Electron Energy Loss Spectra of Graphite, Graphene and ...etsf.polytechnique.fr/system/files/20112008_Hambach_MORE08.pdf · Electron Energy Loss Spectra of Graphite, Graphene and

Electron Energy Loss Spectra of Graphite,Graphene and Carbon Nanotubes: PlasmonDispersion and Crystal Local Field Effects.

Ralf Hambach1, C. Giorgetti1, X. Lopez1, F. Sottile1, F. Bechstedt2,C. Kramberger3 , T. Pichler3, N. Hiraoka4, Y.Q. Cai4, L. Reining1.

1 LSI, Ecole Polytechnique, CNRS-CEA/DSM, Palaiseau, France2 IFTO, Friedrich-Schiller-Universitat Jena, Germany

3 University of Vienna, Faculty of Physics, Vienna, Austria4 National Synchrotron Radiation Research Center, Hsinchu, Taiwan

20. 11. 2008 — MORE08 Vienna/Austria

R. Hambach Collective Excitations in Carbon Systems

What do we want to describe?

collective excitationsI Coulomb interactionI probed by EELS, IXS

k0, E0 k0−

q, E0−

hω

q

contributions to the spectra

I elastic scattering → crystal structureI inelastic scattering → collective excitations

Key quantity: S(q, ω) ∝ −q2={ε−1(q, ω)}


How do we calculate ε−1?

ab initio calculations

1. ground state calculation gives φKSi

2. independent-particle polarisability χ0

3. RPA full polarisability χ = χ0 + χ0vχ

4. dielectric function ε−1 = 1 + vχ

(ε−1: no retardation, no multiple scattering,no Bragg reflection of the incident electron)

χ0

Codes:ABINIT: X. Gonze et al., Comp. Mat. Sci. 25, 478 (2002)DP-code: www.dp-code.org; V. Olevano, et al., unpublished.


www.dp-code.org

Self-Consistent Hartree Potentials

long range v0

I difference between EELS and absorptionI vanishes for large qI vanishes for localised systems

short range v

I crystal local field effects in solidsI depolarisation in finite systems


Outlook

dimensionality

1. induced Hartree potentials in low dimensional systems⇒ linear plasmon dispersion in SWCNT + Graphene

2. assembling nanoobjects⇒ role of interaction

inhomogenitiy

3. crystal local field effects in solids⇒ enhanced anisotropy in Graphite


Single-Wall CNTexperiments


Vertical Aligned SWNT

EEL spectraspecimen

I oriented SWCNTI diameter: 2 nmI nearly isolated

spectroscopy

I angular-res. EELSI resolution:

∆E = 0.2 eV∆q = 0.05 A−1

[ C. Kramberger, M.Rummeli, M. Knupfer, J. Fink, B.Buchner,T. Pichler, IFW Dresden, Germany ]



EEL spectra

π plasmon at 9eV in Graphite




EEL spectra

0 0.2 0.4 0.6 0.84

5

6

7

8

9

(a) Experiment

0momentum transfer q (1/Å)

4en

ergy

loss

(eV

)




I linear π plasmondispersion

I SWNT ⇔ graphene

0 0.2 0.4 0.6 0.84

5

6

7

8

9

(a) Experiment


4en

ergy

loss

(eV

)



Graphenecalculations


ε−1 in isolated nanosystems

IPA: ={ε−1} ∝ ={χ0}→ sum of interband transitions

RPA: ={ε−1} ∝ ={χ}full susceptibility χ = χ0(1− vχ0)−1

contains self-consistent response→ mixing of interband transitions

χ0


IPA: independent particles

energy loss in graphene(in-plane, q = 0.41 A−1)

0 2 4 6 8 10energy loss (eV)

- Im

ε-1

(a

rb. u

.)

IPA=⇒ given by ={χ0}:interpretation in terms ofband-transitions

χ0


IPA: independent particles



- Im

ε-1

(a

rb. u

.)

IPAπ−π∗ at K

bandstructure

M K Γ M-20

-15

-10

-5

0

5

Ene

rgie

(eV

)

π∗

π

σ

σ∗


RPA: random phase approximation



- Im

ε-1

(a

rb. u

.)

