Dirac electrons in topological crystalline insulators
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One of the simplest but universal challenges at the frontier of materials physics is controlling band structure, both for realizing novel phenomena and for practical functionalities. In this talk, I will describe our atomic scale investigations of novel quantum materials called topological crystalline insulators (TCIs). In TCIs, topology and crystal symmetry intertwine to create massless Dirac electrons, which can be described by the same equations used for relativistic particles traveling close to the speed of light. Using Landau level spectroscopy and atomic resolution imaging in TCIs, we have discovered massive Dirac electrons coexisting with massless Dirac electrons. Our findings experimentally demonstrate the unique and extraordinary tunability of Dirac electrons in TCIs, which provides a significant step for realizing fundamentally and practically important quantum states via strain engineering. As the final part of this talk, I will also introduce our recent attempt of combing visualization techniques with epitaxial thin film based quantum materials design.Keywords:
Topological insulator
Helical Dirac fermion
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The concept of topological insulator (TI) has introduced a new point of view to condensed-matter physics, relating a priori unrelated subfields such as quantum (spin, anomalous) Hall effects, spin-orbit coupled materials, some classes of nodal superconductors and superfluid $^3$He, etc. From a technological point of view, topological insulator is expected to serve as a platform for realizing dissipationless transport in a non-superconducting context. The topological insulator 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 contrast the cases of weak and strong TIs. We study robustness of the surface and the bulk of TIs against disorder, addressing the physics of Dirac and Weyl semimetals as a new perspective of research in the field.
Topological insulator
Surface States
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Dirac fermions in condensed matter physics hold great promise for novel fundamental physics, quantum devices and data storage applications. IV-VI semiconductors, in the inverted regime, have been recently shown to exhibit massless topological surface Dirac fermions protected by crystalline symmetry, as well as massive bulk Dirac fermions. Under a strong magnetic field (B), both surface and bulk states are quantized into Landau levels that disperse as B^1/2, and are thus difficult to distinguish. In this work, magneto-optical absorption is used to probe the Landau levels of high mobility Bi-doped Pb0.54Sn0.46Te topological crystalline insulator (111)-oriented films. The high mobility achieved in these thin film structures allows us to probe and distinguish the Landau levels of both surface and bulk Dirac fermions and extract valuable quantitative information about their physical properties. This work paves the way for future magnetooptical and electronic transport experiments aimed at manipulating the band topology of such materials.
Topological insulator
Landau quantization
Massless particle
Surface States
Magnetism
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AbstractA wide range of materials, like d-wave superconductors, graphene, and topological insulators, share a fundamental similarity: their low-energy fermionic excitations behave as massless Dirac particles rather than fermions obeying the usual Schrödinger Hamiltonian. This emergent behavior of Dirac fermions in condensed matter systems defines the unifying framework for a class of materials we call "Dirac materials." In order to establish this class of materials, we illustrate how Dirac fermions emerge in multiple entirely different condensed matter systems and we discuss how Dirac fermions have been identified experimentally using electron spectroscopy techniques (angle-resolved photoemission spectroscopy and scanning tunneling spectroscopy). As a consequence of their common low-energy excitations, this diverse set of materials shares a significant number of universal properties in the low-energy (infrared) limit. We review these common properties including nodal points in the excitation spectrum, density of states, specific heat, transport, thermodynamic properties, impurity resonances, and magnetic field responses, as well as discuss many-body interaction effects. We further review how the emergence of Dirac excitations is controlled by specific symmetries of the material, such as time-reversal, gauge, and spin–orbit symmetries, and how by breaking these symmetries a finite Dirac mass is generated. We give examples of how the interaction of Dirac fermions with their distinct real material background leads to rich novel physics with common fingerprints such as the suppression of back scattering and impurity-induced resonant states.PACS:: 73.20.-r Electron states at surfaces and interfaces73.25.+i Surface conductivity and carrier phenomena73.50.-h Electronic transport phenomena in thin films74.20.-z Theories and models of superconducting state73.22.Pr Electronic structure of graphene75.76.+j Spin transport effects71.55.-i Impurity and defect levelsKeywords: Dirac materialsd-wave superconductorsgraphenetopological insulatorschiralityback scatteringimpurity resonance AcknowledgementsWe are grateful to D. Arovas, D. Abergel, R. Biswas, A.H. Castro Neto, H. Dahal, V. Fal'ko, M. Fogelström, J. Fransson, M. Graf, Z. Huang, P. Hoffmann, M.I. Katsnelson, A.I. Lichtenstein, J. Linder, F. Lombardi, H. Manoharan, J. Moore, N. Nagaosa, K. Scharnberg, Z.X. Shen, Y. Tanaka, O. Tjernberg, A. Yazdani, S.C. Zhang, J.X. Zhu for discussions. This work has been supported by US DOE BES E304, LDRD, University of California UCOP-09-027, the German Research Foundation (DFG) via SFB 668 and SPP 1459, Dirac Materials ERC-DM-321031, and the Swedish Research Council (VR). TOW thanks KITP Santa Barbara for hospitality during a visit where parts of this work were written.For figures with copyright from the American Physical Society: Readers may view, browse, and/or download material for temporary copying purposes only, provided these uses are for noncommercial personal purposes. Except as provided by law, this material may not be further reproduced, distributed, transmitted, modified, adapted, performed, displayed, published, or sold in whole or part, without prior written permission from the American Physical Society.
