Vibrational properties of graphene and graphene nanoribbons Christian Thomsen Institut für Festkörperphysik TU Berlin

1. Vibrational properties of graphene and graphene nanoribbons Christian Thomsen Institut für Festkörperphysik TU Berlin

2. Topics • Nanoribbon vibrations • Graphene under uniaxial strain • Graphene nanoribbons under uniaxial strain • TERS: individual NTs and small bundles

3. Topics • Nanoribbon vibrations • Graphene under uniaxial strain • Graphene nanoribbons under uniaxial strain • TERS: individual NTs and small bundles

4. 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

5. Potential for applications • high mobility • easy to prepare • band-gap engineering

6. Classification Armchair Zigzag N-AGNR N-ZGNR width (number of dimers)‏ edge type („chiral” NR not considered here)‏

7. Wave propagation : continuous : quantized

8. Brillouin zone Brillouin zone of nanoribbons: N discrete lines (N: number of dimers)‏ 6 modes for each line here: 10-AGNR and 10-ZGNR

9. Electronic properties: Armchair NRs => three families of AGNRs, N=3p, N=3p+1, N=3p+2 Son, Cohen, Louie PRL 97, 216803 (2006)‏

10. 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)‏

11. 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

12. 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

13. Width dependence (armchair) E2g

14. LO Softening (armchair) family dependence also in phonon spectrum strong softening of the LO phonon in 3p+2 ribbons

15. 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)

16. 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)

17. 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)

18. 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))

19. Topics • Nanoribbon vibrations • Graphene under uniaxial strain • Graphene nanoribbons under uniaxial strain • TERS: individual NTs and small bundles

20. 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)‏

21. 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)‏

22. Method

23. Electronic band structure under strain

24. Dirac cone at K-point strains shift the Dirac cone but don’t open a gap

25. Phonon band structure under strain

26. Raman spectrum of graphene

27. Shift of the E2g -mode shift rate independent of strain direction

28. Shift of the E2g -mode

29. 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)

30. 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)

31. Double resonance: inner and outer defect- induced D-mode

32. Strained w/ diff. polarizations

33. Topics • Nanoribbon vibrations • Graphene under uniaxial strain • Graphene nanoribbons under uniaxial strain • TERS: individual NTs and small bundles

34. NR-Band gap under strain • band gap for N=13, 14, 15 AGNRs • linear dependence for small strains

35. G+ and G- modes as fct. of strain N=7

36. G- for different NR widths • approaching the dependence of graphene

37. G+ for different NR widths • approaching the dependence of graphene

38. Topics • Nanoribbon vibrations • Graphene under uniaxial strain • Graphene nanoribbons under uniaxial strain • TERS: individual NTs and small bundles

39. 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)

40. TERS setup Laser wavelength 532 nm

41. Tip-enhanced Raman spectra small bundles of individual nanotubes on a silicon wafer

42. Tip-enhanced Raman spectra small bundles of individual nanotubes on a silicon wafer

43. Chirality: Raman spectra The Raman spectrum is divided into • radial breathing mode • defect-induced mode • high-energy mode

44. Tip-enhanced Raman spectra small bundles of individual nanotubes on a silicon wafer N.Peica, CT, J. Maultzsch, Carbon, submitted (2010)

45. Sample-tip distance dependence enhancement factors between 2 103 and 4 104

46. 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)

47. Frequency shifts in TERS shifts of 5 cm -1 observed

48. 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

49. 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

50. Acknowledgments Janina Maultzsch Technische Universität Berlin Nils Rosenkranz Technische Universität Berlin Marcel Mohr Technische Universität Berlin Niculina Peica Technische Universität Berlin