Warping Term Research Articles

Anisotropic Josephson Diode Effect in the Topological Hybrid Junctions with the Hexagonal Warping

Recently the diode effect in superconductivity became an active area of research. In particular, the three-dimensional topological insulators may be one of the most suitable materials to implement the superconducting diodes. It is common to consider only linear and quadratic terms of the topological insulator Hamiltonian in the low energy expansion. Typically the effect of the hexagonal warping is neglected. However, the hexagonal warping can be very significant in consideration of the transport properties of the TI materials, such as Bi2Se3 or Bi2Te3. In this theoretical work we present the study of the Josephson diode effect based on the topological insulator weak link. We address the question of the hexagonal warping influence on the Josephson diode effect. We argue that the warping term leads to the anisotropy of the Josephson diode effect.

Rashba spin–orbit coupling induced rectified currents in monolayer graphene with exchange field and sublattice potential

We investigate the effect of Rashba spin–orbit coupling (RSOC) on photoconductivities of rectified currents in monolayer graphene with exchange field and sublattice potential. The system shows that the photoconductivities of resonant shift and injection current contributions are nonzero, while the photoconductivities of non-resonant shift current contribution are zero. We find that the RSOC induces a warping term, which leads to the nonzero rectified currents. Moreover, the photoconductivities of resonant injection (shift) current contribution are (not) related to the relaxation rate. The similar behavior can be found in other Dirac materials, and our findings provide a way to tune the nonlinear transport properties of Dirac materials.

An Extensible Framework for Short-Term Holiday Load Forecasting Combining Dynamic Time Warping and LSTM Network

Due to the extreme change of human behavior, the load consumption in public holidays fluctuates more significantly compared to general weekdays resulting in the difficulty of hourly holiday load forecasting. The holiday load forecasting is even challenging because the forecast is practically predicted on the nearest workday which might be more than one days prior to the public holiday. In this paper, we propose a Joint Dynamic time warping and LSTM (JDL) framework, to predict the hourly holiday load consumption on the nearest workday which is at least one day before the incoming holiday. The proposed JDL is a hybrid short-term holiday forecasting framework which combines dynamic time warping (DTW) and long-short term memory (LSTM) network. The DTW predicts the load consumption of the nearest workday and any preceding compensatory holiday(s), if any, based on the similar holiday occurrence pattern. LSTM predicts the highly unpredictable load consumption of the target holiday by univariate and multivariate models. Current results show the proposed JDL outperforms others.

Nonlinear photoncurrent in transition metal dichalcogenide with warping term under illuminating of light* *Project supported by the National Natural Science Foundation of China (Grant No. 11504095) and the Science and Technology Program for Institutions of Higher Learning in Shandong Province, China (Grant No. J18KA224).

We investigate the photoconductivities of injection current and the shift current in transition metal dichalcogenide with warping term in the presence of sublattice potential and spin orbit coupling. The system shows the valley photoconductivities of injection current and the photoconductivities of shift current. It is found that the warping term and the geometric tensor play a critical role in the system, which are responsible for the photoconductivities. Due to the interplay between the sublattice potential and the spin orbit coupling, the photoconductivities can be tuned. Furthermore, the effect of warping term on geometric tensor and the amplitude of the photoconductivities are also discussed.

Gapped Dirac cones and spin texture in thin film topological insulator

The protected surface states of topological insulators (TIs) form gapless Dirac cones corresponding non-degenerate eigenstates with helical spin polarisation. The presence of a warping term deforms the isotropic cone of the most simple model into snowflake Fermi surfaces as in Bi2Se3 and Bi2Te3. Their features have been identified in STM quasiparticle interference (QPI) experiments on isolated surfaces. Here we investigate the QPI spectrum for the TI thin-film geometry with finite tunnelling between the surface states. This leads to a dramatic change of spectrum due to gapping and a change in spin texture that should leave distinct signatures in the QPI pattern. We consider both normal and magnetic exchange scattering from the surface impurities and obtain the scattering t-matrix in Born approximation as well as the general closed solution. We show the expected systematic variation of QPI snowflake intensity features by varying film thickness and study, in particular, the influence on backscattering processes. We predict the variation of the QPI spectrum for Bi2Se3 thin films using the observed gap dependence from ARPES results.

