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Magnetotransport study of the mini-Dirac cone in AB-stacked four- to six-layer graphene under perpendicular electric field

Landau levels of AB-stacked four- to six-layer graphene were studied experimentally in the presence of a perpendicular electric field. The Landau levels at a low magnetic field showed a characteristic structure that originated from the mini-Dirac cones created by the perpendicular electric fields. Six-fold or twelve-fold degenerate Landau levels arising from the mini-Dirac cones were observed in the vicinity of the charge neutrality point as a radial structure in a plot of their Landau fan diagrams. The structure of four-layer graphene had approximate electron-hole symmetry near the charge neutrality point, while the five- and six-layer graphene showed asymmetry. Numerical calculations of the dispersion relation and Landau level spectra indicated that trigonal warping played an essential role in forming the experimentally observed Landau level structure in low magnetic fields. In high magnetic fields, trigonal warping becomes less significant and the Landau level spectra become simpler, approaching those that were calculated without considering trigonal warping. The appearance of mini-Dirac cones in a perpendicular electric field would be a common feature of AB-stacked multilayer graphene.


Introduction
Monolayer graphene has a Dirac cones with a massless dispersion relation [1][2][3]. Because the velocity of its electrons is expected to be about 1/300th the speed of light, much faster than electrons in conventional semiconductors even at low carrier densities, graphene is expected to be a candidate materials for next-generation devices [4][5]. Graphene also has a unique property of controllability of its band structure, i.e., stacking the graphene layers dramatically changes the band structure . The low energy band structure of bilayer graphene is approximately parabolic, and the electrons in it can be regarded as massive Dirac fermions [1,[6][7][8]26]. When there are more than three layers, two-different stackings, i.e., ABA and ABC, are possible in terms of the relative positions of the carbon atoms which form dimers in adjacent layers. In AB-stacked graphene (i.e., the ABA stacking), the band structure systematically evolves as the number of layers increases.
The even-odd effect implies that AB-stacked graphene with an even number of layers does not have Dirac cones. However, recent studies have indicated that Dirac cones are created by applying a perpendicular electric field. For example, Morimoto and Koshino [38] theoretically showed that a perpendicular electric field forms an energy gap between the bottoms of the bilayer-like bands, but the gap closes locally as a result of trigonal warping, and thereby mini-Dirac cones form.
Mini-Dirac cones are expected to appear significantly in AB-stacked graphene of many layers. The recently found intrinsic resistance peak structures arising from topological changes in the Fermi surface [27,[29][30] are expected to be partly due to mini-Dirac cones. In AB-stacked four-layer [27,29,31], five-layer [31], six-layer [30], and seven-layer graphene [31], the carrier density dependence of the resistivity shows peak structures specific to the band structures, and maps of the resistivity as a function of perpendicular electric flux density and carrier density show ridge structures. [27,[29][30][31]. Numerical calculations of the dispersion relation in the presence of a perpendicular electric flux density indicated that the conspicuous ridges are due to mini-Dirac cones formed by the field.
A previous study examined the Landau level structure in AB-stacked tetralayers in the presence of perpendicular electric field in relation to the appearance of mini-Dirac cones [27]. Six-fold degenerate Landau levels associated with the mini-Dirac cones are reported. In the current study, we performed a systematic and detailed study on the Landau level structure of AB-stacked four-to six-layer graphene in a perpendicular electric field. We observed that, in a perpendicular electric field, the mini-Dirac appeared in AB-stacked graphene having several layers, and we interpreted the observed Landau level spectra in terms of the shapes of the dispersion relations.
Here, we briefly describe the variation in the dispersion relations arising when a perpendicular electric field is applied. Figure 2 shows plots of numerical calculation for AB-stacked four-to six-layer graphene for ⊥ = 0 (top panels) and 2.4 × 10 −7 cm −2 As (bottom panels). The calculations were based on the effective mass approximation with all the Slonczewski-Weiss-McClure parameters of graphite. The dispersion relations for ⊥ = 0 showed systematic variations, as indicated by the even-odd effect; a linear band can be seen only in the five-layer graphene [12,9,17,25,[27][28][29][30][31][32][33][34][35][36][37][47][48][49], whereas the band structure is rather more complicated near = 0 [26,36,48] than the simple expectation that mono-and bilayer-like bands have linear and parabolic dispersions [9,26,35,37]. In a perpendicular electric field, the dispersion relations become significantly different from those for ⊥ = 0. The field opens energy gaps between the bottoms of the bilayer-like bands, as in the case of bilayer graphene; however, the energy gap tends to close locally and mini-Dirac cone structures appear (left arrows), which have linear dispersion relations over a limited ranges of k near = 0. In addition, for the six-layer case, a few mini-Dirac cones formed at energies away from = 0. One can also recognize cone-like structures exhibiting energy gaps at their vertices, as indicated by the right arrows.
The mini-Dirac cones originate from the trigonal warping inherent to graphene. Trigonal warping locally closes the energy gaps of the bilayer-like bands that are created by a perpendicular electric field [38]. This results in a change in the band mass as well as the shape of the Fermi surface, and induces a topological change of the Fermi surface when the chemical potential and perpendicular electric fields are varied by changing the gate voltages [27,[29][30][31]].

