We explore the scattering of Dirac electrons in a double-gated topological insulator in the presence of magnetic proximity effects and warped surface states. It is found that a magnetic field can shift the Dirac cone in momentum space and deform the constant-energy contour, or opens up a band gap at the Dirac point, depending on the magnetization orientation. The double gate voltage induces quantum wells and/or quantum barriers on the surface of topological insulators, generating surface resonant tunnelling states. It is found that the hexagonal warping effect can increase the electronic transport at high energies when the constant-energy contour exhibits a snowflake shape. The energy-dependent conductances in the parallel and antiparallel magnetic configurations exhibit out-of-phase oscillations due to the quantum interference of propagating waves in the region between the two magnetized segments. Although the conductance spectrum of the double-well structure is higher than that of the double-barrier structure, the magnetoresistance ratio versus the separation distance between the two magnetized barriers exhibits pronounced oscillations due to the resonant tunnelling states. We show that the surface state transport can be controlled by the exchange field and gate voltage without breaking time reversal symmetry, suggesting that the double gated and magnetized topological insulators can be utilized to achieve a large magnetoresistance ratio with a tunable sign.
We investigate the influence of the hexagonal warping on the transport properties of the topological insulators. We study the charge conductivity within Kubo formalism in the first Born approximation using low energy expansion of the Hamiltonian near the Dirac point. The effects of disorder, magnetic field and chemical potential value are analyzed in details. We found that the presence of the hexagonal warping effects significantly the conductivity of the topological insulator. In particular, it gives rise to the growth of the longitudinal conductivity with the increase of the disorder and ani…
We investigate the charge conductivity and current-induced spin polarization on the surface state of a three-dimensional topological insulator by including the hexagonal warping effect of Fermi surface both in classical and quantum diffusion regimes. We present general expressions of conductivity and spin polarization, which are reduced to simple forms for usual scattering potential. Due to the hexagonal warping, the conductivity and spin polarization show an additional quadratic carrier density dependence both for Boltzmann contribution and quantum correction. In the presence of warping term,…
We explore a combined effect of hexagonal warping and of finite effective mass on both the tunneling density of electronic states (TDOS) and structure of Landau levels (LLs) of 3D topological insulators. We find the increasing warping to transform the square-root van Hove singularity into a logarithmic one. For moderate warping an additional logarithmic singularity and a jump in the TDOS appear. This phenomenon is experimentally verified by direct measurements of the local TDOS in Bi$_2$Te$_3$. By combining the perturbation theory and the WKB approximation we calculate the LLs in the presence …
The three-dimensional topological insulator \ce{Bi2Te3} differs from other topological insulators in the \ce{Bi2Se3} family in that the effective Hamiltonian of its surface states on a flat semi-infinite slab requires the addition of a cubic momentum hexagonal warping term in order to reproduce the experimentally measured constant energy contours. In this work, we derive the appropriate effective Hamiltonian for the surface states of a \ce{Bi2Te3} \textit{cylinder} incorporating the corresponding hexagonal warping terms in a cylindrical geometry. We show that at the energy range where the surf…
Ultra-thin 3D topological insulators provide a stage to study the surface physics of such materials by minimizing the bulk contribution. Further, the experimentally verified snowflake like structure of the Fermi surface leads to a hexagonal warping term, and this discourse examines it as a perturbation in the presence of a magnetic field. We find that there are corrections to both energy dispersion and eigenstates which in turn alter the density of states in the presence of a magnetic field. Both the quantum capacitance and the Hall coefficient are evaluated analytically and it is shown here t…
We study superconducting properties of the bulk states of a doped topological insulator. We obtain that the hexagonal warping stabilizes the nematic spin-triplet superconducting phase with $E_u$ pairing and the direction of the nematic order parameter which opens the full gap is the ground state. This order parameter exhibits non-BCS behavior. The ratio of the order parameter to the critical temperature of $Δ(0)/T_c$ differs from the BCS ratio. It depends on the chemical potential and the value of the hexagonal warping. We discuss the relevance of the obtained results for the explanation of th…
