Session V37: Topological Insulators: Magnetoelectric Effect, Control, and Dynamics

A thermodynamic measure of the Magento-electric coupling in the 3D topological insulator

We show that the magneto-electric coupling in 3D (strong) topological insulators is related to a second derivative of the bulk magnetization. The formula we derive is the non-linear response analog of the Streda formula for Hall conductivity (P. Streda, J. Phys. C: Solid State Physics, 15, 22 (1982)), which relates the Hall conductivity to the derivative of the magnetization with respect to chemical potential. Our finding allows one to extract the magneto-electric coefficient by measuring the magnetization, while varying the chemical potential and one more perturbing field. Such an experimental setup could circumvent many of the current difficulties with measuring the magneto-electric response in 3D topological insulators. The relation we find also makes transparent the effect of disorder, contained entirely in the density of states, and changing nothing as long as the system is gapped.   

Magnetoelectric response of a topological surface under the conditions of an imperfect quantum Hall effect

An effective magnetic monopole is induced by an external charge in a topological insulator, and in an ordinary insulator covered by a graphene sheet or another two-dimensional electron system, when it has a perfect surface quantum Hall effect [X.-L. Qi, R. Li, J. Zang, and S.-C. Zhang, Science \textbf{323}, 1184 (2009)]. We discuss the observability of this magnetoelectric response under the realistic conditions of a quantum Hall effect that is imperfect because of finite temperature, disorder, or unintended doping. By generalizing the surface electrodynamics to allow for a finite longitudinal conductivity, we analyze the transient behavior which occurs as the potential from a suddenly introduced external charge is screened. Screening severely limits the experimental time scales on which observation of magnetic-monopole-related phenomena is possible. We estimate the longitudinal conductivity values that are necessary for the monopole to survive for an extended period of time and discuss implications of our findings for other transport properties of the surface.   

Magnetoelectric effect in topological insulator/magnetic layer nanostructure

The topological insulator (TI) surface band structure can be modified by contacting ferromagnetic layers due to the proximity exchange interaction. When the magnetization \textbf{M} is in the in-plane direction, the proximate exchange interaction results in a shift of the Dirac cone in the momentum space, whereas an energy gap can be generated for out-of-plane orientation culminating the maximum value at perpendicular to plane direction. Such opening of energy gap lowers the valence electron energy that can be only partially compensated by increase of the conduction electron energies depending on chemical potential $\mu $. Such correlation of electronic energy and magnetization direction open up a new way toward the electrical manipulation of \textbf{M}. To quantitative estimation of this effect, we provide the thermodynamic potential calculation of the TI surface electrons interacted with proximate ferromagnetic insulator as a function of \textbf{M} rotation by angle \textit{$\theta $} about in-plane axis \textbf{x}. The result can be described as an additive magnetic energy $E=K_{eff}$\textit{($\mu )$sin}$^{2}$\textit{$\theta $} in the form of uniaxial anisotropy, which is induced by interaction with TI surface electrons. In the case of TI Bi$_{2}$Se$_{3}$, the numerical estimations predict the $K_{eff}$\textit{($\mu )$} variation in the range of 1 meV/nm$^{2}$ if \textit{$\mu $} vary over 100 meV. The possible applications of the effect (memory and logic) are discussed.   

Magnetoelectric and thermoelectric transport in graphene and helical metal: Effect of applied electric field

We report on the electrical and thermoelectric transport properties of the surface state of the 3D topological insulator (TI) and graphene in a quantizing magnetic field. An unique feature of these systems is the evolution of the Landau level spectrum as a function of applied in plane electric field. We bench mark out results at small fields by computing conductivity and thermopower within linear response. We find that the universal values of thermopower in the clean limit depend on the gyromagnetic ratio in TIs, providing a clear distinction from graphene. In large electric fields we find an oscillation of conductivity as a function of applied electric field for fixed chemical potential, but not for fixed particle density. Signatures of the Landau level dependence on electric fields are also found in thermopower. These results are suggested as possible probes, in transport measurements of topological surface states.   

