The optical conductivity for the surface excitations for a Topological Insulator as a function of the chemical potential and disorder is considered. Due to the time reversal symmetry the chiral metallic surface states are protected against disorder. This allow to use the averaged single particle Green's function to compute the optical conductivity. We compute the conductivity in the limit of a finite disorder. We find that the conductivity as a function of the chemical potential $μ$ and frequency $Ω$ is given by the universal value $σ(Ω>2μ)= \frac{e^2 π}{8h}$. For frequencies $Ω< μ$ and elastic mean free path $l_{el}=vτ$ which obey $k_{F}l>1$ we obtain the conductivity is given by $σ(Ω<2|μ|)=\frac{e^2}{2h}\frac{k_{F}l_{el}}{(Ωτ)^2+1}$. In the limit of zero disorder we find $σ(μ\neq0,Ω, k_{F}l\rightarrow \infty)=\frac{e^2 π}{2h}|μ|δ(Ω)$.
We study the spin conductivity of the bulk states of three-dimensional topological insulators within Kubo formalism. Spin Hall effect is the generation of the spin current that is perpendicular to the applied voltage. In the case of a three-dimensional topological insulator, applied voltage along $x$ direction generates transverse spin currents along $y$ and $z$ directions with comparable values. We found that a finite non-universal value of the spin conductivity exists in the gapped region due to the inversion of bands. Contribution to the spin conductivity from the vertex corrections enhance…
The lifetime of electrons on edge states of a two-dimensional topological insulator against the background of an allowed two-dimensional band has been determined. It has been shown that this time in the case of scattering on Coulomb impurities can be significantly larger than the mean free time of two-dimensional electrons. As a result, the conductivity of the metallic two-dimensional topological insulator strip can be determined primarily by edge states.
Electron spin-momentum locking together with background magnetic textures can significantly alter the electron transport properties. We theoretically investigate the electron transport at the interface between a topological insulator and a magnetic insulator with magnetic skyrmions on the top. In contrast to the conventional topological Hall effect in normal metals, the skyrmions yield an additional contribution to the anomalous Hall conductivity even in the absence of in-plane magnetic texture, arising from the phase factor characteristic to Dirac electrons acquired at skyrmion boundary.
We consider a disordered topological insulator thin film placed on the top of a ferromagnetic insulator with a perpendicular exchange field $M$ and subjected to a perpendicular electric field. The presence of ferromagnetic insulator causes that bottom surface states of the topological insulator thin film become spin polarized and the electric field provides a potential difference $V$ between the two surface states, resulting in breaking of time-reversal and inversion symmetry in the system. Using Kubo formalism and employing the first Born approximation as well as the self-consistent Born appr…
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 study spin conductivity of the surface states in a thin film of a topological insulator with Kubo formulas. Hybridization between the different sides of the film opens a gap at the Dirac point. We found that in the gapped region spin conductivity remains finite. In the gapless region near the band gap spin conductivity is enhanced. These findings make a thin film of a topological insulator as a promising material for spintronic applications.
Motivated by the quest for experimentally accessible dynamical probes of Floquet topological insulators, we formulate the linear response theory of a periodically driven system. We illustrate the applications of this formalism by giving general expressions for optical conductivity of Floquet systems, including its homodyne and heterodyne components and beyond. We obtain the Floquet optical conductivity of specific driven models, including two-dimensional Dirac material such as the surface of a topological insulator, graphene, and the Haldane model irradiated with circularly or linearly polariz…
Results are presented for the optical Hall conductivity of a Floquet topological insulator (FTI) for an ideal closed quantum system, as well as an open system in a nonequilibrium steady-state with a reservoir. The steady-state, even for the open system, is strongly dependent on the topological phase of the FTI, with certain phases showing a remarkable near-cancellation from pockets of Berry-curvature of opposite signs, leading to a suppressed low-frequency Hall conductivity, that also shows an anomalous temperature dependence, by increasing as the temperature of the reservoir is increased. Suc…
Topological insulators (TI) have attracted extensive research effort due to their insulating bulk states but conducting surface states. However, investigation and understanding of thermal transport in topological insulators, particularly the effect of surface states are lacking. In this work, we studied thickness-dependent in-plane thermal conductivity of Bi2Te2Se TI thin films. A large enhancement of both thermal and electrical conductivity was observed for films with thicknesses below 20 nm, which is attributed to the surface states and bulk-insulating nature of these films. Surprisingly, a …
