De Haas-van Alphen Effect in Pyrolytic and Single-Crystal Graphite

Abstract

The de Haas-van Alphen (DHVA) oscillations have been observed in the differential susceptibility of well-annealed pyrolytic graphite (PG) with the low-frequency field-modulation technique. The results for PG are compared with similar studies of natural single crystal (SCG) by Soule et al., and by the present authors. Majority-electron and majority-hole DHVA periods in PG are respectively equal to the periods in SCG for $0\ifmmode^\circ\else\textdegree\fi{}\ensuremath{\le}\ensuremath{\theta}\ensuremath{\lesssim}50\ifmmode^\circ\else\textdegree\fi{}$, where $\ensuremath{\theta}$ is the angle between the applied field and the $c$ axis of the sample. Observation of a long-period, minority-carrier DHVA oscillation in PG and SCG confirms Soule's discovery of such an oscillation in SCG and provides a sensitive test for the Slonczewski-Weiss band model. The minority-carrier oscillation in PG is qualitatively but not quantitatively similar to the one in SCG. This oscillation is attributed to magnetic energy levels passing through the Fermi level near the Brillouin-zone corner at ${k}_{z}=\frac{\ensuremath{\pi}}{{c}_{0}}$, where for $\ensuremath{\theta}=0\ifmmode^\circ\else\textdegree\fi{}$ there is no extremal area of the Fermi surface at zero field, when spin-orbit interactions are neglected. This interpretation is supported by the calculation of an oscillatory free energy and by the excellent agreement for PG between the predicted values and experimental results for the period, effective mass, period anisotropy, and phase. The phase is not the value which would be predicted from a free-electron model for either a maximal or a minimal cross-sectional area. The existence of minority-carrier oscillations in PG indicates that the hole and electron Fermi surfaces are closed and that the electron surface intersects the hexagonal face of the Brillouin zone. The band parameters of the Slonczewski-Weiss model for PG were evaluated from the DHVA data and the preliminary magnetoreflection measurements of Dresselhaus and Mavroides. A similar analysis for SCG was carried out successfully for the majority-carrier pockets, but did not provide a consistent picture for the minority-carrier surface. The effects of spin-orbit interaction on the Fermi surface are discussed. Neutron and x-ray diffraction studies show that the $c$ axes of the individual crystallites in a PG sample are aligned to no better than \ensuremath{\sim}1\ifmmode^\circ\else\textdegree\fi{}. The effects of this misalignment as well as of the finite crystallite size on the DHVA oscillations are considered.