Trigonal Warping Effect Research Articles

Wavefronts Dislocations of Friedel Oscillations in Graphene: Trigonal Warping Effect

The electron waves of a host system exhibit an oscillating response to an external impurity, namely Friedel oscillations (FOs), extensively studied in two‐dimensional materials. Recently, wavefront dislocations, a new feature of FOs, have been revealed in graphene. However, previous analytical works have been limited to the linear dispersion of graphene. Herein, the fate of wavefront dislocations is investigated numerically in FOs beyond the linear regime. The wavefront dislocations are robust against the trigonal warping effect, crucial for high doping graphene, due to the invariant winding number of the tight‐binding energy band. Furthermore, the opening of the gap, increasing the electronic Fermi wavelength, can highlight the wavefront dislocations blurred by intravalley scattering‐induced short‐range oscillations. These results should be observable using current experimental technology. Therefore, this study not only demonstrates the robust existence of wavefront dislocations in FOs over a wide range of energies but also deepens the understanding of intervalley scattering in graphene and other two‐dimensional valleytronic materials.

Specular Electron Focusing between Gate-Defined Quantum Point Contacts in Bilayer Graphene.

We report multiterminal measurements in a ballistic bilayer graphene (BLG) channel, where multiple spin- and valley-degenerate quantum point contacts (QPCs) are defined by electrostatic gating. By patterning QPCs of different shapes along different crystallographic directions, we study the effect of size quantization and trigonal warping on transverse electron focusing (TEF). Our TEF spectra show eight clear peaks with comparable amplitudes and weak signatures of quantum interference at the lowest temperature, indicating that reflections at the gate-defined edges are specular, and transport is phase coherent. The temperature dependence of the focusing signal shows that, despite the small gate-induced bandgaps in our sample (≲45 meV), several peaks are visible up to 100 K. The achievement of specular reflection, which is expected to preserve the pseudospin information of the electron jets, is promising for the realization of ballistic interconnects for new valleytronic devices.

φ0 -Josephson junction in twisted bilayer graphene induced by a valley-polarized state

Recently, gate-defined Josephson junctions in magic angle twisted bilayer graphene (MATBG) were studied experimentally and highly unconventional Fraunhofer patterns were observed. In this work, we show that an interaction-driven valley-polarized state connecting two superconducting regions of MATBG would give rise to a long-sought-after purely electric controlled $\varphi_0$-junction in which the two superconductors acquire a finite phase difference $\varphi_0$ in the ground state. We point out that the emergence of the $\varphi_0$-junction stems from the valley-polarized state which breaks time-reversal symmetry and trigonal warping effects which break the intravalley inversion symmetry. Importantly, a spatially non-uniform valley polarization order parameter at the junction can explain the highly unconventional Fraunhofer patterns observed in the experiment. Our work explores the novel transport properties of the valley-polarized state and suggests that gate-defined MATBG Josephson junctions could realize the first purely electric controlled $\varphi_0$-junctions with applications in superconducting devices.

Trigonal warping effects on optical properties of anomalous Hall materials

The topological nature of solids is related to the symmetries present in the material, for example, a quantum spin Hall effect can be observed in topological insulators with time-reversal symmetry, while broken time-reversal symmetry may give rise to the presence of an anomalous quantum Hall effect (AHE). Here, we consider the effects of broken rotational symmetry on the Dirac cone of an AHE material by adding trigonal warping terms to the Dirac Hamiltonian. We calculate the linear optical conductivity semianalytically to show how by breaking the rotational symmetry we can obtain a topologically distinct phase. The addition of trigonal warping terms causes the emergence of additional Dirac cones, which when combined has a total Chern number of $\ensuremath{\mp}1$ instead of $\ifmmode\pm\else\textpm\fi{}1/2$. This results in drastic changes in the anomalous Hall and longitudinal conductivity. The trigonal warping terms also activate the higher-order Hall responses which do not exist in a $\mathcal{R}$-symmetric conventional Dirac material. We found the presence of a nonzero second-order Hall current even in the absence of a Berry curvature dipole. This shift current is also unaffected by the chirality of the Dirac cone, which should lead to a nonzero Hall current in time-reversal symmetric systems.

