Geometrical engineering of a two-band Chern insulator in two dimensions with arbitrary topological index

Sticlet, Doru; Piéchon, Frédéric; Fuchs, Jean-Noël; Kalugin, Pavel; Simon, Pascal

doi:10.1103/physrevb.85.165456

Cited by 113 publications

(104 citation statements)

References 45 publications

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“…When k spreads over Brillouin torus, h describes a closed surface Σ. The Chern number can then be viewed as -the (normalized) flux of a magnetic monopole located at the origin through the surface Σ -the number of times the surface Σ wraps around the origin (in particular, it is zero if the origin is « outside » Σ ; more precisely it is the homotopy class of Σ in the punctured space 3 − 0) -the number of (algebraically counted) intersections of a ray coming from the origin with Σ, which is the method used in [29].…”

Section: Berry Curvature and Chern Numbermentioning

confidence: 99%

An introduction to topological insulators

Fruchart

Carpentier

2013

Comptes Rendus. Physique

165

126

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Electronic bands in crystals are described by an ensemble of Bloch wave functions indexed by momenta defined in the first Brillouin Zone, and their associated energies. In an insulator, an energy gap around the chemical potential separates valence bands from conduction bands. The ensemble of valence bands is then a well defined object, which can possess non-trivial or twisted topological properties. In the case of a twisted topology, the insulator is called a topological insulator. We introduce this notion of topological order in insulators as an obstruction to define the Bloch wave functions over the whole Brillouin Zone using a single phase convention. Several simple historical models displaying a topological order in dimension two are considered. Various expressions of the corresponding topological index are finally discussed. RésuméLes bandes électroniques dans un cristal sont définies par un ensemble de fonctions d'onde de Bloch dépendant du moment défini dans la première zone de Brillouin, ainsi que des énergies associées. Dans un isolant, les bandes de valence sont séparées des bandes de conduction par un gap en énergie. L'ensemble des bandes de valence est alors un objet bien défini, qui peut en particulier posséder une topologie non triviale. Lorsque cela se produit, l'isolant correspondant est appelé isolant topologique. Nous introduisons cette notion d'ordre topologique d'une bande comme une obstruction à la définition des fonctions d'ondes de Bloch à l'aide d'une convention de phase unique. Plusieurs modèles simples d'isolants topologiques en dimension deux sont considérés. Différentes expressions des indices topologiques correspondants sont finalement discutées.Keywords: topological insulator, topological band theory, quantum anomalous Hall effect, quantum spin Hall effect, Chern insulator, Kane-Mele insulator Mots-clés : isolant topologique, théorie des bandes topologique, effet Hall quantique anomal, effet Hall quantique de spin, isolant de Chern, isolant de Kane-Mele

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Section: Berry Curvature and Chern Numbermentioning

confidence: 99%

An introduction to topological insulators

Fruchart

Carpentier

2013

Comptes Rendus. Physique

165

126

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“…Although as it is wellknown and seen of Eq.5, the time-reversal symmetry breaks when any magnetization is turned on, system remains chiralpolarized even if out-of-plane magnetization is applied 42 . In fact, by application of z-axis magnetization, as shown in Fig.2, the degeneracy of the bands with opposite chiralities breaks 44 .…”

Section: A Hamiltonianmentioning

confidence: 93%

“…where, α is the chiral index 24,41,42 which has two different signs ± corresponding to the upper and lower block digonal matrices in Eq.3. Here the last term is the mass term which its negative sign leads to topologically non-trivial band structure.…”

Section: A Hamiltonianmentioning

confidence: 99%

Effect of chiral selective tunneling on quantum transport in magnetic topological-insulator thin films

Sabze

Cheraghchi

2017

Phys. Rev. B

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The electronic transport properties in magnetically doped ultra-thin films of topological-insulators is investigated by using Landauer-buttiker formalism. The chiral selective tunneling is addressed in such systems which leads to transport gap and as a consequence current blocking. This quantum blocking of transport occurs when the magnetic states with opposite chirality are aligned energetically. This can be obsereved when an electron tunnels through a barrier or well of magnetic potential induced by the exchange field. It is proved and demonstrated that this chiral transition rule fails when structural inversion asymmetric potential or an in-plane magnetization is turning on. This new finding is useful to interpret quantum transport through topological-insulator thin films especially to shed light on longitudinal conductance behavior of quantum anomalous Hall effect. Besides, one can design electronic devices by means of magnetic topological-insulator thin films based on the chiral selective tunneling leading to negative differential resistance.

