Temperature-driven changes in the Fermi surface of graphite

We report on temperature-dependent size and anisotropy of the Fermi pockets in graphite revealed by magnetotransport measurements. The magnetoresistances (MRs) obtained in fields along the $c$ axis obey an extended Kohler's rule, with the carrier density following the prediction of a temperature-dependent Fermi energy, indicating a change in the Fermi pocket size with temperature. The angle-dependent magnetoresistivities at a given temperature exhibit a scaling behavior. The scaling factor that reflects the anisotropy of the Fermi surface is also found to vary with temperature. Our results demonstrate that temperature-driven changes in Fermi surface can be ubiquitous and need to be considered in understanding the temperature-dependent carrier density and MR anisotropy in semimetals.

Figure 1

Magnetoresistivities of graphite. (a) Magnetic field dependence ${\rho }_{xx}\left(H\right)$ measured at various temperatures. For clarity, we plot only a portion of the ${\rho }_{xx}\left(H\right)$ curves taken at temperatures from $T=2$ to 120 K at intervals of 2 K, 123 to 180 K at intervals of 3 K, 184 to 200 K at intervals of 4 K, and from 205 to 300 K at intervals of 5 K. (b) Temperature dependence ${\rho }_{xx}\left(T\right)$ constructed from ${\rho }_{xx}\left(H\right)$ data. For clarity, we present ${\rho }_{xx}\left(T\right)$ curves at magnetic fields of $H=0$, 0.02, 0.04, 0.08, 0.2, 0.6, 1.4, 2.6, 4.2, 6.2, and 9 T (from bottom to top). Data were taken in magnetic field parallel to the $c$ axis of the crystal.

Figure 2

Extended Kohler's rule of the magnetoresistance (MR). (a) and (d) Magnetic field and temperature dependences of the MR derived from data in Figs. 1 and 1, respectively. (b) and (e) Kohler's rule plots of the data in (a) and (d), respectively. (c) and (f) Extended Kohler's rule plots of the MR curves in (a) and (d), respectively. To be comparable with the scaling results in Fig. S6 in the Supplemental Material [35], theoretical values of $n_T$ were calculated by normalizing the temperature-dependent carrier densities in Fig. S6 in the Supplemental Material [35] to that at $T=300\mathrm{K}$, i.e., $n_T=$ 1 at $T=300\mathrm{K}$. Symbols in (a)–(c) are the same as those in Fig. 1, while the same symbols as those in Fig. 1 are used for (d)–(f) [see legends in (c) and (f), respectively]. Dashed red lines in (c) and (f) represent a power law relationship of $\mathrm{MR}\sim H^{\alpha }$ with $\alpha =1.8$.

Figure 3

Anisotropy of the magnetoresistivity in graphite. (a) ${\rho }_{xx}\left(H\right)$ curves at various angles θ obtained at $T=300\mathrm{K}$. (b) Data in (a) replotted with $H$ scaled by a factor ${\varepsilon }_{\theta }$. (c) Angle dependence of the scaling factor ${\varepsilon }_{\theta }$. Symbols are experimental data, and the solid line is a fit with ${\varepsilon }_{\theta }={(\mathrm{co}{\mathrm{s}}^2\theta +\mathrm{si}{\mathrm{n}}^2\theta /{\gamma }^2)}^{1/2}$ and $\gamma =20$. A schematic in the inset of (c) shows the definition of angle θ.

Figure 4

Temperature dependences of the derived anisotropy factor γ (red open circles) and the magnetoresistance (MR) at $H=0$.2 T (solid line).