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).]

R. Hambach Spatial Resolution in EELS

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).]

R. Hambach Spatial Resolution in EELS

Outlook

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

R. Hambach Spatial Resolution in EELS

Semi-classical TheoryHow do we calculate EELS ?

R. Hambach Spatial Resolution in 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

R. Hambach Spatial Resolution in EELS

EELS: Classical Perturbation

v

b

ϕext

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

R. Hambach Spatial Resolution in EELS

EELS: Classical Perturbation

v

b

ϕextϕind

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

R. Hambach Spatial Resolution in EELS

EELS: Classical Perturbation

v

b

ϕextϕind

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

R. Hambach Spatial Resolution in EELS

EELS: Classical Perturbation

v

b

ϕextϕind

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

⇒ energy loss probability S(b, ω)

R. Hambach Spatial Resolution in EELS

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.

R. Hambach Spatial Resolution in EELS

Spatially Resolved EELSAloof Spectroscopy for Graphene

R. Hambach Spatial Resolution in EELS

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

R. Hambach Spatial Resolution in EELS

Energy Loss for Graphene

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

I atomic resolution (∆b =1.1A)

R. Hambach Spatial Resolution in EELS

Energy Loss for Graphene

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

R. Hambach Spatial Resolution in EELS

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)

∆ϕext = ρext/ε0

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 ′)

R. Hambach Spatial Resolution in EELS

Induced Charge DensityLooking at Plasmons in Graphite

R. Hambach Spatial Resolution in EELS

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

R. Hambach Spatial Resolution in EELS

Graphite: π–Plasmon

Induced charge fluctuations ρind = χϕext

(resolution: 0.8 A, isosurface at 50%)

I plane wave perturbation

ϕext (r , ω) ∝ e−i(ωt−qr)

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

R. Hambach Spatial Resolution in EELS

Graphite: π–Plasmon

Induced charge fluctuations ρind = χϕext

(resolution: 0.8 A, isosurface at 50%)

(partial) ground state density

pz - orbitals sp2 - orbitals

R. Hambach Spatial Resolution in EELS

Graphite: π + σ–Plasmon

Induced charge fluctuations ρind = χϕext

(resolution: 0.8 A, isosurface at 35%)

I plane wave perturbation

ϕext (r , ω) ∝ e−i(ωt−qr)

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

R. Hambach Spatial Resolution in EELS

Graphite: π + σ–Plasmon

Induced charge fluctuations ρind = χϕext

(resolution: 0.8 A, isosurface at 35%)

(partial) ground state density

pz - orbitals sp2 - orbitals

R. Hambach Spatial Resolution in EELS

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)]

R. Hambach Spatial Resolution in EELS

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

R. Hambach Spatial Resolution in EELS

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!

R. Hambach Spatial Resolution in EELS

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

R. Hambach Spatial Resolution in EELS

Energy Loss for an Electron

The energy loss in semi-classical approximation is given by

〈dWdt 〉t =

∫dr〈E ind ·jext〉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.

R. Hambach Spatial Resolution in EELS