Phase Analysis of Quantum Oscillations in Graphite

The quantum de Haas–van Alphen (dHvA) and Shubnikov–de Haas oscillations measured in graphite were decomposed by pass-band filtering onto contributions from three different groups of carriers. Generalizing the theory of dHvA oscillations for 2D carriers with an arbitrary spectrum and by detecting the oscillation frequencies using a method of two-dimensional phase-frequency analysis which we developed, we identified these carriers as (i) minority holes having a 2D parabolic massive spectrum $p_{\perp }^2/2m_{\perp }$, (ii) massive majority electrons with a 3D spectrum and (iii) majority holes with a 2D Dirac-like spectrum $\pm v{p}_{\perp }$ which seems to be responsible for the unusual strongly-correlated electronic phenomena in graphite.

Figure 1

dHvA and SdH oscillations in graphite. Upper panel shows the region of fields $7\mathrm{k}\mathrm{O}\mathrm{e}<H<50\mathrm{k}\mathrm{O}\mathrm{e}$, characteristic for majority carrier oscillations whereas the region of fields in low panel $0.25\mathrm{k}\mathrm{O}\mathrm{e}<H<2.5\mathrm{k}\mathrm{O}\mathrm{e}$ corresponds to minority carrier oscillations.

Figure 2

Spectral intensity of dHvA oscillations of susceptibility $|\chi (\nu )|$ and of SdH oscillations of magnetoresistance $|R(\nu )|$. Peaks $m_{1,2}$, $e_{1,2}$, $h_{1,2}$ correspond to the 1st and 2nd harmonics of oscillations from minority electrons, majority electrons and majority holes. The low and high frequency plots of $|R(\nu )|$ are obtained from different sets of experimental data.

Figure 3

Quantum oscillations of susceptibility $\Delta \chi (H^{-1})$ and of resistance $\Delta R(H^{-1})$ for different groups of carriers, obtained after two-harmonic band-pass filtering of experimental data. Dot lines show the one-harmonic phase fit using the phase factors specified in Table . The sign of $\Delta R$ is inverted to recover the oscillating part of the conductivity $\Delta \sigma$.

Figure 4

Contour plot of the positive part of the phase-shift function $K(\varphi ,\nu )=\mathrm{R}\mathrm{e}{e}^{i\phi }\chi (\nu )$ for minority and majority carries. Position of maxima of $K(\varphi ,\nu )$ determines the oscillation frequencies ${\nu }_i$ and phases ${\varphi }_i$ for different groups of carriers. Upper panel presents the corresponding spectral intensity $|\chi (\nu )|$.