Report

BULK MAGNETIZATION OF GRAPHENE Tight binding approximation: the mobile electrons are always located in the proximity of an atom, and then are conveniently described by the pz atomic orbital of the atoms it touches. : normalized 2 wavefunction for an isolated atom. 1 conduction electron for each C atom in the 2 state. Unit cell (WXYZ) contains 2 atoms ( and ). 1 = 2 = = 2.46 Å (foundamental lattice displacement). The base functions are periodical functions with the same periodicity as the (2D) lattice. k is a wave vector. It defines a reciprocal lattice and acts as a kind of quantum number. A. Barbon, Corso di Magnetochimica. A.A. 2013-14. Graphene = 1 + 2 1 = 2 = 2∙ − 2∙ − Extended wave function THE BAND THEORY OF GRAPHENE Variational principle to obtain the best value of , by substituting the wavefunction in the Schroedinger equation: = 1 + 2 1 + 2 = 1 + 2 By pre-multiplication by 1 ∗ or 2 ∗ and integration we have: 11 21 11 + 12 = 21 + 22 = = = ∗ ∗ = Number of unit cells 12 1 1 ∙ = ∙ 2 ∙ 22 = 1 / + 22 / 2 11 We obtain: = 11 ′ ± 12 ′ A. Barbon, Corso di Magnetochimica. A.A. 2013-14. Graphene ′ = /N : interaction between an or atom with itself 0 ′ : interaction between first neighbors of the same type ( or ) 0 : interaction between first neighbors of opposite type ( and ) = 11 ′ ± 12 ′ 11 ′ = − 20 ′ cos 2 Energy levels as function of ky (kx=0) E − ≈ ±0 1 + 4 cos2 kx=0 Zero band-gap ky A. Barbon, Corso di Magnetochimica. A.A. 2013-14. Graphene Calculation of the density of states: It is possible to show that the number of electronic energy states per atom: N° of free electrons plus positive holes per atom: 1 = − = 30 2 30 2 ∞ 2 0 = 6 3 0 : number of atoms in the lattice 2 : Fermi distribution At room temperature ( = 0.025 eV) the effective number of free electrons ( ), per atom, is = 2.3 ∙ 10−4 N(E) 4 3.5 E f(E) 3 f (E) x 10 2.5 1 exp[( E Ec) / kT ] 1 2 1.5 1 0.5 Ec 0 E A. Barbon, Corso di Magnetochimica. A.A. 2013-14. Graphene 0 50 100 150 Temperature [K] 200 250 300 Magnetic susceptivity: 0 = 2 / ∝ G. Wagoner, Phys. Rev., 118, 647 (1960). A. Barbon, Corso di Magnetochimica. A.A. 2013-14. Graphene