Surface charge conductivity of a topological insulator in a magnetic field: The effect of hexagonal warping

Taskin A.A., Legg H.F., Yang F., Sasaki S., Kanai Y., Matsumoto K., Rosch A., Ando Y.

A prominent feature of topological insulators (TIs) is the surface states comprising of spin-nondegenerate massless Dirac fermions. Recent technical advances have made it possible to address the surface transport properties of TI thin films by tuning the Fermi levels of both top and bottom surfaces. Here we report our discovery of a novel planar Hall effect (PHE) from the TI surface, which results from a hitherto-unknown resistivity anisotropy induced by an in-plane magnetic field. This effect is observed in dual-gated devices of bulk-insulating Bi2−x Sb x Te3 thin films, where the field-induced anisotropy presents a strong dependence on the gate voltage with a characteristic two-peak structure near the Dirac point. The origin of PHE is the peculiar time-reversal-breaking effect of an in-plane magnetic field, which anisotropically lifts the protection of surface Dirac fermions from backscattering. The observed PHE provides a useful tool to analyze and manipulate the topological protection of the TI surface. Topological surface states can lose their protection in many ways but the subtle mechanisms remain far from well understood. Here, Taskin et al. report a novel planar Hall effect in dual-gated Bi2−x Sb x Te3 thin films, originating from anisotropic lifting of time reversal symmetry protection by an in-plane magnetic field.

Chiba T., Takahashi S., Bauer G.E.

We theoretically study the magnetoresistance (MR) of two-dimensional massless Dirac electrons as found on the surface of three-dimensional topological insulators (TIs) that are capped by a ferromagnetic insulator (FI). We calculate charge and spin transport by Kubo and Boltzmann theories, taking into account the ladder-vertex correction and the in-scattering due to normal and magnetic disorder. The induced exchange splitting is found to generate an electric conductivity that depends on the magnetization orientation, but its form is very different from both the anisotropic and the spin Hall MR. The in-plane MR vanishes identically for nonmagnetic disorder, while out-of-plane magnetizations cause a large MR ratio. On the other hand, we do find an in-plane MR and planar Hall effect in the presence of magnetic disorder aligned with the FI magnetization. Our results may help us understand recent transport measurements on TI|FI systems.

Qi S., Qiao Z., Deng X., Cubuk E.D., Chen H., Zhu W., Kaxiras E., Zhang S. ., Xu X., Zhang Z.

The quantum anomalous Hall effect (QAHE) is a fundamental quantum transport phenomenon that manifests as a quantized transverse conductance in response to a longitudinally applied electric field in the absence of an external magnetic field, and it promises to have immense application potential in future dissipationless quantum electronics. Here, we present a novel kinetic pathway to realize the QAHE at high temperatures by n-p codoping of three-dimensional topological insulators. We provide a proof-of-principle numerical demonstration of this approach using vanadium-iodine (V-I) codoped Sb_{2}Te_{3} and demonstrate that, strikingly, even at low concentrations of ∼2%  V and ∼1% I, the system exhibits a quantized Hall conductance, the telltale hallmark of QAHE, at temperatures of at least ∼50  K, which is 3 orders of magnitude higher than the typical temperatures at which it has been realized to date. The underlying physical factor enabling this dramatic improvement is tied to the largely preserved intrinsic band gap of the host system upon compensated n-p codoping. The proposed approach is conceptually general and may shed new light in experimental realization of high-temperature QAHE.

Pan Y., Nikitin A.M., Araizi G.K., Huang Y.K., Matsushita Y., Naka T., de Visser A.

