Thickness-dependent phase transition in graphite under high magnetic field

Various electronic phases emerge when applying high magnetic fields in graphite. However, the origin of a semimetal-insulator transition at $B\simeq 30$ T is still not clear, while an exotic density-wave state is theoretically proposed. In order to identify the electronic state of the insulator phase, we investigate the phase transition in thin-film graphite samples that were fabricated on silicon substrate by a mechanical exfoliation method. The critical magnetic fields of the semimetal-insulator transition in thin-film graphite shift to higher magnetic fields, accompanied by a reduction in temperature dependence. These results can be qualitatively reproduced by a density-wave model by introducing a quantum size effect. Our findings establish the electronic state of the insulator phase as a density-wave state standing along the out-of-plane direction, and help determine the electronic states in other high-magnetic-field phases.

Figure 1

(a) Schematic view of Brillouin zone and Fermi surfaces at $B=0$ in graphite. Electron pockets and hole pockets are formed around K $\left({\mathrm{K}}^{\prime }\right)$ and H $\left({\mathrm{H}}^{\prime }\right)$ points, respectively. Valleys along H-K-H and ${\mathrm{H}}^{\prime }-{\mathrm{K}}^{\prime }-{\mathrm{H}}^{\prime }$ lines are energetically degenerated. The size of Fermi pockets is exaggerated for clarity. (b) Calculated Landau subbands in graphite under magnetic field $B=30$ T along the $c$ axis. Only four Landau subbands are at the Fermi level ${\varepsilon }_F$. Calculation is based on the Slonczewski-Weiss-McClure model with ${\gamma }_3=0$ (Refs. [8, 9, 10]). Red lines indicate the dispersions for the bulk (thick enough) system, and blue markers are for the thin-film system. Width of horizontal light blue and light red bars indicate $k_{B}T$ at low and high temperatures, respectively (see main text). The density-wave state characterized by $q_z=2k_F$ is expected to appear in the bulk system, while it is expected to be unstable in the thin-film system owing to the sparse $k_z$ states.

Figure 2

In-plane resistance as a function of magnetic field along the $c$ axis in (a) 173-nm-thick and (b) 80-nm-thick graphite at $T=1.6,2.0,3.0,4.2$, and 5.4 K. Several dip structures up to 10 T are Shubnikov–de Haas oscillations. In both samples, the critical magnetic field of the semimetal-insulator transition increases with elevating temperatures. Critical fields in thinner samples are higher, and show small temperature dependence.

Figure 3

(a) Phase boundary of the semimetal-insulator transition in $B-T$ plane for each graphite sample, obtained from Fig. 2. The bulk line is taken from Ref. [17]. By reducing sample thickness, the phase boundary shifts to higher magnetic fields, and the slope becomes steep. (b) Simulated phase boundaries for several thicknesses. Two characteristics of the thinner system are qualitatively reproduced.

Figure 4

In-plane resistance as a function of magnetic field along the $c$ axis in 173-nm-thick graphite for several conditions. At $T=4.2$ K, the semimetal-insulator transition can be observed for low current 5 and $10\mu \mathrm{A}$, while larger current of $100\mu \mathrm{A}$ destroys the transition. At $T=2$ K with low current of $5\mu \mathrm{A}$, the transition shifts to the lower magnetic field. Note that the absolute values of the magnetic fields are not reliable (see main text).

Figure 5

Raw ac data in graphite with $d=173$ nm at $T\sim 100$ K, and the dc component extracted from it by the numerical lock-in technique. Magnetic fields are displayed on the right axis. Ascending and descending branches are indicated by broken and solid lines, respectively.

Figure 6

Density-response function ${\chi }^{\left(0\right)}$ as a function of $q_z$ under $B=30$ and 40 T perpendicular to the plate for bulk and 300-, 30-, 20-, and 10-u.c.-thick systems at $T=1,2,5$, and 10 K from top to bottom. For the bulk system, a sharp peak evolves at $q_z=2k_F$ and is rapidly suppressed with elevating temperatures. These peak heights shrink with loss of temperature dependence in thin-film systems.