Noninvasive Probe of Charge Fractionalization in Quantum Spin-Hall Insulators

When an electron with well-defined momentum tunnels into a nonchiral Luttinger liquid, it breaks up into two separate wave packets that carry fractional charges and move in opposite directions. Observing this phenomenon has proven difficult, in part due to single-particle and plasmon backscattering caused by measurement probes. In this talk we propose a topological insulator RC circuit that might be ideally suited for detecting fractional charges directly and in a noninvasive fashion.   

Topological charge pumping effect by the magnetization dynamics on the Surface of Three-Dimensional Topological Insulators

We discuss a current dynamics on the surface of a 3-dimensional topological insulator induced by magnetization precession of a ferromagnet attached. It is found that the magnetization dynamics generates a direct charge current when the precession axis is within the surface plane. This rectification effect is due to a quantum anomaly (parity anomaly) and is topologically protected. The robustness of the rectification effect against first-varying exchange field is confirmed by the explicit calculation, where we adopt the dimensional regularization to remove the divergence which is inevitable in the study on the electromagnetic response of the Dirac system.   

Electric tuning of topological insulator states and their surface scattering

Materials that show the topological insulating (TI) properties have been extensively studied both experimentally and theoretically in recent years, but an application of those materials for electronic devices is still a challenge. In this study we explored a possibility to switch electric conductance of a surface TI states in a Bi$_{2}$Te$_{3}$ thin film. In such films there are two surfaces that have TI states located at them. We have done a theoretical study of the surface states using atomistic description within an empirical tight binding approximation. It was found that applying electric field perpendicular to the surface one can affect spin polarization of the TI states. Analysis of the scattering rates for electrons that occupy TI states close to the Fermi level shows that: a) the surface states are spin polarized under an applied electric field; b) the electron scattering between two surface states depends on whether they occupy the same surface and have parallel spins; c) by varying strength of an applied electric field it is possible to modulate the scattering rate, because at small fields the states are not spin-polarized, while at higher fields the states become spin-polarized, and at very high fields the surface states are significantly coupled to the bulk states.   

Electrical control of the Kondo effect in a helical edge liquid

Magnetic impurities affect the transport properties of the helical edge states of quantum spin Hall insulators by allowing single-electron backscattering. We study such a system in the presence of a Rashba spin-orbit interaction induced by an external electric field, showing that this can be used to control the Kondo temperature, as well as the correction to the electrical conductivity due to the impurity. In particular, the impurity contribution to the dc conductivity can be switched on and off by properly adjusting the strength of the Rashba coupling.   

Nonlinear spectroscopy of non-Abelian Berry curvature

We propose a general scheme to measure the Berry curvatures of energy bands in insulators by standard nonlinear optical spectroscopy. Our method employs optical and terahertz lights to produce a signal. A general calculation shows that the third order response of the solid is directly related to the Berry curvatures of the energy bands. In particular, for a time-reversal invariant system, we get a nonzero effect compared with the linear response methods, which provides information about the underlying non-Abelian Berry curvature. For insulators with rotational symmetry, the response is proportional to the Berry curvature flux of the iso-energy surface, which enables people to determine the topological properties of the energy bands explicitly. The method is applied to the eight-band model of III-V compound semiconductors and gives a quantized susceptibility with some global coefficients.   

The Shockley-model description of the edge states in topological insulators

We show that the edge states in topological insulators can be understood based on the well-known Shockley model, a 1D tight-binding model with two atoms per elementary cell connected via alternating tunneling amplitudes. We generalize the model to a 3D Shockley-like model corresponding to the sequence of layers connected via the tunneling amplitudes dependent on the in-plane momentum $\mathbf{p = (p_x,p_y)}$. The Hamiltonian of the model is a $2\times2$ matrix with the complex off-diagonal matrix element $t(k,\mathbf{p})$ dependent on both $\mathbf{p}$ and the out-of-plane momentum $k$. The equation $t(k,\mathbf{p})=0$ defines vortex lines in the 3D momentum space. We show that the projection of the vortex lines on the 2D momentum space defines a boundary between the regions of $\mathbf{p}$ where the edge states exist or do not exist. The vorticity of the vortex lines determines the crystal sublattice on which the edge states are localized. We illustrate how our approach works for the well-established topological insulator model by Fu, Kane, and Mele. We find that different configurations of the vortex contours are responsible for the topological insulator phases with even or odd numbers of the surface Dirac cones. We discuss how real materials, such as Bi$_2$Se$_3$, can be described by this model.   

