By Toshiaki Enoki, Tsuneya Ando

From a chemistry element, graphene is the extrapolated severe of condensed polycyclic hydrocarbon molecules to limitless dimension. right here, the concept that on aromaticity which natural chemists make the most of is acceptable. fascinating concerns showing among physics and chemistry are reported in nano-sized graphene (nanographene), as we realize the significance of the form of nanographene in realizing its digital constitution. during this booklet, the basic concerns at the digital, magnetic, and chemical homes of condensed polycyclic hyodrocarbon molecules, nanographene and graphene are comprehensively discussed.

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Note that η(T ) is not necessarily the same as η(0). When there is no degeneracy, the state at t = T is completely same as that at t = 0 when the process is suﬃciently slow and adiabatic. 27) and therefore represents phase change corresponding to electron energy. Extra phase ϕ is called the Berry phase. 28) November 22, 2012 14:28 PSP Book - 9in x 6in 02-Toshiaki-Enoki-c02 Electronic States of Monolayer Graphene ¨ and substitute ψ(t) into the time-dependent Schrodinger equation. 29) and on integration we have T ϕ = −i dt ψ[η(t)] 0 dψ[η(t)] .

25) Fsk = √ 1 2 This is the “spin” part of an eigenfunction, obtained by choosing φs (k) = −θ (k)/2 in such a way that the wave function becomes continuous as a function of θ(k). It seems that this Fsk does not change the sign after 2π rotation θ (k) → θ(k) + 2π . However, the wave function changes the sign when we consider Berry’s phase [43, 44]. Let us consider the case that the Hamiltonian contains parameter η and change η from η(0) to η(T ) as a function of time t from t = 0 to t = T and assume that H[η(T )] = H[η(0)].

65]. 1, in the Boltzmann theory, the Hall conductivity in weak magnetic ﬁeld B is given by σx y = −ωc τ (ε F )σ0 , where ωc is the classical cyclotron frequency given by ωc (ε) = eBv 2 /cε. The cyclotron frequency diverges at the Dirac point, showing the presence of singularity there. In spite of this singularity, the Hall coeﬃcient is simply R H = −(ns ec)−1 as in conventional systems. Similar calculations in the self-consistent Born approximation were recently performed for the weak-ﬁeld Hall conductivity [65].