Thermoelectric properties of gapped bilayer graphene

Unlike in conventional semiconductors, both the chemical potential and the band gap in bilayer graphene (BLG) can be tuned via application of external electric field. Among numerous device implications, this property also designates BLG as a candidate for high-performance thermoelectric material. In this theoretical study we have calculated the Seebeck coefficients for abrupt interface separating weakly- and heavily-doped areas in BLG, and for a more realistic rectangular sample of mesoscopic size, contacted by two electrodes. For a given band gap () and temperature (T) the maximal Seebeck coefficient is close to the Goldsmid–Sharp value , the deviations can be approximated by the asymptotic expression , with the electron charge −e, the Boltzmann constant , and . Surprisingly, the effects of trigonal warping term in the BLG low-energy Hamiltonian are clearly visible at few-Kelvin temperatures, for all accessible values of meV. We also show that thermoelectric figure of merit is noticeably enhanced (ZT > 3) when a rigid substrate suppresses out-of-plane vibrations, reducing the contribution from ZA phonons to the thermal conductivity.

Hexagonal warping effect on Majorana zero modes at the ends of superconducting vortex lines in doped strong 3D topological insulators

In a superconducting topological insulator, superconducting vortex line may trap 1-dimensional topological band with Majorana zero modes (MZMs) localized at its ends. In this work, we study the effect of hexagonal warping to the vortex phase transition. We carry out both analytical calculations based on a semiclassical formula and numerical calculations based on full quantum mechanics of Bogliubov de Gennes equation. We find that the hexagonal warping term enlarges the topological region of the vortex line as the chemical potential changes, and leads to the MZMs even in the absence of topological surface states.

Nonlinear response in a noncentrosymmetric topological insulator

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 broken. Here we use a nonlinear response theory based on a generalized Kubo formula to explain the frequency up-conversion in topological materials.

Fundamentally fastest optical processes at the surface of a topological insulator

We predict that a single oscillation of a strong optical pulse can significantly populate the surface conduction band of a three-dimensional topological insulator, Bi2Se3. Both linearly- and circularly-polarized pulses generate chiral textures of interference fringes of population in the surface Brillouin zone. These fringes constitute a self-referenced electron hologram carrying information on the topology of the surface Bloch bands, in particular, on the effect of the warping term of the low-energy Hamiltonian. These electron-interference phenomena are in a sharp contrast to graphene where there are no chiral textures for a linearly-polarized pulse and no interference fringes for circularly-polarized pulse. These predicted reciprocal space electron-population textures can be measured experimentally by time resolved angle resolved photoelectron spectroscopy (TR-ARPES) to gain direct access to non-Abelian Berry curvature at topological insulator surfaces.

Lifshitz transition and thermoelectric properties of bilayer graphene

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 $L>10\,\mu{}$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 quantities characterizing remaining maximum of the Seebeck coefficient still unveil the presence of massless Dirac fermions and make it possible to determine the trigonal warping strength. Behavior of the thermoelectric figure of merit ($ZT$) is also discussed.

Spin injection and helicity control of surface spin photocurrent in a three dimensional topological insulator

A three-dimensional (3D) topological insulator (TI) is a unique quantum phase of matter with exotic physical properties and promising spintronic applications. However, surface spin current in a common 3D TI remains difficult to control and the out-of-plane spin texture is largely unexplored. Here, by means of surface spin photocurrent in Bi2Te3 TI devices driven by circular polarized light, we identify the subtle effect of the spin texture of the topological surface state including the hexagonal warping term on the surface current. By exploring the out-of-plane spin texture, we demonstrate spin injection from GaAs to TI and its significant contribution to the surface current, which can be manipulated by an external magnetic field. These discoveries pave the way to not only intriguing new physics but also enriched spin functionalities by integrating TI with conventional semiconductors, such that spin-enabled optoelectronic devices may be fabricated in such hybrid structures.

