大槻 東巳

The concept of the topological insulator (TI) has introduced a new point of view to condensedmatter physics, relating a priori unrelated subfields such as quantum (spin, anomalous) Hall effects, spin-orbit coupled materials, some classes of nodal superconductors, superfluid He-3, etc. From a technological point of view, TIs are expected to serve as platforms for realizing dissipationless transport in a non-superconducting context. The TI exhibits a gapless surface state with a characteristic conic dispersion (a surface Dirac cone). Here, we review peculiar finite-size effects applicable to such surface states in TI nanostructures. We highlight the specific electronic properties of TI nanowires and nanoparticles, and in this context we contrast the cases of weak and strong TIs. We study the robustness of the surface and the bulk of TIs against disorder, addressing the physics of Dirac and Weyl semimetals as a new research perspective in the field.

A new method utilizing the low energy muon spin rotation (mu SR) technique to investigate the spin structure within the surface layer of topological insulator has been proposed. In order to detect the spin polarization parallel with the surface, one applies a weak field B-par, which, by breaking the time reversal symmetry, is expected to induce a net magnetization perpendicular the surface. This will affect the profile of muon spin depolarization spectra. Preliminary measurements on the topological insulator Bi1.5Sb0.5TeSe2 with tetradymite structure are also shown.

The first near-field optical imaging of light localization in a GaN nanocolumn system was performed. The sample used was a randomly arranged GaN nanocolumn with high aspect ratio. We attached an InGaN single quantum well at the apex of each GaN nanocolumn as an illuminant antenna and observed luminescence from the illuminant using an aperture-type scanning near-field optical microscope. By this technique, we directly obtained optical images of luminescence and its spatial distribution for the GaN nanocolumn system. These images, along with histogram analysis, excitation wavelength dependence, and numerical calculations, offer evidence of Anderson localization of light. (C) 2014 The Japan Society of Applied Physics

The quantum phase transition between the three dimensional Dirac semimetal and the diffusive metal can be induced by increasing disorder. Taking the system of a disordered Z(2) topological insulator as an important example, we compute the single particle density of states by the kernel polynomial method. We focus on three regions: the Dirac semimetal at the phase boundary between two topologically distinct phases, the tricritical point of the two topological insulator phases and the diffusive metal, and the diffusive metal lying at strong disorder. The density of states obeys a novel single parameter scaling, collapsing onto two branches of a universal scaling function, which correspond to the Dirac semimetal and the diffusive metal. The diverging length scale critical exponent nu and the dynamical critical exponent z are estimated, and found to differ significantly from those for the conventional Anderson transition. Critical behavior of experimentally observable quantities near and at the tricritical point is also discussed.

We report a careful finite size scaling study of the metal-insulator transition in Anderson's model of localization. We focus on the estimation of the critical exponent v that describes the divergence of the localization length. We verify the universality of this critical exponent for three different distributions of the random potential: box, normal and Cauchy. Our results for the critical exponent are consistent with the measured values obtained in experiments on the dynamical localization transition in the quantum kicked rotor realized in a cold atomic gas.

We report on conductance fluctuation in quasi-one-dimensional wires made of epitaxial Bi2Se3 thin film. We found that this type of fluctuation decreases as the wire length becomes longer and that the amplitude of the fluctuation is well scaled to the coherence, thermal diffusion, and wire lengths, as predicted by conventional universal conductance fluctuation (UCF) theory. Additionally, the amplitude of the fluctuation can be understood to be equivalent to the UCF amplitude of a system with strong spin-orbit interaction and no time-reversal symmetry. These results indicate that the conductance fluctuation in Bi2Se3 wires is explainable through UCF theory. This work verifies the scaling relationship of UCF in a system with strong spin-orbit interaction. © 2013 American Physical Society.

We report on conductance fluctuation in quasi-one-dimensional wires made of epitaxial Bi2Se3 thin film. We found that this type of fluctuation decreases as the wire length becomes longer and that the amplitude of the fluctuation is well scaled to the coherence, thermal diffusion, and wire lengths, as predicted by conventional universal conductance fluctuation (UCF) theory. Additionally, the amplitude of the fluctuation can be understood to be equivalent to the UCF amplitude of a system with strong spin-orbit interaction and no time-reversal symmetry. These results indicate that the conductance fluctuation in Bi2Se3 wires is explainable through UCF theory. This work verifies the scaling relationship of UCF in a system with strong spin-orbit interaction.

