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Vibrational properties of graphene and graphene nanoribbons Christian Thomsen Institut für Festkörperphysik TU Berlin. Topics. Nanoribbon vibrations Graphene under uniaxial strain Graphene nanoribbons under uniaxial strain TERS: individual NTs and small bundles. Topics.

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slide1

Vibrational properties of graphene and graphene nanoribbons

Christian Thomsen

Institut für Festkörperphysik

TU Berlin

slide2

Topics

  • Nanoribbon vibrations
  • Graphene under uniaxial strain
  • Graphene nanoribbons under uniaxial strain
  • TERS: individual NTs and small bundles
slide3

Topics

  • Nanoribbon vibrations
  • Graphene under uniaxial strain
  • Graphene nanoribbons under uniaxial strain
  • TERS: individual NTs and small bundles
slide4

What are nanoribbons?

Graphite

3D-crystal

sp2-hybridization

stacked planes

Graphene

2D-crystal

single graphite plane

periodic in x-y-plane

Nanoribbon

  • strip of graphene
  • „quasi 1D-crystal“
  • periodic in 1 direction
potential for applications
Potential for applications
  • high mobility
  • easy to prepare
  • band-gap engineering
slide6

Classification

Armchair

Zigzag

N-AGNR

N-ZGNR

width (number of dimers)‏

edge type („chiral” NR not considered here)‏

slide7

Wave propagation

: continuous

: quantized

slide8

Brillouin zone

Brillouin zone of nanoribbons:

N discrete lines (N: number of dimers)‏

6 modes for each line

here: 10-AGNR and 10-ZGNR

slide9

Electronic properties: Armchair NRs

=> three families of AGNRs, N=3p, N=3p+1, N=3p+2

Son, Cohen, Louie PRL 97, 216803 (2006)‏

slide10

Electronic properties: Zigzag NRs

band gap opens for

anti-ferromagnetic

ground state

metallic if spin is not

considered

Son, Cohen, Louie Nature 444, 347 (2006)‏

slide11

Calculational details

  • Siesta: www.uam.es/siesta
  • Kohn-Sham self consistent density functional method
  • norm-conserving pseudopotentials
  • strictly confined atom centered numerical atomic orbitals (NAO) as basis functions
  • phonon calculation: finite differences to obtain force constant matrix
slide12

Fundamental modes & “overtones”

Nanoribbons have 3N modes

E2g corresponds to 0-LO and 0-TO

A wavelength and a wavevector kperp can be assigned to overtones

here: 7-AGNR

||

Interpretation as fundamental modes and overtones

slide14

LO Softening (armchair)

family dependence also in phonon spectrum

strong softening of the LO phonon in 3p+2 ribbons

slide15

Mapping of the overtones

graphene phonon dispersion:

AGNR GKM

ZGNR  GM

Grüneis, et al. PRB 65,155405 (2002)

Mohr, CT et al., PRB 76, 035439 (2007)

Mohr, CT et al., PRB 80, 155418 (2009)

slide16

Mapping of the overtones

Mapping of a

15-AGNR

and a 8-ZGNR onto the graphene dispersion

Grüneis, et al. PRB 65,155405 (2002)

Mohr, CT et al., PRB 76, 035439 (2007)

Mohr, CT et al., PRB 80, 155418 (2009)

slide17

Graphite dispersion

Double resonance:

Grüneis, et al., PRB 65, 155405 (2002)

Reich and CT, Phil. Trans. 362, 2271 (2004)

Inelastic x-ray scattering:

Maultzsch, CT, et al., PRL 92, 075501 (2004)

Mohr, CT et al., PRB 76, 035439 (2007)

unfolding nanoribbons:

Gillen, CT et al., PRB 80, 155418 (2009)

Gillen et al., PRB in print (2010)

slide18

Phonon dispersion

OddN: modes pairwise degenerate

at X-point (zone-folding)

4th acoustic mode („1-ZA“)

(rotational mode)

EvenN: modes pairwise degenerate

at X-point

4th acoustic mode („1-ZA“)

compare: Yamada et al, PRB, 77, 054302 (2008))

slide19

Topics

  • Nanoribbon vibrations
  • Graphene under uniaxial strain
  • Graphene nanoribbons under uniaxial strain
  • TERS: individual NTs and small bundles
uniaxial strain in graphene
Uniaxial strain in graphene