Geometric–material analogy for multiscale modelling of twisted plates

It is well known that the macroscopic behaviour of many engineering materials is strongly affected by the role of underlying microstructure. Currently though, mathematical expressions linking behaviour of large scale structures to the geometry of their microscopic structure are largely lacking. In this respect, establishing quantitative links across different material lengthscales may offer new pathways for engineering design. In the present work an analogy between cross sectional geometrical properties, representing macrostructure, and a material length parameter, representing microstructure, is presented. The analogy is established through the study of a thin plate subject to axial loading undergoing finite displacements from two alternative perspectives. First, we consider a thin elastic plate with a pretwist about the loading axis where a warping term is introduced accounting for the out-of-plane deformation of the cross section. The coupled governing differential equations and the corresponding coupled boundary conditions are explicitly derived employing a classical structural mechanics approach utilising an energy variational statement. Secondly, an axially loaded thin flat plate (i.e. with no pretwist) is studied with strain gradient elasticity theory incorporating only one material length parameter representing the microstructure, in addition to the two classical Lamé stiffness constants. The ensuing analogy emerges by comparison of the governing equations of the two formulations which shows a mathematical expression can be identified, which incorporates both geometric and material length variables, that formalises the link between microscale and macroscale. This mathematical expression, which constitutes the kernel of the proposed multiscale approach, admits a twofold interpretation depending on the assumed independent variable. On the one hand, the proposed multiscale modelling approach suggests that a plate with complex global geometry can be substituted by a structurally - equivalent, flat plate with constitutive relations given by a non - local, strain gradient theory. On the other hand, the material length parameter can be interpreted on a physical basis because for the first time it has been identified as a known function of geometrical features of the structure through simple algebraic relationships for various cross sectional profiles.

Shift charge and spin photocurrents in Dirac surface states of topological insulator

The generation of photocurrent in condensed matter is of main interest for photovoltaic and optoelectronic applications. Shift current, a nonlinear photoresponse, has attracted recent intensive attention as a dominant player of bulk photovoltaic effect in ferroelectric materials. In three dimensional topological insulators $\text{Bi}_2\text{X}_3$ (X: Te, Se), we find that Dirac surface states with a hexagonal warping term carry shift current by linearly polarized light. In addition, shift spin-current is introduced with the time-reversal symmetry breaking perturbation. The estimate for the magnitudes of the shift charge- and spin-currents are 0.13$I_0$ and 0.21$I_0$(nA/m) with the intensity of light $I_0$ measured in $(\text{W}/\text{m}^2)$, respectively, which can offer a useful method to generate these currents efficiently.

Effects of a magnetic Fe monolayer on the structural and surface electronic properties ofSb2Te3

In this paper, we investigate the properties of Fe monolayers deposited on the (111) surface of the topological insulator $\mathrm{S}{\mathrm{b}}_{2}\mathrm{T}{\mathrm{e}}_{3}$ by density functional theory simulations. We consider high-coverage monolayers and assume ferromagnetic configurations. We show that upon relaxation, the Fe atoms partly penetrate into the surface and that their magnetic moments are reduced due to the chemical interaction with the Te and Sb atoms of the two topmost layers. We compute the magnetic anisotropy energies and show that the easy axis is in-plane. We investigate hexagonal warping effects and find that the clean $\mathrm{S}{\mathrm{b}}_{2}\mathrm{T}{\mathrm{e}}_{3}$(111) displays a large warping term. In spite of this, the surface-state gap for in-plane magnetization is below 1 meV. For very high coverages corresponding to three Fe atoms in the unit cell of $\mathrm{S}{\mathrm{b}}_{2}\mathrm{T}{\mathrm{e}}_{3}$(111), no surface Dirac cones are observed except in the case of a metastable model consisting of a hexagonal Fe monolayer lying on top of the substrate, which also exhibits an out-of-plane easy axis. Finally, we discuss the relevance of our paper to recent experiments about magnetic impurities deposited on topological insulators.

