ABC Trilayer Research Articles

Electronic and structural properties of vacancies and hydrogen adsorbates on trilayer graphene

Using ab initio calculations, we study the electronic and structural properties of vacancies and hydrogen adsorbates on trilayer graphene. Those defects are found to share similar low-energy electronic features, since they both remove a pz electron from the honeycomb lattice and induce a defect level near the Fermi energy. However, a vacancy also leaves unpaired electrons on the lattice, which lead to important structural differences and also contribute to magnetism. We explore both ABA and ABC stackings and compare properties such as formation energies, magnetic moments, spin density and the local density of states (LDOS) of the defect levels. These properties show a strong sensitivity to the layer in which the defect is placed and smaller sensitivities to sublattice placing and stacking type. Finally, for the ABC trilayer, we also study how these states behave in the presence of an external field, which opens a tunable gap in the band structure of the non-defective system. The pz defect states show a strong hybridization with band states as the field increases, with reduction and eventually loss of magnetization, and a non-magnetic, midgap-like state is found when the defect is at the middle layer.

Direction-controlled chemical doping for reversible G-phonon mixing in ABC trilayer graphene.

Not only the apparent atomic arrangement but the charge distribution also defines the crystalline symmetry that dictates the electronic and vibrational structures. In this work, we report reversible and direction-controlled chemical doping that modifies the inversion symmetry of AB-bilayer and ABC-trilayer graphene. For the “top-down” and “bottom-up” hole injection into graphene sheets, we employed molecular adsorption of electronegative I2 and annealing-induced interfacial hole doping, respectively. The chemical breakdown of the inversion symmetry led to the mixing of the G phonons, Raman active Eg and Raman-inactive Eu modes, which was manifested as the two split G peaks, G− and G+. The broken inversion symmetry could be recovered by removing the hole dopants by simple rinsing or interfacial molecular replacement. Alternatively, the symmetry could be regained by double-side charge injection, which eliminated G− and formed an additional peak, Go, originating from the barely doped interior layer. Chemical modification of crystalline symmetry as demonstrated in the current study can be applied to other low dimensional crystals in tuning their various material properties.

Band structure of ABC-trilayer graphene superlattice

We investigate the effect of one-dimensional periodic potentials on the low energy band structure of ABC trilayer graphene first by assuming that all the three layers have the same potential. Extra Dirac points having the same electron hole crossing energy as that of the original Dirac point are generated by superlattice potentials with equal well and barrier widths. When the potential height is increased, the numbers of extra Dirac points are increased. The dispersions around the Dirac points are not isotropic. It is noted that the dispersion along the ky direction for kx = 0 oscillates between a non-linear dispersion and a linear dispersion when the potential height is increased. When the well and barrier widths are not identical, the symmetry of the conduction and valence bands is broken. The extra Dirac points are shifted either upward or downward depending on the barrier and well widths from the zero energy, while the position of the central Dirac point oscillates with the superlattice potential height. By considering different potentials for different layers, extra Dirac points are generated not from the original Dirac points but from the valleys formed in the energy spectrum. Two extra Dirac points appear from each pair of touched valleys, so four Dirac points appeared in the spectrum at particular barrier height. By increasing the barrier height of superlattice potential two Dirac points merge into the original Dirac point. This emerging and merging of extra Dirac points is different from the equal potential case.

Theory for optical Hall conductivity in the trilayer graphene in the quantum Hall regime

We theoretically study optical Hall conductivities for ABA and ABC trilayer graphenes. In ABA-stacked trilayer, the resonance spectrum is shown to be a superposition of effective monolayer and bilayer contributions with band gaps, while ABC trilayer exhibits a distinct spectrum peculiar to the cubic-dispersed bands with a strong trigonal warping, where we found the signals associated with low-energy Dirac cones are conspicuous enough to be directly observable owing to a large Lifshitz-transition energy (~ 10 meV). We have also revealed how the presence or absence of the inversion symmetry affects the optical Hall conductivities.

Local spectroscopy of the electrically tunable band gap in trilayer graphene

The stacking order in trilayer graphene plays a critical role in determining the existence of an electric field tunable band gap. We present spatially resolved tunneling spectroscopy measurements of dual gated Bernal (ABA) and rhombohedral (ABC) stacked trilayer graphene devices. We demonstrate that while ABA trilayer graphene remains metallic, ABC trilayer graphene exhibits a widely tunable band gap as a function of electric field. However, we find that charged impurities in the underlying substrate cause substantial spatial fluctuations of the gap size. Our work elucidates the microscopic behavior of trilayer graphene and its consequences for macroscopic devices.

Transport Studies of Dual-Gated ABC and ABA Trilayer Graphene: Band Gap Opening and Band Structure Tuning in Very Large Perpendicular Electric Fields

We report on the transport properties of ABC and ABA stacked trilayer graphene using dual, locally gated field effect devices. The high efficiency and large breakdown voltage of the HfO(2) top and bottom gates enable independent tuning of the perpendicular electric field and the Fermi level over an unprecedentedly large range. We observe a resistance change of 6 orders of magnitude in the ABC trilayer, which demonstrates the opening of a band gap. Our data suggest that the gap saturates at a large displacement field of D ~ 3 V/nm, in agreement with self-consistent Hartree calculations. In contrast, the ABA trilayer remains metallic even under a large perpendicular electric field. Despite the absence of a band gap, the band structure of the ABA trilayer continues to evolve with increasing D. We observe signatures of two-band conduction at large D fields. Our self-consistent Hartree calculation reproduces many aspects of the experimental data but also points to the need for more sophisticated theory.