We propose theoretically a new effect, i.e. nonlinear planar Nernst effect (NPNE), in nonmagnetic topological insulator (TI) Bi2Te3 in the presence of an in-plane magnetic field. We find that the Nernst current scales quadratically with temperature gradient but linearly with magnetic field and exhibits a cosine dependence of the orientation of the magnetic field with respect to the direction of the temperature gradient. The NPNE has a quantum origin arising from the conversion of a nonlinear transverse spin current to a charge current due to a joint result of hexagonal warping effect, spin-mom…
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…
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…
The topological protected electronic states on the surface of a topological insulator can progressively change their Fermi cross-section from circular to a snowflake shape as the chemical potential is increased above the Dirac point because of an hexagonal warping term in the Hamiltonian. Another effect of warping is to change the spin texture which exists when a finite gap is included by magnetic doping, although the in-plane spin component remains locked perpendicular to momentum. It also changes the orbital magnetic moment, the matrix element for optical absorption and the circular dichrois…
Zigzag nanoribbons of monolayer graphene-like two-dimensional materials host spontaneous edge magnetism at the zigzag terminations, whose configuration controls the band gap. In this article, the edge magnetism of zigzag nanoribbons of bilayer hexagonal crystals are studied. The specific models of bilayer graphene and bilayer silicene are both studied. As the gated voltage increases, the total energy level, magnetic structure and band structures of the ground state and the first quasi-stable excited state are tuned. For certain region of the gated voltage, the band gaps of spin up and down are…
We investigate the spin conductivity of topological insulators taking into account both the surface and quasi-two-dimensional bulk states. We apply a low-energy expansion of the Hamiltonian up to the third order in momentum and take into account the vertex corrections arising due to the short range disorder. Hexagonal warping gives rise to the additional anisotropic components in the spin conductivity tensor. Typically, isotropic part of the spin conductivity is larger than anisotropic one. The helical regime for the bulk states, in which the electrons in the Fermi level have the same projecti…
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 …
By solving the Boltzmann transport equation we investigate theoretically the general form of oscillations in the resistivity caused by varying the direction of an applied magnetic field for the case of quasi-two dimensional systems on hexagonal lattices. The presence of the angular magnetoresistance oscillations can be used to map out the topology of the Fermi surface and we study how this effect varies as a function of the degree of interplane warping as well as a function of the degree of isotropic scattering. We find that the angular dependent effect due to in-plane rotation follows the sym…
We study the effect of the Fermi surface anisotropy on the odd-frequency spin-triplet pairing component of the induced pair potential. We consider a superconductor/ ferromagnetic insulator (S/FI) hybrid structure formed on the 3D topological insulator (TI) surface. In this case three ingredients insure the possibility of the odd-frequency pairing: 1) the topological surface states, 2) the induced pair potential, and 3) the magnetic moment of a nearby ferromagnetic insulator. We take into account the strong anisotropy of the Dirac cone in topological insulators when the chemical potential lies …
In this paper we discuss the role of material parameters and external field effects on a thin film topological insulator(TI) in the context of quantum phase transition(QPT). First, we consider an in-plane tilted magnetic field and determine the band structure of the surface states as a function of the tilt angle. We show that the presence of either a hybridization term or hexagonal warping or a combination of both leads to a semi-metal to insulator phase transition which is facilitated by their ${\cal PT}$ symmetry breaking character. We then note that while the introduction of an electric fie…
We study the quantum capacitance in a topological insulator thin film system magnetized in the in-plane direction in the presence of an out-of-plane magnetic field and hexagonal warping. To first order, the modification in quantum capacitance due to hexagonal warping compared to the clean case, where both the in-plane magnetization and hexagonal warping are absent, is always negative, and increases in magnitude monotonically with the energy difference from the charge neutrality point. In contrast, the change in the quantum capacitance due to in-plane magnetization oscillates with the energy in…
Silicene is a monolayer of silicon atoms forming a two-dimensional honeycomb lattice. We investigate the topological properties of a silicene superstructure generated by an external periodic potential. The superstructure is a quantum spin-Hall (QSH) insulator if it is topologically connected to silicene. It is remarkable that two inequivalent K and K' points in the silicene Brillouin zone are identified in certain superstructures. In such a case two Dirac cones coexist at the same Dirac point in the momentum space and they are hexagonally warped by the Coulomb interaction. We carry out a numer…