Jacutingaite family: An efficient platform for coexistence of spin valley Hall effects, valley spin-valve realization, and layer spin crossover

Jacutingaite family, a family of naturally occurring exfoliable minerals (${\mathrm{Pt}}_{2}{\mathrm{HgSe}}_{3}$ and ${\mathrm{Pd}}_{2}{\mathrm{HgSe}}_{3}$) discovered in Brazil, is recently predicted to be the very first large-gap Kane-Mele quantum spin Hall insulators. However, yet to date, the door is locked for realizing versatile spin-valley based phenomena due to inversion symmetry in this family. By exploiting the inversion symmetry using different strategies such as placing the jacutingaite family in the vertical electric field or growing its composites with some suitable well-matched systems, we reveal several promising valley spin based phenomena in the jacutingaite family. Overall, we achieve the coupled spin valley Hall effect, valley spin-valve effect, Rashba spin splitting around the low-symmetric $M$ point, layer crossover accompanied by spin crossover, and selective-excitation of carriers from opposite valleys in this family. More interestingly, swapping of the induced spin splittings, Berry curvatures, and the spin texture between the two valleys occur upon the reversal of electric field from ${E}_{z}>0$ to ${E}_{z}<0$ without destroying the ${\mathcal{Z}}_{2}$ topological hallmarks. Meanwhile, considerable hexagonal trigonal warping effects around the ${K}_{+}$ and ${K}_{\ensuremath{-}}$ valleys are realized. Furthermore, by applying vertical electric field through interfacial coupling with ferroelectric ${\mathrm{III}}_{2}\ensuremath{-}{\mathrm{VI}}_{3}$ film, valley contrasting Berry curvature in the ${\mathrm{Pt}}_{2}{\mathrm{HgSe}}_{3}$ layer with a larger spin splitting in the lower conduction band $(\ensuremath{\sim}188\phantom{\rule{0.16em}{0ex}}\mathrm{meV})$ is achieved in the $+P$ polarization state. By inverting the polarization state from $+P$ to $\ensuremath{-}P$, the Fermi level shifts down into the valence band of the ${\mathrm{Pt}}_{2}{\mathrm{HgSe}}_{3}$ layer $(\ensuremath{\sim}82\phantom{\rule{0.16em}{0ex}}\mathrm{meV})$ revealing a hole doping in the ${\mathrm{Pt}}_{2}{\mathrm{HgSe}}_{3}$ layer. This could establish an effective mechanism to control doping levels in the ${\mathrm{Pt}}_{2}{\mathrm{HgSe}}_{3}$ layer as desirable for achieving superconductivity at finite doping in the jacutingaite family. Our paper provides a path towards integrating valleytronics and spintronics in multivalley materials with broken inversion symmetry.

Surface states and quasiparticle interference in Bernal and rhombohedral graphite with and without trigonal warping

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 surface of semi-infinite ABA and ABC graphite in the presence of impurity scattering. We compare our predictions to experimental STM data of impurity-localized states on the surface of Bernal graphite which appear to be in a good agreement with our calculations.

Evidence of Lifshitz Transition in the Thermoelectric Power of Ultrahigh-Mobility Bilayer Graphene.

Resolving low-energy features in the density of states (DOS) holds the key to understanding a wide variety of rich novel phenomena in graphene-based 2D heterostructures. The Lifshitz transition in bilayer graphene (BLG) arising from trigonal warping has been established theoretically and experimentally. Nevertheless, the experimental realization of its effects on transport properties has been challenging because of its relatively low energy scale (∼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 DOS of BLG. We observed additional entropy features in the vicinity of the charge neutrality point (CNP) in gapped BLG. This apparent violation of the Mott formula can be explained quantitatively by considering the effects of trigonal warping, thereby serving as possible evidence of a Lifshitz transition.

