Trilayer Graphene Research Articles

Robust superconductivity in magic-angle multilayer graphene family.

The discovery of correlated states and superconductivity in magic-angle twisted bilayer graphene (MATBG) established a new platform to explore interaction-driven and topological phenomena. However, despite multitudes of correlated phases observed in moiré systems, robust superconductivity appears the least common, found only in MATBG and more recently in magic-angle twisted trilayer graphene. Here we report the experimental realization of superconducting magic-angle twisted four-layer and five-layer graphene, hence establishing alternating twist magic-angle multilayer graphene as a robust family of moiré superconductors. This finding suggests that the flat bands shared by the members play a central role in the superconductivity. Our measurements in parallel magnetic fields, in particular the investigation of Pauli limit violation and spontaneous rotational symmetry breaking, reveal a clear distinction between the N = 2 and N > 2-layer structures, consistent with the difference between their orbital responses to magnetic fields. Our results expand the emergent family of moiré superconductors, providing new insight with potential implications for design of new superconducting materials platforms.

Investigating the Semi-Analytical Models of Momentum Relaxation Mean Free Time and Path and Ionization Coefficient of Trilayer Graphene Nanoribbon- Based FETs

Trilayer graphene nanoribbon as a superlative semiconductor promises potential applications in the diodes, FETs, gas and biosensors. It is a suitable candidate for nanoelectronic applications due to attractive electronic, mechanical, thermoelectric and optoelectronics properties. The aim of the present paper is to analytically investigate the momentum relaxation mean free time and path and also ionization coefficient of trilayer graphene nanoribbon in nanoscale FETs. The models are derived based on energy band structure and calculating the effective mass, carrier velocity and scattering rate. Furthermore, the effects of interlayer potential energies, temperature and potential difference of layers are also taken into account on the modeled parameters. As a result, the obtained results and findings of this research are in rational agreement with published data, in terms of value and trend. The results and figures of merit for the proposed device showed a promising performance for transistor applications.

Thermodynamic parameters of atomically thin superconductors derived from the upper critical field

The amplitude of the ground-state superconducting energy gap Δ(0) and relative jump in the electronic specific heat at the transition temperature are the primary fundamental parameters of any superconductor. Several well-established techniques are available for measuring these values in bulk samples. However, a limited number of techniques can be used to measure these parameters in atomically thin superconductors. Here, we propose a new approach for extracting Δ(0) and in atomically thin superconductors by utilizing the upper critical field data from perpendicular, B c2,Δ(T) (when a magnetic field is applied perpendicular to the film surface) and parallel, B c2,‖(T) (when a magnetic field is applied in the direction parallel to the film surface), external field directions. The deduced parameters for few-layer-thick Al, Sn, NbSe2, MoS2, magic angle twisted trilayer graphene, and WTe2 are well-matched values expected for strongly and moderately strongly coupled electron-phonon-mediated superconductors. In many reports, the enhancement of B c2,‖(0) above the Pauli–Clogston–Chandrasekhar limiting field (i.e. magnetic field required to break the Cooper pair) in atomically thin superconductors has been explained based on the assumption of exotic pairing mechanisms, for instance, Ising-type pairing. Here, we explain the observed B c2,‖(0) enhancement based on the geometrical enhancement factor that originates from the sample geometry. This approach does not assume the existence of novel exotic pairing mechanisms in atomically thin superconductors.

Charge State Effects in Swift-Heavy-Ion-Irradiated Nanomaterials

The aim of this experimental work was to investigate the influence of the ion beam charge state on damage production in nanomaterials. To achieve this, we employed Raman spectroscopy, atomic force microscopy, and transmission electron microscopy to investigate nanomaterials irradiated by a 23 MeV I beam. We found a significant influence of the ion charge state on damage production in monolayer graphene, but found no evidence of this effect in bilayer and trilayer graphene, nor in graphite. Furthermore, we found no evidence of this effect in CaF2 and SiO2 nanocrystals irradiated with the same ion beam.