Hexagonal warping (HW) in three-dimensional topological insulators is, by now, well-known. We show that non-Hermitian (NH) loss/gain can generate an exceptional HW effect in double Weyl-semimetals (DWSM). This unique feature of DWSMs has distinctive effects on Fermi surface topology. Importantly, in the presence of such a $k^3$ spin orbit coupling mimicking term, the symmetry associated with the DWSMs is changed, leading to four exceptional points, among which two are degenerate. Introducing a driving field removes this degeneracy. The combined action of the NH warping and driving parameters l…
We determine the nature of the magnetic order on the surface of a topological insulator (TI) which develops due to hexagonal warping and the resulting Fermi surface (FS) nesting in the presence of a repulsive Hubbard interaction. For this purpose we investigate the spin susceptibility and derive a Landau theory to compare the different accessible phases. For a nearly hexagonal FS and sufficiently strong interaction the magnetic ground state is formed by a skyrmion lattice, i.e., by a superposition of three helical spin density waves which preserves C$_3$ symmetry. The magnetic ground state is …
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 …
We performed low temperature scanning tunneling microscopy (STM) and spectroscopy (STS) studies on the electronic properties of (R3xR3)R30° phase of silicene on Ag(111) surface. We found the existence of Dirac Fermion chirality through the observation of -1.5 and -1.0 power law decay of quasiparticle interference (QPI) patterns. Moreover, in contrast to the trigonal warping of Dirac cone in graphene, we found that the Dirac cone of silicene is hexagonally warped, which is further confirmed by density functional calculations and explained by the unique superstructure of silicene. Our results de…
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…
Top-gated, few-layer graphene field-effect transistors (FETs) fabricated on thermally-decomposed semi-insulating 4H-SiC substrates are demonstrated. Physical vapor deposited SiO2 is used as the gate dielectric. A two-dimensional hexagonal arrangement of carbon atoms with the correct lattice vectors, observed by high-resolution scanning tunneling microscopy, confirms the formation of multiple graphene layers on top of the SiC substrates. The observation of n-type and p-type transition further verifies Dirac Fermions unique transport properties in graphene layers. The measured electron and hole …
We have performed spin- and angle-resolved photoemission spectroscopy of the topological insulator Pb(Bi,Sb)2Te4 (Pb124) and observed significant out-of-plane spin polarization on the hexagonally warped Dirac-cone surface state. To put this into context, we carried out quantitative analysis of the warping strengths for various topological insulators (Pb124, Bi2Te3, Bi2Se3, and TlBiSe2) and elucidated that the out-of-plane spin polarization Pz is systematically correlated with the warping strength. However, the magnitude of Pz is found to be only half of that expected from the kp theory when th…
We analyze the strong hexagonal warping of the Dirac cone of Bi$_2$Te$_3$ by angle-resolved photoemission. Along $\overlineΓ$$\overline{\rm M}$, the dispersion deviates from a linear behavior meaning that the Dirac cone is warped outwards and not inwards. We show that this introduces an anisotropy in the lifetime broadening of the topological surface state which is larger along $\overlineΓ$$\overline{\rm K}$. The result is not consistent with nesting. Based on the theoretically predicted behavior of the ground-state spin texture of a strongly warped Dirac cone, we propose spin-dependent scatte…
The effects of Coulomb interactions on the electronic properties of bilayer graphene nanoribbons (BGNs) covered by a gate electrode are studied theoretically. The electron density distribution and the potential profile are calculated self-consistently within the Hartree approximation. A comparison to their single-particle counterparts reveals the effects of interactions and screening. Due to the finite width of the nanoribbon in combination with electronic repulsion, the gate-induced electrons tend to accumulate along the BGN edges where the potential assumes a sharp triangular shape. This has…
We demonstrate dual-gated $p$-type field-effect transistors (FETs) based on few-layer tungsten diselenide (WSe$_2$) using high work-function platinum source/drain contacts, and a hexagonal boron nitride top-gate dielectric. A device topology with contacts underneath the WSe$_2$ results in $p$-FETs with $I_{ON}$/$I_{OFF}$ ratios exceeding 10$^7$, and contacts that remain Ohmic down to cryogenic temperatures. The output characteristics show current saturation and gate tunable negative differential resistance. The devices show intrinsic hole mobilities around 140 cm$^2$/Vs at room temperature, an…