Helical Dirac fermion
Topological insulator
Fermion doubling
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The Dirac equation for relativistic electron waves is the parent model for Weyl and Majorana fermions as well as topological insulators. Simulation of Dirac physics in three-dimensional photonic crystals, though fundamentally important for topological phenomena at optical frequencies, encounters the challenge of synthesis of both Kramers double degeneracy and parity inversion. Here we show how type-II Dirac points---exotic Dirac relativistic waves yet to be discovered---are robustly realized through the nonsymmorphic screw symmetry. The emergent type-II Dirac points carry nontrivial topology and are the mother states of type-II Weyl points. The proposed all-dielectric architecture enables robust cavity states at photonic-crystal---air interfaces and anomalous refraction, with very low energy dissipation.
Parity (physics)
Topological insulator
Point reflection
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The engineering of Dirac matter using photonic materials opens unhindered opportunities to explore unconventional transport and novel topological phases. Thanks to the direct optical access to the spatial and momentum wavefunctions and spectrum exciton polaritons in semiconductor microcavities appear as an extraordinary platform to emulate 1D and 2D Hamiltonians, including Dirac Hamiltonians. By etching a GaAs-based microcavity, a honeycomb lattice for polaritons has been fabricated. The lowest two bands of this structure emulate for photons the π and π* bands of graphene. Remarkably, the system also permits exploring orbital degrees of freedom, inaccessible in actual graphene. In the first part of this thesis a polariton emulator is used to address the physics of edge states in a honeycomb lattice. New edge states, with flat and dispersive bands, have been discovered and visualised in orbital graphene.In the second part of the thesis we demonstrate experimentally a method to tailor the Dirac dispersion for photons. By implementing uni-axial strain in the honeycomb lattice, Dirac photons that combine zero, finite and infinite effective masses are created. The experimental and theoretical results here presented open new perspectives for the engineering of interfaces between photonic lattices with different types of Dirac dispersions. Furthermore, the excitonic component of polaritons assures sensitivity to external magnetic fields, providing the possibility to break the time reversal symmetry of the system and to study photonic topological edge states in exotic Dirac cones. Finally, nonlinear Dirac physics can be probed in this system owing to polariton-polariton interactions. The engineering of Dirac matter using photonic materials opens unhindered opportunities to explore unconventional transport and novel topological phases. Thanks to the direct optical access to the spatial and momentum wavefunctions and spectrum exciton polaritons in semiconductor microcavities appear as an extraordinary platform to emulate 1D and 2D Hamiltonians, including Dirac Hamiltonians. By etching a GaAs-based microcavity, a honeycomb lattice for polaritons has been fabricated. The lowest two bands of this structure emulate for photons the π and π* bands of graphene. Remarkably, the system also permits exploring orbital degrees of freedom, inaccessible in actual graphene. In the first part of this thesis a polariton emulator is used to address the physics of edge states in a honeycomb lattice. New edge states, with flat and dispersive bands, have been discovered and visualised in orbital graphene. In the second part of the thesis we demonstrate experimentally a method to tailor the Dirac dispersion for photons. By implementing uni-axial strain in the honeycomb lattice, Dirac photons that combine zero, finite and infinite effective masses are created. The experimental and theoretical results here presented open new perspectives for the engineering of interfaces between photonic lattices with different types of Dirac dispersions. Furthermore, the excitonic component of polaritons assures sensitivity to external magnetic fields, providing the possibility to break the time reversal symmetry of the system and to study photonic topological edge states in exotic Dirac cones. Finally, nonlinear Dirac physics can be probed in this system owing to polariton-polariton interactions.
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Three dimensional (3D) topological insulators are novel states of quantum matter that feature spin-momentum locked helical Dirac fermions on their surfaces and hold promise to open new vistas in spintronics, quantum computing and fundamental physics. Experimental realization of many of the predicted topological phenomena requires finding multi-variant topological band insulators which can be multiply connected to magnetic semiconductors and superconductors. Here we present our theoretical prediction and experimental discovery of several new topological insulator classes in AB2X4(124), A2B2X5(225), MN4X7(147), A2X2X'(221) [A,B=Pb,Ge,Sb,Bi and M,N=Pb,Bi and X,X'=Chalcogen family]. We observe that these materials feature gaps up to about 0.35eV. Multi-variant nature allows for diverse surface dispersion tunability, Fermi surface spin-vortex or textured configurations and spin-dependent electronic interference signaling novel quantum transport processes on the surfaces of these materials. Our discovery also provides several new platforms to search for topological-superconductivity (arXiv:0912.3341v1 (2009)) in these exotic materials.