Nearly flat bands in twisted triple bilayer graphene

We investigate the electronic structure of alternating-twist triple Bernal-stacked bilayer graphene (t3BG) as a function of interlayer coupling $\omega$, twist angle $\theta$, interlayer potential difference $\Delta$, and top-bottom bilayers sliding vector $\boldsymbol{\tau}$ for three possible configurations AB/AB/AB, AB/BA/AB, and AB/AB/BA. The parabolic low-energy band dispersions in a Bernal-stacked bilayer and gap-opening through a finite interlayer potential difference $\Delta$ allows the flattening of bands in t3BG down to $\sim 20$~meV for twist angles $\theta \lesssim 2^{\circ}$ regardless of the stacking types. The easier isolation of the flat bands and associated reduction of Coulomb screening thanks to the intrinsic gaps of bilayer graphene for finite $\Delta$ facilitate the formation of correlation-driven gaps when it is compared to the metallic phases of twisted trilayer graphene under electric fields. We obtain the stacking dependent Coulomb energy versus bandwidth $U/W \gtrsim 1$ ratios in the $\theta$ and $\Delta$ parameter space. We also present the expected $K$-valley Chern numbers for the lowest-energy nearly flat bands.

Evidence for unconventional superconductivity in twisted trilayer graphene.

Magic-angle twisted trilayer graphene (MATTG) has emerged as a moiré material that exhibits strong electronic correlations and unconventional superconductivity1,2. However, local spectroscopic studies of this system are still lacking. Here we perform high-resolution scanning tunnelling microscopy and spectroscopy of MATTG that reveal extensive regions of atomic reconstruction favouring mirror-symmetric stacking. In these regions, we observe symmetry-breaking electronic transitions and doping-dependent band-structure deformations similar to those in magic-angle bilayers, as expected theoretically given the commonality of flat bands3,4. Most notably in a density window spanning two to three holes per moiré unit cell, the spectroscopic signatures of superconductivity are manifest as pronounced dips in the tunnelling conductance at the Fermi level accompanied by coherence peaks that become gradually suppressed at elevated temperatures and magnetic fields. The observed evolution of the conductance with doping is consistent with a gate-tunable transition from a gapped superconductor to a nodal superconductor, which is theoretically compatible with a sharp transition from a Bardeen-Cooper-Schrieffer superconductor to a Bose-Einstein-condensation superconductor with a nodal order parameter. Within this doping window, we also detect peak-dip-hump structures that suggest that superconductivity is driven by strong coupling to bosonic modes of MATTG. Our results will enable further understanding of superconductivity and correlated states in graphene-based moiré structures beyond twisted bilayers5.

Unusual magnetic transition and enhanced diamagnetism in highly oriented pyrolytic graphite with long-range stacking-order

The recent observations of unusual superconductive effects in highly oriented pyrolytic graphite (HOPG) and twisted bi-layer, tri-layer and multilayer graphene materials have attracted an important attention. In this work we present a novel investigation on the relationship between structural ordering and magnetism in HOPG. Orientation dependent X-ray diffraction and temperature dependent magnetometry allowed for the identification of those HOPG samples exhibiting a long-range ordering along the c-axis. ZFC and FC magnetic curves acquired from T ~ 2 K to 300 K revealed a significant temperature-dependent variation of the diamagnetic component. By analyzing the ZFC/FC magnetic-curves (through magnetic moment subtraction methods) we identify the presence of a weak magnetic transition in the T range from 200 K to 2 K. The possible origin of the observed magnetization-trend is interpreted in relation to contributions arising from 1) disorder-rich locally-twisted-interfaces and 2) those internal graphitic sublattices exhibiting negative thermal expansion. Interestingly, magnetization against field measurements acquired at T ~ 50 K (max. field of 150 Oe) revealed the presence of a weak ferromagnetic-like hysteresis. Disappearance of the unusual ZFC/FC signal-splitting was instead found in those magnetic curves acquired from other HOPG samples with higher degree of disorder (short-range-ordering along the c-axis). This second category of samples was found to exhibit a disorder-mediated re-entrant spin-glass-like ferromagnetic behavior.

Dislocation-Induced Energy Dissipation in a Tunable Trilayer Graphene Resonator

In crystalline materials, the creation and modulation of dislocations are often associated with plastic deformation and energy dissipation. Here we report a study on the energy dissipation of a trilayer graphene ribbon resonator. The vibration of the ribbon generates cyclic mechanical loading to the graphene ribbon, during which mechanical energy is dissipated as heat. Measuring the quality factor of the graphene resonator provides a way to evaluate the energy dissipation. The graphene ribbon is integrated with silicon micro actuators, allowing its in-plane tension to be finely tuned. As we gradually increased the tension, we observed, in addition to the well-known resonance frequency increase, a large change in the energy dissipation. We propose that the dominating energy dissipation mechanism shifts over three regions. With small applied tension, the graphene is in elastic region, and the major energy dissipation is through graphene edge folding; as the tension increases, dislocations start to develop in the sample to gradually dominate the energy dissipation; finally, at large enough tension, graphene layers become decoupled and start to slide and cause friction, which induces the more severe energy dissipation. The generation and modulation of dislocations are modeled by molecular dynamics calculation and a method to count the energy loss is proposed and compared to the experiment.

