PDF Publication Title:
Text from PDF Page: 164
1454 Graphene – Synthesis, Characterization, Properties anWdill-bAe-psept-lbiyc-IaN-tTiEoCnHs e2 v2 where gv = gs = 2 are respectively the spin and valley degeneracies (McClure (1956)). The low-energy Landau levels dispersion of bilayer graphene is approximately given by the relation (McCann et al. (2006)): χ(εF) = −gvgs 6πc2 δ(εF), (14) Esn = sh ̄ωc n(n−1), (s = ±, n = 0,1,2,..) (15) with ωc = eB/m∗ the cyclotron frequency associated with the effective mass m∗ of bilayer graphene. The Landau levels energy spacing is now linear in B owing to the usual quadratic energy dispersion of bilayers, see Fig. 9a. The two lowest levels of n = 0 (per spin and valley) appear at zero energy. This amounts to 8-fold degeneracy in total and causes doubling of the Hall conductivity jump at zero electron density, see Fig. 9b. The orbital susceptibility for small Fermi energy becomes (Koshino et al. (2007); Safran et al. (1984)): e2v2 |εF| χ(εF) = −gvgs 4πc2γ −ln γ , (16) 11 which has a logarithmic singularity in contrast to the delta-function in monolayer graphene. The singularity is weaker than in monolayer, since the Landau level spacing is narrower so that the total energy gain in magnetic field at εF = 0 becomes smaller. When increasing the magnetic field amplitude, the energy of the particular Landau level other than zero-energy √ levels crosses over from linear B to dispersion from linear to quadratic. The experimental observation of the magnetotransport and the quantum Hall effect is revealing yet a rich scenario of Landau level spectrum in trilayer graphene (Bao et al. (2011); Kumar et al. (2011); Taychatanapat et al. (2011); Zhang et al. (2011)). For ABA multi-layer graphenes, the Landau level spectra can be again decomposed into a superposition of the monolayer and bilayer subsystems as introduced in section 2. In this case, the physical properties in magnetic fields, such as Hall conductivity and the magnetic susceptibility can be expressed as the summation over components of subsystems (Koshino et al. (2007; 2008)). In trilayer graphene, for example, the spectrum is composed of bilayer and monolayer Landau levels, resulting in a 12-fold degeneracy at zero energy. The effect of the next-nearest interlayer couplings, such as γ2 and γ5 (see Fig. 4 a), are often neglected in the simplest approximation, but become particularly important for the low-energy spectrum near the charge neutrality point (Koshino et al. (2011)). For trilayer, the 12-fold degeneracy is lifted by those couplings, causing a qualitative change in the quantum Hall plateau structure. The Landau spectrum of ABC multilayers is quite different from the one of ABA, where the pair of low-energy flat-bands gives the Landau level sequence (Guinea et al. (2006); Koshino et al. (2009)): B , in accordance with the crossover of the zero-field ( h ̄ ω B ) N Esn =s γN−1 n(n−1)···(n−N+1), (s=±, n=0,1,2,..), (17) 1 where N is the number of layers. Including valley and spin degree of freedom, 4N-fold degenerate Landau levels appear at zero energy.PDF 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 (Standard Web Page)