Hybridization and Field Driven Phase Transitions in Hexagonally Warped Topological Insulators

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 [Formula: see text] symmetry breaking character. We then note that while the introduction of an electric field does not allow for this QPT since it does not break [Formula: see text] symmetry, it can be used in conjunction with a tunneling element to reach a phase transition efficiently. The corresponding critical point is then nontrivially dependent on the electric field, which is pointed out here. Then, we demonstrate that including a hexagonal warping term leads to an immediate [Formula: see text] symmetry violating QPT.

Effective Hamiltonian for surface states of Bi2Te3 nanocylinders with hexagonal warping

The three-dimensional topological insulator differs from other topological insulators in the 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 cylinder incorporating the corresponding hexagonal warping terms in a cylindrical geometry. We show that at the energy range where the surface states dominate, the effective Hamiltonian adequately reproduces the dispersion relation obtained from a full four-band Hamiltonian which describes both the bulk and surface states. As an example application of our effective Hamiltonian, we study the transmission between two collinear cylinders magnetized in different directions perpendicular to their axes. We show that the hexagonal warping term results in a transmission profile between the cylinders which may be of utility in a multiple state magnetic memory bit.

Effective Hamiltonian for surface states of topological insulator thin films with hexagonal warping

The effective Hamiltonian of the surface states on semi-infinite slabs of the topological insulators (TI) Bi2Te3 and Bi2Se3 require the addition of a cubic momentum hexagonal warping term on top of the usual Dirac fermion Hamiltonian in order to reproduce the experimentally measured constant energy contours at intermediate values of Fermi energy. In this work, we derive the effective Hamiltonian for the surface states of a Bi2Se3 thin film incorporating the corresponding hexagonal warping terms. We then calculate the dispersion relation of the effective Hamiltonian and show that the hexagonal warping leads distorts the equal energy contours from the circular cross sections of the Dirac cones.

Effect of perturbative hexagonal warping on quantum capacitance in ultra-thin topological insulators

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 shows it to be 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 that we recover their established forms along with small corrections which preserve the object of treating hexagonal warping perturbatively. In our approach, the established Hall conductivity expression develops several minute correction terms and thus its behavior remains largely unaffected due to warping. The zero-temperature quantum capacitance exhibits Shubnikov–de Haas oscillations with reduced frequencies, with a lowered average capacitance with increased warping of the Fermi surface, while maintaining the usual amplitudes.

The evaluation of non-topological components in Berry phase and momentum relaxation time in a gapped 3D topological insulator

The zero gap surface states of a 3D-topological insulator host Dirac fermions with spin locked to the momentum. The gap-less Dirac fermions exhibit electronic behaviour different from those predicted in conventional materials. While calculations based on a simple linear dispersion can account for observed experimental patterns, a more accurate description of the physics of these systems and a better agreement between experimental data theoretical results can be obtained by including higher order terms in the Hamiltonian. In this work, in presence of a time reversal symmetry breaking external magnetic field and higher order warping term, alteration to the topologically ordained Berry phase of , momentum relaxation time, and the magneto-conductivity tensors is established. The relation between scattering times and the deviations to topological Berry phase of is also emphasized.

Superconductivity in the surface states of a Bi2X3 topological insulator: effects of a realistic model

Superconductivity in the topological surface states is essential to both the surface spectrum of bulk superconducting state and the proximity-induced superconductivity of (X is Se or Te) topological insulators. While previous theories were mostly based on simplified models for the bulk topological insulator and the surface states, the accumulating experiments stimulate us to make an analysis using realistic model for the normal state electronic structures, incorporating terms responsible for particle–hole asymmetry and hexagonal warping. An effective low-energy model for the topological surface states is derived first. Then we identify all the bulk time-reversal-invariant superconducting pairings in the topological insulator that can open a gap in the topological surface states. Many more pairings are found to be able to gap the topological surface states as compared to conclusions based on simplified models. The number of proximity-induced pairing channels in the topological surface states increases by one as a result of the hexagonal warping term, but is not changed by the particle–hole asymmetry term.