We determine the optical response of topological insulator thin films in the presence of a quantizing, external magnetic field. We explicitly take into account hybridization between the states of top and bottom surface. The interplay between hybridization and Zeeman energies gives rise to topological and normal insulator phases and phase transitions between them. The optical response in the two phases and at the phase transition point is investigated. We show that the difference in magneto-optical response can be used to distinguish the topological phase from the normal phase of the system. Fu…
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…
We study the surface conductivity of a three dimensional topological insulator doped with magnetic impurities. The spin-momentum locking of surface electrons makes their scattering from magnetic impurities anisotropic and the standard relaxation time approximation is not applicable. Using the semiclassical Boltzmann approach together with a generalized relaxation time scheme, we obtain closed forms for the relaxation times and analytic expressions for the surface conductivities of the system as functions of the bulk magnetization and the orientation of the aligned surface magnetic impurities. …
The Haldane model on a honeycomb lattice is a paradigmatic example of a system featuring quantized Hall conductivity in the absence of an external magnetic field, that is, a quantum anomalous Hall effect. Recent theoretical work predicted that the anomalous Hall conductivity of massive Dirac fermions can display Shubnikov-de Haas (SdH) oscillations, which could be observed in topological insulators and honeycomb layers with strong spin--orbit coupling. Here, we investigate the electronic transport properties of Chern insulators subject to high magnetic fields by means of accurate spectral expa…
We have measured the differential resistance in a two-dimensional topological insulator (2DTI) in a HgTe quantum well, as a function of the applied dc current. The transport near the charge neutrality point is characterized by a pair of counter propagating gapless edge modes. In the presence of an electric field, the energy is transported by counter propagating channels in the opposite direction. We test a hot carrier effect model and demonstrate that the energy transfer complies with the Wiedemann Franz law near the charge neutrality point in the edge transport regime.
We develop an effective medium model of thermal conductivity that accounts for both percolation and interface scattering. This model accurately explains the measured increase and decrease of thermal conductivity with loading in composites dominated by percolation and interface scattering, respectively. Our model further predicts that strong interface scattering leads to a sharp decrease in thermal conductivity, or an insulator transition, at high loadings when conduction through the matrix is restricted and heat is forced to diffuse through particles with large interface resistance. The accura…
The Hikami-Larkin-Nagaoka (HLN) formula [Prog. Theor. Phys. 63, 707 (1980)] describes the quantum corrections to the magnetoconductivity of a quasi-2D electron gas (quasi-2DEG) with parabolic dispersion. It predicts a crossover from weak localization to antilocalization as a function of the strength of scattering off spin-orbit impurities. Here, we derive the conductivity correction for massless Dirac fermions in 3D topological insulators (TIs) in the presence of spin-orbit impurities. We show that this correction is always positive and therefore we predict weak antilocalization for every valu…
Two-dimensional (2D) massless Dirac electrons appear on a surface of three-dimensional topological insulators. The conductivity of such a 2D Dirac electron system is studied for strong topological insulators in the case of the Fermi level being located at the Dirac point. The average conductivity $\langleσ\rangle$ is numerically calculated for a system of length $L$ and width $W$ under the periodic or antiperiodic boundary condition in the transverse direction, and its behavior is analyzed by applying a finite-size scaling approach. It is shown that $\langleσ\rangle$ is minimized at the clean …
In this work we analytically investigate the longitudinal optical conductivity of the C$_{2v}$ symmetric topological insulator. The conductivity expressions at $ T = 0 $ are derived using the Kubo formula and expressed as a function of the ratio of the Dresselhaus and Rashba parameters that characterize the low-energy Hamiltonian. We find that the longitudinal inter-band conductivity vanishes when Dresselhaus and Rashba parameters are equal in strength, also called the persistent spin helix state. The calculations are extended to obtain the frequency-dependent real and imaginary components of …