Angle-tunable two-photon absorption in twisted graphene systems

We theoretically study the two-photon absorption of twisted bilayer, trilayer and double bilayer graphene with Bernal stacking and demonstrate the layer and twist angle dependent two-photon absorption properties based on both the density functional theory calculations and our developed continuum model for small angle range. The expressions of two-photon absorption for above mentioned three twisted graphene layers around Dirac points are obtained. For small rotation angles which are less than 15 ° , the rotation angle well tunes the two-photon absorption coefficient in terms of the angle-dependent renormalized Fermi velocity. • Energy bands of twisted graphene systems around K point were investigated using DFT calculation. • Two-photon absorption spectra in twisted graphene systems were simulated on the basis of the second-order perturbation theory. • Twist angle effectively tuned the two-photon absorption coefficient within the small angle range ( θ < 15 ° ).

Nontrivial effect of out-of-plane acoustic phonon mode on limiting room temperature conductivity of ABA-stacked trilayer graphene

The ABA-stacked trilayer graphene (TLG) has a parabolic dispersion band spanning the Fermi level, apart from the linear dispersion band which is almost a duplicate of that in monolayer graphene (MLG). Such a band structure implies that the TLG is a metal and seemingly has a larger conductivity than MLG due to the additional contributions of the parabolic bands when the linear dispersion bands in TLG and MLG host the same number of carriers. However, our numerical calculations indicate that the conductivity of TLG is smaller than that in MLG roughly by 20% when a moderate carrier doping drives the Fermi energy away from the Dirac point by at least 50 meV. To explain such a disagreement between the band structure and the conductivity, we find that the out-of-plane atomic vibration, i.e., the so-called $ZA$ phonon mode in the long-wavelength region, plays a critical role in limiting the conductivity of TLG due to the strong interband scattering. In contrast to MLG, such a $ZA$ phonon mode is completely decoupled from the electron-phonon scattering due to the symmetry incompatibility.

Longitudinal conductivity in ABC-stacked trilayer graphene under irradiating of linearly polarized light

We study the effect of linearly polarized light on the band structure and longitudinal conductivity in ABC-stacked trilayer graphene. The linearly polarized light can induce a pair of additional points in ABC-stacked trilayer graphene, where conduct and valence bands touch. The locations of these points are determined by the amplitude of the light. Furthermore, the layer pseudospin polarization can be controlled by the light. When the Fermi energy locates at Dirac points, i.e., E f = 0, the longitudinal conductivity shows resonance phenomena when the light is present. Away from the Dirac points, the longitudinal conductivity is unchanged as varying E f for weak light field at larger Fermi energy, and the amplitude of longitudinal conductivity can be controlled by tuning the light field amplitude. Moreover, the effect of linearly polarized light on resonance phenomena in k-cubic Rashba–Dresselhaus system under the irradiating of linearly polarized light is discussed.

Moiré band structures of the double twisted few-layer graphene

Very recently, unconventional superconductivity has been observed in the double twisted trilayer graphene (TLG), where three monolayer graphene (MLG) are stacked on top of each other with two twist angles [J. M. Park, et al., Nature 590, 249 (2021); Z. Hao, et al., Science 371, 1133 (2021); X. Zhang, et al., Phys. Rev. Lett.127, 166802 (2021)]. When some of MLGs in the double twisted TLG are replaced by bilayer graphene (BLG), we get a new family of double twisted moire heterostructure, namely double twisted few layer graphene (DTFLG). In this work, we theoretically investigate the moire band structures of the DTFLGs with diverse arrangements of MLG and BLG. We find that, depending on the relative rotation direction of the two twist angles (alternate or chiral twist) and the middle van der Waals (vdW) layer (MLG or BLG), a general (X+Y+Z)-DTFLG can be classified into four categories, i.e. (X+1+Z)-ATFLG, (X+2+Z)-ATFLG, (X+1+Z)-CTFLG and (X+2+Z)-CTFLG, each of which has its own unique band structure. Here, X, Y, Z denote the three vdW layers, i.e. MLG or BLG. Interestingly, the (X+1+Z)-ATFLGs have a pair of perfect flat bands at the magic angle about $1.54^\circ$ coexisting with a pair of linear or parabolic bands, which is quite like the double twisted TLG. Meanwhile, when the twist angle is smaller than a "magic angle" $1.70^\circ$, the (X+2+Z)-CTFLGs can have two isolated narrow bands at $E_f$ with band width less than 5 meV. The influence of electric field and the topological features of the moire bands have been studied as well. Our work indicates that the DTFLGs, especially the (X+1+Z)-ATFLG and (X+2+Z)-CTFLG, are promising platforms to study the moire flat band induced novel correlation and topological effects.