AbstractRecently it was demonstrated that Sr intercalation provides a new route to induce superconductivity in the topological insulator Bi2Se3. Topological superconductors are predicted to be unconventional with an odd-parity pairing symmetry. An adequate probe to test for unconventional superconductivity is the upper critical field,Bc2. For a standard BCS layered superconductorBc2shows an anisotropy when the magnetic field is applied parallel and perpendicular to the layers, but is isotropic when the field is rotated in the plane of the layers. Here we report measurements of the upper critical field of superconducting SrxBi2Se3crystals (Tc = 3.0 K). Surprisingly, field-angle dependent magnetotransport measurements reveal a large anisotropy ofBc2when the magnet field is rotated in the basal plane. The large two-fold anisotropy, while six-fold is anticipated, cannot be explained with the Ginzburg-Landau anisotropic effective mass model or flux flow induced by the Lorentz force. The rotational symmetry breaking ofBc2indicates unconventional superconductivity with odd-parity spin-triplet Cooper pairs (Δ4-pairing) recently proposed for rhombohedral topological superconductors, or might have a structural nature, such as self-organized stripe ordering of Sr atoms.

Xu Y., Miotkowski I., Chen Y.P.

Topological insulators are a novel class of quantum matter with a gapped insulating bulk, yet gapless spin-helical Dirac fermion conducting surface states. Here, we report local and non-local electrical and magneto transport measurements in dual-gated BiSbTeSe2 thin film topological insulator devices, with conduction dominated by the spatially separated top and bottom surfaces, each hosting a single species of Dirac fermions with independent gate control over the carrier type and density. We observe many intriguing quantum transport phenomena in such a fully tunable two-species topological Dirac gas, including a zero-magnetic-field minimum conductivity close to twice the conductance quantum at the double Dirac point, a series of ambipolar two-component half-integer Dirac quantum Hall states and an electron-hole total filling factor zero state (with a zero-Hall plateau), exhibiting dissipationless (chiral) and dissipative (non-chiral) edge conduction, respectively. Such a system paves the way to explore rich physics, ranging from topological magnetoelectric effects to exciton condensation. Novel physics of topological aspects are obscured due to lack of effective way to manipulate topological particles. Here, Xu et al. demonstrate independent control of Dirac fermions on top and bottom surfaces of BiSbTeSe2flakes by dual-gating, which suggests a way to manipulate exotic particles.

Liu C., Zhang S., Qi X.

The quantum anomalous Hall effect is defined as a quantized Hall effect realized in a system without an external magnetic field. The quantum anomalous Hall effect is a novel manifestation of topological structure in many-electron systems and may have potential applications in future electronic devices. In recent years, the quantum anomalous Hall effect was proposed theoretically and realized experimentally. In this review article, we provide a systematic overview of the theoretical and experimental developments in this field.

Peng X., Yang Y., Singh R.R., Savrasov S.Y., Yu D.

To date, spin generation in three-dimensional topological insulators is primarily modelled as a single-surface phenomenon, attributed to the momentum-spin locking on each individual surface. In this article, we propose a mechanism of spin generation where the role of the insulating yet topologically non-trivial bulk becomes explicit: an external electric field creates a transverse pure spin current through the bulk of a three-dimensional topological insulator, which transports spins between the top and bottom surfaces. Under sufficiently high surface disorder, the spin relaxation time can be extended via the Dyakonov–Perel mechanism. Consequently, both the spin generation efficiency and surface conductivity are largely enhanced. Numerical simulation confirms that this spin generation mechanism originates from the unique topological connection of the top and bottom surfaces and is absent in other two-dimensional systems such as graphene, even though they possess a similar Dirac cone-type dispersion. Future spintronic devices may exploit topological insulators, bulk-insulating materials possessing conductive surface states with orthogonally-locked electronic spin and momentum. Here, the authors propose a mechanism by which bulk spin currents drive surface spin accumulation in such a material.

Repin E.V., Burmistrov I.S.

We explore a combined effect of hexagonal warping and a finite effective mass on both the tunneling density of electronic surface states and the structure of Landau levels 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 tunneling density of surface states appear. By combining the perturbation theory and the WKB approximation, we calculate the Landau levels in the presence of hexagonal warping. We predict that due to the degeneracy removal, the evolution of Landau levels in the magnetic field is drastically modified.

