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Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Ab Initio calculations of electronic excitationsCarbon Nanotubes and Graphene layer systems

Francesco Sottile

Laboratoire des Solides IrradiesEcole Polytechnique, Palaiseau - France

European Theoretical Spectroscopy Facility (ETSF)

Strasbourg, 27 August 2008

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Outline

1 Introduction

2 Electron Energy Loss SpectroscopyLinear response within DFT

3 Applications: Nanotubes and Graphene

4 Perspectives

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Outline

1 Introduction

2 Electron Energy Loss SpectroscopyLinear response within DFT

3 Applications: Nanotubes and Graphene

4 Perspectives

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Introduction

Theoretical Spectroscopy Group (ETSF)

Results on Nanotubes and Graphene:

Coordinator: Christine GiorgettiRalf HambachXochitl LopezFederico IoriV.Olevano, A. Marinopoulos, L. Reining, F. SottileExperiments: Thomas Pichler group (Dresden)

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Outline

1 Introduction

2 Electron Energy Loss SpectroscopyLinear response within DFT

3 Applications: Nanotubes and Graphene

4 Perspectives

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Spectroscopy: Electron Scattering

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Spectroscopy: Electron Scattering

Energy Loss Function

d2σ

dΩdE∝ Im

ε−1

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Spectroscopy: Electron Scattering

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Spectroscopy: Electron Scattering

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Spectroscopy: Electron Scattering

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Spectroscopy: Electron Scattering

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Spectroscopy: Electron Scattering

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Spectroscopy: Electron Scattering

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Outline

1 Introduction

2 Electron Energy Loss SpectroscopyLinear response within DFT

3 Applications: Nanotubes and Graphene

4 Perspectives

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Linear Response Approach

System submitted to an external perturbation

Vtot = ε−1Vext

Vtot = Vext + Vind

E = ε−1D

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Linear Response Approach

System submitted to an external perturbation

Vtot = ε−1Vext

Vtot = Vext + Vind

E = ε−1D

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Linear Response Approach

System submitted to an external perturbation

Vtot = ε−1Vext

Vtot = Vext + Vind

E = ε−1D

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Linear Response Approach

Definition of polarizability

not polarizable ⇒ Vtot = Vext ⇒ ε−1 = 1polarizable ⇒ Vtot 6= Vext ⇒ ε−1 6= 1

ε−1 = 1 + vχ

χ is the polarizability of the system

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Linear Response Approach

Definition of polarizability

not polarizable ⇒ Vtot = Vext ⇒ ε−1 = 1polarizable ⇒ Vtot 6= Vext ⇒ ε−1 6= 1

ε−1 = 1 + vχ

χ is the polarizability of the system

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Linear Response Approach

Definition of polarizability

not polarizable ⇒ Vtot = Vext ⇒ ε−1 = 1polarizable ⇒ Vtot 6= Vext ⇒ ε−1 6= 1

ε−1 = 1 + vχ

χ is the polarizability of the system

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Linear Response Approach

Definition of polarizability

not polarizable ⇒ Vtot = Vext ⇒ ε−1 = 1polarizable ⇒ Vtot 6= Vext ⇒ ε−1 6= 1

ε−1 = 1 + vχ

χ is the polarizability of the system

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Linear Response Approach

Polarizability

interacting system δn = χδVext

non-interacting system δnn−i = χ0δVtot

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Linear Response Approach

Polarizability

interacting system δn = χδVext

non-interacting system δnn−i = χ0δVtot

Single-particle polarizability

χ0 =∑ij

φi (r)φ∗j (r)φ

∗i (r

′)φj(r′)

ω − (εi − εj)

hartree, hartree-fock, dft, etc.

