Graphene Photonics and Optoelectronics

A.C. Ferrari
2016 Lasers Congress 2016 (ASSL, LSC, LAC)   unpublished
E lectrons propagating through the bidimensional structure of graphene have a linear relation between energy and momentum, and thus behave as massless Dirac fermions 1-3 . Consequently, graphene exhibits electronic properties for a two-dimensional (2D) gas of charged particles described by the relativistic Dirac equation, rather than the non-relativistic Schrödinger equation with an eff ective mass 1,2 , with carriers mimicking particles with zero mass and an eff ective 'speed of light' of
more » ... d of light' of around 10 6 m s -1 . Graphene exhibits a variety of transport phenomena that are characteristic of 2D Dirac fermions, such as specifi c integer and fractional quantum Hall eff ects 4,5 , a 'minimum' conductivity of ~4e 2 /h even when the carrier concentration tends to zero 1 , and Shubnikov-de Haas oscillations with a π phase shift due to Berry's phase 1 . Mobilities (μ) of up to 10 6 cm 2 V -1 s -1 are observed in suspended samples. Th is, combined with near-ballistic transport at room temperature, makes graphene a potential material for nanoelectronics 6,7 , particularly for high-frequency applications 8 . Graphene also shows remarkable optical properties. For example, it can be optically visualized, despite being only a single atom thick 9,10 . Its transmittance (T) can be expressed in terms of the fi ne-structure constant 11 . Th e linear dispersion of the Dirac electrons makes broadband applications possible. Saturable absorption is observed as a consequence of Pauli blocking 12,13 , and nonequilibrium carriers result in hot luminescence 14-17 . Chemical and physical treatments can also lead to luminescence 18-21 . Th ese properties make it an ideal photonic and optoelectronic material. Electronic and optical properties Electronic properties. Th e electronic structure of singlelayer graphene (SLG) can be described using a tight-binding Hamiltonian 2,3 . Because the bonding and anti-bonding σ-bands are well separated in energy (>10 eV at the Brillouin zone centre Γ), they can be neglected in semi-empirical calculations, retaining only the two remaining π-bands 3 . Th e electronic wavefunctions from diff erent atoms on the hexagonal lattice overlap. However, any such overlap between the p z (π) and the s or p x and p y orbitals is strictly zero by symmetry. Consequently, the p z electrons, which form the π-bonds, can be treated independently from the other valence electrons. Within this π-band approximation it is easy to describe the electronic spectrum of the total Hamiltonian and to obtain the dispersion relations E ± (k x , k y ) restricted to firstnearest-neighbour interactions only: T = (1 + 0.5πα) -2 ≈ 1 -πα ≈ 97.7% ( 3) where α = e 2 /(4πε 0 ħc) = G 0 /(πε 0 c) ≈ 1/137 is the fi ne-structure constant 11 . Graphene only refl ects <0.1% of the incident light in the visible region 11 , rising to ~2% for ten layers 9 . Th us, we can take
doi:10.1364/assl.2016.atu3a.1 fatcat:x5hjvxn4bvg4jj766rjih5jlfq