Wang J., Lian B., Zhang S.

The search for topologically non-trivial states of matter has become an important goal for condensed matter physics. Here, we give a theoretical introduction to the quantum anomalous Hall (QAH) effect based on magnetic topological insulators in two-dimensions (2D) and three-dimensions (3D). In 2D topological insulators, magnetic order breaks the symmetry between the counter-propagating helical edge states, and as a result, the quantum spin Hall effect can evolve into the QAH effect. In 3D, magnetic order opens up a gap for the topological surface states, and chiral edge state has been predicted to exist on the magnetic domain walls. We present the phase diagram in thin films of a magnetic topological insulator and review the basic mechanism of ferromagnetic order in magnetically doped topological insulators. We also review the recent experimental observation of the QAH effect. We discuss more recent theoretical work on the coexistence of the helical and chiral edge states, multi-channel chiral edge states, the theory of the plateau transition, and the thickness dependence in the QAH effect.

Kandala A., Richardella A., Kempinger S., Liu C., Samarth N.

When a three-dimensional ferromagnetic topological insulator thin film is magnetized out-of-plane, conduction ideally occurs through dissipationless, one-dimensional (1D) chiral states that are characterized by a quantized, zero-field Hall conductance. The recent realization of this phenomenon, the quantum anomalous Hall effect, provides a conceptually new platform for studies of 1D transport, distinct from the traditionally studied quantum Hall effects that arise from Landau level formation. An important question arises in this context: how do these 1D edge states evolve as the magnetization is changed from out-of-plane to in-plane? We examine this question by studying the field-tilt-driven crossover from predominantly edge-state transport to diffusive transport in Crx(Bi,Sb)2−xTe3 thin films. This crossover manifests itself in a giant, electrically tunable anisotropic magnetoresistance that we explain by employing a Landauer–Büttiker formalism. Our methodology provides a powerful means of quantifying dissipative effects in temperature and chemical potential regimes far from perfect quantization. When magnetized out-of-plane, three-dimensional ferromagnetic topological insulator thin films exhibit the quantum anomalous Hall effect. Here, the authors follow the evolution of this dissipationless chiral edge transport effect as the magnetization is brought in-plane under an applied magnetic field.

Proskurin I., Ogata M., Suzumura Y.

We investigate a longitudinal conductivity of a two-dimensional relativistic electron gas with a tilted Dirac cone in magnetic field. It is demonstrated that the conductivity behaves differently in the directions parallel and perpendicular to the tilting of the cone. At high magnetic fields the conductivity at non-zero Landau levels in the direction perpendicular to the tilting modifies non-trivially, in contrast to the parallel case. At zero temperature the crossover of the conductivity at the Dirac point from high to low magnetic field is studied numerically. It is found that that the tilting produces anisotropy of the conductivity which changes with the magnetic field which is different from the anisotropy coming from the Fermi velocity. We also discuss the conductivity at finite temperatures and finite magnetic fields which can be directly compared with the experiments in $\alpha$-(BEDT-TTF)$_2$I$_3$ organic conductor. We find that the tilting does not affect so much the magnetic-filed dependence of the conductivity except for the prefactor. We discuss the interpretation of recent experimental data and make some proposals to detect the effect of the tilting in future experiments.

Chang C., Zhao W., Kim D.Y., Zhang H., Assaf B.A., Heiman D., Zhang S., Liu C., Chan M.H., Moodera J.S.

