Graphite as a bose metal

Graphite as a bose metal

Yakov Kopelevich

Instituto de Física ''Gleg Wataghin'', Universidade Estadual de Campinas, Unicamp 13083-970, Campinas, São Paulo, Brasil

ABSTRACT

Although a considerable amount of the research work has been done on graphite, its physical properties are still not well understood, and novel phenomena such as the magnetic-field-driven metal-insulator transition (MIT), the quantum Hall effect, ferromagnetic and superconducting correlations have recently been revealed. Theoretical analysis suggests that the MIT in graphite is the condensed-matter realization of the magnetic catalysis phenomenon known in relativistic theories of (2 + 1) - dimensional Dirac fermions (DF), i. e. that the applied field opens an insulating gap in the spectrum of DF, associated with the electron-hole pairing. On the other hand, we demonstrate in this paper that a two parameter scaling analysis proposed by Das and Doniach [D. Das and S. Doniach, Phys. Rev. B 64, 134511 (2001)] to characterize the magnetic-field-tuned Bose metal - insulator transition can be well applied to the MIT observed in graphite. We discuss the possibility that the MIT in graphite is associated with the transition between Bose metal and excitonic insulator states.

The apparent metal-insulator transition (MIT) in two-dimensional (2D) electron (hole) systems which takes place either varying carrier concentration or applying a magnetic field H has attracted a broad research interest [1]. Recently, a similar MIT driven by a magnetic field applied perpendicular to basal planes has been reported for graphite [2 - 5], and attributed to the metal - excitonic insulator transition [6-8]. In this paper we demonstrate that the MIT in graphite can also be understood as a Bose metal - insultor transition (BM-IT). In particular, it is found that the two parameter scaling analysis proposed by Das and Doniach [9] to characterize the BM-IT in 2D systems can be well applied to the MIT measured in graphite. We speculate that the MIT in graphite is associated with the transition between Bose metal and excitonic insulator states.

Magnetotransport measurements have been performed on several well-characterized [4, 5] quasi-2D highly oriented pyrolitic graphite (HOPG) samples obtained from the Research Institute "Graphite" (Moscow) and the Union Carbide Co. Here, we present the results of the basal-plane resistance R_b(H, T) measurements obtained on two HOPG samples with the room temperature, zero-field out-of-plane/basal-plane resistivity ratio r_c/ r_b = 5 ×10⁴ and r_b(T = 300K) = 3 mWcm and r_b(T = 300K) = 5 mWcm for the samples labeled respectively as HOPG-UC and HOPG-3. Low-frequency (f = 1 Hz) and dc standard four-probe magnetoresistance measurements were performed on samples with dimensions 4 × 4 × 1.2mm³ (HOPG-3) and 5 × 5 × 1mm³ (HOPG-UC) in the temperature interval 2K < T < 300K using different 9 T-magnet He-cryostats. For the measurements, silver past electrodes were placed on the sample surface, while the resistivity values were obtained in a geometry with an uniform current distribution through the sample cross section. All resistance measurements were performed in the Ohmic regime in both H || c-axis and H ^ c-axis applied magnetic field configurations.

The transition from metallic- (dR_b/dT > 0) to insulator-like (dR_b/dT < 0) behavior of the basal-plane resistance R_b(T, H) driven by a magnetic field applied perpendicular to the graphene planes has been observed for all studied graphite samples. Fig. 1 presents R_b(T, H) measured for the HOPG-UC sample. As can be seen from Fig. 1, R_b(T) has a metallic character at zero or low enough fields. As the applied field exceeds H ~ 0.5 ¼ 1 kOe, R_b(T) becomes insulating-like, suggesting the occurrence of MIT driven by the magnetic field. Fig. 1 shows that R_b(T) goes through a shallow minimum at the field-dependent temperature T_min(H > H_c), where H_c is a threshold field below which the metallic state of graphite is preserved, and Fig. 2 gives a detailed view of the resistance minimum. It has been demonstrated [5, 10] that the magnetic field component perpendicular to basal graphite planes solely drives the MIT imposing certain constrains on theoretical approaches to the MIT.

