 12 organic compounds, and 21 inorganicfrom experimental optical data for electronenergies from 50 eV to 30 keV.

2. The modified Bethe equation fits the opticalIMFPs well over the entire energy range.

3. The TPP-2M equation provides reasonableestimates of IMFPs over the entire energyrange (although there were large deviations for graphite, diamond, Cs, GaAs, and LiF).

4. IMFPs from elastic-peak electron spectroscopyexperiments generally agree well with the optical IMFPs.

5. Simple analytical formulae have been developed to provide useful estimates of effective attenuation lengths (for film-thickness measurements), mean escape depths, and information depths (for estimates of surface sensitivity). These formulae enable convenient corrections for elastic-scattering effects (for 50).

6. The NIST SESSA database provides physical data (including IMFPs from the TPP-2M equation) for energies up to 20 keV. A planned enhancement will enable simulations of XPS spectra with polarized X-rays.

100

101

102

103

101 102 103 104

42 Elemental Solids

Inel

astic

Mea

n Fr

ee P

ath

()


43

Optical IMFPs (points) for Li, Be, graphite, diamond, glassy C, and Na

101 102 103 104 105

Inel

astic

Mea

n Fr

ee P

ath

()


Li

Be

Graphite

Diamond

GlassyCarbon

Na

101

102

103

101

102

103

100

100

100

100

100

100

100

Solid lines show fits with the modified Bethe equation

Dashed lines show IMFPs from the TPP-2M equation

44

Optical IMFPs (points) for Mg, Al, Si, K, Sc, and Ti



101 102 103 104 105

Inel

astic

Mea

n Fr

ee P

ath

()


Mg

Al

Si

K

Sc

Ti

101

102

103

101

102

103

100

100

100

100

100

100

100

45

Optical IMFPs (points) for V, Cr, Fe, Co, Ni, and Cu



101 102 103 104 105

Inel

astic

Mea

n Fr

ee P

ath

()


V

Cr

Fe

Co

Ni

Cu

101

102

103

101

102

103

100

100

100

100

100

100

100

46

Optical IMFPs (points) for Ge, Y, Zr, Nb, Mo, and Ru



101 102 103 104 105

Inel

astic

Mea

n Fr

ee P

ath

()


Ge

Y

Zr

Nb

Mo

Ru

101

102

103

101

102

103

100

100

100

100

100

100

100

47

Optical IMFPs (points) for Rh, Pd, Ag, In, Sn, and Cs



101 102 103 104 105

Inel

astic

Mea

n Fr

ee P

ath

()


Rh

Pd

Ag

In

Sn

Cs

101

102103

101

102

103

100

100

100100

100

100

100

104

48

Optical IMFPs (points) for Gd, Tb, Dy, Hf, Ta, and W



101 102 103 104 105

Inel

astic

Mea

n Fr

ee P

ath

()


Gd

Tb

Dy

Hf

Ta

W

101

102

103

101

102

103

100

100

100

100

100

100

100

49

Optical IMFPs (points) for Re, Os, Ir, Pt, Au, and Bi



101 102 103 104 105

Inel

astic

Mea

n Fr

ee P

ath

()


Re

Os

Ir

Pt

Au

Bi

101

102

103

101

102

103

100

100

100

100

100

100

100

50

The Inelastic Mean Free Path (IMFP): Theory, Experiment, and ApplicationsS. Tanuma, C. J. Powell*, D. R. Penn*, and K. Goto**National Institute for Materials Science (NIMS)*National Institute of Standards and Technology (NIST)**National Institute of Advanced Industrial Science and Technology Chubu Slide Number 22. Calculation of IMFPs from optical data Conditions and materials for IMFP calculations Evaluations of Energy-Loss Functions with Sum RulesEnergy loss function and sum-rule calculations for organic compounds(e.g., polyethylene)Slide Number 7Sum-rule results for elemental solids and inorganic compoundsSum-rule results for organic compounds Calculated IMFPs for 42 elemental solidsCalculated IMFPs for compoundsModified Bethe equation for IMFPAnalysis of energy dependence of IMFPs by Fano Plots: 50 eV to 30 000 eVFano plots for Si and RuFano plots for -carotene and polyethyleneFano plots for Al2O3 and GaAsResults of Modified Bethe equation fits to calculated IMFPs over the 50 eV to 30,000 eV rangeTPP-2M equation for IMFPsComparison of calculated IMFPs for Si and Ru with TPP-2M IMFPsComparison of calculated IMFPs for -carotene and polyethylene with TPP-2M IMFPsComparison of calculated IMFPs for Al2O3 and GaAs with TPP-2M IMFPsRMS differences between calculated IMFPs (from optical energy-loss functions) and TPP-2M IMFPs over the 50 eV to 30,000 eV energy range: Summary of IMFP calculationsSummary (continued)3. Experimental Determination of Electron Inelastic Mean Free Paths by Elastic-Peak Electron Spectroscopy (EPES)Slide Number 26Measurements of EPIs by absolute CMAMeasured elastic-peak intensity ratios (with respect to Ni) as a function of electron energyCalculation of Elastic-Peak Intensities from Monte Carlo SimulationsCalculation of Elastic-Peak Intensity Ratio to Ni ReferenceIMFPs determined from EPI ratios for Au, Ag, Si, and CuIMFPs determined from EPI ratios for Cr, Fe, Ga, and GraphiteIMFPs determined from EPI ratios for Mo, Pt, Zn, Ta, and WAnalysis of IMFPs with Fano PlotRMS difference of IMFPs determined by EPES from optical IMFPs (Penn algorithm) and IMFPs from the TPP-2M equation SummarySlide Number 37Slide Number 38Slide Number 39Slide Number 40Slide Number 41Slide Number 42Slide Number 43Slide Number 44Slide Number 45Slide Number 46Slide Number 47Slide Number 48Slide Number 49Slide Number 50

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The Inelastic Mean Free Path (IMFP): Theory, Experiment, and Applicationsbnl.gov/ps/nsls/workshops/2009/haxpes/files/Powell_Tanuma.pdf · The Inelastic Mean Free Path (IMFP): Theory,