PDF Publication Title:
Text from PDF Page: 153
ElelcetrcontircoTrnanicspoTrtrParonpesrtpieosorftFPewr-oLapyeerGrtriaephsenoefMaFteriawls-LayerGrapheneMaterials 143 Where N is the number of cells considered, k is the wave vector, din is a lattice translation vector and φi(r − din) is the pz local atomic orbital. The π-orbitals electronic band structure is therefore a solution of the Schrödinger equation: Hφ(k) = E(k)φ(k) (3) where H is the Hamiltonian for an electron in the atomic potential given by the atoms in the graphene lattice and φ(k) is a linear combination of Bloch wavefunctions which for simplicity wecanwriteintheformφ=aφA+bφB,whereφA andφB aregivenbyEq. 2andaandb are two coefficients. To solve Eq. 3 we need to find the matrix elements Hij =< φi|H|φj > of the Hamiltonian and the overlaps between the Bloch wavefunctions < φi|φj >. We start from noticing that if the two carbon atoms forming the graphene sublattice are energetically equivalent, the onsite energies of the sublattice A and B are equivalent (HAA = HBB) and without loss of generality we can set this energy equal to zero. The solution of Eq.3 is further simplified if we consider that the most significant hopping parameter is the first neighbour (γ0 ≈ 2.8eV, Castro Neto et al. (2009)) and that HBA is simply the complex conjugate of HAB (HBA = HAB∗). Therefore we can calculate the integral HAB =< φA|H|φB > where each atom is surrounded by three neighbours with relative coordinates (1/3, 1/3), (1/3,-2/3), (-2/3, 1/3). The term HAB reads: H = γ (eik·(a1 +a2 ) +eik·(a1 −2a2 ) +eik·(−2a1 +a2 )). (4) AB033 33 33 We can now project the solution of Eq. 3 onto < φA| and < φB| to obtain the system: aHAA + bHAB = E(k)a bHAB = E(k)a aHBA + bHBB = E(k)b ⇒ aHAB∗ = E(k)b (5) which has non-zero solutions for the coefficients a and b only if its secular determinant is zero. This condition leads to the energy dispersion of the graphene π-orbitals: √ E(kx,ky)=±γ0 1+4cos( 3rky)cos(rkx)+4cos2(rkx) (6) √222 3. The energy distribution vanishes at six points in the reciprocal lattice space where r = a0 with coordinates ±2π/r(1/ 3, 1/3), ±2π/r(0, 2/3), ±2π/r(−1/ 3, 1/3), see Fig. 1b. In these six K-space points the valence and conduction band touch one another, but only two of these points are independent. These are commonly indicated by K and K’ and also known as valleys. The electronic states close to the Fermi level (E=0) are readily described by a Taylor expansion of the energy dispersion in Eq. 6 at a chosen K point, from the eigenvalues of the effective Hamiltonians HK and HK′ for the high symmetry K and K’ points: √√ 0 kx + iky 0 0 kx − iky k −ik x y k +ik x y ( 7 ) 0 H K = h ̄ v F where both Hamiltonians operate on φ resembles the spinor wavefunctions in quantum electrodynamics, where the index of the spin corresponds to the sublattice for graphene and is referred to as pseudospin. In conclusion, the charge carriers in graphene are mass-less Dirac electrons obeying a linear energy dispersion: E(k) = ±vF|k| (8) a n d H K ′ = − h ̄ v F = (φA , φB )T . This two-component wave functionPDF Image | GRAPHENE SYNTHESIS CHARACTERIZATION PROPERTIES
PDF Search Title:
GRAPHENE SYNTHESIS CHARACTERIZATION PROPERTIESOriginal File Name Searched:
Graphene-Synthesis.pdfDIY PDF Search: Google It | Yahoo | Bing
Salgenx Redox Flow Battery Technology: Power up your energy storage game with Salgenx Salt Water Battery. With its advanced technology, the flow battery provides reliable, scalable, and sustainable energy storage for utility-scale projects. Upgrade to a Salgenx flow battery today and take control of your energy future.
| CONTACT TEL: 608-238-6001 Email: greg@infinityturbine.com | RSS | AMP |