Spatial Resolution in Electron Energy-Loss
Spectroscopy
Ralf Hambach1,2 , C. Giorgetti1,2 , F. Sottile1,2 , Lucia Reining1,2 .
1
LSI, Ecole Polytechnique, CEA/DSM, CNRS, Palaiseau, France
2 European Theoretical Spectroscopy Facility (ETSF)
23. 03. 2009 — DPG09, Dresden
R. Hambach
Spatial Resolution in EELS
What is EELS?
angular resolved
broad beam geometry
I
momentum transfer q, energy loss ~ω
I
resolution: ∆q ≈ 0.05 Å−1 , ∆~ω ≈ 0.2eV
k0 , E0
I
k0
, E0
−q
ω
− h̄
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
E0
I
highly focussed beam
I
impact parameter b, energy loss ~ω
I
resolution: ∆b ≈ 1 Å, ∆~ω ≈ 0.5eV
b
E0 − h̄ω
EELS for Si/SiO2
[M. Couillard et. al., PRB 77, 085318 (2008).]
R. Hambach
Spatial Resolution in EELS
Outlook
1. Methods and Theory
2. Spatially Resolved EELS: Graphene
3. Charge fluctuations in Graphite
R. Hambach
Spatial Resolution in EELS
Semi-classical Theory
How do we calculate EELS ?
R. Hambach
Spatial Resolution in EELS
EELS: Classical Perturbation
1. electron flying along straight line
ρext (r , t) = −eδ r − (b + vt)
2. external potential (no retardation)
∆ϕext = ρext /ε0
3. linear response of the medium
v
b
ρind = χϕext ,
−1 = 1+v χ
4. energy loss ofZ the electron
h dW
dt it =
R. Hambach
dr hE ind ·j ext it
Spatial Resolution in EELS
EELS: Classical Perturbation
1. electron flying along straight line
ρext (r , t) = −eδ r − (b + vt)
2. external potential (no retardation)
ϕext
∆ϕext = ρext /ε0
3. linear response of the medium
v
b
ρind = χϕext ,
−1 = 1+v χ
4. energy loss ofZ the electron
h dW
dt it =
R. Hambach
dr hE ind ·j ext it
Spatial Resolution in EELS
EELS: Classical Perturbation
1. electron flying along straight line
ρext (r , t) = −eδ r − (b + vt)
2. external potential (no retardation)
ϕext
ϕind
∆ϕext = ρext /ε0
3. linear response of the medium
v
b
ρind = χϕext ,
−1 = 1+v χ
4. energy loss ofZ the electron
h dW
dt it =
R. Hambach
dr hE ind ·j ext it
Spatial Resolution in EELS
EELS: Classical Perturbation
1. electron flying along straight line
ρext (r , t) = −eδ r − (b + vt)
2. external potential (no retardation)
ϕext
ϕind
∆ϕext = ρext /ε0
3. linear response of the medium
v
b
ρind = χϕext ,
−1 = 1+v χ
4. energy loss ofZ the electron
h dW
dt it =
R. Hambach
dr hE ind ·j ext it
Spatial Resolution in EELS
EELS: Classical Perturbation
1. electron flying along straight line
ρext (r , t) = −eδ r − (b + vt)
2. external potential (no retardation)
ϕext
ϕind
∆ϕext = ρext /ε0
3. linear response of the medium
v
b
ρind = χϕext ,
−1 = 1+v χ
4. energy loss ofZ the electron
h dW
dt it =
dr hE ind ·j ext it
⇒ 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 φKS
i
2. independent-particle polarisability χ0
3. RPA full polarisability χ = χ0 + χ0 v χ
non-local response of crystals
I
ρind (r, t) =
R
χ(r , r 0 ; t − t 0 )ϕext (r 0 , t 0 )
Codes:
A BINIT: 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
χ0
Spatially Resolved EELS
Aloof Spectroscopy for Graphene
R. Hambach
Spatial Resolution in EELS
Energy Loss for Graphene
Electron parallel to sheet
(super-cell with 30 Å vac)
Energy loss for different pos. b
14
b=0 Å
b = 0.6 Å
b = 1.2 Å
b = 1.8 Å
b=6 Å
I
non-dispersive modes
I
surface plasmon: 6.2eV
I
further excitations at
20eV and 32eV
S(b,ω) [arb.u.]
