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36 Electronic Band Structure of Graphene and σ ≡ (σx,σy). In this four-spinor representation, the first two com- ponents represent the lattice components at the K point and the last two components those at the K′ point. We emphasise that one must clearly dis- tinguish both types of isospin: (a) the sublattice isospin is represented by the Pauli matrices σj, where “spin up” corresponds to the component on one sublattice and “spin down” to that on the other one. A rotation within the SU(2) sublattice isospin space yields the band indices λ = ±, and the band index (or band isospin) is, thus, intimitely related to the sublattice isospin. (b) The valley isospin, which is described by a second set of Pauli matrices τj, the z-component of which appears in the Hamiltonian (2.36), is due to the twofold valley degeneracy and is only indirectly related to the presence of two sublattices. The eigenstates of the Hamiltonian (2.36) are the four-spinors 1 0 ξ=+ 1 λeiφq ξ=− 1 0 Ψk,λ = √2 0 where we have, now, and Ψk,λ = √2 1 0 φq =arctanqy . qx Density of states The electronic density of states ρ(ǫ) counts the number of quantum states in the vicinity of a fixed energy ǫ. It turns out to be a very useful quantity when discussing the electric transport in the diffusive regime. It may be obtained from the total number of states below the energy ǫ, Zǫ 0 where we have chosen the electronic π∗ band, by derivation with respect to the energy 1 ∂N+ ρ(ǫ) = A ∂ǫ , (2.40) , −λeiφq (2.37) (2.38) Nλ=+ = g X = A k/ǫk+ ≤ǫ dǫ′ρ(ǫ′), (2.39)PDF Image | Physical Properties of Graphene
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