Circularly Polarized Light on Graphene with Trigonal Warping

In this article, we theoretically investigate the electronic band structure of monolayer graphene in the presence of trigonal warping and photo-induced effects. The total Hamiltonian of the system has been written and the optical absorption of circularly polarized light for the high frequency regime have been modelled by the Haldane interaction. The relation between trigonal warp aspects and optical absorption of circularly polarized light has been overviewed through the model. The main aim of this investigation is to assess the effects of trigonal warp and photo induced-effects which show for the high frequency regime that the edge states can be controlled by tuning the amplitude of the light. Additionally, theoretically analyzed the versatile electronic properties of trigonal warped-graphene under circularly polarized light. We conclude that photo-induced effect which induced circularly polarized light leads to the opening of energy gap between valance and conduction bands while raises electron-hole asymmetry in the system.

Effect of trigonal warping on the Berry curvature and valley/spin Hall effects in monolayer MoS2

Recently, the nonlinear optical processes study on monolayer MoS2 has revealed the existence of a trigonal warping in the band structure which distorts the Fermi surface from a circle at low energies to a triangle at higher energies. Here we theoretically demonstrate that the trigonal warping induce important modifications on the Berry curvature, orbital magnetic moment, and valley Hall conductance in monolayer MoS2. The analytical results show that both Berry curvature and orbital magnetic moment are trigonally warped in contrast to the isotropic case without warping, which is consistent with the numerical calculations. The valley Hall conductance is reduced by the trigonal warping due to the warped Berry curvatures, whereas the spin Hall conductance remains nearly intact. Our findings are useful in explaining the valley and spin Hall data on monolayer MoS2 and other transition metal dichalcogenides.

Theory of second-order excitonic nonlinearities in transition metal dichalcogenides

Single layers of transition metal dichalcogenides exhibit tightly bound excitons with large oscillator strength making them to promising materials for future optoelectronic applications. Despite that the linear response of these ultrathin materials is extensively studied, a profound understanding of the excitonic second-order nonlinear optical response, including trigonal warping and Berry curvature effects, is lacking so far. To bridge this gap, in this paper we report a microscopic model for the second-order nonlinear susceptibility tensor of these materials. Based on a $kp$ description of the underlying many-particle Hamiltonian we are able to address excitonic features as well as trigonal warping effects, spin-valley coupling, and also Berry curvature to the susceptibility tensor. We present explicit calculation of the latter for the exemplary materials ${\mathrm{MoS}}_{2}$ and ${\mathrm{WS}}_{2}$ on silicon substrate and encapsulated in hBN. For instance we predict resonant susceptibility for second-harmonic generation $0.5--1.3\phantom{\rule{4pt}{0ex}}\mathrm{nm}/\mathrm{V}$ and double-resonant susceptibility $10--30\phantom{\rule{4pt}{0ex}}\mathrm{n}\mathrm{m}/\mathrm{V}$ for the difference and sum-frequency generation processes. Our results are in good agreement with available experimental findings in the literature.

Trigonal warping and photo-induced effects on zone boundary phonon in monolayer graphene

We have reported the electronic band structure of monolayer graphene when the combined effects arising from the trigonal warp and highest zone-boundary phonons having A1g symmetry with Haldane interaction which induced photo-irradiation effect. On the basis of our model, we have introduced a diagonalization to solve the associated Fröhlich Hamiltonian. We have examined that, a trigonal warping effect is introduced on the K and K' points, leading to a dynamical band gap in the graphene electronic band spectrum due to the electron-A1g phonon interaction and Haldane mass interaction. Additionally, the bands exhibited an anisotropy at this point. It is also found that, photo-irradiation effect is quite smaller than the trigonal warp effects in the graphene electronic band spectrum. In spite of this, controllability of the photo induced effects by the Haldane mass will have extensive implications in the graphene.