An almost ideal quantum anomalous Hall state is observed in (Bi,Sb)Te films doped with vanadium. This state is reached without the application of a polarizing magnetic film, making these materials interesting for low-power electronic applications. The discovery of the quantum Hall (QH) effect led to the realization of a topological electronic state with dissipationless currents circulating in one direction along the edge of a two-dimensional electron layer under a strong magnetic field1,2. The quantum anomalous Hall (QAH) effect shares a similar physical phenomenon to that of the QH effect, whereas its physical origin relies on the intrinsic spin–orbit coupling and ferromagnetism3,4,5,6,7,8,9,10,11,12,13,14,15,16. Here, we report the experimental observation of the QAH state in V-doped (Bi,Sb)2Te3 films with the zero-field longitudinal resistance down to 0.00013 ± 0.00007h/e2 (~3.35 ± 1.76 Ω), Hall conductance reaching 0.9998 ± 0.0006e2/h and the Hall angle becoming as high as 89.993° ± 0.004° at T = 25 mK. A further advantage of this system comes from the fact that it is a hard ferromagnet with a large coercive field (Hc > 1.0 T) and a relative high Curie temperature. This realization of a robust QAH state in hard ferromagnetic topological insulators (FMTIs) is a major step towards dissipationless electronic applications in the absence of external fields.

Banerjee K., Son J., Deorani P., Ren P., Wang L., Yang H.

Absence of backscattering and occurrence of weak antilocalization are two characteristic features of topological insulators. We find that the introduction of defects results in the appearance of a negative contribution to magnetoresistance (MR) in the topological insulator $\mathrm{BiSbTeS}{\mathrm{e}}_{2}$ at temperatures below 50 K. Our analysis shows that the negative MR originates from an increase in the density of defect states created by the introduction of disorder, which leaves the surface states unaffected. We find a decrease in the magnitude of the negative MR contribution with increasing temperature and a robustness of the topological surface states to external disorder.

Siu Z.B., Jalil M.B., Tan S.G.

A hexagonal warping term has been proposed recently to explain the experimentally observed 2D equal energy contours of the surface states of the topological insulator Bi2Te3. Differing from the Dirac fermion Hamiltonian, the hexagonal warping term leads to the opening up of a band gap by an in-plane magnetization. We study the transmission between two Bi2Te3 segments subjected to different in-plane magnetizations and potentials. The opening up of a bandgap and the accompanying displacement and distortion of the constant energy surfaces from their usual circular shapes by the in-plane magnetizations, modify the transverse momentum overlap between the two Bi2Te3 segments and strongly modulate the transmission profile. The strong dependence of the TI surface state transport of Bi2Te3 on the magnetization orientation of an adjacent ferromagnetic layer may potentially be utilized in, e.g., a memory readout application.

Fan Y., Upadhyaya P., Kou X., Lang M., Takei S., Wang Z., Tang J., He L., Chang L., Montazeri M., Yu G., Jiang W., Nie T., Schwartz R.N., Tserkovnyak Y., et. al.

Recent demonstrations of magnetization switching induced by in-plane current in heavy metal/ferromagnetic heterostructures (HMFHs) have drawn great attention to spin torques arising from large spin–orbit coupling (SOC). Given the intrinsic strong SOC, topological insulators (TIs) are expected to be promising candidates for exploring spin–orbit torque (SOT)-related physics. Here we demonstrate experimentally the magnetization switching through giant SOT induced by an in-plane current in a chromium-doped TI bilayer heterostructure. The critical current density required for switching is below 8.9 × 104 A cm−2 at 1.9 K. Moreover, the SOT is calibrated by measuring the effective spin–orbit field using second-harmonic methods. The effective field to current ratio and the spin-Hall angle tangent are almost three orders of magnitude larger than those reported for HMFHs. The giant SOT and efficient current-induced magnetization switching exhibited by the bilayer heterostructure may lead to innovative spintronics applications such as ultralow power dissipation memory and logic devices. Heterostructures consisting of ferromagnets and heavy metals have become a focus of interest because their strong spin–orbit coupling allows for efficient current-induced magnetization switching phenomena. Now, a magnetically doped topological insulator bilayer is shown to display a range of appealing characteristics for current-induced magnetization switching, including a significantly enhanced efficiency.

Zhang Z.B.

