Transport measurements through a few-electron circular quantum dot in bilayer graphene display bunching of the conductance resonances in groups of four, eight and twelve. This is in accordance with the spin and valley degeneracies in bilayer graphene and an additional threefold 'minivalley degeneracy' caused by trigonal warping. For small electron numbers, implying a small dot size and a small displacement field, a two-dimensional s- and then a p-shell are successively filled with four and eight electrons, respectively. For electron numbers larger than twelve, as the dot size and the displacement field increase, the single-particle ground state evolves into a three-fold degenerate minivalley ground state. A transition between these regimes is observed in our measurements and can be described by band-structure calculations. Measurements in magnetic field confirm Hund's second rule for spin filling of the quantum dot levels, emphasizing the importance of exchange interaction effects.
In van der Waals multilayers of triangular lattice, trigonal warping occurs universally due to the interlayer hopping. We theoretically investigate the effect of trigonal warping upon distinctive topological phases, like the quantum anomalous Hall effect (QAHE) and the quantum valley Hall effect (QVHE). Taking Bernal-stacked bilayer graphene as an example, we find that the trigonal warping plays a crucial role in the formation of QAHE in large exchange field and/or interlayer potential difference by inducing extra band inversion points at momentum further away from high symmetric point. The pr…
I suggest to use the effect of Coulomb drag between two closely positioned graphite monolayers (graphene sheets) for experimental measurement of the strength of weak non-linearities of the spectrum in graphene. I consider trigonal warping as a representative mechanism responsible for the drag effect. Since graphene is relatively defect-free, I evaluate the drag conductivity in the ballistic regime and find that it is proportional to the fourth power of the warping strength.
We study the superlattice minibands produced by the interplay between moiré pattern induced by hexagonal BN substrate on graphene layer and the interlayer coupling in bilayer graphene with Bernal stacking (BLG). We compare moiré miniband features in BLG, where they are affected by the interlayer asymmetry of BLG-hBN heterostructure and trigonal warping characteristic for electrons in Bernal-stacked bilayers with those found in monolayer graphene.
Monolayer Graphene contains two inequivalent local minimum, valleys, located at $K$ and $K'$ in the Brillouin zone. There has been considerable interest in the use of these two valleys as a doublet for information processing. Herein I propose a method to resolve valley currents spatially, using only a weak magnetic field. Due to the trigonal warping of the valleys, a spatial offset appears in the guiding centre co-ordinate, and is strongly enhanced due to collimation. This can be exploited to spatially separate valley states. Based on current experimental devices, spatial separation is possibl…
The stability of the zero-energy Landau levels in bilayer graphene against the chiral symmetric disorder is examined in the presence of the trigonal warping. Based on the tight-binding lattice model with a bond disorder correlated over several lattice constants, it is shown that among the four Landau levels per spin and per valley, two Landau levels exhibit the anomalous sharpness as in the absence of the trigonal warping, while the other two are broadened, yielding split peaks in the density of states. This can be attributed to the fact that the total chirality in each valley is +2 or -2, whi…
In the present study, the anisotropic resistivity of the monolayer graphene has been obtained in semiclassical regime beyond the Dirac point approximation. In particular, detailed investigations were made on the dependence of conductivity on the Fermi energy. At low energies, in the vicinity of the Dirac points, band energy of the monolayer graphene is isotropic at the Fermi level. Meanwhile, at the intermediate Fermi energies anisotropic effects such as trigonal warping is expected to be the origin of the anisotropic resistivity. However, besides the band anisotropy there also exists an other…
We investigate Fabry-Perot interference in an ultraclean carbon nanotube resonator. The conductance shows a clear superstructure superimposed onto conventional Fabry-Perot oscillations. A sliding average over the fast oscillations reveals a characteristic slow modulation of the conductance as a function of the gate voltage. We identify the origin of this secondary interference in intervalley and intravalley backscattering processes which involve wave vectors of different magnitude, reflecting the trigonal warping of the Dirac cones. As a consequence, the analysis of the secondary interference …
