Effective Confining Potential of Quantum States in Disordered Media

Arnold, Douglas N.; David, Guy; Jerison, David; Mayboroda, Svitlana; Filoche, Marcel

doi:10.1103/physrevlett.116.056602

Cited by 117 publications

(176 citation statements)

References 19 publications

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“…The localization landscape is a new tool in the study of Anderson localization, pioneered in 2012 by Filoche and Mayboroda [1], which has since stimulated much computational and conceptual progress [2][3][4][5][6][7][8][9][10]. The "landscape" of a Hamiltonian H is a function u(r) that provides an upper bound for eigenstates ψ at energy E > 0: |ψ(r)|/|ψ| max ≤ E u(r), |ψ| max = max r |ψ(r)|.…”

Section: Introduction -mentioning

confidence: 99%

Localization landscape for Dirac fermions

et al. 2020

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In the theory of Anderson localization, a landscape function predicts where wave functions localize in a disordered medium, without requiring the solution of an eigenvalue problem. It is known how to construct the localization landscape for the scalar wave equation in a random potential, or equivalently for the Schrödinger equation of spinless electrons. Here we generalize the concept to the Dirac equation, which includes the effects of spin-orbit coupling and allows to study quantum localization in graphene or in topological insulators and superconductors. The landscape function u(r) is defined on a lattice as a solution of the differential equation H u(r) = 1, where H is the Ostrowsky comparison matrix of the Dirac Hamiltonian. Random Hamiltonians with the same (positive definite) comparison matrix have localized states at the same positions, defining an equivalence class for Anderson localization. This provides for a mapping between the Hermitian and non-Hermitian Anderson model.

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Section: Introduction -mentioning

confidence: 99%

Localization landscape for Dirac fermions

et al. 2020

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“…1/u, acts as an "effective" confining potential seen by the localized eigenstates. 28 In other words, the quantum properties of a disordered potential such as quantum wave interference, quantum confinement and tunneling are directly translated into 1/u in the form of a classical potential. An important property of the function 1/u (or u) is that its crest (respectively, valley) lines partition the domain into the localization subregions, i.e., the regions of lower energy solutions of the Schrödinger equation (LL1, Ref.…”

mentioning

confidence: 99%

Localization landscape theory of disorder in semiconductors. II. Urbach tails of disordered quantum well layers

Piccardo

et al. 2017

Phys. Rev. B

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Urbach tails in semiconductors are often associated to effects of compositional disorder. The Urbach tail observed in InGaN alloy quantum wells of solar cells and LEDs by biased photocurrent spectroscopy is shown to be characteristic of the ternary alloy disorder. The broadening of the absorption edge observed for quantum wells emitting from violet to green (indium content ranging from 0 to 28%) corresponds to a typical Urbach energy of 20 meV. A 3D absorption model is developed based on a recent theory of disorder-induced localization which provides the effective potential seen by the localized carriers without having to resort to the solution of the Schrödinger equation in a disordered potential. This model incorporating compositional disorder accounts well for the experimental broadening of the Urbach tail of the absorption edge. For energies below the Urbach tail of the InGaN quantum wells, type-II well-to-barrier transitions are observed and modeled. This contribution to the below bandgap absorption is particularly efficient in near-UV emitting quantum wells. When reverse biasing the device, the well-to-barrier below bandgap absorption exhibits a red shift, while the Urbach tail corresponding to the absorption within the quantum wells is blue shifted, due to the partial compensation of the internal piezoelectric fields by the external bias. The good agreement between the measured Urbach tail and its modeling by the new localization theory demonstrates the applicability of the latter to compositional disorder effects in nitride semiconductors.

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“…To prove the estimates for T (1) and T (2) we argue as follows. By the definition ofG h(x),d(x) (x, y; x ′ , y ′ ), we see that T (0) f (x, y) satisfies the equation…”

Section: The Approximation Of the Torsion Function By Vmentioning

confidence: 99%

The Torsion Function of Convex Domains of High Eccentricity

Beck

2019

Potential Anal

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The torsion function of a convex planar domain Ω has convex level sets, but explicit formulae are known only for rectangles and ellipses. Here we study the torsion function on convex planar domains of high eccentricity. We obtain an approximation for the torsion function by viewing the domain as a perturbation of a rectangle in order to define an approximate Green's function for the Laplacian. For a class of convex domains we use this approximation to establish sharp bounds on the Hessian and the infinitesimal shape of the level sets around its maximum. We also use these results to compare the behaviour of the torsion function and the first eigenfunction of the Dirichlet Laplacian around their respective maxima.

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Effective Confining Potential of Quantum States in Disordered Media

Cited by 117 publications

References 19 publications

Localization landscape for Dirac fermions

Localization landscape for Dirac fermions

Localization landscape theory of disorder in semiconductors. II. Urbach tails of disordered quantum well layers

The Torsion Function of Convex Domains of High Eccentricity

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