Topological insulator
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Realization of photonic charge-2 Dirac point by engineering super-modes in topological superlattices
Abstract Quite recently, an unconventional variety of fourfold linear band degeneracy points has been discovered in certain condensed-matter systems. Contrary to standard 3-D Dirac monopoles, these quadruple points known as the charge-2 Dirac points are characterized by nonzero topological charges, which can be exploited to delve into hitherto unknown realms of topological physics. Here, we report on the experimental realization of a charge-2 Dirac point by deliberately engineering hybrid topological states, called super-modes, in a 1-D optical superlattice system with synthetic dimensions. Utilizing direct reflection and transmission measurements, we propose the existence of the synthetic charge-2 Dirac point in the visible region. We also show an experimental approach to manipulating two spawned Weyl points possessing equal charge. Topological end modes resulting from the charge-2 Dirac point can be delicately controlled within truncated superlattices, opening a pathway to rationally engineer local fields with intense enhancement.
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Dirac points lie at the heart of many fascinating phenomena in condensed matter physics, from massless electrons in graphene to the emergence of conducting edge states in topological insulators. At a Dirac point, two energy bands intersect linearly and the particles behave as relativistic Dirac fermions. In solids, the rigid structure of the material sets the mass and velocity of the particles, as well as their interactions. A different, highly flexible approach is to create model systems using fermionic atoms trapped in an optical lattice, a method which so far has only been applied to explore simple lattice structures. In my talk I will report on the creation of Dirac points with adjustable properties in a tunable honeycomb optical lattice [1]. Using momentum-resolved interband transitions, we observe a minimum band gap inside the Brillouin zone at the position of the Dirac points. We exploit the unique tunability of our lattice potential to adjust the effective mass of the Dirac fermions by breaking the inversion symmetry of the lattice. Moreover, changing the lattice anisotropy allows us to move the position of the Dirac points inside the Brillouin zone. When increasing the anisotropy beyond a critical limit, the two Dirac points merge and annihilate each other. We map out this topological transition in lattice parameter space and find excellent agreement with ab initio calculations. Our results not only pave the way to model materials where the topology of the band structure plays a crucial role, but also provide the possibility to explore many-body phases resulting from the interplay of complex lattice geometries with interactions.
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The Dirac equation was able to unite relativity with quantum mechanics and successfully describe the behaviour of spin-1/2 particles. Over the past few decades it has been discovered that the electronic properties of many materials is also governed by emergent physics akin to that described by the Dirac equation, these materials are known a ``Dirac materials''. Often there is a deep connection between topology and the appearance of linearly dispersing electronic bands which result in a material's Dirac-like physics.
In this thesis we will investigate the impact disorder has on Dirac materials. In particular we will be interested in the theoretical description of transport properties - such as the electrical conductivity of a material - that result from their Dirac physics. Dirac materials provide a fascinating playground for the study of novel quantum mechanical phenomena, both theoretically and in the lab. As such, many of the examples in this thesis are the product of close theoretical and experimental collaborations.
We begin this thesis with a detailed overview of the ever-growing class of materials which obey a Dirac-like description and by introducing many of the concepts used in later chapters. Having done this we turn to a discussion of disorder. Of particular importance will be that Dirac electrons are protected from back-scattering off impurity potentials that retain the symmetries protecting the Dirac point. We will use our knowledge of disordered Dirac materials to calculate the conductivity of the surface of a topological insulator.
In the second half of this thesis we will discuss three novel phenomena which we theoretically describe and have been experimentally observed in Dirac materials: Firstly, we will discuss how it is possible to enable back-scattering in a Dirac material, in a controlled manner. We will see that this is achieved by the application of a magnetic field in the plane of a topological insulator's surface which leads to an anisotropy of magnetoresistance and, associated to this, a planar Hall effect. Secondly, we will discuss confinement of Dirac surface states on a very thin nanowire. We will show that the quantisation of the wave-function around the wire leads to oscillatory behaviour of the resistivity that has also been experimentally observed. Finally, we turn to 3d Dirac semi-metals, we will show that their quasi-1d physics in a strong magnetic field leads to a magnetoresistivity that is strongly dependent on the angle of the applied magnetic field when there are multiple Fermi-surfaces in the Brillouin zone.
Topological insulator
Helical Dirac fermion
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Helical Dirac fermion
Massless particle
Bilayer graphene
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