We theoretically study the finite-size effects in the dynamical response of a quantum anomalous Hall insulator in the disk geometry. Semi-analytic and numerical results are obtained for the wavefunctions and energies of the disk within a continuum Dirac Hamiltonian description subject to a topological infinite mass boundary condition. Using the Kubo formula, we obtain the frequency-dependent longitudinal and Hall conductivities and find that optical transitions between edge states contribute dominantly to the real part of the dynamic Hall conductivity for frequency values both within and beyon…
Recent work has extended topological band theory to open, non-Hermitian Hamiltonians, yet little is understood about how non-Hermiticity alters the topological quantization of associated observables. We address this problem by studying the quantum anomalous Hall effect (QAHE) generated in the Dirac surface states of a 3D time-reversal-invariant topological insulator (TI) that is proximity-coupled to a metallic ferromagnet. By constructing a contact self-energy for the ferromagnet, we show that in addition to generating a mass gap in the surface spectrum, the ferromagnet can introduce a non-Her…
The anomalous Hall effect is caused by magnetic textures such as skyrmions. We derive an analytical formula of the Hall conductivity on the surface of a topological insulator up to third order in magnetization, $\boldsymbol{M}(\boldsymbol{x})$, based on a perturbative approach. We identify the magnetic textures that contribute to the Hall conductivity up to third order in magnetization and second order in spatial differentiation. We treat magnetization as a perturbation to calculate the Hall conductivity for each magnetic texture based on the linear response theory. Furthermore, we estimate th…
In this Book Chapter (invited) we briefly review the basic concepts defining topological insulators and focus on elaborating on the key experimental results that revealed and established their symmetry protected (SPT) topological nature. We then present key experimental results that demonstrate magnetism, Kondo insulation, mirror chirality or topological crystalline order and superconductivity in spin-orbit topological insulator settings and how these new (bulk insulating) phases of matter arise through topological quantum phase transitions from Bloch band insulators via topological or Dirac s…
This paper is focused on the the variable-range hopping of electrons in semiconductor nanocrystal (NC) films below the critical doping concentration $n_c$ at which it becomes metallic. The hopping conductivity is described by the Efros-Shklovskii law which depends on the localization length of electrons. We study how the localization length grows with the doping concentration $n$ in the film of touching NCs. For that we calculate the electron transfer matrix element $t(n)$ between neighboring NCs for two models when NCs touch by small facets or just one point. We study two sources of disorder:…
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 stren…
Despite the broadband response, limited optical absorption at a particular wavelength hinders the development of optoelectronics based on Dirac fermions. Heterostructures of graphene and various semiconductors have been explored for this purpose, while non-ideal interfaces often limit the performance. The topological insulator is a natural hybrid system, with the surface states hosting high-mobility Dirac fermions and the small-bandgap semiconducting bulk state strongly absorbing light. In this work, we show a large photocurrent response from a field effect transistor device based on intrinsic…
Problem of nonlinear conductivity within relaxation time approximation in noncentrosymmetric insulators
Department of Physical Science, Ritsumeikan University（立命馆大学物理科学系）；Department of Applied Physics, The University of Tokyo（东京大学应用物理系）；Department of Liberal Arts and Basic Sciences, College of Industrial Technology, Nihon University（日本大学工业技术学院文理基础科学系）
The spectrum of massless Dirac electrons on the side surface of a three-dimensional weak topological insulator is significantly affected by whether the number of unit atomic layers constituting the sample is even or odd; it has a finite-size energy gap in the even case while it is gapless in the odd case. The conductivity of such a two-dimensional Dirac electron system with quenched disorder is calculated when the Fermi level is located at the Dirac point. It is shown that the conductivity increases with increasing disorder and shows no clear even-odd difference when the aspect ratio of the sy…
Polyvinyl-pyrrolidone (PVP) loaded with different fractions of dispersed nano-graphene platelets (NGP) were studied by Broadband Dielectric Spectroscopy in the frequency range from 1 mHz to 1 MHz. Complex permittivity and dynamic ac conductivity as a function of frequency, temperature and composition were explored. The concentration-dependent insulator-to-conductor transition was traced through dependence of the dc conductivity and the onset of the dispersive ac ac conductivity. The temperature evolution of the dielectric spectra, below and above the fractional threshold exhibits different dyn…
