{'fraction_non_alphanumeric': 0.051614543605390285, 'fraction_numerical': 0.018560894991100942, 'mean_word_length': 4.368516833484986, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Dirac and Normal Fermions in Graphite and Graphene: Implications to the Quantum Hall Effect

Spectral analysis of Shubnikov de Haas (SdH) oscillations of magnetoresistance and of Quantum Hall Effect (QHE) measured in quasi-2D highly oriented pyrolytic graphite (HOPG) [Phys. Rev. Lett. 90, 156402 (2003)] reveals two types of carriers: normal (massive) electrons with Berry phase 0 and Dirac-like (massless) holes with Berry phase pi. We demonstrate that recently reported integer- and semi-integer QHE for bi-layer and single-layer graphenes take place simultaneously in HOPG samples.

Spectral analysis of the Shubnikov de Haas (SdH) magnetoresistance oscillations and of the Quantum Hall Effect (QHE) measured in quasi-2D highly oriented pyrolytic graphite (HOPG) [Phys. Rev. Lett. 90, 156402 (2003)] reveals two types of carriers: normal (massive) electrons with Berry phase 0 and Dirac-like (massless) holes with Berry phase π. We demonstrate that recently reported integerand semi-integer QHE for bi-layer and single-layer graphenes take place simultaneously in HOPG samples. Observations of the quantum Hall effect (QHE) and magnetic-field-driven metal-insulator transition in quasi-2D highly oriented pyrolytic graphite (HOPG) [1,2] indicate that much in the physics of graphite had been missed in the past and had triggered a considerable interest in graphite once again. Analyzing the quantum de Haas van Alphen (dHvA) and Shubnikov de Haas (SdH) oscillations in bulk HOPG samples we experimentally proved [3] that, besides the conventional electronic charge carriers with the massive spectrum E = p 2 /2m, massless (2+1)-dimensional fermions (holes) with Dirac-like linear spectrum E = ±v|p| and non-trivial Berry phase π do exist in graphite. It is likely that these Dirac-type carriers are responsible for the strongly correlated phenomena predicted by theory [4,5,6]. The results of [3] were confirmed by the recent angle resolved photoemission spectroscopy (ARPES) experiments performed on HOPG [7,8], that provide an unambiguous experimental evidence for coexistence of Dirac-like holes and normal (massive) electrons located at H and K poins of the Brillouin zone, respectively.
Very recently, remarkable progress in the technology of ultrathin films consisting of one [9,10], two [11] or several [12] graphite monolayers (graphenes) was achieved. Analysis of unconventional half-integer QHE and Berry phase π of SdH oscillations in graphene led to the conclusion that charge carriers in graphene have the nature of Dirac-like massless fermions [9,10]. At the same time, bilayer graphene demonstrates a new type of the integer QHE in which the last (zero-level) plateau is missing, and therefore the charge carriers are described by the chiral fermions having the Berry phase 2π and the energy spectrum is represented by two touching parabolas E(p) = ±p 2 /2m [11].
In the present work we apply our filtering and phase analysis procedures [3] to analyze the SdH oscillations and the QHE effect in HOPG aiming to clarify how these effects correlate with those studied in single and bilayer graphene systems. In particular, we demonstrate that normal and Dirac-like fermions seen in the bulk graphite are responsible respectively for the integer and semi-integer precursors of the QHE. To check the independence of these conclusions on the windowing and frequency filtration cut-off we used an alternative, insensitive to filtration procedure method of two-dimensional phase-frequency analysis, described in detail in [3].
We analyze the field dependencies of the basal-plane resistance R xx (B) and the Hall resistance R xy (B) reported in Ref. [1] for the sample labeled as HOPG-3 (noting that the HOPG-UC sample, similar to HOPG-3, has been studied in Ref. [3]).