Using a reformulated Kubo formula we calculate the zero-energy minimal conductivity of bilayer graphene taking into account the small but finite trigonal warping. We find that the conductivity is independent of the strength of the trigonal warping and it is three times as large as that without trigonal warping, and six times larger than that in single layer graphene. Although the trigonal warping of the dispersion relation around the valleys in the Brillouin zone is effective only for low energy excitations, our result shows that its role cannot be neglected in the zero-energy minimal conducti…
We study the electronic band structure of monolayer graphene when Rashba spin-orbit coupling is present. We show that if the Rashba spin-orbit coupling is stronger than the intrinsic spin-orbit coupling, the low energy bands undergo trigonal-warping deformation and that for energies smaller than the Lifshitz energy, the Fermi circle breaks up into separate parts. The effect is very similar to what happens in bilayer graphene at low energies. We discuss the possible experimental implications, such as threefold increase of the minimal conductivity for low electron densities, the wavenumber depen…
We calculate the form of quasiparticle interference patterns in bilayer graphene within a low-energy description, taking into account perturbatively the trigonal warping terms. We introduce four different types of impurities localized on the A and B sublattices of the first and the second layer, and we obtain closed-form analytical expressions both in real and Fourier spaces for the oscillatory corrections to the local density of states generated by the impurities. Finally, we compare our findings with recent experimental and semi-analytical T-matrix results from arXiv:2104.10620 and we show t…
We use a combined ab-initio calculations and k.p theory based approach to derive a low-energy effective Hamiltonian for monolayer MoS2 at the K point of the Brillouin zone. It captures the features which are present in first-principles calculations but not explained by the theory of Xiao et al. [Phys Rev Lett 108, 196802 (2012)], namely the trigonal warping of the valence and conduction bands, the electron-hole symmetry breaking, and the spin-splitting of the conduction band. We also consider other points in the Brillouin zone which might be important for transport properties. Our findings lea…
We develop an analytical mode-matching technique for the tight-binding model to describe electron transport across graphene P-N junctions. This method shares the simplicity of the conventional mode-matching technique for the low-energy continuum model and the accuracy of the tight-binding model over a wide range of energies. It further reveals an interesting phenomenon on a sharp P-N junction: the disappearance of the well-known Klein tunneling (i.e., perfect transmission) at normal incidence and the appearance of perfect transmission at oblique incidence due to trigonal warping at energies be…
We use an exact analytical technique [Phys. Rev. B \textbf{101}, 115405 (2020), Phys. Rev. B \textbf{102}, 165117 (2020)] to recover the surface Green's functions for Bernal (ABA) and rhombohedral (ABC) graphite. For rhombohedral graphite we recover the predicted surface flat bands. For Bernal graphite we find that the surface state spectral function is similar to the bilayer one, but the trigonal warping effects are enhanced, and the surface quasiparticles have a much shorter lifetime. We subsequently use the T-matrix formalism to study the quasiparticle interference patterns generated on the…
We show that the valley Chern number of the low energy band in twisted double bilayer graphene can be tuned through two successive topological transitions, where the direct bandgap closes, by changing the electric field perpendicular to the plane of the graphene layers. The two transitions with Chern number changes of -3 and +1 can be explained by the formation of three satellite Dirac points around the central Dirac cone in the moiré Brillouin zone due to the presence of trigonal warping. The satellite cones have opposite chirality to the central Dirac cone. Considering the overlap of the ban…
Bilayer graphene (two coupled graphitic monolayers arranged according to Bernal stacking) is a two-dimensional gapless semiconductor with a peculiar electronic spectrum different from the Dirac spectrum in the monolayer material. In particular, the electronic Fermi line in each of its valleys has a strong p -> -p asymmetry due to a trigonal warping, which suppresses the weak localization effect. We show that weak localisation in bilayer graphene may be present only in devices with pronounced intervalley scattering, and we evaluate the corresponding magnetoresistance.