Kumar R., Bajracharya P., Haghi Ashtiani P., Paxson R., Kolagani R., Budhani R.C.

The measurements of anisotropic magnetoresistance (AMR), planar Hall effect (PHE) and temperature-dependent conductivity in materials with strong spin-orbit coupling yield valuable information about charge carrier scattering processes, localization effects, and band topology. Although electronic structure calculations establish the sesqui-chalcogenide $\mathrm{S}{\mathrm{b}}_{2}\mathrm{T}{\mathrm{e}}_{3}$ a topological insulator (TI), detailed measurements of AMR and PHE are valuable to address the manifestations of the band topology on charge carrier transport in this system. Here, we report on measurements of the longitudinal and Hall resistivity, ${\ensuremath{\rho}}_{xx}$ and ${\ensuremath{\rho}}_{xy}$, respectively, of the $\mathrm{S}{\mathrm{b}}_{2}\mathrm{T}{\mathrm{e}}_{3}$ films of varied crystallinity over a wide phase space of temperature ($T$), magnetic field (B), and the angle between B and charge current density (J). The films exhibit semiconducting or metallic behavior depending on their crystallinity. The epitaxial films on (0001) sapphire grown at $150{\phantom{\rule{0.16em}{0ex}}}^{\ensuremath{\circ}}\mathrm{C}$ are metallic with a hole carrier density $({n}_{h})$ and mobility $({\ensuremath{\mu}}_{h})$ of $\ensuremath{\sim}{10}^{19}\phantom{\rule{4pt}{0ex}}\mathrm{c}{\mathrm{m}}^{\ensuremath{-}3}$ and $\ensuremath{\sim}{10}^{2}\phantom{\rule{0.16em}{0ex}}{\mathrm{cm}}^{2}\phantom{\rule{0.16em}{0ex}}{\mathrm{V}}^{\ensuremath{-}1}\phantom{\rule{0.16em}{0ex}}{\mathrm{s}}^{\ensuremath{-}1}$, respectively, at room temperature. The conduction in the semiconducting film exhibits a Shklovskii-Efros (SE)-type variable range hopping (VRH) at very low temperature, with a transition to the Mott type VRH at $T\ensuremath{\ge}30$ K. The SE-type VRH is characterized by a Coulomb gap of 0.3 meV and a localization length of \ensuremath{\approx}12 nm, which matches with the average crystallite size in these disordered films. While signatures of weak antilocalization are seen in the magnetoresistance (MR) of epitaxial films at $T\ensuremath{\le}20$ K, the MR at $T$ > 20 K agrees with Kohler's rule when corrected for the temperature variation of carrier density. The epitaxial films are characterized by a negative AMR and PHE which varies quadratically with magnetic field, but the orbital plots of ${\ensuremath{\rho}}_{xy}$ vs ${\ensuremath{\rho}}_{xx}$ negate the presence of a chiral anomaly in transport. The amplitude of MR anisotropy for $100{\phantom{\rule{0.16em}{0ex}}}^{\ensuremath{\circ}}\mathrm{C}$ grown $\mathrm{S}{\mathrm{b}}_{2}\mathrm{T}{\mathrm{e}}_{3}$ film is $\ensuremath{\approx}102\mathrm{n}\mathrm{\ensuremath{\Omega}}\mathrm{m}$ at 300 K, which is an order of magnitude larger than in $\mathrm{B}{\mathrm{i}}_{2}\mathrm{T}{\mathrm{e}}_{3}$ and potentially important for the development of AMR-based sensors.

Ba J., Wang Y., Duan H., Deng M., Wang R.

