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2009 CARBON ALLOTROPESMagnetotransport to extract the spin gap for charged excitations in graphiteIn a magnetic field the properties of graphite are remarkablywell described by the Slonczewski, Weiss and Mc-Clure (SWM) band structure calculations. While orbitaleffects have been extensively used to caliper the Fermi surface,the more subtle spin effects have received less attention.Indeed, the well documented movement of the Fermienergy in a magnetic field seriously complicates the extractionthe spin gap (g-factor) from the magnetotransport data.Recent advances in experimental techniques, in particularthe vastly increased desktop computing power available fordiagonalizing the SWM Hamiltonian, makes it timely torevisit this problem, extending previous measurements tohigher magnetic fields and lower temperatures.The oscillatory component of longitudinal resistance ∆R xxas a function of the magnetic field from B = 0 − 28 T forvaries orientations between θ = 0 ◦ and θ = 90 ◦ is shownin figure. 27(a). Quantum oscillations due to the majorityelectrons and holes Fermi surfaces are clearly observed.For increasing tilt angles the quantum oscillations shift as1/cos(θ) to higher magnetic field demonstrating the quasi2D nature of graphite. The experimentally observed splitting∆B = B ↓ − B ↑ , where B ↑ and B ↓ are the magneticfield positions of the spin up and spin down features, isplotted as a function of the mean total magnetic field positionB m = (B ↓ + B ↑ )/2 for the n = 1 electron and holefeatures (figure.27(b)). The failure of ∆B to follow a simplequadratic behavior is an experimental signature that themovement of the Fermi energy must be taken into accountwhen extracting the g-factor.Figure 27: (a) ∆R xx as a function of the magnetic field for differentangles (0 ◦ ≤ θ ≤ 90 ◦ ). (b) Magnetic field splitting ∆B as afunction of the total magnetic field B m . The thick line is calculatedfor the n = 1 electron Landau band with g s = 2.53 takinginto account the movement of the Fermi energy. The thin line isthe expected parabolic behaviour if E f is constant and requiresand unreasonably large value of g s = 6.5In order to extract the g-factor, we use the SWM band structuremodel.The effect of the in-plane magnetic field can beincorporated into the SWM model through an effective spingap ∆ s = g e f f µ B B ⊥ where the real g-factor g s = g e f f cos(θ).In graphite, E f moves with the applied perpendicular magneticfield as carriers are transferred between the electronand hole pockets. The Fermi level has to be calculated selfconsistentlyassuming the sum of the electron and hole concentrationsis constant, n− p = n 0 . At each angle, the effectivespin gap is found for which the SWM model gives thecorrect magnetic field position for the crossing of the spinup and spin down Landau band with the Fermi energy.Figure.28 shows the result for the spin gap ∆ s extractedfrom the SWM calculations as a function of the total magneticfield for the n = 1 hole and the n = 1 − 4 electronLandau bands. The spin gap is similar for both the electronand holes Landau bands at a given total magnetic field.∆ s increases linearly with magnetic field and a linear fit to∆ s = g s µ B B m (solid line) for both electron and hole Landaubands gives g s = 2.53 ± 0.07. The enhancement comparedto the electron spin resonance value can be attributedto electron-electron interactions.Figure 28: (a) Spin gap ∆ s for the electron and hole Landaubands, obtained by fitting the SWM model to ∆B, as a functionof the magnetic field. (b) The corresponding SWM g-factors foreach data point. The thick dashed line corresponds to g s = 2.53while the thin dashed lines correspond to the anisotropic electronspin resonance g-factors.J. M. Schneider, M. Orlita, M. Potemski and D. K. MaudeN. A. Goncharuk, P. Vasek, P. Svoboda and Z. Vyborny (Academy of Science, Prague)21