Experimental
Graphene was prepared by mechanically exfoliating high-quality Kish graphite crystal with adhesive tape [1]. The number of layers and stacking were determined by combining information from different methods of measurement. The number of layers in the graphene flakes was roughly determined from the color signal intensity of the digitized image. The shape of the Raman D' band spectra was measured for many graphene flakes for which the number of layers was determined by atomic force microscopy (AFM) topography. The number of layers and the stacking of graphene samples were determined by referring the spectra shapes [32,[39][40][41][42][43]. We also checked the number of layers and stacking via the Landau level structure [27][28]32]. and dimethilpolysiloxane (PDMS) [46]. The bottom gate electrode was a P ++ doped Si substrate covered with SiO2 (300nm thick). The top gate electrode was formed by placing thick graphene on the encapsulated graphene stack. The sample geometry was defined by reactive ion etching using a mixture of low pressure CF4 and O2 gas. The geometry of the effective sample area was identical to that of the top gate electrode. Electrical contact with the graphene was attained by using the edge-contact technique [46].
The top and bottom gate electrodes were used to control the carrier density and perpendicular electric field independently. The carrier density induced in the graphene is thus given by the sum of the carriers induced by the gate voltages: where and are the specific capacitance of the top and bottom gate electrodes and, 0 and 0 are for the offset voltages associated with the gate electrode The difference in the charge density induced by the two electrode is proportional to the perpendicular electric flux density ⊥ , which is given by Here, is about 108 aF/μm, and / varies depending on thickness of top gate insulator (h-BN); it typically ranged from 3 to 5.