Motivated by recent experiments observing the nonlinear planar Hall effect (NPHE) in nonmagnetic topological materials, we employ the density matrix method to consider all the intraband and interband transitions. This gives a deeper insight for the different mechanisms of NPHE on the same footing beyond the semiclassical theory. Under broken time-reversal symmetry, besides the usual Berry curvature dipole (BCD) contribution, there exists the quantum metric (QM) induced NPHE, which includes the intrinsic and extrinsic components, and exists even within the band gap. This QM term extends the Berry-connection polarizability (BCP) theory which captures only the intrinsic contribution and cannot be applied to the case with finite scattering and nonzero frequency. Moreover, we reveal that the underlying physics of BCP originates essentially from the combination of three interband transitions (injection, shift, and anomalous), very differently from the BCD which is only contributed by an anomalous mechanism. We compare different mechanisms by calculating the NPHE on the surface states of topological insulators and find that the NPHE from different mechanisms exhibits different dependence on the in-plane magnetic field and the chemical potential. Our theory provides an alternative perspective to understand the complicated lineshapes of the NPHE observed near the Dirac point.

Hattori Y., Sagisaka K., Yoshizawa S., Tokumoto Y., Edagawa K.

Topological surface states of Bi-doped ${\mathrm{PbSb}}_{2}{\mathrm{Te}}_{4} [\mathrm{Pb}{({\mathrm{Bi}}_{0.20}{\mathrm{Sb}}_{0.80})}_{2}{\mathrm{Te}}_{4}]$ are investigated through analyses of quasiparticle interference (QPI) patterns observed by scanning tunneling microscopy. Interpretation of the experimental QPI patterns in the reciprocal space is achieved by numerical QPI simulations using two types of surface density of states produced by density functional theory calculations or the Fu's surface state model [Phys. Rev. Lett. 103, 266801 (2009)]. We found that the Dirac point (DP) of the surface state appears in the bulk band gap of this material and, with the energy being away from the DP, the isoenergy contour of the surface state is substantially deformed or separated into segments due to hybridization with bulk electronic states. These findings provide a more accurate picture of topological surface states, especially at energies away from the DP, providing valuable insight into the electronic properties of topological insulators.

Wang C.M., Du Z.Z., Lu H., Xie X.C.

Recently, the planar Hall effect has attracted tremendous interest. In particular, an in-plane magnetization can induce an anomalous planar Hall effect with a $2\ensuremath{\pi}/3$ period for hexagon-warped energy bands. This effect is similar to the anomalous Hall effect resulting from an out-of-plane magnetization. However, this anomalous planar Hall effect is absent in the planar Hall experiments. Here, we explain its absence, by performing a calculation that includes not only the Berry curvature mechanism, as those in the previous theories, but also the disorder contributions. The conventional $\ensuremath{\pi}$-period planar Hall effect will occur if the mirror-reflection symmetry is broken, which buries the anomalous one. This is because the anomalous planar Hall effect is of the higher order with respect to the small $h/({E}_{F}\ensuremath{\tau})$, when compared to the conventional planar Hall effect, with ${E}_{F}$ being the Fermi energy and $\ensuremath{\tau}$ the relaxation time. We show that an in-plane strain can enhance the anomalous Hall conductivity and changes the period from $2\ensuremath{\pi}/3$ to $2\ensuremath{\pi}$. We propose a scheme to extract the hidden anomalous planar Hall conductivity from the experimental data. Our work will be helpful in detecting the anomalous planar Hall effect and could be generalized to understand mechanisms of the planar Hall effects in a wide range of materials.

Stepina N.P., Bazhenov A.O., Shumilin A.V., Kuntsevich A.Y., Kirienko V.V., Zhdanov E.S., Ishchenko D.V., Tereshchenko O.E.