G.D. Mahan Many Particle Physics (Plenum, New York, 1990)

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Linear Response Approach

Polarizability

interacting system δn = χδVext

non-interacting system δnn−i = χ0δVtot

χ0 =∑ij

φi (r)φ∗j (r)φ

∗i (r

′)φj(r′)

ω − (εi − εj)

i

unoccupied states

occupied states

j

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Linear Response Approach

Polarizability

interacting system δn = χδVext

non-interacting system δnn−i = χ0δVtot

m

Density Functional Formalism

δn = δnn−i

δVtot = δVext + δVH + δVxc

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Linear Response Approach

Polarizability

χδVext = χ0 (δVext + δVH + δVxc)

χ = χ0

(1 +

δVH

δVext+

δVxc

δVext

)δVH

δVext=

δVH

δn

δn

δVext= vχ

δVxc

δVext=

δVxc

δn

δn

δVext= fxcχ

with fxc = exchange-correlation kernel

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Linear Response Approach

Polarizability

χδVext = χ0 (δVext + δVH + δVxc)

χ = χ0

(1 +

δVH

δVext+

δVxc

δVext

)δVH

δVext=

δVH

δn

δn

δVext= vχ

δVxc

δVext=

δVxc

δn

δn

δVext= fxcχ

with fxc = exchange-correlation kernel

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Linear Response Approach

Polarizability

χδVext = χ0 (δVext + δVH + δVxc)

χ = χ0

(1 +

δVH

δVext+

δVxc

δVext

)δVH

δVext=

δVH

δn

δn

δVext= vχ

δVxc

δVext=

δVxc

δn

δn

δVext= fxcχ

χ = χ0 + χ0 (v + fxc) χwith fxc = exchange-correlation kernel

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Linear Response Approach

Polarizability

χδVext = χ0 (δVext + δVH + δVxc)

χ = χ0

(1 +

δVH

δVext+

δVxc

δVext

)δVH

δVext=

δVH

δn

δn

δVext= vχ

δVxc

δVext=

δVxc

δn

δn

δVext= fxcχ

χ =[1− χ0 (v + fxc)

]−1χ0

with fxc = exchange-correlation kernel

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Linear Response Approach

Polarizability

χδVext = χ0 (δVext + δVH + δVxc)

χ = χ0

(1 +

δVH

δVext+

δVxc

δVext

)δVH

δVext=

δVH

δn

δn

δVext= vχ

δVxc

δVext=

δVxc

δn

δn

δVext= fxcχ

χ =[1− χ0 (v + fxc)

]−1χ0

with fxc = exchange-correlation kernel

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Linear Response Approach

Polarizability χ in TDDFT

1 DFT ground-state calc. → φi , εi [Vxc ]

2 φi , εi → χ0 =∑

ij

φi (r)φ∗j (r)φ∗

i (r′)φj (r′)

ω−(εi−εj )

3

δVH

δn= v

δVxc

δn= fxc

variation of the potentials

4 χ = χ0 + χ0 (v + fxc) χ

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Linear Response Approach

Polarizability χ in TDDFT

1 DFT ground-state calc. → φi , εi [Vxc ]

2 φi , εi → χ0 =∑

ij

φi (r)φ∗j (r)φ∗

i (r′)φj (r′)

ω−(εi−εj )

3

δVH

δn= v

δVxc

δn= fxc

variation of the potentials

4 χ = χ0 + χ0 (v + fxc) χ

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Linear Response Approach

Polarizability χ in TDDFT

1 DFT ground-state calc. → φi , εi [Vxc ]

2 φi , εi → χ0 =∑

ij

φi (r)φ∗j (r)φ∗

i (r′)φj (r′)

ω−(εi−εj )

3

δVH

δn= v

δVxc

δn= fxc

variation of the potentials

4 χ = χ0 + χ0 (v + fxc) χ

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Linear Response Approach

Polarizability χ in TDDFT

1 DFT ground-state calc. → φi , εi [Vxc ]

2 φi , εi → χ0 =∑

ij

φi (r)φ∗j (r)φ∗

i (r′)φj (r′)

ω−(εi−εj )