IPAπ−π∗ at KRPA

I given by ={χ}:no interpretation byband-transitions

I contributions from KI mixing of dispersion


RPA: random phase approximation



- Im

ε-1

(a

rb. u

.)

IPAπ−π∗ at KRPAwithout "K"

I given by ={χ}:no interpretation byband-transitions

I contributions from KI mixing of dispersion


Plasmon dispersion

01-Im(1/ε) [arb. u.]

0

2

4

6

8

10

Ene

rgy

loss

[eV

]

0 0.2 0.4 0.6 0.8 1momentum transfer q (1/Å)

0

2

4

6

8

10

π-pl

asm

on p

ositi

on -

Ene

rgy

(eV

)

RPAIPA

q=0.408 1/Å

1P. Longe, and S. M. Bose, Phys. Rev. B 48, 18239 (1993)2F. L. Shyu and M. F. Lin, Phys. Rev. B 62,8508 (2000)


SWNT vs. Graphenecomparison


SWCNT vs. Graphene

0 0.2 0.4 0.6 0.84

5

6

7

8

9

VASWCNT

(a) Experiment

0 0.2 0.4 0.6 0.8 1

(b) Calculation


4

ener

gy lo

ss (

eV)


SWCNT vs. Graphene

0 0.2 0.4 0.6 0.84

5

6

7

8

9

VASWCNT

(a) Experiment

0 0.2 0.4 0.6 0.8 1

graphene-1L

(b) Calculation


4

ener

gy lo

ss (

eV)


Role of Interactions

0 0.2 0.4 0.6 0.84

5

6

7

8

9

VASWCNTbulk-SWCNT

(a) Experiment

0 0.2 0.4 0.6 0.8 1

graphene-1L

(b) Calculation


4

ener

gy lo

ss (

eV)



0 0.2 0.4 0.6 0.84

5

6

7

8

9

VASWCNTbulk-SWCNTgraphite

(a) Experiment

0 0.2 0.4 0.6 0.8 1

graphene-1Lgraphite

(b) Calculation


4

ener

gy lo

ss (

eV)



0 0.2 0.4 0.6 0.84

5

6

7

8

9


(a) Experiment

0 0.2 0.4 0.6 0.8 1

graphene-1Lgraphene-2Lgraphite

(b) Calculation


4

ener

gy lo

ss (

eV)


Conclusions

SWCNT ⇐⇒ graphene

I isolated SWCNT ⇐⇒ graphene-1LI bundled SWCNT ⇐⇒ graphene-2L

graphene

I induced Hartree potentials importantI picture of independent transitionsI mixing of transitions/dispersions→ leads to linear dispersion

0 0.2 0.4 0.6 0.84

5

6

7

8

9


(a) Experiment

0 0.2 0.4 0.6 0.8 1


(b) Calculation


4

ener

gy lo

ss (

eV)


- Im

ε-1

(a

rb. u

.)

IPAπ−π∗ at KRPAwithout "K"


Single-Wall CNTcalculations [Xochitl Lopez]


(3,3) Single Wall Carbon nanotube

Experiment: oriented SWCNT(Diameter 20 A, nearly isolated)

Calculation: (3,3) SWCNT(Diameter 4 A, low interaction)


Graphitecalculation: revealing an angular anomalyexplication: in terms of crystal local field effectsexperiment: verification by inelastic x-ray scattering


On-axis: continuous

inelastic elastic

2 4 6 8 10energy loss (eV)

0

1

2

3

4

5

S(q

,ω)

(1/ keV

)

3

8/3

7/3

2

5/3

4/3

1

2/3

1/3

0

q1=0 q

3

c-a

xis

I energy loss S(q, ω)in graphite (AB)

I q along c-axisI weak dispersion1

I and off-axis?

1Y. Q. Cai et. al., Phys. Rev. Lett. 97, 176402 (2006)R. Hambach Collective Excitations in Carbon Systems

On-axis: continuous

inelastic elastic


0

1

2

3

4

5

S(q

,ω)

(1/ keV

)

3

8/3

7/3

2

5/3

4/3

1

2/3

1/3

0

q1=0 q

3

c-a

xis

I energy loss S(q, ω)in graphite (AB)

I q along c-axisI weak dispersion1

I and off-axis?