The existence of strong trigonal warping around the K point for the low energy electronic states in multilayer (N$\geq$2) graphene films and graphite is well established. It is responsible for phenomena such as Lifshitz transitions and anisotropic ballistic transport. The absolute orientation of the trigonal warping with respect to the center of the Brillouin zone is however not agreed upon. Here, we use quasiparticle scattering experiments on a gated bilayer graphene/hexagonal boron nitride heterostructure to settle this disagreement. We compare Fourier transforms of scattering interference m…
We present a mainly analytical study of the entanglement spectrum of Bernal-stacked graphene bilayers in the presence of trigonal warping in the energy spectrum. Upon tracing out one layer, the entanglement spectrum shows qualitative geometric differences to the energy spectrum of a graphene monolayer. However, topological quantities such as Berry phase type contributions to Chern numbers agree. The latter analysis involves not only the eigenvalues of the entanglement Hamiltonian but also its eigenvectors. We also discuss the entanglement spectra resulting from tracing out other sublattices. A…
The adhesion of graphene on slightly lattice-mismatched surfaces, for instance of hexagonal boron nitride (hBN) or Ir(111), gives rise to a complex landscape of sublattice symmetry-breaking potentials for the Dirac fermions. Whereas a gap at the Dirac point opens for perfectly lattice-matched graphene on hBN, we show that the small lattice incommensurability prevents the opening of this gap and rather leads to a renormalized Dirac dispersion with a trigonal warping. This warping breaks the effective time reversal symmetry in a single valley. On top of this a new set of massless Dirac fermions …
Due to low dimensionality, the controlled stacking of the graphene films and their electronic properties are susceptible to environmental changes including strain. The strain-induced modification of the electronic properties such as the emergence and modulation of bandgaps crucially depends on the stacking of the graphene films. However, to date, only the impact of strain on electronic properties of Bernal and AA-stacked bilayer graphene has been extensively investigated in theoretical studies. Exploiting density functional theory and tight-binding calculation, we investigate the impacts of in…
The electronic band structure of ABC-stacked multilayer graphene is studied within an effective mass approximation. The electron and hole bands touching at zero energy support chiral quasiparticles characterized by Berry's phase N pi for N-layers, generalizing the low-energy band structure of monolayer and bilayer graphene. We investigate the trigonal-warping deformation of the energy bands and show that the Lifshitz transition, in which the Fermi circle breaks up into separate parts at low energy, reflects Berry's phase N pi. It is particularly prominent in trilayers, N=3, with the Fermi circ…
Using the transfer matrix in the angular-momentum space we investigate the impact of trigonal warping on magnetotransport and scaling properties of a ballistic bilayer graphene in the Corbino geometry. Although the conductivity at the charge-neutrality point and zero magnetic field exhibits a one-parameter scaling, the shot-noise characteristics, quantified by the Fano factor $\mathcal{F}$ and the third charge-transfer cumulant $\mathcal{R}$, remain pseudodiffusive. This shows that the pseudodiffusive transport regime in bilayer graphene is not related to the universal value of the conductivit…
We study the effect of trigonal warping on the focussing of electrons by n-p junctions in graphene. We find that perfect focussing, which was predicted for massless Dirac fermions, is only preserved for one specific sample orientation. In the general case, trigonal warping leads to the formation of cusp caustics, with a different position of the focus for graphene's two valleys. We develop a semiclassical theory to compute these positions and find very good agreement with tight-binding simulations. Considering the transmission as a function of potential strength, we find that trigonal warping …
We report ultra-strong high-order nonlinear multiphoton processes in monolayer MoS$_2$ (1L-MoS$_2$): the third harmonic is thirty times stronger than the second harmonic, and the fourth harmonic is comparable to the second harmonic. We find that second and third harmonic processes are strongly dependent on elliptical polarization, which can be used to selectively tune harmonic generation with different orders. We explain this by calculating the nonlinear response functions of 1L-MoS$_2$ with a continuum-model Hamiltonian and quantum-mechanical diagrammatic perturbation theory, highlighting the…
We analyze the electronic structure of topological surface states in the family of magnetic topological insulators MnBi$_{2n}$Te$_{3n+1}$. We show that, at natural-cleavage surfaces, the Dirac cone warping changes its symmetry from hexagonal to trigonal at the magnetic ordering temperature. In particular, an energy splitting develops between the surface states of same band index but opposite surface momenta upon formation of the long-range magnetic order. As a consequence, measurements of such energy splittings constitute a simple protocol to detect the magnetic ordering via the surface electr…
Bilayer silicene has richer physical properties than bilayer graphene due to its buckled structure together with its trigonal symmetric structure. The buckled structure arises from a large ionic radius of silicon, and the trigonal symmetry from a particular way of hopping between two silicenes. It is a topologically trivial insulator since it carries a trivial $\mathbb{Z}_{2}$ topological charge. Nevertheless, its physical properties are more akin to those of a topological insulator than those of a band insulator. Indeed, a bilayer silicene nanoribbon has edge modes which are almost gapless an…
In this work we show that the trigonal warping of the electronic bands in bilayer graphene dramatically modifies the behavior of the one-dimensional modes topologically confined due to an inhomogeneous bias that changes sign across a channel. The topologically protected states are present but their behavior is disrupted from the predicted in the isotropic approximation. We present detailed studies of the electronic properties of the 1D channel in function of the orientation of the channel.