As seen from Fig. 1, R xy (B) demonstrates several QHE-like plateaus at high fields, also observed in various HOPG samples by other groups [13,14]. However, unlike the conventional QHE, R xx (B) does not drop to zero at the positions of these plateaus. Only weak SdH oscillations ∆R xx (B) [3] are seen superimposed on al- most linear R xx (B) dependence. This can be explained either by imperfection of the QHE itself or by the existence of a background resistance of unknown origin. It is interesting to note that QHE with similar behavior of R xx (B) has been reported for Bechgaard salts [15,16] and for (Bi 1−x Sb x ) 2 Te 3 layered semiconductors [17,18]. Because quantum oscillations are periodic functions of the inverse field B −1 , the inverse-field spectral analysis is the appropriate tool to discriminate between different groups of carriers involved in oscillations. Fig. 2 presents the spectral intensity of SdH oscillations of the resistance R xx (B −1 ) and the Hall conductance G xy (B −1 ) ≡ R −1 xy (B −1 ), (in both cases polynomial background has been subtracted) together with analyzed in [3] spectral intensity of dHvA oscillations of magnetic susceptibility χ(B −1 ), (χ = dM/dB). Fig. 2 shows that two peaks in the susceptibility spectrum observed at 4.68T and at 6.41T are also resolved in the spectrum of G xy (B −1 ). The higher-frequency magnetoresistance peak manifests itself as the right-shoulder structure of R xx (B −1 ) spectrum. From comparative analysis of dHvA and SdH oscillations in [3] we identified these peaks as originating from normal (electrons) and Dirac (holes) carriers. Here, we re-affirm this conclusion in a different way, namely by means of the comparative analysis of longitudinal and Hall conductance oscillations, which is based on the separate study of the contributions from both types of carriers. Fig. 3 (curves a and b) shows the raw data of G xy (B −1 ) and ∆R xx (B −1 ) (after background signal subtraction) in which the normal electron contribution dominates in the magnetotransport quantum oscillations [3]. Extraction of the Dirac-like carriers signal from these data was the most challenging task that we performed  [10] and carrier concentration (ν = n/n0 with n0 = 1.5 · 10 12 cm −2 at B = 14T )(dotted line d) [9] its harmonics (see Fig. 2) and using the pass-band filtering of ∆R xx (B −1 ) with the corresponding hole frequency windowing. The resulting contributions from Dirac-like carriers are presented in Fig. 4 (curves a and b). Note that the sign of ∆R xx is reversed in order to present the oscillating part of the conductivity ∆σ xx , because in our case of ρ xx ≫ ρ xy and ∆σ xx is related to ∆R xx as: The situation is just opposite to the conventional QHE case with giant oscillations of R xx (B) when ρ xx ≪ ρ xy and σ xx behaves in the same way as R xx . Note also that the amplitude of SdH oscillations associated with Diraclike fermions (Fig. 4, curve b) is three times smaller than that for normal carriers (Fig. 3, curve b).
To facilitate the comparative analysis of the results obtained for various graphite (graphene) samples, we present all the data as a function of the normalized filling factor ν = B 0 /B, where B 0 corresponds to the SdH oscillation frequency. The conductance G xy is normalized by G 0xy , (≃ 28Ω −1 for electrons and 19Ω −1 for holes) corresponding to the step between subsequent QHE plateaus. Fig. 3 (curve c) includes SdH oscillations reported for an 8µm thick HOPG sample taken from the inset of Fig.  4 in [13] after the polynomial background signal subtraction. The coincidence of the SdH oscillation frequency, phase and the asymmetric form of the signal with those measured in our HOPG sample suggests that the same Fermi surface electron pocket is responsible for SdH oscillations in both samples.
Before we proceed with the data analysis, we note that Landau Level (LL) quantization spectra for normal and Dirac-like carriers are different. In the normal carrier case the equidistant LLs E n = (e /m ⊥ c)B(n + 1/2) [19] are separated by the gap E 0 = e B/2m ⊥ c from E = 0 whereas in the Dirac-like case E n = ±v 2e Bn/c [20], and the Lowest Landau Level (LLL) is located exactly at E 0 = 0. This leads to two important experimental consequences.