A global phase diagram of disordered weak and strong topological insulators is established numerically. As expected, the location of the phase boundaries is renormalized by disorder, a feature recognized in the study of the so-called topological Anderson insulator. Here, we report unexpected quantization, i.e., robustness against disorder of the conductance peaks on these phase boundaries. Another highlight of the work is on the emergence of two subregions in the weak topological insulator phase under disorder. According to the size dependence of the conductance, the surface states are either robust or "defeated" in the two subregions. The nature of the two distinct types of behavior is further revealed by studying the Lyapunov exponents. © 2013 American Physical Society.

A global phase diagram of disordered weak and strong topological insulators is established numerically. As expected, the location of the phase boundaries is renormalized by disorder, a feature recognized in the study of the so-called topological Anderson insulator. Here, we report unexpected quantization, i.e., robustness against disorder of the conductance peaks on these phase boundaries. Another highlight of the work is on the emergence of two subregions in the weak topological insulator phase under disorder. According to the size dependence of the conductance, the surface states are either robust or "defeated" in the two subregions. The nature of the two distinct types of behavior is further revealed by studying the Lyapunov exponents.

We report observations of lasing phenomena in the regularly arranged InGaN/GaN nanocolumns. We show several types of laser actions, namely random lasing and distributed feedback lasing resulting from the randomness of the sample configurations. © OSA 2013.

We conducted numerical and experimental studies on Anderson localization of light in two-dimensional random systems of semiconductor columns. We investigated finite size scaling of the localization effect as a function of localization length and system size from the calculation. We therefore obtained a localization parameter map. We also report observations of random lasing in the semiconductor samples. We show that the occurrence of the laser action have a strong relationship to the localization length by comparisons of the map with the experimental results

We have estimated the critical exponent describing the divergence of the localization length at the metal-quantum spin Hall insulator transition. The critical exponent for the metal-ordinary insulator transition in quantum spin Hall systems is known to be consistent with that of topologically trivial symplectic systems. However, the precise estimation of the critical exponent for the metal-quantum spin Hall insulator transition proved to be problematic because of the existence, in this case, of edge states in the localized phase. We have overcome this difficulty by analyzing the second smallest positive Lyapunov exponent instead of the smallest positive Lyapunov exponent. We find a value for the critical exponent nu = 2.73 +/- 0.02 that is consistent with that for topologically trivial symplectic systems.

The nontrivialness of a topological insulator (TI) is characterized either by a bulk topological invariant or by the existence of a protected metallic surface state. Yet, in realistic samples of finite size, this nontrivialness does not necessarily guarantee the gaplessness of the surface state. Depending on the geometry and on the topological indices, a finite-size energy gap of different nature can appear, and, correspondingly, exhibit various scaling behaviors of the gap. The spin-to-surface locking provides one such gap-opening mechanism, resulting in a power-law scaling of the energy gap. Weak and strong TIs show different degrees of sensitivity to the geometry of the sample. As a noteworthy example, a strong TI nanowire of a rectangular-prism shape is shown to be more gapped than that of a weak TI of precisely the same geometry. DOI: 10.1103/PhysRevB.86.245436 PACS number(s): 73.22.-f, 73.20.At, 72.80.Sk

The Anderson transition is a disorder driven quantum phase transition between metallic and insulating phases. In contrast to the common belief that two dimensional (2D) systems are always insulating and that the Anderson transition does not occur in 2D, in certain universality classes 2D systems can be metallic. We review the recent development of the theory of the Anderson transition in 2D. There are ten universality classes: three Wigner-Dyson classes, three chiral universality classes, and four Bogoliubov-de Gennes classes. We report results for critical exponents and distributions of conductance for the symplectic universality class. We emphasize that, on the one hand, the existence of a topological insulating phase does not alter the value of the critical exponent, while on the other, it strongly affects the form of the conductance distribution at the transition.

The possibility of Anderson localization of light in random dielectric systems has been discussed over the last three decades [1]. However, theoretical and experimental studies for the light localization in random media are much more difficult than those in photonic crystals. Therefore, observation of the light localization has been realized only recently [2]. As a result, it is fair to say the wave localization phenomenon is not well understood. © 2011 IEEE.

