Quantification of Atomic Vibrations in Mono-Layer Graphene
Christopher S. Allen¹, Emanuela Liberti¹, Judy S. Kim¹, Qiang Xu², Ye Fan¹, Kuang He¹, Alex W. Robertson¹, Henny W. Zandbergen³, Jamie H. Warner¹ and Angus I. Kirkland¹ ¹ | Department of Materials, University of Oxford, United Kingdom ² | DENSsolutions, The Netherlands ³ | Kavli Institute of Nanoscience, Delft University of Technology, The Netherlands We have measured the mean square amplitude of both in- and out-of-plane lattice vibrations for mono-layer graphene at temperatures ranging from ∼100 K to 1300 K. The amplitude of lattice vibrations was calculated from data extracted from selected area electron diffraction patterns recorded across a known temperature range with over 80 diffraction peaks measured per diffraction pattern. Using an analytical Debye model, we have also determined values for the maximum phonon wavelength that can be supported by a mono-layer graphene crystal and the magnitude of quantum mechanical zero point vibrations. For in-plane phonons, the quantum mechanical zero point contribution dominates the measured atomic displacement at room temperature, whereas for out-of-plane modes, thermally populated phonons must be considered. We find a value for the maximum phonon wavelength sampled that is several orders of magnitudes smaller than the physical crystallite size.
A. W. Robertson, G.-D. Lee, K. He, Y. Fan, C. S. Allen, S. Lee, H. Kim, E. Yoon, H. Zheng, A. I. Kirkland, J. H. Warner. Partial Dislocations in Graphene and Their Atomic Level Migration Dynamics Nano Letters,DOI: 10.1021/acs.nanolett.5b02080 (2015) C. S. Allen, E. Liberti, J. S. Kim, Q. Xu, Y. Fan, K. He, A. W. Robertson, H. W. Zandbergen, J. H. Warner, A. I. Kirkland. Temperature dependence of atomic vibrations in mono-layer graphene. Journal of Applied Physics, 118, 7, 074302 (2015) K. He, A. W. Robertson, C. Gong, C. S. Allen, Q. Xu, H. Zandbergen, J. C. Grossmanm, A. I. Kirkland, J. H. Warner. Controlled formation of closed-edge nanopores in graphene. Nanoscale, 7, 11602-11610 (2015) K. He, A. W. Robertson, Y. Fan, C. S. Allen, Y. C. Lin, K. Suenaga, A. I. Kirkland, J. H. Warner. Temperature dependence of the reconstruction of zigzag edges in graphene. ACS Nano, 9, 5, 4786-4795 (2015) C. S. Allen, G. Liu, Y. Chen, A. W. Robertson, K. He, K. Porfyrakis, J. Zhang, G. A. D. Briggs, J. H. Warner. Optically enhanced charge transfer between C60 and single-wall carbon nanotubes in hybrid electronic devices. Nanoscale, 6, 572-580, (2014). A. W. Robertson, B. Montanari, K. He, C. S. Allen, Y. A. Wu, N. Harrison, A. I. Kirkland, J. H. Warner. Structural Reconstruction of the Graphene Monovacancy. ACS Nano, 7, 5, 4495–4502, (2013). A. W. Robertson, B. Montanari, K. He, J. Kim, C. S. Allen, Y. A. Wu, J. Olivier, J. Neethling, N. Harrison, A. I. Kirkland, J. H. Warner. Dynamics of Single Fe Atoms in Graphene Vacancies. Nano Letters, 13, 4, 1468-1475, (2013). A. W. Robertson, C. S. Allen, Y. A. Wu, K. He, J. Olivier, J. Neethling, A. I. Kirkland, J. H. Warner. Spatial Control of Defect Creation in Graphene at the Nanoscale. Nature Communications, 3, 1144, (2012). C. S. Allen, A. W. Robertson, A. I. Kirkland, J. H. Warner. The Identification of Inner Tube Defects in Double-Wall Carbon Nanotubes. Small, 8, 24, 3810-3815 ,(2012). Y. A. Wu, Y. Fan, S. Speller, G. L. Creeth, J. T. Sadowski, K. He, A. W. Robertson, C. S. Allen, J. H. Warner. Large Single Crystals of Graphene on Melted Copper Using Chemical Vapor Deposition. ACS Nano, 6, 6, 5010-5017, (2012). Y. S. Chaudhary, T. W. Woolerton, C. S. Allen, J. H. Warner, E. Pierce, S. W. Ragsdale, F. A. Armstrong. Visible light-driven CO2 reduction by enzyme coupled CdS nanocrystals. Chemical Communications, 48, 58-60 (2012). C. S. Allen, Y. Ito, A. W. Robertson, H. Shinohara, J. H. Warner. Two-dimensional coalescence dynamics of encapsulated metallofullerenes in carbon nanotubes. ACS Nano, 5, 12, 10084-10089, (2011). C. S. Allen, M. D. Elkin, G. Burnell, C. Zhang, S. Hofmann, J. Robertson, B. J. Hickey. Transport measurements on carbon nanotubes structurally characterized by electron diffraction. Physical Review B, 84, 115444, (2011). C. S. Allen, C. Zhang, G. Burnell, A. P. Brown, J. Robertson, B. J. Hickey. A review of methods for the accurate determination of the chiral indices of carbon nanotubes from electron diffraction patterns. Carbon, 49, 4961-4971, (2011). C. S. Allen, M. D. Elkin, C. Zhang, S. Hofmann, G. Burnell, J. Robertson, B. J. Hickey. Device fabrication with precisely placed carbon nanotubes of known chiral vector. Journal of Physics: Conference Series, 241, 012082 (2010). C. Zhang, F. Yan, C. S. Allen, B. C. Bayer, S. Hofmann, B. J. Hickey, D. Cott, G. Zhong, J. Robertson. Growth of vertically-aligned carbon nanotube forests on conductive cobalt disilicide support. Journal of Applied Physics, 108, 024311, (2010). J. E. McKendry, C. S. Allen, K. Critchley, M. L. Gorzny, A. S. Walton, S. D. Evans. Magnetic field enhanced nano-tip fabrication for four-probe STM studies. Nanotechnology, 19, 085201, (2008). A. S. Walton, C. S. Allen, K. Critchley, M. L. Gorzny, J. E. McKendry, R. M. D. Brydson, B. J. Hickey, S. D. Evans. Four-probe electrical transport measurements on individual metallic nanowires. Nanotechnology, 18, 065205, (2007).
C. S. Allen and J. H. Warner. Chapter 3.1 Properties of Graphene – Electronic properties in Graphene Fundamentals and Emergent Applications, Elsevier 2013. C. S. Allen and J. H. Warner. Chapter 5.8 Characterisation Techniques – Hall mobility and Field-effect Mobility in Graphene Fundamentals and Emergent Applications, Elsevier 2013.