{'fraction_non_alphanumeric': 0.04875040571243103, 'fraction_numerical': 0.018889970788704965, 'mean_word_length': 4.420830401125968, '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': 4, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Stacking-enriched magneto-transport properties of few-layer graphenes

The quantum Hall effects in the sliding bilayer graphene and AAB-stacked trilayer system are investigated by the Kubo formula and the generalized tight-binding model. The various stacking configurations can greatly diversify the magnetic quantization and thus create the rich and unique transport properties. The quantum conductivities are very sensitive to the Fermi energy and magnetic-field strength. The diverse features cover the specific non-integer conductivities, the integer conductivities with the distinct steps, the splitting-created reduction and complexity of quantum conductivity, a vanishing or non-zero conductivity at the neutral point, and the well-like, staircase, composite, and abnormal plateau structures in the field-dependencies. Such stacking-dependent characteristics mainly originate from the crossing, anticrossing and splitting Landau-level energy spectra and three kinds of quantized modes.

neighboring and next-neighboring interlayer atomic interactions, respectively, create the separated Dirac cone and parabolic bands, and the valley splitting of the latter. At low energy, this further leads to six quantized LLs with double spin degeneracy and thus the QHE step of 2e 2 /h. However, the higher-energy QHE in ABA stacking could be regarded as the superposition of those of monolayer and AB bilayer. [47][48][49] The ABC-stacked trilayer graphene, on the other hand, presents the important differences in the main features of QHE compared with the ABA trilayer system. The quantum Hall conductivity is quantized as a sequence of σ xy = 4(±|m | ± 1/2)e 2 /h in the absence of the σ xy = ±2e 2 /h, ±4e 2 /h, and ±8e 2 /h plateaus. [48,51] Specifically, a σ xy = 12e 2 /h step appears near zero energy, being associated with the n = 0, 1 and 2 LLs of the first group due to the surfacelocalized flat bands. [52] In fact, the fourfold spin and valley degeneracy is retained for the ABC trilayer due to the inversion symmetry, resulting in the QHE step height of 4e 2 /h.
The above-mentioned interesting phenomena of electronic transport properties in graphene opens the door to exploring the configuration-enriched QHE in other layered graphenes, such as AAB-stacked trilayer and sliding bilayer systems.
In this work, the linear Kubo formula, combined with the generalized tight-binding model, is used to investigate the unusual QHE in few-layer graphenes with high-and low-symmetry stacking configurations. The developed model would be very useful in the identification of the magneto-electronic selection rules under the static case; that is, the available transition channels in magneto-transport property could be examined thoroughly.
The dependencies of quantum conductivity on the Fermi energy (E F ) and magnetic-field strength are explored in detail. This study shows that the feature-rich LLs can create the extraordinary magneto-transport properties. The sliding bilayer systems, with three kinds of LLs present the unusual QHE, covering the integer and non-integer conductivities, the zero and non-vanishing conductivities at the neutral point, the well-like, staircase and composite quantum structures, and the different step heights. Furthermore, the reduced conductivity, the complex plateaus and the abnormal structures are revealed in the AAB-stacked trilayer graphene, with the splitting and perturbed LLs. These results are deduced to be dominated by the crossing and anti-crossing energy spectra, and the spatial oscillation modes. The predictions on the diverse quantum conductivities could be experimentally verified, as done for other layered graphene. [2,3,44,45,47,48] 2 Models and Methods The generalized tight-binding model is developed to investigate the essential properties of layered graphenes in external fields. The low-energy Hamiltonian is mainly built from the 2p z -orbital tight-binding functions in a unit cell. The two sublattices in the l-th (l = 1, 2, ..., N ) layer are denoted as A l and B l , where N is the number of graphene layers. The interlayer distance and the C-C bond length are, respectively, d 0 = 3.37Å and b 0 = 1.42Å.