Electron states with the spin-momentum-locked Dirac dispersion at the surface of a three-dimensional (3D) topological insulator are known to lead to weak antilocalization (WAL), i.e., low temperature and low-magnetic-field quantum interference-induced positive magnetoresistance (MR). In this work, we report on the MR measurements in ${(\mathrm{Bi},\mathrm{Sb})}_{2}{(\mathrm{Te},\mathrm{Se})}_{3}$ 3D topological insulator thin films epitaxially grown on Si(111), demonstrating an anomalous WAL amplitude. This anomalously high amplitude of WAL cannot be explained by parabolic or linear MR and indicates the existence of an additional MR mechanism. Another supporting observation is not linear in the classically weak magnetic field Hall effect in the same films. The increase of the low-field Hall coefficient, with respect to the higher-field value, reaches $10%$. We consistently explain both transport features within a two-liquid model, where the mobility of one of the components strongly drops in a weak magnetic field. We argue that this dependence may arise from the Zeeman-field-induced gap opening mechanism.

Weng Z., Ba J., Ke Y., Duan H., Deng M., Wang R.

We study in-plane magnetotransport for the interface of a ferromagnet insulator (FMI) and a topological insulator (TI), taking into account thermal fluctuation effect of the magnetization in the FMI. It is found that the planar Hall effect (PHE) can display complicated angular dependencies beyond the standard paradigm ${\ensuremath{\sigma}}_{xx}\ensuremath{\sim}({\ensuremath{\sigma}}_{\ensuremath{\parallel}}\ensuremath{-}{\ensuremath{\sigma}}_{\ensuremath{\perp}}){cos}^{2}\ensuremath{\theta}$ and ${\ensuremath{\sigma}}_{xy}=({\ensuremath{\sigma}}_{\ensuremath{\parallel}}\ensuremath{-}{\ensuremath{\sigma}}_{\ensuremath{\perp}})sin\ensuremath{\theta}cos\ensuremath{\theta}$, where ${\ensuremath{\sigma}}_{\ensuremath{\parallel}(\ensuremath{\perp})}$ is the longitudinal conductivity in the direction parallel (vertical) to the magnetic field and $\ensuremath{\theta}$ denotes the relative angle between the applied electric and magnetic fields. The rich structures in angular dependence of the conductivities can arise from the cooperation and competition effect between multiple coexisting mechanisms, e.g., impurity and magnon scattering, band anisotropy, as well as reconstruction of the electronic structure by magnetic fields. Consequently, in addition to the relative angle $\ensuremath{\theta}$, the actual direction of the external fields can be crucial to the PHE, dependent on the specific properties of a material. As a result, the planar Hall conductivity can be nonvanishing even for parallel construction of the applied electric and magnetic fields. Our findings are based on but not limited to the FMI/TI heterostructure, which would be helpful to understand the emergent PHEs in topological materials.

Khokhlov D.A., Akzyanov R.S., Rakhmanov A.L.

Zhou Y., Duan H., Wu Y., Deng M., Wang L., Culcer D., Wang R.

Motivated by recent experiments observing a large antidamping spin-orbit torque (SOT) on the surface of a three-dimensional topological insulator, we investigate the origin of the current-induced SOT beyond linear response theory. We find that a strong antidamping SOT arises from intraband transitions in the nonlinear response and does not require interband transitions as is the case in linear transport mechanisms. The joint effect of warping and an in-plane magnetization generates a nonlinear antidamping SOT which can exceed the intrinsic one by several orders of magnitude, depending on the warping parameter and the position of the Fermi energy, and exhibits a complex dependence on the azimuthal angle of the magnetization. This nonlinear SOT provides an alternative explanation of the observed giant SOT in recent experiments.

Ghadiri H., Saffarzadeh A.

The hexagonal warping effect on transport properties and Goos-H\"anchen (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 derive expressions for calculating lateral shifts in the presence of warping effect. We study the dependence of transmission probabilities and GH shifts of transmitted beams on system parameters in detail by giving an explanation for the appearance of large peaks in the lateral shifts corresponding to their transmission peaks. It is shown that the separation between two transmitted beams through their different GH shifts can be as large as a few micrometers, which is large enough to be observed experimentally. Finally, we propose a method to measure the GH shift of electron beams based on the transverse magnetic focusing technique in which, by tuning an applied magnetic field, a detectable resonant path for electrons can be induced.