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…
This is a numerical study of thermoelectric properties of ballistic bilayer graphene in the presence of trigonal warping term in the effective Hamiltonian. We find, in the mesoscopic samples of the length $L>10\,μ$m at sub-Kelvin temperatures, that both the Seebeck coefficient and the Lorentz number show anomalies (the additional maximum and minimum, respectively) when the electrochemical potential is close to the Lifshitz energy, which can be attributed to the presence of the van Hove singularity in a bulk density of states. At higher temperatures the anomalies vanish, but measurable quantiti…
Resolving low-energy features in the density of states (DOS) holds the key to understanding wide variety of rich novel phenomena in graphene based 2D heterostructures. Lifshitz transition in bilayer graphene (BLG) arising from trigonal warping has been established theoretically and experimentally. Nevertheless, the experimental realization of its effects on the transport properties has been challenging because of its relatively low energy scale ($\sim 1$ meV). In this work, we demonstrate that the thermoelectric power (TEP) can be used as an effective probe to investigate fine changes in the D…
We perform an analytical study of the energy and entanglement spectrum of non-interacting fermionic bilayer honeycomb lattices in the presence of trigonal warping in the energy spectrum, on-site energy difference and uniform magnetic field. Employing single particle correlation functions, we present an explicit form for layer-layer entanglement Hamiltonian whose spectrum is entanglement spectrum. We demonstrate that in the absence of trigonal warping, at zero on-site energy difference exact correspondence is established between entanglement spectrum and energy spectrum of monolayer which means…
The optical conductivity of graphene, bilayer graphene, and graphite in quantizing magnetic fields is studied. Both dynamical conductivities, longitudinal and Hall's, are analytically evaluated. The conductivity peaks are explained in terms of electron transitions. We have shown that trigonal warping can be considered within the perturbation theory for strong magnetic fields larger than 1 T and in the semiclassical approach for weak fields when the Fermi energy is much larger than the cyclotron frequency. The main optical transitions obey the selection rule with \Deltan = 1 for the Landau numb…
We describe the weak localization correction to conductivity in ultra-thin graphene films, taking into account disorder scattering and the influence of trigonal warping of the Fermi surface. A possible manifestation of the chiral nature of electrons in the localization properties is hampered by trigonal warping, resulting in a suppression of the weak anti-localization effect in monolayer graphene and of weak localization in bilayer graphene. Intervalley scattering due to atomically sharp scatterers in a realistic graphene sheet or by edges in a narrow wire tends to restore weak localization re…
We show how the trigonal warping effect in doped graphene can be used to produce fully valley polarized currents. We propose a device that acts both as a beam splitter and a collimator of these electronic currents. The result is demonstrated trough an optical analogy using two dimensional photonic crystals.
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 neu…
We investigate the semiclassical electronic transport properties of the bilayer silicene-like system in the presence of charged impurity. The trigonal warping due to the interlayer hopping, and its effect to the band structure of bilayer silicene is discussed. Besides the trigonal warping, the external field also gives rise to the anisotropic effect of the mobility (at finite temperature) which can be explored by the Boltzmann theory. The dynamical polarization as well as the scattering wave vector-dependent screening within random phase approximation are very important in determining the scat…
We study relaxation of an excited electron in the conduction band of intrinsic graphene at zero temperature due to production of interband electron-hole pairs. The electronic band curvature, being anisotropic because of trigonal warping, is shown to suppress relaxation for a range of directions of the initial electron momentum. For other directions, relaxation is allowed only if the curvature exceeds a finite critical value; otherwise, a non-decaying quasiparticle state is found to exist.
We propose an implementation of a valley selective electronic Veselago lens in bilayer graphene. We demonstrate that in the presence of an appropriately oriented potential step, low-energy electrons radiating from a point source can be re-focused coherently within the same band. The phenomenon is due to the trigonal warping of the band structure that leads to a negative refraction index. We show that the interference pattern can be controlled by an external mechanical strain.
