Christian Thomsen

Vibrational properties of graphene and graphene

nanoribbons

Christian ThomsenInstitut für Festkörperphysik

TU Berlin

Christian Thomsen

Topics

Nanoribbon vibrations

Graphene under uniaxial strain

Graphene nanoribbons under uniaxial strain

TERS: individual NTs and small bundles

Christian Thomsen

Topics

Nanoribbon vibrations

Graphene under uniaxial strain

Graphene nanoribbons under uniaxial strain

TERS: individual NTs and small bundles

Christian Thomsen

Graphite

Graphene

Nanoribbon strip of graphene

• „quasi 1D-crystal“ periodic in 1 direction

2D-crystal single graphite plane periodic in x-y-plane

3D-crystal sp2-hybridization stacked planes

What are nanoribbons?

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Potential for applications

high mobility

easy to prepare

band-gap engineering

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ClassificationZigzagArmchair

width (number of dimers) edge type („chiral” NR not considered here)

N-AGNR N-ZGNR

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Wave propagation

: continuous

: quantized

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Brillouin zone

Brillouin zone of nanoribbons:

N discrete lines (N: number of dimers)

6 modes for each line

here: 10-AGNR and 10-ZGNR

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Electronic properties: Armchair NRs

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

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

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Electronic properties: Zigzag NRs

band gap opens for anti-ferromagneticground state

metallic if spin is notconsidered

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

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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

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Fundamental modes & “overtones”

Interpretation as fundamental modes and 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

||

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Width dependence (armchair)

E2g

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LO Softening (armchair)

family dependence also in phonon spectrum

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

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Mapping of the overtonesgraphene phonon dispersion:

AGNR KM

ZGNR M

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

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

Grüneis, et al. PRB 65,155405 (2002)

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Mapping of the overtonesMapping of a 15-AGNRand a 8-ZGNR onto the graphene dispersion

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

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

Grüneis, et al. PRB 65,155405 (2002)

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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)

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Phonon dispersion

Odd N: modes pairwise degenerate at X-point (zone-folding)

4th acoustic mode („1-ZA“)(rotational mode)

Even N: modes pairwise degenerate at X-point

4th acoustic mode („1-ZA“)

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

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Topics

Nanoribbon vibrations

Graphene under uniaxial strain

Graphene nanoribbons under uniaxial strain

TERS: individual NTs and small bundles

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Mohiuddin, Ferrari et al,. PRB 79, 205433 (2009)Huang, Heinz et al., PNAS 106, 7304 (2009)

Uniaxial strain in graphene

Polarized measurements reveal orientation of graphene sample

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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)

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Method

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Electronic band structure under strain

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Dirac cone at K-point

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

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Phonon band structure under strain

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Raman spectrum of graphene

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Shift of the E2g -mode

shift rate independent of strain direction

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Shift of the E2g -mode

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Ni et al., ACS Nano 2, 2301 (2008)Mohiuddin, Ferrari et al. PRB 79, 205433 (2009)Huang, Heinz et al., PNAS 106, 7304 (2009)

Comparison with experiments

excellent agreement with Mohiuddin/Ferrari

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

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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.

qphonon varies strongly with incident photon energy.

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

V 2

p h

E

k

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Double resonance: inner and outer

defect- induced D-mode

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Strained w/ diff. polarizations

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Topics

Nanoribbon vibrations

Graphene under uniaxial strain

Graphene nanoribbons under uniaxial strain

TERS: individual NTs and small bundles

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NR-Band gap under strain

band gap for N=13, 14, 15 AGNRs

linear dependence for small strains

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G+ and G- modes as fct. of strain

N=7

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G- for different NR widths

approaching the dependence of graphene

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approaching the dependence of graphene

G+ for different NR widths

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Topics

Nanoribbon vibrations

Graphene under uniaxial strain

Graphene nanoribbons under uniaxial strain

TERS: individual NTs and small bundles

Christian Thomsen

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)

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TERS setup

Laser wavelength 532 nm

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Tip-enhanced Raman spectra

small bundles of individual nanotubes on a silicon wafer

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Tip-enhanced Raman spectra

small bundles of individual nanotubes on a silicon wafer

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Chirality: Raman spectra

100 200 1400 1500 1600

Inte

nsity

(ar

b. u

nits

)

Raman Shift (cm-1)

HEM

D

RBM

SWNT The Raman spectrum is divided into

• radial breathing mode

• defect-induced mode

• high-energy mode

21

RBM Cd

C

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Tip-enhanced Raman spectra

small bundles of individual nanotubes on a silicon wafer

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

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Sample-tip distance dependence

enhancement factors between 2 103 and 4 104

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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)

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Frequency shifts in TERS

shifts of 5 cm -1 observed

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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

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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

Christian Thomsen

Acknowledgments

Janina Maultzsch Technische Universität Berlin

Nils Rosenkranz Technische Universität Berlin

Marcel Mohr Technische Universität Berlin

Niculina Peica Technische Universität Berlin