Intrinsic and substrate induced spin-orbit interaction in chirally stacked trilayer graphene

We present a combined group-theoretical and tight-binding approach to calculate the intrinsic spin-orbit coupling (SOC) in ABC stacked trilayer graphene. We find that compared to monolayer graphene, a larger set of d orbitals (in particular the d_{z^2} orbital) needs to be taken into account. We also consider the intrinsic SOC in bilayer graphene, because the comparison between our tight-binding bilayer results and the density functional computations of Ref.[40] allows us to estimate the values of the trilayer SOC parameters as well. We also discuss the situation when a substrate or adatoms induce strong SOC in only one of the layers of bilayer or ABC trilayer graphene. Both for the case of intrinsic and externally induced SOC we derive effective Hamiltonians which describe the low-energy spin-orbit physics. We find that at the K point of the Brillouin zone the effect of Bychkov-Rashba type SOC is suppressed in bilayer and ABC trilayer graphene compared to monolayer graphene.

Mechanical properties of multilayer boron nitride with different stacking orders

The mechanical properties of multilayer boron nitride (BN) with various stacking orders are characterized by ab initio calculations. Based on the density functional theory (DFT), the values of elastic constants of AB bilayer, ABA and ABC trilayer and graphite-like BN in the harmonic elastic deformation range are revealed. DFT calculations are performed in the context of generalized gradient approximation (GGA) and by adopting the Perdew–Burke–Ernzerhof (PBE) exchange correlation. This investigation shows that Young’s modulus and Poisson’s ratio of multilayer BN are lower than those of monolayer BN. A comparative study on the mechanical properties of multilayer BN with different stacking orders is presented. It is indicated that the mechanical properties of multilayer BN are dependent on the number of layers and their stacking order.

Faraday rotation in bilayer and trilayer graphene in the quantum Hall regime

Optical Hall conductivity, as directly related to Faraday rotation, is theoretically studied for bilayer and trilayer graphene. In bilayer graphene, the trigonal warping of the band dispersion greatly affects the resonance structures in Faraday rotation not only in the low-energy region where small Dirac cones emerge, but also in the higher-energy parabolic bands as a sequence of satellite resonances. In ABA-stacked trilayer, the resonance spectrum is a superposition of effective monolayer and bilayer contributions with band gaps, while ABC trilayer exhibits a distinct spectrum peculiar to the cubic-dispersed bands with a strong trigonal warping, where the signals associated with low-energy Dirac cones should be directly observable owing to a large Lifshitz transition energy ($\ensuremath{\sim}10\phantom{\rule{0.28em}{0ex}}\mathrm{meV}$).

Vacancy induced zero energy modes in graphene stacks: The case of ABC trilayer

The zero energy modes induced by vacancies in ABC-stacked trilayer graphene are investigated. Depending on the position of the vacancy, a new zero energy solution is realised, different from those obtained in multilayer compounds with Bernal stacking. The electronic modification induced in the sample by the new vacancy states is characterised by computing the local density of states and their localisation properties are studied by the inverse participation ratio. We also analyse the situation in the presence of a gap in the spectrum due to a perpendicular electric field.

Hund's rules for theN=0Landau levels of trilayer graphene

The $N=0$ Landau levels of ABC and ABA trilayer graphene both have approximate 12-fold degeneracies that are lifted by interactions to produce strong quantum Hall effects at all integer filling factors between $\ensuremath{\nu}=\ensuremath{-}6$ and $\ensuremath{\nu}=6$. We discuss similarities and differences between the strong-magnetic-field weak-disorder physics of the two trilayer cases, and between trilayer and bilayer cases. These differences can be used to identify the stacking order of high-quality trilayer samples by studying their quantum Hall effects.

Stacking-order dependent transport properties of trilayer graphene

We report markedly different transport properties of ABA- and ABC-stacked trilayer graphenes. Our experiments in double-gated trilayer devices provide evidence that a perpendicular electric field opens an energy gap in the ABC trilayer, while it causes the increase of a band overlap in the ABA trilayer. In a perpendicular magnetic field, the ABA trilayer develops quantum Hall plateaus at filling factors of \nu = 2, 4, 6... with a step of \Delta \nu = 2, whereas the inversion symmetric ABC trilayer exhibits plateaus at \nu = 6 and 10 with 4-fold spin and valley degeneracy.

Integer Quantum Hall Effect in Trilayer Graphene

By using high-magnetic fields (up to 60 T), we observe compelling evidence of the integer quantum Hall effect in trilayer graphene. The magnetotransport fingerprints are similar to those of the graphene monolayer, except for the absence of a plateau at a filling factor of ν=2. At a very low filling factor, the Hall resistance vanishes due to the presence of mixed electron and hole carriers induced by disorder. The measured Hall resistivity plateaus are well reproduced theoretically, using a self-consistent Hartree calculations of the Landau levels and assuming an ABC stacking order of the three layers.