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“…We now ask whether the driving can induce gapped phases with |c 1 | > 1. One way to address this question is to notice that in order to obtain larger Chern numbers, it is necessary to have more than one pair of Dirac points [23]. Previous studies have shown that multiphoton resonances, occurring when the Floquet bands overlap, can induce additional pairs of Dirac points, provided that TRS holds [20,24].…”

mentioning

confidence: 99%

“…Following the elegant method of the Brouwer degree developed in Ref. [23], the Chern number (of the lower quasienergy band) reads…”

mentioning

confidence: 99%

Engineering anomalous quantum Hall plateaus and antichiral states with ac fields

2014

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We investigate the ac electric field induced quantum anomalous Hall effect in honeycomb lattices and derive the full phase diagram for arbitrary field amplitude and phase polarization. We show how to induce antichiral edge modes as well as topological phases characterized by a Chern number larger than 1 by means of suitable drivings. In particular, we find that the Chern number develops plateaus as a function of the frequency, providing a time-dependent analog to the ones in the quantum Hall effect. Introduction. The realization of different topological states of matter is one of the major challenges for both fundamental reasons and technological perspectives. Several of these states have been originally predicted in the honeycomb lattice, whose Dirac-like band structure brings the system at a critical point of a topological phase transition. There, the development of gaps by different mechanisms results in a variety of topological phases [1,2]. In this line, the quantum spin Hall phase was first obtained in HgTe quantum wells [3,4], where the topological phase transition was controlled by the thickness of the quantum well. Likewise, the quantum anomalous Hall effect (QAHE)-for which time-reversal symmetry (TRS) is broken in the absence of a magnetic field-was originally proposed by Haldane in the honeycomb lattice [1]. However, despite the success of the theoretical model, it has been very difficult to achieve experimentally, until very recently in doped topological insulators [5].Simultaneously, different techniques have been proposed to externally control the topological properties of a system. One of the most promising consists in periodically driving the system in order to achieve a topological phase transition [6][7][8][9][10][11]. Proposals with a time periodic driving in semiconductors [6,12], optical lattices [13], or graphene [7,[14][15][16][17][18] have been suggested to achieve various dynamical generalizations of static topological phases, called Floquet topological phases [12], and have been recently observed in photonic crystals [19]. Importantly for our purposes, the previous studies for the honeycomb lattice were restricted either to numerical calculations in finite-size systems, where the physical mechanism driving the topological phase transitions was not clear, or to very high frequencies and low-energy approximations, where most of the phase diagram remained unknown.In the present work, we derive the full phase diagram of periodically driven honeycomb lattices in the QAHE regime, by explicitly calculating the Chern number of the Floquet bands. Our model is valid for arbitrary field frequency, amplitude, and polarization and therefore goes beyond the single Dirac cone description and the rotating wave approximation. Surprisingly, we find that a clockwise driving may also lead to counterclockwise (or antichiral) edge states, and demonstrate that their appearance is linked to two distinct band inversion mechanisms. Moreover, we show how to induce topological phases with Chern numbers larger tha...

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Geometrical engineering of a two-band Chern insulator in two dimensions with arbitrary topological index

Cited by 113 publications

References 45 publications

An introduction to topological insulators

An introduction to topological insulators

Effect of chiral selective tunneling on quantum transport in magnetic topological-insulator thin films

Engineering anomalous quantum Hall plateaus and antichiral states with ac fields

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