Arabikhah M., Saffarzadeh A.

Based on a self-consistent $t$-matrix approximation, we explore the influence of magnetic and nonmagnetic doping on the surface electronic states and conductivity of topological insulators. We show that warping parameter has a crucial impact on the density of states and dc conductivity of the doped surfaces. As the warping strength is increased, the surface density of states at high energies is suppressed and the resonant states induced by impurities in the vicinity of the Dirac point gradually disappear. It is found that nonmagnetic impurities break electron-hole symmetry at low warping strength, while the symmetry remains unchanged when the surface is magnetically doped. Our findings reveal that surface conductivity can be controlled by tuning the doping, the direction of external magnetic field and that of impurity magnetic moments. Also, the surface conductivity features in topological insulators with warped energy dispersions are not significantly affected by the presence of impurities compared to that of materials with circular energy contour.

Khokhlov D.A., Akzyanov R.S.

We theoretically investigate quasiparticle interference in superconducting topological insulators with the nematic order parameter. This order parameter spontaneously breaks the rotational symmetry of the crystal. Such rotational symmetry breaking is visible in the quasiparticle interference picture both in coordinate and momentum spaces. For a small bias voltages quasiparticle interference incommensurate with the crystal symmetry and shows nematic behavior. If the bias voltage is comparable with the value of the order parameter interference picture is similar to the interference picture of the normal state. Interference patterns are sensitive to the orientation of the nematicity. We compare our results with the existing experimental data.

Rao W., Zhou Y., Wu Y., Duan H., Deng M., Wang R.

Recently, the linear and nonlinear planar Hall effect (PHE) on the topological insulators (TIs) surface has been extensively studied in experiments. To explain this phenomenon, various microscopic mechanisms are proposed theoretically, and one has to employ different mechanisms to separately understand the linear and nonlinear PHE even for the same system. Here, we study the planar magnetic resistance effect in TI thin film and find that a peculiar anisotropic scattering and a spin valve structure with respect to the PHE can be caused by the tilt and shift of Dirac cones, respectively, which are induced by the combination of spin-momentum locking of surface states and an in-plane magnetic field. The tilt and shift effects can act as the origin of both the linear and nonlinear PHE by distorting the spin texture of surface states or forming the spin polarization. These two mechanisms interplay and dominate, respectively, in strong coupling (thin TI) and decoupling (thick TI) between bottom and top surfaces. For thick TI film, we show that both the linear and nonlinear PHEs induced by the tilt effect can recover the results observed in recent experiments. Our theory provides a perspective to understand the origin of both linear and nonlinear PHE observed in recent experiments.

Imai Y., Yamakage A., Kohno H.

We study spin-orbit torques and transport properties of Dirac electrons on the surface of ferromagnetic topological insulators. A focus is on the effects of deviation from the ideal Dirac model with linear dispersion, which takes an additional scalar form, quadratic in wave vector, and breaks particle-hole symmetry. This term removes some peculiar features of the linear Dirac model, and, combined with in-plane magnetization, gives rise to in-plane anisotropies. We study in detail the chemical potential dependence of longitudinal/transverse conductivities and reactive/dissipative spin-orbit torques, including the out-of-plane spin polarization. It is found that the in-plane anisotropy is generally stronger for $p$-type carriers than $n$-type carriers (for positive curvature of the quadratic term). It is also found that the in-plane anisotropy of the dissipative spin-orbit torque changes sign across the Dirac point. Because of the similarity of the model, we also study the magnetic Rashba model focusing on its Dirac-like parameter region. In this model, with an in-plane magnetization, a curious discontinuity is found at the Dirac point in the current-induced out-of-plane spin polarization.

Bhalla P.

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 the ratio of the non-linear to the linear conductivities. The theory is tested for a hexagonally warped and gapped topological insulator. Results show enhancement in the longitudinal and Hall effects on increasing the warping strength while show opposite behavior with the surface gap.