Thermoelectric efficiency of holey topological insulators

We study the thermoelectric properties of three-dimensional topological insulators with many holes (or pores) in the bulk. We show that at high density of these holes the thermoelectric figure of merit, $ZT$, can be large due to the contribution of the conducting surfaces and the suppressed phonon thermal conductivity. The maximum efficiency can be tuned by an induced gap in the surface states dispersion through tunneling or external magnetic fields. The large values of $ZT$, much higher than unity for reasonable parameters, make this system a strong candidate for applications in heat management of nanodevices, especially at low temperatures.   

Spin-charge Dynamics On Surface States of Topological Insulators and 2DEG

We study the spin-charge dynamics on the surface of a topological insulator and spin-orbit coupled two dimensional electron gas. A new approach is developed to study the spin-charge dynamics even valid in the very strong spin-orbit coupling regime where the spin splitting energy due to SOIs is at the same order to the Fermi energy. The effects of Coulomb interaction and external field are also considered. We predict the fast oscillation of spin polarization perpendicular and parallel to the surface. Based on our theory, we provide a scheme to measure the transport property of the surface state isolated from the bulk contribution.   

Ultrastrong coupling cavity QED of the magnetic cyclotron transition in a 2D electron gas: massive versus Dirac fermions

We show that the cyclotron transition of a two-dimensional electron gas can be ultrastrongly coupled to a cavity photon mode. The ratio between the vacuum Rabi frequency $\Omega_0$ and the cyclotron frequency $\omega_0$ can be much larger than $1$ for large filling factor $\nu$ of the Landau levels (the normalized coupling $\Omega_0/\omega_0$ scales as $\sqrt{\alpha \, n_{QW} \nu }$, where $\alpha$ is the fine structure constant and $n_{QW}$ is the number of quantum wells). We present a comprehensive cavity QED theory both for semiconductors with massive electrons\cite{hagenmuller1,scalari} and graphene with Dirac fermions\cite{hagenmuller2}. We show the dramatic impact on the quantum ground state and excitation properties, drawing the comparison between the two different types of 2D electron gas. \\[4pt] [1] D. Hagenm\"{u}ller, S. De Liberato, and C. Ciuti, Phys. Rev. B 81, 235303 (2010) and references therein.\\[0pt] [2] Experiments demonstrating ultrastrong coupling of the cyclotron transition in a GaAs-system in the THz regime have been reported, see G. Scalari \textit{et al}., submitted.\\[0pt] [3] D. Hagenm\"{u}ller and C. Ciuti, submitted.   

Topological Floquet Spectrum in Three Dimensions via a Two-Photon Resonance

A recent theoretical work [Nature Phys., 7, 490 (2011)] has demonstrated that external non-equilibrium perturbations may be used to convert a two-dimensional semiconductor, initially in a topologically trivial state, into a Floquet topological insulator. Here, we develop a non-trivial extension of these ideas to three-dimensional systems. In this case, we show that a two-photon resonance may provide the necessary twist needed to transform an initially unremarkable band structure into a topological Floquet spectrum. We provide both an intuitive, geometrical, picture of this phenomenon and also support it by an exact solution of a realistic lattice model that upon irradiation features single topological Dirac modes at the two-dimensional boundary of the system. It is shown that the surface spectrum can be controlled by choosing the polarization and frequency of the driving electromagnetic field. Specific experimental realizations of a three-dimensional Floquet topological insulator are proposed.