(i) SdH oscillations of conductivity [3] (where A(B) is the non-oscillating amplitude) acquire the phase factor either γ = 1/2 or γ = 0 for normal and Dirac carriers, respectively. The additional phase factor δ governed by the curvature of Fermi surface is quite small: |δ| < 1/8 (in 2D δ = 0) and can be neglected.
(ii) QHE plateaus occur either in integer (normal carriers) or in semi-integer (Dirac fermions) way and can be expressed via the phase factor γ as [21,22]: where µ = ∓1 for electrons/holes, and g s is the (iso)-spin degeneracy factor. The phase factor γ is defined for an arbitrary spectrum E(p) by the quasiclassical quantization condition of the Fermi surface cross-section S(E f ) = (n + γ)2πeB/ c (n ≫ 1). This factor is uniquely related to the topological Berry phase Φ B = kπ acquired by a fermion, moving around S(E f ) [23]: γ equals either 1/2 for even k (normal carriers) or 0 for odd k (Dirac-like fermions). For example, the two-parabola spectrum of bilayer graphite E(p) = ±p 2 /2m ⊥ is quantized in magnetic field as: E n = ±(e /m ⊥ c)B n(n − 1) [24]. This gives an unconventional doubly degenerate LLL with E 0 = E 1 = 0 but for higher n the system recovers the behavior of normal carriers having γ = 1/2 (with n → n − 1 in (3)).
We obtain the phase factors γ and δ by plotting the inverse field values at conductivity oscillation maxima (minima) as a function of their number (number -1/2), as shown in Fig. 5. The linear extrapolation of data points to B −1 = 0 unambiguously determines the phase factors: γ = 1/2, δ ≃ −1/8 for data depicted in Fig.  3 (curve b) and γ = 0, δ = 0 for data shown in Fig. 4 (curve b). Thus, we identify here normal (3D) and Diraclike (2D) charge carriers in agreement with our previous results obtained for HOPG-UC sample [3].
Curves (a) in Figs. 3 and 4 provide evidence for the staircase behavior (quantization) of G xy for both normal and Dirac-like fermions. As expected for QHE regime, the minima in longitudinal conductivity σ xx coincide with the plateau positions in G xy .
In the case of normal carriers , Fig. 3 (curve a), the quantization of G xy corresponds to the integer QHE, as follows from Eq. (3) for γ = 1/2. In the same Fig. 3 we presented the QHE staircases of σ xy , reported in [11] for bilayer graphite as function of both inverse field B −1 (Fig. 3, curve d) and carrier concentration n (Fig. 3, curve e). The scales of B −1 and n were normalized to the corresponding oscillation periods of σ xx . The Hall conductivity was normalized to the step between subsequent plateaus in σ xy (B). The comparative analysis indicates that all three dependencies, i. e. G xy (B −1 ) for HOPG, σ xy (B −1 ) and σ xy (n) for bilayer graphene are equally well fit to the integer QHE theory with γ = 1/2. We stress, however, that our data don't allow us to distinguish between the conventional integer QHE with Berry phase 0 and the chiral one with Berry phase 2π, proposed for bilayer graphite [11]. This is because both models imply γ = 1/2 and the same integer QHE staircase at LL n > 1. The only difference between them is the absence of zero level plateau for chiral fermions whose (non)existence we cannot verify at the moment.
To conclude, the results reported in this letter give new insight on the behavior of Dirac and normal fermions in graphite. Namely, we demonstrate the coexistence of QHE precursors associated with normal electrons and Dirac-like holes. To the best of our knowledge this is the first observation of the simultaneous occurrence of integer and semi-integer quantum Hall effects. This work was supported by Brazilian scientific agencies FAPESP, CNPq, CAPES, and French agency COFE-CUB. We thank to Prof. M. G. Karkut for the useful discussions.