1Y. Q. Cai et. al., Phys. Rev. Lett. 97, 176402 (2006)R. Hambach Collective Excitations in Carbon Systems

Off-axis: discontinuous!

inelastic elastic


0

1

2

3

4

5

S(q

,ω)

(1/ k

eV)

3

8/3

7/3

2

5/3

4/3

1

2/3

1/3

0

q1=0 q3

c-a

xis


0

1

2

3

4

5

S(q

,ω)

(1/ k

eV)

3

8/3

7/3

2

5/3

4/3

1

2/3

1/3

0

q1=1/8 (~0.37 1/Å) q3


What is the origin?

RPA with full v RPA with v = 0


0

1

2

3

4

5

S(q

,ω)

(1/ k

eV)

3

8/3

7/3

2

5/3

4/3

1

2/3

1/3

0

q1=1/8 (with LFE)


3

8/3

7/3

2

5/3

4/3

1

2/3

1/3

0

q1=1/8 (without LFE)q3q3


Crystal Local Field Effects

dielectric function in crystals

I recall: ε−1 = 1 + vχ, χ = χ0 + χ0vχ

I RPA: ε = 1− vχ0

I ε is a matrix: ε(q, q′;ω) =(εGG′(qr , ω)

)I energy loss function (EELS, IXS)

S(q, ω) ∝ −={ε−1GG(qr , ω)}, q = qr+G

⇒ mixing of all transitions in χ0

⇒ crystal local field effects (LFE)

χ0


Physical picture of LFE: Dipoles

Two ways of understanding crystal local field effects:

dipole pictureperturbation induced dipoles

ϕt sc ++ϕikk

I induced local fieldsI crystal structure important


Physical picture of LFE: Coupled modes

plane wave pictureperturbing mode induced mode

eiq · r --ei(q+G) · rscll

I Bragg-reflection inside the crystalI couples modes with same qr

I εGG′(qr ) describes coupling δϕi (G)δϕt (G′)

⇒ key for understanding the discontinuity


Simple 2×2 model for LFE

I dominant coupling between the twomodes 0 and G = (0, 0, 2)

0BBBB@ε00 ... ε0G ......

...εG0 ... εGG ......

...

1CCCCA −→(

ε00 ε0GεG0 εGG

)

we introduce an effective 2×2-matrix ε

I Remember: the loss function was

S(q, ω) ∝ −={ε−1GG(qr , ω)}


Simple 2×2 model for LFE

I inverting the effective 2×2-matrix ε

ε−1GG(qr , ω) = 1

εGG+ εG0ε0G

(εGG)2 ε−100 (qr , ω)

without LF correction ...

(known as two plasmon-band model2)

2L. E. Oliveira, K. Sturm, Phys. Rev. B 22, 6283 (1980).R. Hambach Collective Excitations in Carbon Systems

1. Recurring excitations


0

1

2

3

4

5

S(q

,ω)

(1/ k

eV)

3

8/3

7/3

2

5/3

4/3

1

2/3

1/3

0

q1=1/8 (~0.37 1/Å) q3

ε−1GG(qr , ω) = 1

εGG+ εG0ε0G

(εGG)2 ε−100 (qr , ω)

S(qr +G) = SNLF(qr +G) + f ·S(qr )

coupling of excitations of momentumqr 1. Brillouin zone

qr + G higher Brillouin zone

⇒ reappearance3 of the anisotropicspectra from q → 0

3K. Sturm, W. Schulke, J. R. Schmitz, Phys. Rev. Lett. 68, 228 (1992).R. Hambach Collective Excitations in Carbon Systems



0

1

2

3

4

5

S(q

,ω)

(1/ k

eV)

3

8/3

7/3

2

5/3

4/3

1

2/3

1/3

0

q1=1/8 (~0.37 1/Å) q3

ε−1GG(qr , ω) = 1

εGG+ εG0ε0G

(εGG)2 ε−100 (qr , ω)








0

1

2

3

4

5

S(q

,ω)

(1/ k

eV)

