Spatial Resolution in Electron Energy-Loss Spectroscopy...Spatial Resolution in Electron Energy-Loss...

Spatial Resolution in Electron Energy-LossSpectroscopy

Ralf Hambach1,2, C. Giorgetti1,2, F. Sottile1,2, Lucia Reining1,2.

1 LSI, Ecole Polytechnique, CEA/DSM, CNRS, Palaiseau, France2 European Theoretical Spectroscopy Facility (ETSF)

23. 03. 2009 — DPG09, Dresden

R. Hambach Spatial Resolution in EELS

What is EELS?

angular resolved

I broad beam geometryI momentum transfer q, energy loss ~ωI resolution: ∆q ≈ 0.05 A−1 ,∆~ω ≈ 0.2eV

k0 ,E

0

k0 −

q, E

0 −hω

q

EELS for graphite(π and π + σ plasmon)[M. Vos, PRB 63, 033108 (2001).]


What is EELS?

spatially resolved

I highly focussed beamI impact parameter b, energy loss ~ωI resolution: ∆b ≈ 1 A,∆~ω ≈ 0.5eV

E0

E0 − hω

b

EELS for Si/SiO2

[M. Couillard et. al., PRB 77, 085318 (2008).]


Outlook

1. Methods and Theory2. Spatially Resolved EELS: Graphene3. Charge fluctuations in Graphite


Semi-classical TheoryHow do we calculate EELS ?


EELS: Classical Perturbation

v

b

1. electron flying along straight line

ρext (r , t) = −eδ(r − (b + v t)

)2. external potential (no retardation)

∆ϕext = ρext/ε0

3. linear response of the medium

ρind = χϕext , ε−1 =1+vχ

4. energy loss of the electron

〈dWdt 〉t =

∫dr〈E ind ·jext〉t



v

b

ϕext


ρext (r , t) = −eδ(r − (b + v t)






〈dWdt 〉t =




v

b

ϕextϕind


ρext (r , t) = −eδ(r − (b + v t)






〈dWdt 〉t =




v

b

ϕextϕind


ρext (r , t) = −eδ(r − (b + v t)






〈dWdt 〉t =


⇒ energy loss probability S(b, ω)


EELS: Quantum Mechanical Response

ab initio calculations (DFT)

1. ground state calculation gives φKSi

2. independent-particle polarisability χ0

3. RPA full polarisability χ = χ0 + χ0vχ

non-local response of crystals

I ρind (r , t) =∫χ(r , r ′; t − t ′)ϕext (r ′, t ′)

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 Sci. 25, 478 (2002)DP-code: www.dp-code.org; V. Olevano, et al., unpublished.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

Spatially Resolved EELSAloof Spectroscopy for Graphene


Energy Loss for Graphene

Electron parallel to sheet(super-cell with 30 A vac)

I non-dispersive modesI surface plasmon: 6.2eVI further excitations at

20eV and 32eV

Energy loss for different pos. b

0 5 10 15 20 25 30 35Energy Loss ω [eV]

0

2

4

6

8

10

12

14

S(b

,ω)

[arb

.u.]

b = 0 Åb = 0.6 Åb = 1.2 Åb = 1.8 Åb = 6 Å

x 4



I spatial distribution of the lossprobability for6eV, 22eV and 32eV

I atomic resolution (∆b =1.1A)



I exponential decay with bI delocalization

-2 0 2 4 6 8Impact Parameter b [1/Å]

2

4

6

8

10

S(b

,ω)

[arb

.u.]

E= 6 eVE=22 eVE=32 eV


Outlook

I Cerenkov radiationI non-local effectsI experiments 1. electron flying along straight line

ρext (r , t) = −eδ`r − (b + v t)

´2. external potential (no retardation)


3. linear response of the mediumρind = χϕext , ε−1 =1+vχ

4. energy loss of the electron〈 dW

dt 〉t =R

dr〈E ind ·jext 〉t5. non-local response χ(r , r ′, t − t ′)


Induced Charge DensityLooking at Plasmons in Graphite


Graphite: π–Plasmon

Induced charge fluctuations ρind = χϕext

I plane wave perturbation

ϕext (r , ω) ∝ e−i(ωt−qr)

I |q| = 0.74 A−1 , λ = 8.5 AI ~ω = 9eV




(resolution: 0.8 A, isosurface at 50%)



I |q| = 0.74 A−1 , λ = 8.5 AI ~ω = 9eV





(partial) ground state density

pz - orbitals sp2 - orbitals


Graphite: π + σ–Plasmon





I |q| = 0.74 A−1 , λ = 8.5 AI ~ω = 30eV


Graphite: π + σ–Plasmon



(partial) ground state density

pz - orbitals sp2 - orbitals


Outlook

I realistic external potential→ electron in graphite

I relation with plasmonsI experimental accessibility

9MeV gold ion in water(from IXS measurements)[P. Abbamonte, PRL (92), 237401 (2004)]


Summary

Spatially resolved EELS

1. atomic resolution in EELS calls forab-initio calculations

2. first tests for graphene layer

induced charge density

3. visualisation of density fluctuations


Summary

Spatially resolved EELS

1. atomic resolution in EELS calls forab-initio calculations

2. first tests for graphene layer

induced charge density

3. visualisation of density fluctuations

Thank you for your attention!


Relation with Plasmons

I Def. Plasmon: normal modes of the intrinsic electronsεE tot = Eext !

= 0, ⇐⇒ <ε = 0I plane wave perturbation excites normal mode(s)

ρind = χϕext

0 10 20 30 40 50Energy Loss ω [eV]

0

1

2

3

4

5

6

EE

LS

EELS, S(q,ω)

max [ρind(q,ω)]

q = 0.74 1/Å, in-plane


Energy Loss for an Electron

The energy loss in semi-classical approximation is given by

〈dWdt 〉t =


=

∫dωω

∫∫dqdq′ ϕext (−q,−ω)iχ(q,q′;ω)ϕext (q′, ω),

and for an electron flying through a crystal

〈dWdt 〉t = −

∞∫0

dωω8π2e2∫∫

dq⊥dq′⊥=[χ(q,q′;ω)eib(q⊥−q′⊥)

q2q′2],

where q(′) = q(′)⊥ + (w/v)ez is split into a part perpendicular

and parallel to the direction of motion of the electron.


Spatial Resolution in Electron Energy-Loss Spectroscopy...Spatial Resolution in Electron Energy-Loss...

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