In disordered materials, the combination of multiple light scattering and optical interference induces the localization of light. This phenomenon is called Anderson localization which is widely observed in electron systems with random potentials. Experimental studies of Anderson localization of light have been performed over the last three decades [1]. However, most of those studies attempted to secure the evidence of light localization by macroscopic observation of the light scattered by disordered materials. Recently, we proposed the direct observation of light localization using GaN nanocolumn samples by near-field scanning optical microscopy (NSOM), and presented the evidence of the light localization by a histogram analysis of NSOM image and by showing the wavelength dependence of the position of the localized spots in NSOM images [2,3]. We also observed random lasing in GaN nanocolumn samples and discussed the lasing property from a viewpoint of Anderson localization [4]. In this study, we extracted the two-dimensional (2D) spatial dependence from the NSOM image to estimate the localization length in GaN nanocolumns. © 2011 IEEE.

In the two-dimensional random system composed of a disordered array of a dielectric cylindrical column ensemble, Anderson localization of light is possible. We show localization parameter maps for the light localization adopting parameters of gallium nitride nanocolumn samples, which consist of random arrays of parallel nanosized columnar semiconductor crystals. The maps indicate parametric dependence of the localization characteristics on the light frequency, the radius of the columns, and the filling fraction of the columns. To obtain the maps, we have simulated temporal light diffusion in random media using the two-dimensional finite-difference time-domain method and analyzed the simulation results by Fourier transformation. We conclude that the main mechanism for localization varies continuously with the column filling fraction from Mie resonance of single column to Bragg-like diffraction of the column ensemble.

In the two-dimensional random system composed of a disordered array of dielectric cylindrical columns, Anderson localization of light occurs. To obtain frequency dependence of the light localization characteristics, we have simulated temporal diffusion of electromagnetic waves in such a random system adopting parameters of actual nanosized semiconductor samples with a high filling fraction of the columns, using the finite-difference time-domain method. We have investigated diffusion length, autocorrelation function of light energy density, and time variation in total energy within the system at several frequencies. We obtain universal behavior of light localization phenomenon as a function of the light localization length and system size, from which we estimate frequency dependence of the localization length. In addition, we show that the frequency dependence of the localization effect depends on the degree of wave interference due to Bragg-like diffraction, rather than on the magnitude of the light scattering cross section of a single scatterer.

We study numerically the charge conductance distributions of disordered quantum spin-Hall (QSH) systems using a quantum network model. We have found that the conductance distribution at the metal-QSH insulator transition is clearly different from that at the metal-ordinary insulator transition. Thus the critical conductance distribution is sensitive not only to the boundary condition but also to the presence of edge states in the adjacent insulating phase. We have also calculated the point-contact conductance. Even when the two-terminal conductance is approximately quantized, we find large fluctuations in the point-contact conductance. Furthermore, we have found a semicircular relation between the average of the point-contact conductance and its fluctuation.

We report observations of random laser action in self-organized GaN nanocolumns. We have measured three samples with different filling fractions and investigated the dependence of the lasing property on the random configuration of nanocolumns. Numerical calculations based on a finite-difference time-domain method have also been performed and the comparison with the experimental results shows a clear relationship between the strength of light localization and the occurrence of random laser action. (C) 2010 American Institute of Physics. [doi:10.1063/1.3495993]

This chapter describes the progress made during the past three decades in the finite size scaling analysis of the critical phenomena of the Anderson transition. The scaling theory of localization and the Anderson model of localization are briefly sketched. The finite size scaling method is described. Recent results for the critical exponents of the different symmetry classes are summarised. The importance of corrections to scaling are emphasised. A comparison with experiment is made, and a direction for future work is suggested.

We present frequency dependence of Anderson localization of light in finite-sized two-dimensional random systems. We have investigated time variation of total energy within a system composed of dielectric columns, by adopting parameters of actual nano-sized semiconductor crystals with a high filling fraction of the columns. We have succeeded in obtaining single parameter scaling of the localization effect. In addition, we show that the frequency dependence of the localization in the system depends on the degree of wave interference due to Bragg-like diffraction, rather than on the magnitude of the light scattering cross section for a single column.

