Millikelvin de Haas–van Alphen and magnetotransport studies of graphite

Recent studies of the electronic properties of graphite have produced conflicting results regarding the positions of the different carrier types within the Brillouin zone, and the possible presence of Dirac fermions. In this paper we report a comprehensive study of the de Haas–van Alphen, Shubnikov–de Haas, and Hall effects in a sample of highly orientated pyrolytic graphite, at temperatures in the range 30$\hspace{0.5em}$mK to 4$\hspace{0.5em}$K and magnetic fields up to 12$\hspace{0.5em}$T. The transport measurements confirm the Brillouin-zone locations of the different carrier types assigned by Schroeder, Dresselhaus, and Javan, Phys. Rev. Lett. 20, 1292 (1968): electrons are at the $K$ point, and holes are near the $H$ points. We extract the cyclotron masses and scattering times for both carrier types from the temperature- and magnetic-field-dependences of the magneto-oscillations. Our results indicate that the holes experience stronger scattering and hence have lower mobility than the electrons. We utilize phase-frequency analysis and intercept analysis of the $1/B$ positions of magneto-oscillation extrema to identify the nature of the carriers in graphite, whether they are Dirac or normal (Schrödinger) fermions. These analyses indicate normal holes and electrons of indeterminate nature.

Figure 1

Brillouin zone of graphite. Electron and hole pockets formed by the $\pi$ bands are centered along the zone edges $\mathit{HKH}$.

Figure 2

Graphical representation of the total phases $\phi =(\delta -\gamma )\hspace{0.5em}\left[2\pi \right]$ determined by the authors listed in Table ; (a) for high-frequency carriers, and (b) for low-frequency carriers. The phase starts at zero at the top of each circle and increases in the clockwise direction. In the case of the measurements of Ref. 6, (N) refers to their results on natural graphite and (H) refers to their results on HOPG.

Figure 3

Raw magnetometry data (change in magnetometer capacitance vs magnetic field) measured at 30$\hspace{0.5em}$mK. The lower left inset shows the detail at low magnetic fields. The upper inset is a schematic picture of the millikelvin torsion-balance magnetometer. The sample magnetisation causes a twist of the rotor which results in an imbalance in the differential capacitor formed by the fixed and rotor plates.

Figure 4

Hall resistance $R_{\mathit{xy}}$ plotted as a function of magnetic field at 30$\hspace{0.5em}$mK. Downward pointing features (marked H) occur when the Fermi energy passes through hole Landau levels; upward ones (E) occur when the Fermi energy passes through electron Landau levels.

Figure 5

Oscillatory magnetic susceptibility $\Delta \chi$ as a function of inverse magnetic field at 30$\hspace{0.5em}$mK. The arrows indicate the ranges of $B^{-1}$ used to obtain the FFTs of Fig. 6.

Figure 6

FFT of the 30$\hspace{0.5em}$mK magnetic susceptibility (Fig. 5) using data in the range from 0.27$\hspace{0.5em}$T${}^{-1}$ to 5$\hspace{0.5em}$T${}^{-1}$, showing the presence of hole oscillations with a fundamental field of 4.36$\hspace{0.5em}$T, marked ‘H’, and electron oscillations of approximately 5.97$\hspace{0.5em}$T, marked ‘E’. The second harmonics of the two carriers are also visible at twice their fundamental frequencies. To illustrate the magnetic field dependences of the two sets of oscillations, the inset (a) shows the FFT from 0.27$\hspace{0.5em}$T${}^{-1}$ to 2$\hspace{0.5em}$T${}^{-1}$ (‘High field’ on Fig. 5) where hole and electron peaks have roughly equal amplitude, while inset (b) shows the FFT for the data range from 2$\hspace{0.5em}$T${}^{-1}$ to 10$\hspace{0.5em}$T${}^{-1}$ (‘Low field’ on Fig. 5) demonstrating the dominance of the electron oscillations at low $B$.

Figure 7

dHvA oscillations at a range of temperatures. Oscillations at high $B^{-1}$ are predominantly due to electrons and these disappear rapidly as temperature increases leaving only the hole oscillations at low $B^{-1}$. Traces for successive temperatures are offset for clarity.

Figure 8

FFT plots of the dHvA oscillations using data in the range from 0.27$\hspace{0.5em}$T${}^{-1}$ to 2$\hspace{0.5em}$T${}^{-1}$, for a range of temperatures. The high-frequency (electron, E) peak is suppressed more rapidly with temperature than the low-frequency (hole, H) peak.

Figure 9

Intercept phase analysis of the 30$\hspace{0.5em}$mK dHvA oscillations. (a) Analysis of the oscillations of electrons. Peak position vs Landau-level index $n$ is plotted up to $5\hspace{0.5em}$T${}^{-1}$. The intercept (see inset) indicates a phase of $-0.20\pm 0.05$$\hspace{0.5em}\left[2\pi \right]$. The open circles show how the low-index raw data deviate from the straight line. (b) Analysis of the hole oscillation extrema (peaks and troughs) indicating a phase of $-0.48\pm 0.06$$\hspace{0.5em}\left[2\pi \right]$, consistent with 2D SFs.

Figure 10

FFT phase analysis of the 30$\hspace{0.5em}$mK dHvA results for electrons—a phase of $-0.25\pm 0.25$$\hspace{0.5em}\left[2\pi \right]$ is observed. The error arises from the uncertainty in determining the peak position.

Figure 11

$R_{\mathit{xx}}$ as a function of magnetic field at 30$\hspace{0.5em}$mK. The inset shows the quadratic dependence of $R_{\mathit{xx}}$ at low magnetic fields predicted by the Drude model, from which the product of the electron and hole mobilities may be deduced.

Figure 12

(a) $-\Delta R_{\mathit{xx}}$ (proportional to $\Delta {\sigma }_{\mathit{xx}}$) as a function of inverse magnetic field at 30$\hspace{0.5em}$mK. The inset is the FFT; the low frequency (hole) peak and its first harmonic are visible but the high frequency (electron) peak is not. (b) Intercept phase analysis of the SdH oscillation extrema (peaks and troughs). Only holes could be analysed and first-harmonic filtered data were used as discussed in the text, yielding a phase of $-0.48\pm 0.01$$\hspace{0.5em}\left[2\pi \right]$, consistent with 2D SF carriers. Raw extrema are also shown but exhibit systematic shifts to the right and to the left of the filtered data due to the presence of higher harmonics.