Spin-valley coherent phases of the ν=0 quantum Hall state in bilayer graphene

Bilayer graphene (BLG) offers a rich platform for broken symmetry states stabilized by interactions. In this work we study the phase diagram of BLG in the quantum Hall regime at filling factor $\nu=0$ within the Hartree-Fock approximation. In the simplest non-interacting situation this system has eight (nearly) degenerate Landau levels near the Fermi energy, characterized by spin, valley, and orbital quantum numbers. We incorporate in our study two effects not previously considered: (i) the nonperturbative effect of trigonal warping in the single-particle Hamiltonian, and (ii) short-range SU(4) symmetry-breaking interactions that distinguish the energetics of the orbitals. We find within this model a rich set of phases, including ferromagnetic, layer-polarized, canted antiferromagnetic, Kekul\'e, a "spin-valley entangled" state, and a "broken U(1) $\times$ U(1)" phase. This last state involves independent spontaneous symmetry breaking in the layer and valley degrees of freedom, and has not been previously identified. We present phase diagrams as a function of interlayer bias $D$ and perpendicular magnetic field $B_{\perp}$ for various interaction and Zeeman couplings, and discuss which are likely to be relevant to BLG in recent measurements. Experimental properties of the various phases and transitions among them are also discussed.

Velocity renormalization in graphene: The role of trigonal warping and electron–phonon coupling effects

We investigate the combined effects of trigonal warping and electron–phonon interactions on the renormalization of the Fermi velocity in graphene. We present an analytical solution to the associated Fröhlich Hamiltonian describing the interaction of doubly degenerate-optical phonon modes of graphene with electrons in the presence of trigonal warp within the framework of Lee–Low–Pines theory. On the basis of our model, it is analytically shown that in addition to its renormalization, Fermi velocity exhibits strong anisotropy due to the trigonal warping. It is also found that in the regime where the trigonal warp starts, distortion of energy bands emerges due to electron–phonon coupling, and the bands exhibit strong anisotropy.

Infinitesimal base transformations method for calculating the k.p Hamiltonian of monolayer MoS2

In this work, we present the study of monolayer MoS2 band structure using k.p method based on infinitesimal basis transformations perturbation theory. We also study the effect of spin-orbit coupling on the band structure. We have calculated the low energy k.p Hamiltonian and derived the dispersion relations for the last valance band and the first conduction band at K and –K points of the first Brillouin zone. The results clearly show the electron-hole asymmetry and trigonal warping effects. The results also indicate the spin-splitting of the first conduction band.

Diffraction catastrophes and semiclassical quantum mechanics for Veselago lensing in graphene

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 splits the single Dirac peak into two distinct peaks, leading to valley polarization. We obtain the transmission curves from tight-binding simulations and find that they are in very good agreement with the results of a billiard model that incorporates trigonal warping. Furthermore, the positions of the transmission maxima and the scaling of the peak width are accurately predicted by our semiclassical theory. Our semiclassical analysis can easily be carried over to other Dirac materials, which generally have different Fermi surface distortions.

Optical selection rules for excitonic Rydberg series in the massive Dirac cones of hexagonal two-dimensional materials

We investigate the optical transition selection rules for excitonic Rydberg series formed in massive Dirac cones. The entanglement of the exciton envelop function with the pseudospin texture leads to anomalous selection rules for one-photon generation of excitons, where $d$-orbitals can be excited with the opposite helicity selection rule from the $s$-orbitals in a given valley. The trigonal warping effects in realistic hexagonal lattices further renders more excited states bright, where $p$-orbitals can also be accessed by one-photon excitation with the opposite valley selection rules to the $s$-orbitals. The one-photon generation of exciton in the various states and the intra-excitonic transition between these states are both dictated by the discrete in-plane rotational symmetry of the lattices, and our results show that in hexagonal 2D materials the symmetry allowed transitions are enabled when trigonal warping effects are included in the massive Dirac fermion model. In monolayer transition metal dichalcogenides where excitons can be generated by visible light and intra-excitonic transitions can be induced by infrared light, we give the strength of these optical transitions, estimated using modified hydrogen-like envelope functions combined with the optical transition matrix elements between the Bloch states calculated at various $k$ points.