The various Hamiltonian matrices are dominated by the intralayer and the interlayer atomic interactions. For sliding bilayer graphene, the stacking configurations can be transformed according to AA (δ = 0) → AB (δ = b 0 ) → AA (δ = 1.5b 0 ) by the relative shift between two graphene layers along the armchair direction (x). The strength of the hopping integrals i,j between the lattice sites i and j is dependent on the distance and the angle of two 2p z orbitals obeying the following relation [43]: (1) 0 = 2.7 eV and 1 = 0.48 eV are, respectively, the intralayer nearest-neighbor and interlayer vertical hopping integrals, d the position vector connecting two lattice sites, and = 0.184b 0 the decay length. [39] For trilayer graphenes, AAB stacking is chosen for a model theoretical study. In this system, the A atoms in three layers have the same (x,y) coordinates, while the B atoms on the third layer are projected at the hexagonal centers of the other two layers. There exist 10 kinds of atom-atom interactions in creating the unusual energy dispersions; that is, they are responsible for the oscillatory, sombrero-shaped and parabolic bands. [40] |α is the LL state with energy E α , S the area of the enlarged unit cell, f α,β the Fermi-Dirac distribution functions, andu x the velocity operator alongx. The matrix elements of the velocity operators, which determine the available inter-LL transitions, are evaluated from the gradient approximation [54] α|u x |β = 1 α| ∂H ∂k x |β This approximation has been successfully used to investigate the optical spectra of carbon-related systems. The velocity matrix elements are dominated by the intralayer nearest-neighbor interaction, [41] so that the quantized mode of the initial state on the A l sublattice must be identical to that of the final state on the B l sublattice. [42] Apparently, the well-behaved, perturbed and undefined LLs of layered graphenes, accompanied with various magnetic selection rules, [39] are expected to exhibit the rich and unique quantum conductivities.   cially, there appears a very narrow plateau near zero energy ( Fig. 4(a)). Such plateau is observable at the sufficiently large field strength. It is induced by a small energy spacing between the n 1 = 0 and n 1 = 1 LLs of the first group. This energy spacing is monotonically dependent on the field strength, as shown in Fig. 2(d). It will be infinitesimal and even disappears below the critical field strength (B 0 < 20 T, e.g., 12T in the inset), leading to a double step in the Hall conductivity. These results are in agreement with the previous predictions and measurements. [44][45][46] At higher energy range (E F > 0.35 eV; Fig. 4 the quantum conductivity is dominated by two groups of LLs (blue and red lines). There are four categories of electronic transitions, including the intragroup (n c and intergroup ones (n c 1 → n c 2 , n c 2 → n c 1 ). [55] The number of steps is significantly enhanced as shown in Fig. 4(b); furthermore, the change in plateau widths no longer follows a simple relation with E F . Specifically, the available intergroup transitions, with the same quantized mode on the A l (B l ) sublattice (Eq. (3)), only happen in two LLs with a large energy spacing. They only make negligible contributions, since the quantum Hall conductivity is inversely proportional to the square of energy difference between the initial and final states (Eq. (2)). This is in great contrast with the frequency-dependent optical excitation spectra. [55] That is to say, the QHE in AB bilayer graphene can be considered as the superposition of the two intragroup transition channels.   Fig. 6(b)). As can be predicted from the crossing behavior of the B 0 -dependent LL spectrum (Fig. 2(d)), the weaker the field strength, the more frequently will the double-height steps appear. The quantum conductivities are diversified by the shift-dependent stacking configurations. The lower-symmetry bilayer system of δ = b 0 /8 shows the unusual step structure, as clearly indicated in Fig. 7(a). The low-lying plateaus are quantized as a special sequence of (-9, -4, -3, 4, 7), leading to the exclusive step heights of 5e 2 /h, e 2 /h, 7e 2 /h, and 3e 2 /h.
Such odd-integer steps are associated with the LLs of n c 2 = 2, 3, 4, and 5 (red lines) and n v 1 = 2, 3, 4 and 5 (blue lines). These LLs are quantized from the hybridized electronic states near the eye-shape region of the band structure (inset in Fig. 2(b)). There exist extra inter-group electronic transitions among these LLs which satisfy the selection rules of ∆n = 0 & ± 2, e.g., n c 2 = 3 → n v 1 = 3, n c 2 = 4 → n v 1 = 2, and others. Especially, only the transitions associated with ∆n = 2 rule will contribute a positive quantum conductivity.
As a result, the low-energy QHE is mainly determined by the intra-and inter-group LL transition channels. When E F crosses zero from a negative value, the n c 2 = 3 becomes oc-  Specifically, the high-symmetry AA stacking presents the non-integer plateaus ( Fig.   7(b)), being in great contrast to AA bilayer ( Fig. 3(a)). Two groups of well-behaved LLs, which are quantized from the tilted Dirac cones (Fig. 2(f)), only contribute the intragroup LL transitions of ∆n = ±1 to quantum conductivity. When an extra n c,v 1 LL becomes occupied in the increase of E F , the quantum conductivity (blue arrows in Fig. 7(b)) is enhanced by a height of 3.8e 2 /h. On the other hand, the occupation of one n c,v 2 LL can create a 4.2e 2 /h-height step (red arrows). The non-integer QHE might be associated with the tilted Dirac cones, in which the localization centers of LLs gradually change with the increasing energy. [39] Also, the non-fixed centers could induce a small conductivity of 0.2e 2 /h near the neutrality point.
