Thermal properties of graphene: Fundamentals and applications
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Introduction Graphene is a two-dimensional (2D) material, formed of a lattice of hexagonally arranged carbon atoms. The term graphene is typically applied to a single layer of graphite, although common references also exist to bilayer or trilayer graphene. (See the introductory article in this issue.) Most thermal properties of graphene are derived from those of graphite and bear the imprint of the highly anisotropic nature of this crystal.1 For instance, the in-plane covalent sp2 bonds between adjacent carbon atoms are among the strongest in nature (slightly stronger than the sp3 bonds in diamond), with a bonding energy2 of approximately 5.9 eV. By contrast, the adjacent graphene planes within a graphite crystal are linked by weak van der Waals interactions2 (∼50 meV) with a spacing3 of h ≈ 3.35 Å. Figure 1a displays the typical ABAB (also known as Bernal) stacking of graphene sheets within a graphite crystal. The strong and anisotropic bonding and the low mass of the carbon atoms give graphene and related materials unique thermal properties. In this article, we survey these unusual properties and their relation to the character of the underlying lattice vibrations. We examine both the specific heat and thermal conductivity of graphene and related materials and the conditions for achieving ballistic, scattering-free heat flow. We also
investigate the role of atomistic lattice modifications and defects in tuning the thermal properties of graphene. Finally, we explore the role of heat conduction in potential device applications and the possibility of architectures that allow control over the thermal anisotropy.
Phonon dispersion of graphene To understand the thermal properties of graphene, one must first inspect the lattice vibrational modes (phonons) of the material. The graphene unit cell, marked by dashed lines in Figure 1a, contains N = 2 carbon atoms. This leads to the formation of three acoustic (A) and 3N – 3 = 3 optical (O) phonon modes, with the dispersions4–7 shown in Figure 1b. The dispersion is the relationship between the phonon energy E or frequency ω (E = ħω, where ħ is the reduced Planck constant) and the phonon wave vector q. Longitudinal (L) modes correspond to atomic displacements along the wave propagation direction (compressive waves), whereas transverse (T) modes correspond to in-plane displacements perpendicular to the propagation direction (shear waves). In typical three-dimensional (3D) solids, transverse modes can have two equivalent polarizations, but the unique 2D nature of graphene allows out-of-plane atomic displacements, also known as flexural (Z) phonons.
Eric Pop, University of Illinois at Urbana-Champaign; [email protected] Vikas Varshney, Air Force Research Laboratory; [email protected] Ajit K. Roy, Air Force Research Laboratory; [email protected] DOI: 10.1557/mrs.2012.203
© 2012 Materials Research Society
MRS BULLETIN • VOLUME 37 • DECEMBER 2012 • www.mrs.org/bulletin
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THERMAL PROPERTIES OF GRAPHENE: FUNDAMENTALS AND APPLICATIONS
Figure 1. (a) Schematic of th
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