Topological phase transition from trigonal warping in van der Waals multilayers

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 presence of trigonal warping shrinks the phase space of QAHE and QVHE, leading to the emergence of valley-polarized QAHE with high Chern numbers ranging from $ \mathcal{C}=-7 $ to $ 7 $. These results suggest that the universal trigonal warping may play important role when the Bloch states at momentum away from high-symmetric points are involved.

Two-dimensional transmission through modified Pöschl-Teller potential in bilayer graphene

We study two-dimensional transport of quasiparticles in bilayer graphene through a modified P\oschl-Teller potential, a potential with hyperbolic profile. Both scattering by well and tunneling through a barrier are investigated. At normal incidence the oscillating and evanescent wave states are completely decoupled, so that only one mode participates in transmission. At oblique incidence a combined contribution of the two modes leads to reduction or enhancement of transmission probability. In both cases the chirality can forbid transition to oscillating modes existing within the well or barrier region, which results in a cloak effect. In the barrier tunneling it appears at normal incidence. In the scattering by the well, however, it arises at an oblique incidence as a consequence of coupling between the positive- and negative-energy states, which we interpret as a secondary cloak effect. When a gap is present in the barrier tunneling, because of the coupling, the transmission probability at and near normal incident angles is greatly enhanced. We also examine the effect of trigonal warping in the barrier tunneling and find valley-dependent transmissions and broadening of resonant-transmission regions. As a result, the resonance curve of conductance has a thicker width and a lower peak-to-bottom contrast than the case without trigonal warping. We demonstrate these phenomena with numerical simulations and explain the results using the chirality-dependent transition probability and semiclassical argument. The numerical results are also compared with those of the corresponding rectangular potential. We observe that the two models display large quantitative differences in their two-dimensional transport.

Trigonal warping effect on velocity and transverse confinement length of topologically confined states in bilayer graphene

In bilayer graphene an interface between regions with opposite sign of the interlayer asymmetry gap leads to the spatial confinement of low-energy chiral modes [I. Martin, Y. M. Blanter, and A. F. Morpurgo, Phys. Rev. Lett. 100, 036804 (2008)]. Here we study the influence of trigonal warping on the properties of the interface-guided stats and find that in the vicinity of the band edges the group velocity and transverse confinement length exhibit a nonmonotonic dependence on the crystallographic orientation of the crystal with respect to the interface.

Influence of uniaxial strain on the linear optical spectra in the metallic single-walled carbon nanotubes

The linear absorption spectra of metallic zigzag single-walled carbon nanotubes (SWNTs) have been theoretically studied under the uniaxial strain by using the standard formulas of Orr and Ward. Due to the trigonal warping effect, the linear absorption spectra of M11 and M22 transitions are dominated by two major peaks, which come from the split Mii− and Mii+ excitons with different band index q. As the uniaxial strain is applied, it is interesting to find that the split peaks will overlap at one point of the uniaxial strain, and the splitting is zero at this point. Hence, we can also describe this variation tendency as the size of splitting, which first decreases to zero and then increases with increasing the uniaxial strain, based on which a supplemented tool is offered to detect the deformation degree of a metallic SWNT under uniaxial strain. In addition, the linear absorption spectra of the bands that nearest to the Fermi level have also been calculated, displaying an increase with the increase of the uniaxial strain, which can offer some useful information for the THz applications. The results obtained here are expected to be confirmed by the future experiment.