Vibrational properties of
graphene and graphene
nanoribbons
Christian Thomsen
Institut 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
What are nanoribbons?
Graphite
Graphene
2D-crystal
single graphite plane
periodic in x-y-plane
3D-crystal
sp2-hybridization
stacked planes
Nanoribbon
strip of graphene
• „quasi 1D-crystal“
periodic in 1 direction
Christian Thomsen
Potential for applications
 high mobility
 easy to prepare
 band-gap engineering
Christian Thomsen
Classification
Armchair
N-AGNR
Zigzag
N-ZGNR
width (number of dimers)
edge type („chiral” NR not considered here)
Christian Thomsen
Wave propagation
: continuous
: quantized
Christian Thomsen
Brillouin zone
Brillouin zone of nanoribbons:
N discrete lines (N: number of dimers)
6 modes for each line
here: 10-AGNR and 10-ZGNR
Christian Thomsen
Electronic properties: Armchair NRs
=> three families of AGNRs, N=3p, N=3p+1, N=3p+2
Son, Cohen, Louie PRL 97, 216803 (2006)
Christian Thomsen
Electronic properties: Zigzag NRs
metallic if spin is not
considered
band gap opens for
anti-ferromagnetic
ground state
Son, Cohen, Louie Nature 444, 347 (2006)
Christian Thomsen
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
Christian Thomsen
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
Christian Thomsen
Width dependence (armchair)
E2g
Christian Thomsen
LO Softening (armchair)
family dependence also in phonon
spectrum
strong softening of the LO phonon in
3p+2 ribbons
Christian Thomsen
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)
Christian Thomsen
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)
Christian Thomsen
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)
Christian Thomsen
Phonon dispersion
Odd N: modes pairwise degenerate
at X-point (zone-folding)
4th acoustic mode („1-ZA“)
(rotational mode)
compare: Yamada et al, PRB, 77, 054302
(2008))
Even N: modes pairwise degenerate
at X-point
4th acoustic mode („1-ZA“)
Christian Thomsen
Topics
 Nanoribbon vibrations
 Graphene under uniaxial strain
 Graphene nanoribbons under uniaxial strain
 TERS: individual NTs and small bundles
Christian Thomsen
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)
Christian Thomsen
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)
Christian Thomsen
Method
Christian Thomsen
Electronic band structure under strain
Christian Thomsen
Dirac cone at K-point
strains shift the
Dirac cone but
don’t open a
gap
Christian Thomsen
Phonon band structure under strain
Christian Thomsen
Raman spectrum of graphene
Christian Thomsen
Shift of the E2g -mode
shift rate
independent of
strain direction
Christian Thomsen
Shift of the E2g -mode
Christian Thomsen
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)
Christian Thomsen
D and 2D mode: Double resonance
 The particular band structure of
CNTs allows an incoming
resonance at any energy.
E
V2
ph
 The phonon scatters the
electron resonantly to the other
band.
k
 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)
Christian Thomsen
Double resonance: inner and outer
defectinduced
D-mode
Christian Thomsen
Strained w/ diff. polarizations
Christian Thomsen
Topics
 Nanoribbon vibrations
 Graphene under uniaxial strain
 Graphene nanoribbons under uniaxial strain
 TERS: individual NTs and small bundles
Christian Thomsen
NR-Band gap under strain
 band gap for
N=13, 14, 15
AGNRs
 linear
dependence
for small
strains
Christian Thomsen
G+ and G- modes as fct. of strain
N=7
Christian Thomsen
G- for different NR widths

approaching
the
dependence
of graphene
Christian Thomsen
G+ for different NR widths

approaching
the
dependence
of graphene
Christian Thomsen
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 sampletip 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)
Christian Thomsen
TERS setup
Laser wavelength 532 nm
Christian Thomsen
Tip-enhanced Raman spectra
small bundles of individual nanotubes on a silicon wafer
Christian Thomsen
Tip-enhanced Raman spectra
small bundles of individual nanotubes on a silicon wafer
Christian Thomsen
In te n sity (a rb . u n its)
Chirality: Raman spectra
SW NT
HEM
RBM
The Raman spectrum is
divided into
• radial breathing
mode
D
• defect-induced mode
• high-energy mode
100 200
1400
1500
1600
R a m a n S h ift (cm -1 )
 RBM 
C1
d
C2
Christian Thomsen
Tip-enhanced Raman spectra
small bundles
of individual
nanotubes on
a silicon wafer
N.Peica, CT, J. Maultzsch, Carbon,
submitted (2010)
Christian Thomsen
Sample-tip distance dependence
enhancement factors between
2 103 and 4 104
Christian Thomsen
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)
Christian Thomsen
Frequency shifts in TERS
shifts of 5 cm -1 observed
Christian Thomsen
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
Christian Thomsen
Conclusions
•
Vibrations of graphene nanoribbons
•
•
Uniaxial strain in graphene
•
•
mapping of overtones on graphene (graphite)
dispersion
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
Christian Thomsen
Descargar

Christian Thomsen