## Transport in Graphene Nanostructures

Graphene is a truly two-dimensional (2-D) crystal consisting of a honeycomb-like hexagonal lattice of carbon atoms, as illustrated in Figure 21.1a. ™e carbon atoms form strong covalent bonds by three in-plane sp2 hybridized orbitals, whereas the fourth valence electron of the 2s22p2 orbitals of carbon, assigned to the perpendicular p∙∙ orbital can move freely in plane forming the so-called π-electron system [sai98]. ™e delocalized π-electronic states are responsible for electrical conductance and make graphene, in contrast to sp3-hybridized insulating diamond, to a gapless semiconductor. In particular, the π-electronic system in combination with the hexagonal lattice, where two carbon atoms sit in the unit cell (see A and B in Figure 21.1a, where the dashed lines mark the unit cell), lead to a number of unique electronic properties of graphene [wal47, cas07, gei07]. ™e electronic band structure can be approximated by a nearest-neighbor tight-binding approach of linear combinations of the perpendicular p∙∙ orbitals. According to

Wallace [wal47], this leadsto Ek E ik ri i i

= ± − ⋅

(illustrated in Figure 21.1b), where γi ≈ 2.8 eV [rei02] are the socalled nearest-neighbor hopping integrals, E0 is the energy of the bare p∙∙ orbital, and r

⃯ i are the vectors pointing to the three A

atoms neighboring each B atom (see arrows in Figure 21.1a). ™e two freely moving valence electrons (per unit cell) completely šll the (valence) π-band and leave the (conduction) π*-band unšlled resulting in a point-like Fermi surface (see Figure 21.1b). ™erefore, in the near vicinity of the Fermi energy, the band structure of graphene [wal47, mac57] can be linearized, leading to two cone-like structures centered at the two inequivalent (so-called) K and K′ points at the corners of the also hexagonal Brillouin zone (see Figure 21.1b and cross sections in Figure 21.2a). Consequently, the dynamics of charge carriers in graphene can be described in the near vicinity of the Fermi energy by a linear dispersion relation, E v=

F | |κ (and

κ = −k K( ))′ , where

the carriers behave like massless particles with a constant Fermi velocity vF ≈ 106 m/s, about 300 times smaller than the speed of

21.1 Introduction ........................................................................................................................... 21-1 21.2Transport Properties of Bulk Graphene............................................................................. 21-3

21.6 Summary ...............................................................................................................................21-22 References.........................................................................................................................................21-22

light [cas06, kat06, kat07]. Additionally, the presence of the two sublattices (A and B in Figure 21.1a), due to two carbon atoms per unit cell, allows to express the wavefunction for a unit cell by ϕ = cAϕA + cBϕB, where ϕA,B are the p∙∙ wave functions at the A, B site and the two component vector (cA, cB) forms the so-called pseudospin in graphene. Indeed, in close analogy to neutrino physics [and99], the dynamics of electrons in the near vicinity of the K points can be fully described by the Dirac-Weyl Hamiltonian, H = vFσ . p, where σ are the 2-D Pauli spin matrices and p is the momentum operator. Here, the graphene pseudospin takes the role of “real” spin and this analogy actually gives rise to the pseudospin terminology. ™e pseudospin up (or down) state is related to an A-B symmetric (or antisymmetric) wave function. Since the symmetry of these wave functions is

a function of the k ⃯

-vector direction, helicity (h = σ . p/2 |p|) becomes a good quantum number h1,2 = ±1/2 [and98, and99, mce99]. ™erefore, electrons (holes) around the K-point have a positive (negative) helicity. ™is implies that σ has its two eigenvalues either in the direction (+) or opposite (−) to the momentum p (as illustrated in Figure 21.2a). In other words, charge carriers in graphene can be described in terms of 2-D massless Dirac fermions [nov05, zha05, zho06, bos07]. ™us, the carriers can be considered as behaving analogously to relativistic particles but with a strongly reduced velocity allowing for the observation of quantum electrodynamic (QED)–specišc phenomena in a solid-state environment [cas07].