Eigenenergies of a non-Hermitian system without parity-time symmetry are complex in general. Here, we show that the chiral boundary states of higher-dimensional (two-dimensional and three-dimensional) non-Hermitian topological insulators without parity-time symmetry can be Hermitian with real eigenenergies under certain conditions. Our approach allows one to construct Hermitian chiral edge and hinge states from non-Hermitian two-dimensional Chern insulators and three-dimensional second-order topological insulators, respectively. Such Hermitian chiral boundary channels have perfect transmission…
We propose a different route to time-reversal invariant Weyl semimetals employing multilayer heterostructures comprising ordinary "trivial" insulators and nontrivial insulators with \textit{pairs} of protected Dirac cones on the surface. We consider both the case of weak topological insualtors, where surface Dirac cones are pinned to time-reversal invariant momenta, and of topological crystalline insulators with unpinned surface Dirac cones. For both realizations we explain phenomenologically how the proposed construction leads to the emergence of a Weyl semimetal phase. We further formulate e…
Chern insulator or quantum anomalous Hall state is a topological state with integer Hall conductivity but in absence of Landau level. It had been well established on various two-dimensional lattices with periodic structure. Here, we report similar Chern insulators can also be realized on the quasicrystal with $5$-fold rotational symmetry. Providing the staggered flux through plaquettes, we propose two types of quasicrystalline Chern insulators. Their topological characterizations are well identified by the robustness of edge states, non-zero real-space Chern number, and quantized conductance. …
The Dirac mass of a two-dimensional QAH insulator is directly related to the parity anomaly of planar quantum electrodynamics, as shown initially in Phys. Rev. Lett. 52, 18 (1984). In this work, we connect the additional momentum-dependent Newtonian mass term of a QAH insulator to the parity anomaly. By calculating the effective action, we reveal that the Newtonian mass term acts like a parity-breaking element of a high-energy regularization scheme. As such, it is directly related to the parity anomaly. In addition, the calculation of the effective action allows us to determine the finite freq…
The third-rank tensor of the static spin Hall conductivity is investigated for two-dimensional (2D) topological insulators by electronic structure calculations. Its seeming quantization is numerically demonstrated for highly symmetric systems independent of the gap size. 2D crystals with hexagonal and square Bravais lattice show similar effects, while true rectangular translational symmetry yields conductivity values much below the quantum $e^2/h$. Field-induced lifting the inversion symmetry does not influence the quantum spin Hall state up to band inversion but the conductivity quantization.…
Linear temperature dependence of conductivity in Si two-dimensional electrons near the apparent metal-to-insulator transition
在Si中的高迁移率二维电子系统中，在表观金属-绝缘体转变的临界密度$n_c=0.32\times10^{11}$ cm $^{-2}$附近，电导率在费米温度附近显示线性温度（$T$）依赖性。当$σ_0$（从线性T依赖性推断的T=0电导率）作为密度的函数绘制时，可以看到具有不同$σ_0(n)$关系的两个区域，表明存在两个不同的阶段。有趣的是，这两种状态之间的急剧转变与$n_c$一致，并且转变的$σ_0$是$\sim$ $e^2/h$，即每平方的量子电导。当T=0时，数据偏离了线性$σ(T)$关系，并讨论了Si样品中可能的渗流型相变。
The discovery and realization of topological insulators, a phase of matter which hosts metallic boundary states when the $d$-dimension insulating bulk is confined to ($d-1$)-dimensions, led to several potential applications. Recently, it was shown that protected topological states can manifest in ($d-2$)-dimensions, such as hinge and corner states for three- and two-dimensional systems, respectively. These nontrivial materials are named higher-order topological insulators (HOTIs). Here we show a connection between spin Hall effect and HOTIs using a combination of {\it ab initio} calculations a…
We are honored to dedicate this article to Emmanuel Rashba on the occasion of his 95 birthday. In the ideal disorder-free situation, a two-dimensional band gap insulator has an activation energy for conductivity equal to half the band gap, $Δ$. But transport experiments usually exhibit a much smaller activation energy at low temperature, and the relation between this activation energy and $Δ$ is unclear. Here we consider the temperature-dependent conductivity of a two-dimensional narrow gap semiconductor on a substrate containing Coulomb impurities, mostly focusing on the case when amplitude o…
Quantum transport in magnetic topological insulators reveals the strong interplay between the magnetism and topology of electronic band structures. A recent experiment on magnetically doped topological insulator Bi2Se3 thin films showed the anomalous temperature dependence of the magnetoconductivity while their field dependence presents a clear signature of weak anti-localization [Tkac et al., Phys. Rev. Lett. 123, 036406(2019)]. Here we demonstrate that the tiny mass of the surface electrons induced by the bulk magnetization leads to a temperature-dependent correction to the πBerry phase, and…