3

δVH

δn= v

δVxc

δn= fxc

variation of the potentials

4 χ = χ0 + χ0 (v + fxc) χ

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Linear Response Approach

Polarizability χ in TDDFT

1 DFT ground-state calc. → φi , εi [Vxc ]

2 φi , εi → χ0 =∑

ij

φi (r)φ∗j (r)φ∗

i (r′)φj (r′)

ω−(εi−εj )

3

δVH

δn= v

δVxc

δn= fxc

variation of the potentials

4 χ = χ0 + χ0 (v + fxc) χ

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Linear Response Approach

RPA and other approximations

fxc =

δVxc

δn“any” other function fxc = 0 7→ RPA

Local field effects

χ =(1− χ0v

)−1χ0 ; χ0

GG′

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Linear Response Approach

RPA and other approximations

fxc =

δVxc

δn“any” other function fxc = 0 7→ RPA

Local field effects

χ =(1− χ0v

)−1χ0 ; χ0

GG′

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Linear Response Approach

RPA and other approximations

fxc =

δVxc

δn“any” other function fxc = 0 7→ RPA

Local field effects

χ =(1− χ0v

)−1χ0 ; χ0

GG′

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Outline

1 Introduction

2 Electron Energy Loss SpectroscopyLinear response within DFT

3 Applications: Nanotubes and Graphene

4 Perspectives

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Actual work at the Theoretical Spectroscopy Group

EELS of semiconductors

IXS and CIXS of semiconductors and metals

EELS of nanotubes and graphene layers

EELS and IXS of strongly correlated systems (Hf, V oxydes)

RIXS spectroscopy

User projects

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Actual work at the Theoretical Spectroscopy Group

EELS of semiconductors

IXS and CIXS of semiconductors and metals

EELS of nanotubes and graphene layers

EELS and IXS of strongly correlated systems (Hf, V oxydes)

RIXS spectroscopy

User projects

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

EELS of nanotubes: plasmon dispersion

Questions

theoretical understanding of electronic excitations of SWNTplasmon dispersion

SWNT and graphene. Strong connection and analysis

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

EELS of nanotubes: plasmon dispersion

VA-SWCNT

diameter: 2nm

nearly isolated

Kramberger, Hambach, Giorgetti, Rummeli, Knupfer, Fink, Buchner,

Reining, Einsarsson, Maruyama, Sottile, Hannewald, Olevano, Marinopoulos,

Pichler, Phys. Rev. Lett. 100, 196803 (2008)

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

EELS of nanotubes: plasmon dispersion

2nm is big!!

linear dispersionreminds us the Dirac

cone

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

EELS of nanotubes: plasmon dispersion

2nm is big!!

linear dispersionreminds us the Dirac

cone

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Numerical simulations

ab-initio calculations

DFT ground-state calculations (LDA)

Independant Particles polarizability: χ0

RPA Full polarisability: χ =[1− χ0υ

]−1χ0

Dielectric function ε−1 = 1 + vχ

energy loss function −Imε−1(q, ω)

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Independent particle picture

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

0 2 4 6 8 10energy loss (eV)

- Im

ε-1

(a

rb. u

.)

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

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Independent particle picture

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

0 2 4 6 8 10energy loss (eV)

- 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∗

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

RPA: random phase approx.

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

0 2 4 6 8 10energy loss (eV)

- Im

ε-1

(a

rb. u

.)

IPAπ−π∗ at KRPA given by χ:

no interpretation byband-transitions

contributions from K

mixing of transitions

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

RPA: random phase approx.

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

0 2 4 6 8 10energy loss (eV)

- Im

ε-1

(a

rb. u

.)

IPAπ−π∗ at KRPAwithout "K"

given by χ:no interpretation byband-transitions

contributions from K

mixing of transitions

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Plasmon dispersion

01-Im(1/ε) [arb. u.]