As for the undefined LLs in δ = 6b 0 /8 and 11b 0 /8 bilayer stackings (Figs. 8(a) and 8(b)), their quantum conductivities are worthy of a detailed examination. The former presents the n c,v 1 conduction and valence LLs, with the perturbed modes, in the low-lying spectral region under the absence of the n c,v 2 LLs (Fig. 2(c)). At higher/deeper energy range where the n c,v 2 LLs come to exist, two groups of LLs are continuously anticrossed together, as shown in the inset of Fig. 8(a). The localization modes of each LL cannot be well defined, since they vary with the field strength ( Fig. 2(c)). Such highly degenerate electronic states without a specific main mode are classified as the undefined LLs. [39] It is noted that n c,v 2 in Fig. 8(a) (also in Fig. 8(b)) only denotes the ordering of LLs, but not the quantum It might have some narrow plateaus (black arrows in Fig. 8(b)), e.g., the plateau at B 0 ∼ 28 T arising from the n c 1 = 13 and n c 2 = 3 LLs. Such plateaus directly reflect the very weak anticrossings.

Quantum Hall effect in trilayer graphene
In AAB-stacked trilayer graphene, the oscillatory energy bands of the first pair (inset in Fig. 2(g)) are magnetically quantized into the unique LLs. Especially, the low-lying LLs, with perturbed modes, present the frequently crossing and anticrossing behaviors, as clearly shown in Figs. 2(g) and 2(h). All the LLs are doubly degenerate, since the destruction of the z-plane mirror symmetry leads to the non-equivalence of two valleys (or two distinguishable localization centers of 1/6 and 2/6 in an enlarged unit cell; details in Ref. 40). Apparently, the feature-rich LLs will induce the unusual QHE. The E F -dependent quantum Hall conductivity exhibits the step height of 2e 2 /h instead of the usual one of 4e 2 /h and the irregular widths of plateaus ( Fig. 9(a)). At low energy, there also exist certain double-height steps of 4e 2 /h. Those steps are closely related to the crossings of entangled LLs around the neutrality point (Figs. 2(g) and 2(h)). As a result of the splitting energy spectra, the LLs localized at 1/6 and 2/6 centers are very different in the quantum conductivities. As E F approaches zero energy, a plateau appears at B 0 = 40 T for the 1/6 center (blue curve in Fig. 9(b)), but two neighboring steps correspond to the 2/6 center (red curve). The absence/presence of the step structure depends on whether the anti-crossing LLs could survive there. For example, at the 1/6 center, the anti-crossing n c,v 1 = 0 and n c 1 = 3 LLs away from E F = 0 (inset in Fig. 10(a)) accounts for the absence of zero-energy step. On the other hand, at the 2/6 center, the n c,v 1 = 0 and n v 1 = 1 LLs continuously become occupied at E F = −3 meV and 0, so that the dominant transition channels of n v 1 = 1 → n c,v 1 = 0 and n c,v 1 = 0 → n c 1 = 1 create the neighboring narrow-width steps. Concerning the neutrality point, the competition between these two localization centers lead to zero conductivity. In short, the splitting and anti-crossing energy spectra have generated the complex quantum structures with the half-reduced conductivities ( Fig.   9(a)). The B 0 -dependent Hall conductivity, being associated with the low-lying LLs, present the extraordinary quantum structures (Fig. 10(c)). The abnormal plateaus are absent in the sliding bilayer system (Figs. 5, 6 & 8). They mainly originate from the intragroup an-ticrossing LLs (or the oscillatory energy bands). For the 1/6 center, the positive/negative quantum conductivity appears and then disappears during a certain range of B 0 , as shown in Fig. 10  There exist certain important differences in quantum conductivities among the AAB-, ABA-and ABC-stacked trilayer graphenes. The step height of 2e 2 /h in a wide energy range, the localization-dependent plateau structures, and the irregular B 0 -dependence are absent in the latter two systems, since they have the higher-symmetry stacking configurations. In general, ABA and ABC have the monolayer-like steps with 4e 2 /h height except that the unusual quantum conductivities appear at a narrow energy range covering the neutrality point. [47,48,51] Such differences clearly illustrate the stacking-diversified QHE; that is, the stacking-dependent atomic interactions dominates the magnetic quantization and thus the magneto-transport properties.

Concluding Remarks
We have used the generalized tight-binding model and the Kubo formula to investigate the unusual QHE in high-and low-symmetry few-layer graphenes, especially for the identification of the selection rules from the former. This method could be further developed to explore the emergent 2D materials, e.g., silicene, germanene, tinene, phosphorene, and MoS 2 . The Fermi-energy-and magnetic-field-dependent quantum Hall conductivities are greatly diversified by various stacking configurations, since they might arise from the intragroup and intergroup LL transitions with/without the monolayer-like and extra selection rules. Three kinds of LLs and the crossing, anti-crossing and spitting energy spectra are responsible for the diverse transport properties, such as, the integer and non-integer conductivities, the splitting-created reduction and complexity of quantum conductivity, the vanishing and non-zero conductivities at the neutrality point, the well-like, staircase and composite plateau structures, the distinct step heights, and the abnormal field dependence.