Polarized measurements reveal orientation of graphene sample

Mohiuddin, Ferrari et al,. PRB 79, 205433 (2009)‏

Huang, Heinz et al., PNAS 106, 7304 (2009)‏

slide21

Calculational details

  • www.quantum-espresso.org
  • Kohn-Sham selfconsistent density functional method
  • norm-conserving pseudopotentials
  • plane-wave basis
  • phonon calculation: linear response theory / DFBT(Density Functional Perturbation Theory)‏
dirac cone at k point
Dirac cone at K-point

strains shift the Dirac cone but don’t open a gap

shift of the e 2g mode
Shift of the E2g -mode

shift rate independent of strain direction

comparison with experiments
Comparison with experiments
  • excellent agreement with Mohiuddin/Ferrari

Mohr, CT, et al., Phys. Rev. B 80, 205410 (2009)

Ni et al., ACS Nano 2, 2301 (2008)

Mohiuddin, Ferrari et al. PRB 79, 205433 (2009)

Huang, Heinz et al., PNAS 106, 7304 (2009)

d and 2d mode d ouble resonance

qphonon varies strongly with incident photon energy.

D and 2D mode: Double resonance
  • The particular band structure of CNTs allows an incoming resonance at any energy.
  • The phonon scatters the electron resonantly to the other band.
  • A defect scatters the electron elastically back to where it can recombine with the hole.

CT and Reich, Phys. Rev. Lett. 85, 5214 (2000)

double resonance inner and outer
Double resonance: inner and outer

defect- induced D-mode

slide33

Topics

  • Nanoribbon vibrations
  • Graphene under uniaxial strain
  • Graphene nanoribbons under uniaxial strain
  • TERS: individual NTs and small bundles
nr band gap under strain
NR-Band gap under strain
  • band gap for N=13, 14, 15 AGNRs
  • linear dependence for small strains
g for different nr widths
G- for different NR widths
  • approaching the dependence of graphene
g for different nr widths37
G+ for different NR widths
  • approaching the dependence of graphene
slide38

Topics

  • Nanoribbon vibrations
  • Graphene under uniaxial strain
  • Graphene nanoribbons under uniaxial strain
  • TERS: individual NTs and small bundles
tip enhanced raman spectra
Tip-enhanced Raman spectra
  • find specific nanotubes, previously identified with AFM
  • observe the RBM as a function of position along the nanotube
  • study frequency shifts as a function of sample-tip distance

Hartschuh et al., PRL (2003) and Pettinger et al., PRL (2004)

N.Peica, CT, J. Maultzsch, JRS, submitted (2010)

N. Peica, CT et al., pss (2009)

slide40

TERS setup

Laser wavelength 532 nm

tip enhanced raman spectra41
Tip-enhanced Raman spectra

small bundles of individual nanotubes on a silicon wafer

tip enhanced raman spectra42
Tip-enhanced Raman spectra

small bundles of individual nanotubes on a silicon wafer

chirality raman spectra
Chirality: Raman spectra

The Raman spectrum is divided into

  • radial breathing mode
  • defect-induced mode
  • high-energy mode
tip enhanced raman spectra44
Tip-enhanced Raman spectra

small bundles of individual nanotubes on a silicon wafer

N.Peica, CT, J. Maultzsch, Carbon, submitted (2010)

sample tip distance dependence
Sample-tip distance dependence

enhancement factors between 2 103 and 4 104

rbm spectra
RBM spectra
  • RBM can be observed even if not visible in the far-field spectrum
  • identified (17,6), (12,8), (16,0), and (12,5) semiconducting NTs from experimental Kataura plots

Popov et al. PRB 72, 035436 (2005)

frequency shifts in ters
Frequency shifts in TERS

shifts of 5 cm -1 observed

frequency shifts in ters48
Frequency shifts in TERS
  • possible explanation of the small shifts are
    • in terms of the double-resonance Raman process of the D and 2D modes (CT, PRL 2000)
    • deformation through the tip approach
    • sensitive reaction of the electronic band structure
conclusions
Conclusions
  • Vibrations of graphene nanoribbons
    • mapping of overtones on graphene (graphite) dispersion
  • Uniaxial strain in graphene
    • comparison to experiments
  • TERS specta of individual NTs
    • large enhancement factors
    • NTs identified
    • possible observation of small frequency shifts
acknowledgments
Acknowledgments

Janina Maultzsch Technische Universität Berlin

Nils Rosenkranz Technische Universität Berlin

Marcel Mohr Technische Universität Berlin

Niculina Peica Technische Universität Berlin