Alternating twisted multilayer graphene: generic partition rules, double flat bands, and orbital magnetoelectric effect

Recently the alternating twisted trilayer graphene is discovered to exhibit unconventional superconductivity, which motivates us to study the electronic structures and possible correlation effects for this class of alternating twisted multilayer graphene (ATMG) systems. In this work we consider generic ATMG systems with M-L-N stacking configurations, in which the M (L) graphene layers and the L (N) layers are twisted by an angle θ (−θ). Based on analysis from a simplified k⋅p model approach, we derive generic partition rules for the low-energy electronic structures, which exhibit various band dispersions including two pairs of flat bands and flat bands co-existing with various gapless Fermionic excitations. For a mirror-symmetric ATMG system with doubled flat bands, we further find that Coulomb interactions may drive the system into a state with intertwined electric polarization and orbital magnetization orders, which can exhibit an interaction-driven orbital magnetoelectric effect.

Tunable excitons in rhombohedral trilayer graphene

Trilayer graphene is receiving an increasing level of attention due to its stacking--dependent magnetoelectric and optoelectric properties, and its more robust ferromagnetism relative to monolayer and bilayer variants. Additionally, rhombohedral stacked trilayer graphene presents the possibility of easily opening a gap via either an external electric field perpendicular to the layers, or via the application of external strain. In this paper, we consider an external electric field to open a bandgap in rhombohedral trilayer graphene and study the excitonic optical response of the system. This is done via the combination of a tight binding model with the Bethe--Salpeter equation, solved semi--analytically and requiring only a simple numerical quadrature. We then discuss the valley--dependent optical selection rules, followed by the computation of the excitonic linear optical conductivity for the case of a rhombohedral graphene trilayer encapsulated in hexagonal boron nitride. The tunability of the excitonic resonances via an external field is also discussed, together with the increasing localization of the excitonic states as the field increases.

Tunable multi-bands in twisted double bilayer graphene

The bandstructure of a material, playing an important role in its electron transport property, is usually governed by the lattice configuration. Materials with a field-effect tunable band, such as bilayer [1] and rhombohedral trilayer graphene [2, 3], are more flexible for electronic applications. Here, on dual-gated twisted double bilayer graphene (TDBG) samples with small twist angle around 1∘, we observe vertical electric-field-tunable bandstructures at multiple moiré fillings with bandgap values continuously varying from zero to tens of mili-electron volts. Moreover, within the first moiré filling on both electron and hole sides, the carrier transport deviates from Fermi liquid behavior, with measured resistivity exhibiting linear temperature dependence between 1.5 K and 50 K. Furthermore, under a vertical magnetic field, the coupling between the two bilayer graphene layers can also be turned on and off by a displacement field. Our results suggest TDBG with small twist angle is a platform for studying the evolution of multiple electric field tunable moiré bands and the resulting emergent correlated electronic phases.

Competing orders and cascade of degeneracy lifting in doped Bernal bilayer graphene

Motivated by recent experiments [H. Zhou et al., Science 375, 774 (2022) and S. C. de la Barrera et al., arXiv:2110.13907], here we propose a general mechanism for valley and/or spin degeneracy lifting of the electronic bands in doped Bernal bilayer graphene, subject to electric displacement ($D$) fields. A $D$-field induced layer polarization (LP), when accompanied by a Hubbard repulsion-driven layer antiferromagnet (LAF) and next-nearest-neighbor repulsion-driven quantum anomalous Hall (QAH) orders, lifts the fourfold degeneracy of electronic bands, yielding a quarter metal for small doping, as also observed in ABC trilayer graphene. With the disappearance of the QAH order, electronic bands recover twofold valley degeneracy, thereby forming a conventional or compensated (with majority and minority carriers) half metal at moderate doping, depending on the relative strength of LP and LAF. At even higher doping and for a weak $D$ field only LAF survives and the Fermi surface recovers fourfold degeneracy. We also show that a pure repulsive electronic interaction mediated triplet $f$-wave pairing emerges from a parent correlated nematic liquid or compensated half metal when an in-plane magnetic field is applied to the system.