12
10
8
6
4
2
x4
0
0
R. Hambach
5
10 15 20 25
Energy Loss ω [eV]
Spatial Resolution in EELS
30
35
Energy Loss for Graphene
R. Hambach
I
spatial distribution of the loss
probability for
6eV, 22eV and 32eV
I
atomic resolution (∆b = 1.1 Å)
Spatial Resolution in EELS
Energy Loss for Graphene
I
exponential decay with b
I
delocalization
S(b,ω) [arb.u.]
10
E= 6 eV
E=22 eV
E=32 eV
8
6
4
2
-2
R. Hambach
0
2
4
6
Impact Parameter b [1/Å]
Spatial Resolution in EELS
8
Outlook
I
Cerenkov radiation
I
non-local effects
I
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 medium
ρind = χϕext ,
−1 = 1+v χ
4. energy lossRof the electron
h dW
i = dr hE ind ·j ext it
dt t
5. non-local response χ(r, r 0 , t − t 0 )
R. Hambach
Spatial Resolution in EELS
Induced Charge Density
Looking 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 )
R. Hambach
I
|q| = 0.74 Å−1 , λ = 8.5 Å
I
~ω = 9eV
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 Å−1 , λ = 8.5 Å
I
~ω = 9eV
(resolution: 0.8 Å, isosurface at 50%)
R. Hambach
Spatial Resolution in EELS
Graphite: π–Plasmon
Induced charge fluctuations ρind = χϕext
(partial) ground state density
pz - orbitals
(resolution: 0.8 Å, isosurface at 50%)
R. Hambach
Spatial Resolution in EELS
sp2 - orbitals
Graphite: π + σ–Plasmon
Induced charge fluctuations ρind = χϕext
I
plane wave perturbation
ϕext (r , ω) ∝ e−i(ωt−qr)
I
|q| = 0.74 Å−1 , λ = 8.5 Å
I
~ω = 30eV
(resolution: 0.8 Å, isosurface at 35%)
R. Hambach
Spatial Resolution in EELS
Graphite: π + σ–Plasmon
Induced charge fluctuations ρind = χϕext
(partial) ground state density
pz - orbitals
(resolution: 0.8 Å, isosurface at 35%)
R. Hambach
Spatial Resolution in EELS
sp2 - orbitals
Outlook
I
realistic external potential
→ electron in graphite
I
relation with plasmons
I
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 for
ab-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 for
ab-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 electrons
!
E tot = E ext = 0, ⇐⇒ < = 0
I
plane wave perturbation excites normal mode(s)
ρind = χϕext
6
q = 0.74 1/Å, in-plane
5
EELS, S(q,ω)
ind
max [ρ (q,ω)]
EELS
4
3
2
1
0
0
10
20
30
40
Energy Loss ω [eV]
R. Hambach
50
Spatial Resolution in EELS
Energy Loss for an Electron
The energy loss in semi-classical approximation is given by
Z
dW
h dt it = drhE ind ·j ext it
Z
ZZ
= dωω
dqdq 0 ϕext (−q, −ω)iχ(q, q 0 ; ω)ϕext (q 0 , ω),
and for an electron flying through a crystal
h dW
dt it
Z∞
ZZ
h χ(q, q 0 ; ω)eib(q ⊥ −q 0⊥ ) i
2 2
0
= − dωω8π e
dq ⊥ dq ⊥ =
,
q 2 q 02
0
0
(0 )
where q ( ) = q ⊥ + (w/v )e z is split into a part perpendicular
and parallel to the direction of motion of the electron.
R. Hambach
Spatial Resolution in EELS