0

2

4

6

8

10E

nerg

y lo

ss [e

V]

0 0,2 0,4 0,6 0,8 1

momentum 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

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

SWCNT vs. Graphene

0 0.2 0.4 0.6 0.84

5

6

7

8

9

VASWCNT

(a) Experiment

0momentum transfer q (1/Å)

4

ener

gy lo

ss (

eV)

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

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

0momentum transfer q (1/Å)

4

ener

gy lo

ss (

eV)

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

SWCNT vs. Graphene

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

0momentum transfer q (1/Å)

4

ener

gy lo

ss (

eV)

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

SWCNT vs. Graphene

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-1Lgraphene-2L

(b) Calculation

0momentum transfer q (1/Å)

4

ener

gy lo

ss (

eV)

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

SWCNT vs. Graphene: Conclusions

Graphene can be studied to get quantitative information aboutVA-SWNT

Vice-versa is also true!

Bulk (bundled) nanotubes can be studied using double layer graphene

High q measurements are applicable to probe intrinsic properties ofindividual objects within bulk arrays.

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

SWCNT vs. Graphene: Conclusions

Graphene can be studied to get quantitative information aboutVA-SWNT

Vice-versa is also true!

Bulk (bundled) nanotubes can be studied using double layer graphene

High q measurements are applicable to probe intrinsic properties ofindividual objects within bulk arrays.

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

SWCNT vs. Graphene: Conclusions

Graphene can be studied to get quantitative information aboutVA-SWNT

Vice-versa is also true!

Bulk (bundled) nanotubes can be studied using double layer graphene

High q measurements are applicable to probe intrinsic properties ofindividual objects within bulk arrays.

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

SWCNT vs. Graphene: Conclusions

Graphene can be studied to get quantitative information aboutVA-SWNT

Vice-versa is also true!

Bulk (bundled) nanotubes can be studied using double layer graphene

High q measurements are applicable to probe intrinsic properties ofindividual objects within bulk arrays.

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Outline

1 Introduction

2 Electron Energy Loss SpectroscopyLinear response within DFT

3 Applications: Nanotubes and Graphene

4 Perspectives

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Ab initio simulation of electronic excitations

Advantages and limits√

reliable√

predictive

× cumbersome

Actual developments in the group

multiwall nanotubes - stacking of graphene layers (1 postdoc)

towards more complex systems - strongly correlated (2 postdocs)

different spectroscopies (X-ray ?) (1 postdoc)

spatial resolution EELS (PhD thesis)

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Ab initio simulation of electronic excitations

Advantages and limits√

reliable√

predictive

× cumbersome

Actual developments in the group

multiwall nanotubes - stacking of graphene layers (1 postdoc)

towards more complex systems - strongly correlated (2 postdocs)

different spectroscopies (X-ray ?) (1 postdoc)

spatial resolution EELS (PhD thesis)

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Discontinuity of the loss function

2 4 6 8 10energy loss (eV)

0

1

2

3

4

5S

(q,ω

) (1

/ keV

) 3

8/3

7/3

2

5/3

4/3

1

2/3

1/3

0

q1=0 q

3

k0

k

q

elastic inelastic

energy loss S(q, ω) ingraphite (AB)

q along c-axis

for multiple Brillouinzones

discontinuity:→ dispersion→ peak vanishes

Ab Initio calculations of electronic excitations Francesco Sottile

Introduction Electron Energy Loss Spectroscopy Applications: Nanotubes and Graphene Perspectives

Discontinuity of the loss function

2 4 6 8 10energy loss (eV)

0

1

2

3

4

5S

(q,ω

) (1

/ keV

) 3

8/3

7/3

2

5/3

4/3

1

2/3

1/3

0

q1=1/8 (~0.37 1/Å) q3

k0

k

q

elastic inelastic

energy loss S(q, ω) ingraphite (AB)

q along c-axis

for multiple Brillouinzones

discontinuity:→ dispersion→ peak vanishes

Ab Initio calculations of electronic excitations Francesco Sottile