Disorder and interference: localization phenomena

Localization of the helical edge states in quantum spin Hall insulators requires breaking time reversal invariance. In experiments this is naturally implemented by applying a weak magnetic field B. We propse a model based on scattering theory that describes the localization of helical edge states due to coupling to random magnetic fluxes. We find that the localization length is proportional to B −2 when B is small, and saturates to a constant when B is sufficiently large. We estimate especially the localization length for the HgTe/CdTe quantum wells with known experimental parameters. PACS numbers: 73.20.Fz, The prediction and discovery of quantum spin Hall insulators (QSHIs) [1, 1, 2, 2] has opened a door to an unexpected category of topological phases in condensed matter [5][6][7][8], and revealed a new route to investigations of edge/boundarystate physics [9][10][11]. Although the prototypes of QSHIs [1, 1] are mainly based on two copies of quantum Hall insulators, which have been investigated for more than three decades [12,13], it was soon realized that the fundamental importance of time reversal invariance (TRI) distinguishes the two systems in a profound way [2]. Indeed, QSHIs, unlike the Zclassified quantum Hall insulators [14], belong to a class of two-dimensional time-reversal-invariant Z 2 topological insulators [2]. The defining feature of QSHIs, as its name suggests, is a pair of helical edge states that persist in the bulk insulating gap of the system [1, 1, 2, 2, 9].The topological power of QSHIs lies precisely in the robustness of the helical edge states against generic perturbations due to unavoidable disorder in every experimental setup, unless TRI is broken. In the presence of both TRI breaking and disorder, the helical edge states will be localized, and the general framework of Anderson's localization theory applies [15]. Nevertheless, the localization of the helical edge states distinguishes itself from conventional one-dimensional localization when the focus is placed on the crucial role TRI plays in the problem. This point becomes especially relevant as TRI can be broken continuously, for instance, by turning on a magnetic field gradually. Indeed, the sensibility of transport though helical edge states to weak magnetic field has been demonstrated experimentally in the measurement of magneto-conductance in topologically nontrivial HgTe/CdTe quantum wells [2, 3]. Related theoretical analyses have been carried out that include the interplay between TRI-breaking and disorder, but mainly consider magnetic impurities [17,18], or bulk random potential combined with magnetic field [19]. A transparent edge theory that focuses on the magnetic-field-dependent localization of the helical edge states, however, is still missing.In this paper we propose a model that explicitly addresses the question on how the localization of helical edge states occurs as a weak magnetic field is gradually turned on. Our model is based on the scattering theory of edge states in the presence of generic edge disorder. In...

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“…The inverse localization length γ = 1/ loc is defined in terms of the scaling variable as: [21][22][23] …”

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confidence: 99%

Magnetic-Field-Induced Localization in 2D Topological Insulators

Delplace

Büttiker

2012

Phys. Rev. Lett.

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“…The transfer matrix is largely determined by the symmetries of the scattering problem. 57,58 Unitarity or current conservation implies det M n = 1 = |r n | 2 + |t n | 2 . Thus the total reflection and transmission probabilities can be expressed as R n = |r n | 2 = sin 2 φ n and T n = |t n | 2 = cos 2 φ n , and we find it useful to parameterize M n as…”

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confidence: 99%

Localization behavior of Dirac particles in disordered graphene superlattices

2012

Self Cite

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Graphene superlattices (GSLs), formed by subjecting a monolayer graphene sheet to a periodic potential, can be used to engineer band structures and, from there, charge transport properties, but these are sensitive to the presence of disorder. The localization behavior of massless 2D Dirac particles induced by weak disorder is studied for both scalar-potential and vector-potential GSLs, computationally as well as analytically by a weak-disorder expansion. In particular, it is investigated how the Lyapunov exponent (inverse localization length) depends on the incidence angle to a 1D GSL. Delocalization resonances are found for both scalar and vector GSLs. The sharp angular dependence of the Lyapunov exponent may be exploited to realize disorder-induced filtering, as verified by full 2D numerical wave packet simulations.

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“…A transfer matrix method is a very effective tool for the numerical computation of the localization length [25]. The formalism can be also used to find analytically the delocalized modes that are appearing in systems with the correlated binary disorder.…”

Section: Delocalized Modes For N -Mersmentioning

confidence: 99%

“…3 we present numerically calculated Anderson localization lengths (using the standard transfer matrix method, see e.g. [25]) for the 4-mer and the mean occupation 0.55, the strength of disorder is set to V 0 = 0.4. The red dashed line corresponds to the system with uniform tunnelings t = 1 while the black solid line is obtained for t = 0.9.…”

Section: Off-diagonal Correlated Disordermentioning

confidence: 99%

Role of Correlations and Off-Diagonal Terms in Binary Disordered One-Dimensional Systems

et al. 2015

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We investigate one dimensional tight binding model in the presence of a correlated binary disorder. The disorder is due to the interaction of particles with heavy immobile other species. Off-diagonal disorder is created by means of a fast periodic modulation of interspecies interaction. The method based on transfer matrix techniques allows us to calculate the energies of extended modes in the correlated binary disorder. We focus on N -mer correlations and regain known results for the case of purely diagonal disorder. For off-diagonal disorder we find resonant energies. We discuss ambiguous properties of those states and compare analytical results with numerical calculations. Separately we describe a special case of the dual random dimer model.

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Disorder and interference: localization phenomena

Abstract: This is v3 of arXiv:1005.0915 with minor corrections marked in color. Communications from B. Nowak and C. Texier are gratefully acknowledged.

Cited by 48 publications

References 117 publications

Magnetic-Field-Induced Localization in 2D Topological Insulators

Magnetic-Field-Induced Localization in 2D Topological Insulators

Localization behavior of Dirac particles in disordered graphene superlattices

Role of Correlations and Off-Diagonal Terms in Binary Disordered One-Dimensional Systems

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