In two-dimensional electron gas (2DEG), spatial gradient of effective magnetic field due to spin orbit interaction yields spin dependent force. By taking this advantage, Stern-Gerlach spin filter in 2DEG has been proposed for generating spin polarized currents without any external magnetic fields and ferromagnetic materials [Phys. Rev. B 72, 041308(R) (2005)]. In order to demonstrate the spin filtering effect, detection of spin polarized electrons becomes of crucial importance. Here, we propose an electrical detection of spin filtering by introducing an in-plane magnetic field in mesoscopic Stern-Gerlach spin filter. In-plane magnetic field induces spin polarized electrons due to Zeeman splitting, generating the imbalance between up-spin and down-spin currents after the spin separation. Calculated spin separation angle becomes 10 degrees based on experimentally accessible parameters. Time evolution of wave packet shows the spin separation as well as the charge imbalance under the in-plane magnetic field. By fabricating Y-branch shaped narrow wire structure with two split gate electrodes at the junction, spin filtering effect can be detected as the magnitude difference of each branch currents. Gate bias dependence of each branch current was measured in B-ex = +/- 15 T at T = 4.2 K.

We study the transport properties of disordered two-dimensional electron systems with a perfectly conducting channel. We introduce an asymmetric Chalker-Coddington network model and numerically investigate the point-contact conductance. We find that the behavior of the conductance in this model is completely different from that in the symmetric model. Even in the limit of a large distance between the contacts, we find a broad distribution of conductance and a non-trivial power law dependence of the averaged conductance on the system width. Our results are applicable to systems such as zigzag graphene nano-ribbons where the numbers of left- and right-going channels are different.

We report an estimate nu = 2.593 [2.587,2.598] of the critical exponent of the Chalker-Coddington model of the integer quantum Hall effect that is significantly larger than previous numerical estimates and in disagreement with experiment. This suggests that models of noninteracting electrons cannot explain the critical phenomena of the integer quantum Hall effect.

We show pseudogap maps of Anderson localization of light adopting the parameters of self-organized nanocolumn samples, which consist of random arrays of parallel nanosized columnar semiconductor crystals. The maps indicate the parametric dependence of the localization effect. To obtain the maps, we simulated light propagation in open random media using the two-dimensional finite-difference time-domain method and analyzed the simulation results by Fourier transformation. We found that the shape of the pseudogaps is close to the one of bandgaps in photonic crystals. We conclude that strong localization of light occurs because of interference by average Bragg diffraction, not strong Mie resonant peaks.

We study the transport properties of disordered electron systems that contain perfectly conducting channels. Two quantum network models that belong to different universality classes, unitary and symplectic, are simulated numerically. The perfectly conducting channel in the unitary class can be realized in zigzag graphene nano-ribbons and that in the symplectic class is known to appear in metallic carbon nanotubes. The existence of a perfectly conducting channel leads to novel conductance distribution functions and a shortening of the conductance decay length.

The two-terminal conductance of quantum Hall wires in the presence of spatially correlated disorder is investigated numerically. It is found that the conductance plateau transitions shift to higher energies than the corresponding Landau subband centers. By performing the analysis on individual samples, it is found that the critical energy of the plateau transition depends on the disorder realization, but the distance between successive critical energies is independent of the sample and is equal to the bulk Landau level separation. (C) 2007 Elsevier B.V. All rights reserved.

The quantum phase diagram of disordered quantum wires in a strong magnetic field is reviewed. For uncorrelated disorder potential the 2-terminal conductance, as calculated with the numerical transfer matrix method, shows zero temperature discontinuous transitions between exactly integer plateau values and zero. This is explained by the dimensional crossover of the bulk localisation length, which drives a transition from delocalised to localised edge states. In the thermodynamic limit, fixing the aspect ratio of the wire, there is a transition from the one dimensional chiral metal of extended edge states to localisation along the wire. In the vicinity of this chiral metal insulator transition (CMIT), states are identified which are superpositions of edge states with opposite chirality. The bulk contribution of such states is found to decrease with increasing wire width. Based on exact diagonalisation results for the eigenstates and their participation ratios, we conclude that these states are characteristic for the CMIT, and have the appearance of nonchiral edges states. Thereby these states are distinguishable from other states in the quantum Hall wire, namely, extended edge states, two-dimensionally (2D) localized, quasi-1D localized, and 2D critical states. In the presence of spatially correlated random potential we find with the numerical transfermatrix method that the potential correlation results in a shift of quantized conductance plateaus in long wires proportional to the strength of the random potential. This shift is found to be insensitive to the strength of magnetic fields and the same for all plateaus. A semiclassical explanation of this effect is proposed. We conclude with an outlook on modfications of the quantum phase diagram due to the spin degree of freedom of the electrons and their interactions. We discuss the stability of the phase diagram at finite temperature. © 2008 Wiley-VCH Verlag GmbH &amp Co. KGaA.