We consider the transport properties of topological insulators surface states in the presence of uncorrelated point-like disorder, both in the classical and quantum regimes. The transport properties of those two-dimensional surface states depend strongly on the amplitude of the hexagonal warping of their Fermi surface. It is shown that a perturbative analysis of the warping fails to describe the transport in experimentally available topological insulators, such as Bi2Se3 and Bi2Te3. Hence we develop a fully non-perturbative description of these effects. In particular, we find that the dependen…
Electrolyte gating is a powerful technique for accumulating large carrier densities in surface two-dimensional electron systems (2DES). Yet this approach suffers from significant sources of disorder: electrochemical reactions can damage or alter the surface of interest, and the ions of the electrolyte and various dissolved contaminants sit Angstroms from the 2DES. Accordingly, electrolyte gating is well-suited to studies of superconductivity and other phenomena robust to disorder, but of limited use when reactions or disorder must be avoided. Here we demonstrate that these limitations can be o…
We investigate the lattice and electronic structures of the bulk and surface of the prototypical layered topological insulators Bi$_2$Se$_3$ and Bi$_2$Te$_3$ using ab initio density functional methods, and systematically compare the results of different methods of including van der Waals (vdW) interactions. We show that the methods utilizing semi-empirical energy corrections yield accurate descriptions of these materials, with the most precise results obtained by properly accounting for the long-range tail of the vdW interactions. The bulk lattice constants, distances between quintuple layers …
Diffusion currents are theoretically examined in two-dimensional Dirac materials, such as those of the transition metal dichalcogenides (TMD) family. The transversal effects are analogues of the valley Hall (VHE) and photogalvanic (PGE) transport phenomena in case when the electron driving force is not an electric field but a gradient of electron density distribution in the sample. The latter can be created by a finite-sized laser spot or by the injection of electrons from other materials. We develop the theory of diffusive VHE effect assuming the anisotropic electron-short-range-impurity skew…
The surface of a three-dimensional topological insulator (TI) hosts two-dimensional massless Dirac fermions (DFs), the gapless and spin-helical nature of which yields many exotic phenomena, such as the immunity of topological surface states (TSS) to back-scattering. This leads to their high transmission through surface defects or potential barriers. Quantum corrals, previously elaborated on metal surfaces, can act as nanometer-sized electronic resonators to trap Schrödinger electrons by quantum confinement. It is thus intriguing, concerning their peculiar nature, to put the Dirac electrons of …
The hexagonal warping effect on transport properties and Goos-Hänchen (GH) lateral shift of electrons on the surface of a topological insulator with a potential barrier is investigated theoretically. Due to the warped Fermi surface for incident electron beams, we can expect two propagating transmitted beams corresponding to the occurrence of double refraction. The transmitted beams have spin orientations locked to their momenta so one of the spin directions rotates compared to the incident spin direction. Based on a low-energy Hamiltonian near the Dirac point and considering Gaussian beams, we…
Silicene have been proposed to host multiple topological phases depending on the environmental parameters such as gated voltage and proximity induced exchange field. Two typical topological phase are quantum spin Hall phase and quantum anomalous Hall phase. We found that in the presence of antiferromagnetic exchange field and (or) gated voltage, the bilayer silicene host a topological phase that the Chern numbers of the two spin are different in both sign and magnitude, .i.e the anomalous Hall effects of the two spins are imbalanced. The topological phase is generated by triple band inversion …
We investigate the spin-dependent thermoelectric effect of graphene flakes with magnetic edges in the ballistic regime. Employing static, respectively, dynamic mean-field theory we first show that magnetism appears at the zigzag edges for a window of Coulomb interactions that increases significantly with increasing flake size. We then use the Landauer formalism in the framework of the non-equilibrium Green's function method to calculate the spin and charge currents in magnetic hexagonal graphene flakes by varying the temperature of the junction for different flake sizes. While in non-magnetic …
We study the photocurrent response of topological insulator surface states to circularly polarized light for arbitrary oblique incidence. We describe the surface states within a Dirac model, including several perturbations such as hexagonal warping, nonlinear corrections to the mode velocity, and applied magnetic fields. We find that the photogalvanic current is strongly suppressed for the usual orbital coupling, prompting us to include the weaker Zeeman coupling. We find that the helicity-independent photocurrent dominates over the helicity-dependent contributions.