3

8/3

7/3

2

5/3

4/3

1

2/3

1/3

0

q1=1/8 (~0.37 1/Å) q3

ε−1GG(qr , ω) = 1

εGG+ εG0ε0G

(εGG)2 ε−100 (qr , ω)






2. Strength of coupling


0

1

2

3

4

5

S(q

,ω)

(1/ k

eV)

3

8/3

7/3

2

5/3

4/3

1

2/3

1/3

0

q1=1/8 (~0.37 1/Å) q3

ε−1GG(qr , ω) = 1

εGG+ εG0ε0G

(εGG)2 ε−100 (qr , ω)

strength of coupling εG0 depends on:I angle ∠(qr , qr +G) andI structure factor ∝ density nG

⇒ enhances the angular anomaly

θ′η

(0,0,2

)q


Experimental verificationby inelastic x-ray scattering


IXS experiments

c-a

xis


2

4

6

8

10

12

14

S(q

,ω)

(1/ k

eV)

3

17/6

16/6

15/6

14/6

13/6

2

11/6

10/6

9/6

8/6

7/6

1

q1=1/8 (~0.37 1/Å) q3

I IXS experimentsN. Hiraoka

(Spring8, Taiwan)I elastic tail removedI uniform scaling


IXS experiments

c-a

xis


2

4

6

8

10

12

14

S(q

,ω)

(1/ k

eV)

3

17/6

16/6

15/6

14/6

13/6

2

11/6

10/6

9/6

8/6

7/6

1

q1=1/8 (~0.37 1/Å) q3


2

4

6

8

10

12

14

S(q

,ω)

(1/ k

eV)

3

17/6

16/6

15/6

14/6

13/6

2

11/6

10/6

9/6

8/6

7/6

1

q1=1/8 (~0.37 1/Å) q3

I IXS experimentsN. Hiraoka

(Spring8, Taiwan)I elastic tail removedI uniform scaling


Conclusions

graphite

I angular anomaly close to Bragg reflectionsI originates from local field effects (coupling to 1. BZ):

1. spectrum from q → 0 reappears (direction of qr )2. coupling εG0(qr ) enforces anisotropy

other systems

I all systems with strong local field effectse. g. layered systems, 1D structures

⇒ caution with loss experiments close to Bragg reflections


Summary

dimensionality

1. strong mixing of transitions (largeenergy range) in low dimensionalsystems (→ linear plasmon dispersion)

2. interactions important for small q

inhomogenity

3. discontinouity in S(q, ω) close to Braggreflections (result of plasmon bands)

0 0.2 0.4 0.6 0.84

5

6

7

8

9


(a) Experiment

0 0.2 0.4 0.6 0.8 1


(b) Calculation


4

ener

gy lo

ss (

eV)


0

1

2

3

4

5

S(q

,ω)

(1/ k

eV)

3

8/3

7/3

2

5/3

4/3

1

2/3

1/3

0

q1=1/8 (~0.37 1/Å) q3


Thank you for your attention!

publications:Linear Plasmon Dispersion in Single-Wall Carbon Nanotubes and theCollective Excitation Spectrum of GrapheneC. Kramberger, R. H., C. Giorgetti, M. Rummeli, M. Knupfer, J. Fink,B. Buchner, Lucia Reining, E. Einarsson, S. Maruyama, F. Sottile,K. Hannewald, V. Olevano, A. G. Marinopoulos, and T. Pichler[Phys. Rev. Lett. 100, 196803 (2008)]

Anomalous Angular Dependence of the Dynamic Structure Factornear Bragg Reflections: GraphiteR. H., C. Giorgetti, N. Hiraoka, Y. Q. Cai, F. Sottile, A. G. Marinopoulos,F. Bechstedt, and Lucia Reining[accepted by Phys. Rev. Lett. (2008)]

codes:ABINIT: X. Gonze et al., Comp. Mat. Sci. 25, 478 (2002)DP-code: www.dp-code.org; V. Olevano, et al., unpublished.


www.dp-code.org

Electron Energy Loss Spectra of Graphite, Graphene and ...etsf.polytechnique.fr/system/files/20112008_Hambach_MORE08.pdf · Electron Energy Loss Spectra of Graphite, Graphene and

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