Quantum transport properties in quantum Hall wires in the presence of spatially correlated random potential are investigated numerically. It is found that the potential correlation reduces the localization length associated with the edge state, in contrast to the naive expectation that the potential correlation increases it. The effect appears as the sizable shift of quantized conductance plateaus in long wires, where the plateau transitions occur at energies much higher than the Landau band centers. The scale of the shift is of the order of the strength of the random potential and is insensitive to the strength of magnetic fields. Experimental implications are also discussed.

A Hall effect due to spin chirality in mesoscopic systems is investigated. We consider a four-terminal Hall system including local spins with geometry of a vortex domain wall, where strong spin chirality appears near the center of the vortex. The Fermi energy of the conduction electrons is assumed to be comparable to the exchange coupling energy where the adiabatic approximation ceases to be valid. Our results show a Hall effect where a voltage drop and a spin current arise in the transverse direction, which is shown to survive in the presence of weak disorder. The similarity between this Hall effect and the conventional spin Hall effect in systems with spin-orbit interaction is pointed out.

A Hall effect due to the spin chirality in mesoscopic systems is investigated numerically. We consider 4-terminal Hall systems including local spins with vortex domain wall geometry, where strong spin chirality appears near the vortex center. The Fermi energy of the conduction electrons is assumed to be comparable to the exchange coupling energy where the adiabatic approximation cannot be applied. Our results show Hall effect where voltage drop and spin current arise in transverse direction.

Electron transport phenomena in disordered electron systems with spin-orbit coupling in two dimensions and below are studied numerically. The scaling hypothesis is checked by analyzing the scaling of the quasi-1D localization length. A logarithmic increase of the mean conductance is also confirmed. These support the theoretical prediction that the two-dimensional metal in systems with spin-orbit coupling has a perfect conductivity. Transport through a Sierpinski carpet is also reported. (c) 2006 Elsevier B.V. All rights reserved.

Conductance fluctuations in two-dimensional random magnetic fields are investigated numerically in the case where the mean and the fluctuation of the random magnetic fields are of the same order. The conductance is evaluated by means of the Landauer formula. It is found that for a system with edge states, the conductance fluctuation exhibits clearly a Shubnikov-de Haas type oscillation in the weak field regime. (c) 2006 Elsevier B.V. All rights reserved.

We investigate the possibility of an Anderson transition below two dimensions in disordered systems of noninteracting electrons with symplectic symmetry. Numerical analysis of energy level statistics and conductance statistics on Sierpinski carpets with spin-orbit coupling indicates the occurrence of an Anderson transition below two dimensions.

We investigate numerically the spin polarization of the current in the presence of Rashba spin-orbit interaction (RSOI) in a 3-terminal conductor. We use equation-of-motion method to simulate the time evolution of the wave packet and focus on single-channel transport. A T-shaped conductor with uniform RSOI proposed by Kiselev and Kim and a Y-shaped conductor with nonuniform RSOI are considered. In the T-shaped conductor, the strength of RSOI is assumed to be uniform. We have found that the spin polarization becomes nearly 100% with little loss of conductance for sufficiently strong spin-orbit coupling. This is due to the spin-dependent group velocity of electrons at the junction which causes the spin separation. In the Y-shaped conductor, the strength of RSOI is modulated perpendicular to the charge Current. A spatial gradient of effective magnetic field due to the nonuniform RSOI causes the Stern-Gerlach type spin separation. The direction of the polarization is perpendicular to the Current and parallel to the spatial gradient. Again almost 100% spin polarization can be realized by this spin separation. (c) 2005 Elsevier B.V. All rights reserved.