Irradiation of the strong light on the material leads to numerous non-linear effects that are essential to understand the physics of excited states of the system and for optoelectronics. Here, we study the non-linear thermoelectric effect due to the electric and thermal fields applied on a non-centrosymmetric system. The phenomenon arises on the Fermi surface with the transitions of electrons from valence to conduction bands. We derive the formlism to investigate these effects and find that the non-linearity in these effects namely non-linear Seebeck and non-linear Peltier effects depends on t…
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…
A gated Hall-bar device is made from an epitaxially grown, free-standing InSb nanosheet on a hexagonal boron nitride (hBN) dielectric/graphite gate structure and the electron transport properties in the InSb nanosheet are studied by gate-transfer characteristic and magnetotransport measurements at low temperatures. The measurements show that the carriers in the InSb nanosheet are of electrons and the carrier density in the nanosheet can be highly efficiently tuned by the graphite gate. The mobility of the electrons in the InSb nanosheet is extracted from low-field magneotransport measurements …
We study the effect of electron-electron interaction on the surface resistivity of three-dimensional (3D) topological insulators. In the absence of umklapp scattering, the existence of the Fermi-liquid ($T^2$) term in resistivity of a two-dimensional (2D) metal depends on the Fermi surface geometry, in particular, on whether it is convex or concave. On doping, the Fermi surface of 2D metallic surface states in 3D topological insulators of the Bi$_2$Te$_3$ family changes its shape from convex to concave due to hexagonal warping, while still being too small to allow for umklapp scattering. We sh…
We report on quantum capacitance measurements of high quality, graphite- and hexagonal boron nitride encapsulated Bernal stacked trilayer graphene devices. At zero applied magnetic field, we observe a number of electron density- and electrical displacement-tuned features in the electronic compressibility associated with changes in Fermi surface topology. At high displacement field and low density, strong trigonal warping gives rise to emergent Dirac gullies centered near the corners of the hexagonal Brillouin and related by three fold rotation symmetry. At low magnetic fields of $B=1.25$~T, th…
We report the fabrication and measurement of top gated epitaxial graphene p-n junctions where exfoliated hexagonal boron nitride (h-BN) is used as the gate dielectric. The four-terminal longitudinal resistance across a single junction is well quantized at the von Klitzing constant R_K with a relative uncertainty of 10-7. After the exploration of numerous parameter spaces, we summarize the conditions upon which these devices could function as potential resistance standards. Furthermore, we offer designs of programmable electrical resistance standards over six orders of magnitude by using extern…
Electrostatic control of Li+ density and transport rate in double-gated van der Waals devices
University of Manchester（曼彻斯特大学）；National Graphene Institute, University of Manchester（国家石墨烯研究所，曼彻斯特大学）；Universidade Federal do Ceará（塞阿拉联邦大学）；Nanjing University of Information Science and Technology（南京信息工程大学）；Universiteit Antwerpen（安特卫普大学）
Nonlinear phenomena are inherent in most systems in nature. Second or higher-order harmonic generations, three-wave and four-wave mixing are typical phenomena in nonlinear optics. To obtain a nonzero signal for second-harmonic generation in the long-wavelength limit ($q\rightarrow0$), the breaking of inversion symmetry is required. In topological materials, a hexagonal warping term which breaks the rotation symmetry of the Fermi surface is observed by angular-resolved photo-emission spectroscopy (ARPES). If a gap opens (e.g., by doping with magnetic impurities) the inversion symmetry will be b…
The electron scattering from periodic line defects on the surface of topological insulators with hexagonal warping effect is investigated theoretically by means of a transfer matrix method. The influence of surface line defects, acting as structural ripples on propagation of electrons are studied in two perpendicular directions due to the asymmetry of warped energy contour under momentum exchange. The transmission profiles and the details of resonant peaks which vary with the number of defects and the strength of their potentials are strongly dependent on the direction in which the line defect…
Understanding strongly interacting electrons enables the design of materials, nanostructures and devices. Developing this understanding relies on the ability to tune and control electron-electron interactions by, e.g., confining electrons to atomically thin layers of 2D crystals with reduced screening. The interplay of strong interactions on a hexagonal lattice with two nonequivalent valleys, topological moments, and the Ising-like spin-orbit interaction gives rise to a variety of phases of matter corresponding to valley and spin polarized broken symmetry states. In this work we describe a hig…
The energy-momentum relationship of electrons on the surface of an ideal "Hydrogen-Atom" Topological Insulator forms a cone - a Dirac cone, which, when warped and distorted (no longer described by the Dirac equation), can lead to unusual phenomena such as enhanced electronic interference around defects (observed in STM) and a magnetically ordered broken symmetry surface with lost topological protection. A detailed spin-texture and hexagonal warping maps on Bi2Te3 are presented in this context as a Viewpoint on L.Fu, Phys. Rev. Lett. \textbf{103}, 266801 (2009).