The characteristic feature of the band structure of a single graphene layer is that there are two isolated points in the first Brillouin zone where the band dispersion is linear E(k) = v|k| (v = v_F ~ 10⁶m/s is the Fermi velocity), so that the electronic states can be described in terms of Dirac equations in two dimensions similar to, for instance, quasi-particles near the gap nodes in superconducting cuprates. The theoretical analysis [6 - 8] suggests that the MIT in graphite is the condensed-matter realization of the magnetic catalysis (MC) phenomenon known in relativistic theories of (2 + 1) - dimensional Dirac fermions. According to the theory, the magnetic field applied perpendicular to the graphene planes opens an insulating gap in the spectrum of Dirac fermions, associated with an electron-hole pairing, leading to the excitonic insulator state below a transition temperature T_ce(H) which can be identified with the T_min(H). As shown in Refs. [5, 10], T_min(H) in the vicinity of H_c can be well described by theoretically predicted dependencies T_min(H) ~ (H – H_c)^1/2 [6] or T_min(H) ~ [1 - (H_c/H)²]H^1/2 [8], indeed. The second expression for the T_min(H) corresponds to the formula (64) of Ref. [8], T_ce ~ (1 – )H^1/2, obtained within the MC theory, where n_b = 2pcn_2D/N_f|eH| º H_c/H is the filling factor, N_f is the number of fermion species (N_f = 2 for graphite), and n_2D is the 2D carrier density. We note a two orders of magnitude difference between the predicted value for m₀H_c ~ 2.5 T (n_2D = n_3Dd ~ 10¹¹cm^–2, n_3D ~ 3 × 10¹⁸ cm^–3 and d = 3.35 Å, the distance between graphene planes) [8] and experimental value m₀H_c = 0.02¼0.04 T. The discrepancy can be understood, however, assuming that the Coulomb coupling given by a dimensionless parameter g = 2pe²/Î₀v (Î₀ is the dielectric constant) drives the system very close to the excitonic instability [6, 7]. In this case, the threshold field H_c can be well below the estimated value of 2.5 T. The above analysis together with the experimental evidence that only ''orbital'' effects drive the MIT [5, 10] support the theoretical expectations of the field-induced excitonic insulator state in graphite.

At the same time, our R_b(T,H) data resemble very much the resistance behavior occurring in superconducting thin films (see, e. g., Ref. [11]). It has been shown in Refs. [3, 5] that the scaling analysis of the magnetic-field-induced superconductor-insulator quantum phase transition (SIT) can be equally well applied to the MIT observed in graphite. According to the scaling theory [12], the resistance in the critical regime of the zero-temperature SIT is given by the equation R(d, T) = R_crf(|d|/T^1/zn), where R_cr is the resistance at the transition, f (|d|/T^1/zn) a scaling function such that f(0) = 1; z and n are critical exponents, and d = H – H_cr the deviation of a variable parameter from its critical value. However, at low enough temperatures, where the resistance R_b(T) saturates, a clear deviation from the scaling takes place in both superconducting films [11] and graphite [3, 5]. More recently, it has been suggested that the SIT measured, e. g., in Mo-Ge films [11] is in fact a Bose metal - insulator transition, and a two parameter scaling formula RT^1+2/z /d^2b = f (d/T^1/zn), has been proposed to characterize it [9], where b = n(z + 2)/2. This analysis implies the existence of a non-superfluid liquid of Cooper pairs (Bose metal) in the zero-temperature limit. In Fig. 4 we use this approach to analyze the data obtained for HOPG-3 sample (see Fig. 3). As Fig. 4 illustrates, this new scaling formula works very well in the whole studied temperature interval, taking H_cr = 1140 Oe, z = 1, and n = 2/3.

The obtained results suggest an interesting possibility, namely an occurrence of the magnetic-field-driven Bose metal - excitonic insulator transition (BM-EIT). We assume that the sample nano-scale inhomogeneities [13] play a crucial role in this phenomenon, leading to co-existing metallic and excitonic insulator regions in graphite at H = 0 [6, 7]. Then, the Cooper pairs can be formed as a result of the electron-exciton interaction during a tunneling event of electrons from metallic regions into the insulating ones [14, 15]. Notice, that R_b(T) at H = 0 can be best described by the equation R_b(T) = R₀ + R₁exp(—E_a/k_BT), where E_a » 35 K is the effective activation energy [5]. The thermally-activated metallic behavior of R_b(T) is consistent with the formation of the superconducting gap in the single particle spectrum, indeed. On the other hand, the high enough applied field opens the excitonic gap in most of the sample and thus drives the BM - EIT.

This work has been supported by FAPESP and CNPq. The author is grateful to D. V. Khveshchenko, S. Doniach, I. A. Shovkovy, P. Esquinazi, and I. A. Luk'yanchuk for a discussion, and J. H. S. Torres and R. R. da Silva for an assistance.

[13] Unpublished.

Received on 31 July, 2003

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Publication Dates

• Publication in this collection
  25 Nov 2005
• Date of issue
  Dec 2003

History

• Received
  31 July 2003