We demonstrate the possibility of existence of indirect moving Wannier-Mott excitons in graphene. Electron-hole binding is conditioned by the trigonal warping of conic energy spectrum. The binding energies are found for the lowest exciton states. These energies essentially depend on the value and direction of exciton momentum and vanish when the exciton momentum tends to the conic points. The ways to observe the exciton states are discussed. The opportunity of experimental observation of zero-gap excitons by means of external electron scattering is examined.
We calculate the spin relaxation time of mobile electrons due to spin precession between random impurity scattering (D'yakonov-Perel' mechanism) in electrically gated bilayer graphene analytically and numerically. Due to the trigonal warping of the bandstructure, the spin relaxation time exhibits an interesting non-monotonic behavior as a function of both the Fermi energy and the interlayer bias potential. Our results are in good agreement with recent four-probe measurements of the spin relaxation time in bilayer graphene and indicate the possibility of an electrically-switched spin device.
Electron group velocity for graphene under uniform strain is obtained analitically by using the Tight-Binding approx- imation. Such closed analytical expressions are useful in order to calculate electronic, thermal and optical properties of strained graphene. These results allow to understand the behavior of electrons when graphene is subjected to strong strain and nonlinear corrections, for which the usual Dirac approach is not longer valid. Some particular cases of uni- axial and shear strain were analized. The evolution of the electron group velocity indicates a break up of the trigonal war…
We study the ballistic conductivity of bilayer graphene in the presence of symmetry-breaking terms in effective Hamiltonian for low-energy excitations, such as the trigonal-warping term ($γ_3$), the electron-hole symmetry breaking interlayer hopping ($γ_4$), and the staggered potential ($δ_{AB}$). Earlier, it was shown that for $γ_3\neq{}0$, in the absence of remaining symmetry-breaking terms (i.e., $γ_4=δ_{AB}=0$), the conductivity ($σ$) approaches the value of $3σ_0$ for the system size $L\rightarrow{}\infty$ (with $σ_0=8e^2/(πh)$ being the result in the absence of trigonal warping, $γ_3=0$)…
We present a theoretical study on the orbital magnetism in multilayer graphenes within the effective mass approximation. The Hamiltonian and thus susceptibility can be decomposed into contributions from sub-systems equivalent to monolayer or bilayer graphene. The monolayer-type subband exists only in odd layers and exhibits a delta-function susceptibility at $E_F=0$. The bilayer-type subband appearing in every layer number gives a singular structure in the vicinity of $E_F=0$ due to the trigonal warping as well as a logarithmic tail away from $E_F=0$. The integral of the susceptibility over en…
We study the renormalization of the effective mass and trigonal warping of bilayer graphene by the electron-electron interaction. One consequence of such a renormalization in the low-energy bands of a bilayer crystal consists of a small reduction of the critical density of the Lifshitz transition (the crossover between the single-pocket and four-pocket topology of the Fermi surface).
We report on the effects of laser illumination on the electronic properties of bilayer graphene. By using Floquet theory combined with Green's functions we unveil the appeareance of laser-induced gaps not only at integer multiples of $\hbar Ω/2$ but also at the Dirac point with features which are shown to depend strongly on the laser polarization. Trigonal warping corrections are shown to lead to important corrections for radiation in the THz range, reducing the size of the dynamical gaps. Furthermore, our analysis of the topological properties at low energies reveals that when irradiated with…
The sequence of the zeroth Landau levels (LLs) between filling factors $ν$=-6 to 6 in ABA-stacked trilayer graphene (TLG) is unknown because it depends sensitively on the non-uniform charge distribution on the three layers of ABA-stacked TLG. Using the sensitivity of quantum Hall data on the electric field and magnetic field, in an ultraclean ABA-stacked TLG sample, we quantitatively estimate the non-uniformity of the electric field and determine the sequence of the zeroth LLs. We also observe anticrossings between some LLs differing by 3 in LL index, which result from the breaking of the cont…
We study the static and dynamic screening of gapped bilayer graphene. Unlike previous works we use the 4-band model instead of the simplified 2-band model. We find that there are important qualitative differences between the dielectric screening function obtained using the 2-band and that obtained using the 4-band model. In particular within the 4-band model in the presence of a band-gap the static screening exhibits Kohn anomalies that are absent within the 2-band model. Moreover, using the 4-band model, we are able to examine the effect of trigonal warping (absent in the 2-band model) on the…