Twisted double ABC-stacked trilayer graphene with weak interlayer coupling

Magnetotransport properties of single- and double-rhombohedral trilayer graphene (TLG) are studied. Double TLG is obtained by folding single TLG, yielding a twist angle of 4\ifmmode^\circ\else\textdegree\fi{}. Raman spectroscopic measurements confirm ABC stacking order and the presence of interlayer coupling. The Landau fan diagram exhibits independently superimposed features of each constituent TLG, indicating weak interlayer coupling. As a result, we show that the current framework for twisted systems can be extended to TLG.

Kohn-Luttinger superconductivity and intervalley coherence in rhombohedral trilayer graphene

The recent experimental discovery of spin-valley symmetry breaking phases and superconductivity in rhombohedral trilayer graphene has generated much excitement. This research attempts to provide theoretical understanding of these phases. A prominent feature of the system is the existence of van Hove singularities in the electronic band structure. Under interactions, these electronic states exhibit a high tendency toward intervalley coherence (IVC) ordering. Upon doping, the IVC order can melt, and the low-lying IVC fluctuations can mediate superconductivity with spin-singlet pairing.

Correlated Insulators, Semimetals, and Superconductivity in Twisted Trilayer Graphene

Motivated by recent experiments indicating strong superconductivity and intricate correlated insulating and flavor-polarized physics in mirror-symmetric twisted trilayer graphene, we study the effects of interactions in this system close to the magic angle, using a combination of analytical and numerical methods. We identify asymptotically exact correlated many-body ground states at all integer filling fractions $\nu$ of the flat bands. To determine their fate when moving away from these fine-tuned points, we apply self-consistent Hartree-Fock numerics and analytic perturbation theory, with good agreement between the two approaches. This allows us to construct a phase diagram for the system as a function of $\nu$ and the displacement field, the crucial experimental tuning parameter of the system, and study the spectra of the different phases. The phase diagram is dominated by a correlated semimetallic intervalley coherent state and an insulating sublattice-polarized phase around charge neutrality, $\nu=0$, with additional spin-polarization being present at quarter ($\nu=-2$) or three quarter ($\nu=+2$) fillings of the quasi-flat bands. We further study the superconducting instabilities emerging from these correlated states, both in the absence and in the additional presence of electron-phonon coupling, also taking into account possible Wess-Zumino-Witten terms. In the experimentally relevant regime, we find triplet pairing to dominate, possibly explaining the observed violation of the Pauli limit. Our results have several consequences for experiments as well as future theoretical work and illustrate the rich physics resulting from the interplay of almost flat bands and dispersive Dirac cones in twisted trilayer graphene.

Interplay of band geometry and topology in ideal Chern insulators in the presence of external electromagnetic fields

Ideal Chern insulating phases arise in two-dimensional systems with broken time-reversal symmetry. They are characterized by having nearly-flat bands, and a uniform quantum geometry -- which combines the Berry curvature and quantum metric -- and by being incompressible. In this work, we analyze the role of the quantum geometry in ideal Chern insulators aiming to describe transport in presence of external out-of-plane magnetic and electric fields. We firstly show that in the absence of external perturbations, novel Berry connections appear in ideal Chern insulating phases. Secondly, we provide a detailed analysis of the deformation of the quantum geometry once weak out-of-plane magnetic fields are switched on. The perturbed Berry curvature and quantum metric provide an effective quantum geometry, which is analyzed in the insulating regime and provides an application of our novel connections. The conditions under which the Girvin-MacDonald-Platzman algebra is realized in this situation are discussed. Furthermore, an investigation of electrical transport due to the new effective quantum geometry is presented once an electric field is added. Restricting to the case of two bands in the metallic regime the quantum metric appears as measurable quantum mechanical correction in the Hall response. Our findings can be applied, for instance, to rhombohedral trilayer graphene at low energies.