Band-contact lines in the electron energy spectrum of graphite

We discuss the known experimental data on the phase of the de Haas–van Alphen oscillations in graphite. These data can be understood if one takes into account that four band-contact lines exist near the HKH edge of the Brillouin zone of graphite.

Figure 1

Dependences of the bands ${\epsilon }_i$ $(i=1–4)$ on $\mathbf{k}$ near the edge HKH in graphite. Shown are the dependences ${\epsilon }_i\left(k_z\right)$ at $k_{\perp }=0$ and the dependences of ${\epsilon }_i$ on $k_{\perp }=\sqrt{k_x^2+k_y^2}$ at some characteristic values of $k_z$. The dashed line marks the position of the Fermi level ${\epsilon }_F$. For clarity, in the construction of the figure the parameter ${\gamma }_1$ has been used, which is 20 times smaller than that of Table II. For realistic values of ${\gamma }_1$ the point $P_0$ is much closer to the point $H$ than in the figure. The inset shows the Brillouin zone of graphite and its characteristic points.

Figure 2

Sketch of the Fermi surface (half of it) and of the band-contact lines in graphite. The accidental contact of the bands ${\epsilon }_2\left(\mathbf{k}\right)$ and ${\epsilon }_3\left(\mathbf{k}\right)$ occurs along the solid lines, while the dashed lines mark the accidental contact of the bands ${\epsilon }_3\left(\mathbf{k}\right)$ and ${\epsilon }_4\left(\mathbf{k}\right)$. The same bands are in contact along the HKH axis due to the symmetry of the crystal. All the lines merge at the point $P_0$. Shown are also the maximum cross sections of the electron $\left(S_e\right)$ and hole $\left(S_h\right)$ majorities and of the hole minority $\left(S_m\right)$ for the magnetic field along the HKH axis. A part of the Fermi surface where the electron and hole majorities touch is presented in an enlarged scale on the right; the band-contact lines pass through the conical features of the Fermi surface.