Study of the Shubnikov-de Haas Effect. Determination of the Fermi Surfaces in Graphite

Abstract

Measurements of the oscillatory magnetoresistance of a high-quality graphite single crystal were made for all angles $\theta$ between the magnetic field and the $c$ axis, for magnetic fields up to 24 kG, and for temperatures from 1.22 to 4.22°K. The results were analyzed by a least-squares fitting to a generalized Landau formula. Oscillations due to electrons were observed for all orientations (including H ⊥ c, where the amplitude dropped by a factor ${10}^5$), proving that the electron Fermi surfaces are closed. Although oscillations due to holes were not observed beyond $\theta \simeq 84°$, indirect arguments show that the hole Fermi surfaces are also closed. Both electron and hole surfaces are elongated along the $c$ axis and have anisotropy ratios of 12.1±1.4 and about 17, respectively. The electron surface is approximately ellipsoidal, whereas the hole surface is similar except for extended ends giving it a diamond-like shape. The results are consistent with a moderate degree of trigonal asymmetry about the $c$ axis. Comparison between the electron density found from the volume of the electron Fermi surfaces and that determined previously from the nonoscillatory galvanomagnetic data confirms the theoretical prediction that there are four electron Fermi surfaces in the Brillouin zone. More indirect arguments show that there are two hole surfaces. Consideration of the size and location of these surfaces along the six zone edges parallel to the $c$ axis leads to a new determination of $\Delta \simeq -0.12\mathrm{}$ eV for the band parameter which represents the difference of potential between the two types of atomic sites in the graphite lattice. Analysis of the temperature and magnetic field dependence of the oscillatory amplitude yields effective mass values in the basal plane of $(0.039\mathrm{}\pm 0.001)m_0$ for electrons and $(0.057\mathrm{}\pm 0.002)m_0$ for holes. These masses show an orientation dependence that is consistent with the derived Fermi surface anisotropies. The large amplitude and asymmetric shape of the oscillations in the magnetoconductivity, measured for H∥c at 1.26 and 4.22°K, are accurately described by the theory of Adams and Holstein. However, there is an unexplained monotonic variation with magnetic field in the total magnetoconductivity. The effective change in temperature due to collision broadening $\Delta T$ is about 5 times greater than that estimated from the conductivity relaxation time. This discrepancy in $\Delta T$ is qualitatively explained and is related directly to the fact, established from the data of Berlincourt and Steele, that the $\Delta T$ found from magnetoresistance oscillations is greater than that found from susceptibility oscillations on the same sample.