AB-stacked four-layer graphene
Although the Landau level structure for AB-stacked four-layer graphene has been reported in Ref. [27], we briefly show results not only for providing complementary information, but also illustrating the systematic evolution with respect to the number of layers. Figure 3  [ 27,29]. Fig. 3(b) is a replot of resistivity against and ⊥ , which shows conspicuous resistance ridge structures specific to the AB-stacked four-layer graphene [27,29] The formation of mini-Dirac cones was verified by making magnetotransport measurements at = 4.2 K . The longitudinal magnetoresistance ( ) showed Shubnikov-de Haas oscillations, which were complicated because of beating due to different bands. Figure 4 (a) is a map of as a function of carrier density and magnetic field for ⊥ = 3.8 × 10 −7 cm −2 As. Figure 4(b) shows a map of / , which allows one to observe small variations in magnetoresistance. The Landau in the presence of the perpendicular electric fields shows a significant variation in shape from those at zero perpendicular electric field (Figs. 4(d) and 4(e)).
Next, let us examine the difference between the Landau level structures in the absence and presence of a perpendicular electric field. For ⊥ = 0, the zero-mode Landau levels are discernible as vertically elongated beating structures at ∼ −0.2, and 0.8 × 10 12 cm −2 (shown by the arrows in Fig. 4(d)); these are approximately reproduced by the numerically calculated density of states (Fig 4(f)). It can be seen that the level structure near | | < 1 × 10 12 cm −2 is asymmetric between the electron and the hole regime. The above-mentioned structure changes drastically in the perpendicular electric The bundles converge to single levels as → 0T. As shown in the lower panel of Fig. 7 (a), there are vertices of mini-Dirac cone structures ( , ℎ , and ) at those energies. Two bundles with three-fold degenerate Landau levels (except for spin) originate from , which is a vertex of a mini-Dirac cone. The vertices ℎ and are those of a mini-Dirac cone-like structure which has an energy gap between the vertices. A bundle with three-fold degenerate Landau levels (except for spin) stems from each of and ℎ . These three-fold degenerate Landau levels can be interpreted as zero-mode Landau levels associated with the vertices of the mini-Dirac cones, as in the case of four-layer graphene.
Besides the above bundles, there is a nondegenerate (except for spin) zero-mode Landau level associated with vertex , which is located at = 0. This principally originates from the monolayer-like band, which forms sharp peak structures at = 0 for ⊥ = 0.
In the dispersion relation shown in Fig. 7(a) ). An electron-like Landau level with index one is also discernible as in Fig. 7(a). However an energy gap associated with the bundle is expected to appear only at low magnetic fields ( < 0.5T), so it would have been barely visible in the experiment.
A correspondence between the zero-mode Landau levels and the vertices of the mini-Dirac cones can also be found in the case of ⊥ = 0. The monolayer-like band forms gapped Dirac cones with vertices ℎ and ( Fig. 7(b)). The energies correspond to those of non-degenerate (except for spin) zero-mode Landau levels arising from the monolayer-like band. As for the bilayer-like bands, the zero-mode Landau levels approximately stem from the bottoms.

Mini-Dirac cones in AB-stacked six-layer graphene
Although the band structure of the six-layer sample was much more complicated than the above-mentioned results, characteristic mini-Dirac cones structures still appeared in the Landau fan diagrams. Figures 8(a)  Nonetheless, the characteristic V-shaped structure is visible in the numerically calculated density of states as well as the experimental maps of (marked by A in Fig.   9(b)).
What is particular to the six-layer graphene as compared with samples with fewer layers is that the mini-Dirac cones are expected to form not only in the vicinity of = 0, but also at larger values of | | [30]. The mini-Dirac cones with vertices , and ℎ are in the electron regime, while those in the hole regime are smaller ones with vertices and . The mini-Dirac cones in the electron regime form V-shaped structure in the map of resistivity around = 1.5 × 10 12 cm −2 (marked by A in Fig. 9(b)), which is similar to the structure formed in the vicinity of = 0 in the AB-stacked four-layer graphene (Figs. 4(a)-4(c)). This should be because the structure of the dispersion relation between ≈ 5 and 40 meV (lower panel of Fig. 9(a)) are qualitatively the same as that of the four-layer graphene except for the presence of an extra bilayer-like band with larger wave numbers. In both six-and four-layer graphene, there are a gapless and a gapped mini-Dirac cone (the vertices of , and ℎ in Fig. 9(a) correspond to , ℎ , and in Fig. 5(a)). Near the energy of these vertices, zero-mode Landau levels of mini-Dirac cones appear as vertically elongated levels at low magnetic fields (marked with B in Fig 9(b)).
As for the hole regime, one such structure associated vertices and is visible in the experimental result as well as in the numerical simulation (marked C in Fig 9(b)). This means that the shape of the dispersion relation between ≈ −30 and − 10 meV significantly differs from the four-layer case.

Discussion
Trigonal warping plays an essential role in forming mini-Dirac cones in AB-stacked multilayer layer graphene in a perpendicular electric field. If we assume no trigonal warping, rotationally symmetric energy gaps form at energies where the bilayer-like Modification of graphene's band structure has been one of the most interesting topics in the graphene research. Graphene itself does not have band gaps, and to overcome this disadvantage for practical applications, various methods of forming a band gap have been proposed for mono-or bilayer graphene, e.g., making narrow wires or antidot structures [55][56][57][58][59][60][61][62] or applying perpendicular electric fields to bilayer graphene [63][64][65][66][67][68][69].
On the other hand, for AB-stacked graphene with numerous layers, the bands become semi-metallic [12,15, 17, 25, 32, 35-36, 48, 70], which is not a preferred property for electronic devices. As has been revealed by the present work, making mini-Dirac cones in multilayer graphene via a perpendicular electric field might lead to a way of controlling the band structure of multilayer graphene for the practical application.

Summary
Landau level in AB-stacked graphene with four to six layers were studied in terms of the formation of mini-Dirac cones in a perpendicular electric field. The samples commonly showed a mini-Dirac cone structure in a perpendicular electric field, and the zero-mode Landau levels of the mini-Dirac cones formed at the energies of the vertices of the mini-Dirac-cones at low magnetic fields. Levels with larger Landau indices were also observed.
Trigonal warping of the band structure was essential to forming the mini-Dirac cone structure in the Landau levels at low magnetic fields. At high magnetic fields, the Landau levels arising from the mini-Dirac cones tended to disappear and changed into ones having simpler structures that were qualitatively the same as those without trigonal warping.
A Calculation of the dispersion relation and Landau levels The dispersion relations were calculated by using a Hamiltonian based on the tightbinding model using all the Slonczewski-Weiss-McClure parameters [12,15,26,36,48].
The wave functions were expanded in plane waves, and the eigenvalues were calculated numerically. In order to consider the effect of the perpendicular electric field, we calculated potential distributions associated with the carrier distributions induced by the screening of the carriers, which were in turn induced by top and bottom gate voltages, by using the methods described in Refs. [30][31][32]. Carrier screening has been studied since before the advent of graphene [71][72][73][74][75][76][77]. Visscher et al. calculated screening effect by using Thomas-Fermi approximations. Guinea studied it within the framework of the random phase approximation. Koshino performed self-consistent calculation of the band structures and distribution of carriers induced by external electric fields [74]. Theoretical studies [71][72][73][74] and experiments [75][76][77] estimate the screening length to be less than a few layers of graphite. We have used model distributions of carrier screening where carriers are induced by application of a gate voltage: for each gate electrode, the induced carriers decay exponentially with a relaxation length of in the direction perpendicular to the two-dimensional plane. The dielectric constant, ϵ/ϵ 0 = 2, was used to calculate the electrostatic potential. The electrostatic potential for each layer was added to the diagonal elements of the Hamiltonian. The Landau levels were calculated following Ref. [11]. The wave functions were expanded in Landau functions, and the energy eigenvalues were numerically calculated. The screening length is a fitting parameters in our calculations. In the previous paper reporting intrinsic resistance peak (ridge) structures in AB-stacked multilayer graphene, we chose 0.45 nm [31]. In this work, we used the                ‫ܦ‬ ୄ = 1.72 × 10 ି cm ିଶ As ‫ܦ‬ ୄ = 1.72 × 10 ି cm ିଶ As ‫ܦ‬ ୄ = 0 cm ିଶ As ‫ܦ‬ ୄ = 0 cm ିଶ As ‫ܦ‬ ୄ = 0 cm ିଶ As ߩ ௫௫ ሺOhmሻ ‫ܤ‬ ሺTሻ ‫ܱܵܦ‬ ሺarb.units.ሻ