Landau levels in 2D materials using Wannier Hamiltonians obtained by first principles
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Abstract:
We present a method to calculate the Landau levels and the corresponding edge states of two dimensional (2D) crystals using as a starting point their electronic structure as obtained from standard density functional theory (DFT). The DFT Hamiltonian is represented in the basis of maximally localized Wannier functions. This defines a tight-binding Hamiltonian for the bulk that can be used to describe other structures, such as ribbons, provided that atomic scale details of the edges are ignored. The effect of the orbital magnetic field is described using the Peierls substitution in the hopping matrix elements. Implementing this approach in a ribbon geometry, we obtain both the Landau levels and the dispersive edge states for a series of 2D crystals, including graphene, Boron Nitride, MoS$_2$ , Black Phosphorous, Indium Selenide and MoO$_3$ . Our procedure can readily be used in any other 2D crystal, and provides an alternative to effective mass descriptions.Keywords:
Wannier function
Hamiltonian (control theory)
Landau quantization
Ribbon
Crystal field theory
Tight binding
We discuss how to construct tight-binding models for ultracold atoms in honeycomb potentials, by means of the maximally localized Wannier functions (MLWFs) for composite bands introduced by Marzari and Vanderbilt [Phys. Rev. B 56, 12847 (1997)]. In particular, we work out the model with up to third-nearest neighbors, and provide explicit calculations of the MLWFs and of the tunneling coefficients for the graphenelike potential with two degenerate minima per unit cell. Finally, we discuss the degree of accuracy in reproducing the exact Bloch spectrum of different tight-binding approximations, in a range of typical experimental parameters.
Tight binding
Ultracold atom
Wannier function
Maxima and minima
Honeycomb
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Maximally localized Wannier functions are localized orthogonal functions that can accurately represent given Bloch eigenstates of a periodic system at a low computational cost, thanks to the small size of each orbital. Tight-binding models based on the maximally localized Wannier functions obtained from different systems are often combined to construct tight-binding models for large systems such as a semi-infinite surface. However, the corresponding maximally localized Wannier functions in the overlapping region of different systems are not identical, and this discrepancy can introduce serious artifacts to the combined tight-binding model. Here, we propose two methods to seamlessly stitch two different tight-binding models that share some basis functions in common. First, we introduce a simple and efficient method: (i) finding the best matching maximally localized Wannier function pairs in the overlapping region belonging to the two tight-binding models, (ii) rotating the spin orientations of the two corresponding Wannier functions to make them parallel to each other, and (iii) making their overall phases equal. Second, we propose a more accurate and generally applicable method based on the iterative minimization of the difference between the Hamiltonian matrix elements in the overlapping region. We demonstrate our methods by applying them to the surfaces of diamond, GeTe, Bi$_2$Se$_3$, and TaAs. Our methods can be readily used to construct reliable tight-binding models for surfaces, interfaces, and defects.
Tight binding
Wannier function
Hamiltonian (control theory)
Minification
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Tight-binding models for ultracold atoms in optical lattices can be properly defined by using the concept of maximally localized Wannier functions for composite bands. The basic principles of this approach are reviewed here, along with different applications to lattice potentials with two minima per unit cell, in one and two spatial dimensions. Two independent methods for computing the tight-binding coefficients - one ab initio, based on the maximally localized Wannier functions, the other through analytic expressions in terms of the energy spectrum - are considered. In the one dimensional case, where the tight-binding coefficients can be obtained by designing a specific gauge transformation, we consider both the case of quasi resonance between the two lowest bands, and that between s and p orbitals. In the latter case, the role of the Wannier functions in the derivation of an effective Dirac equation is also reviewed. Then, we consider the case of a two dimensional honeycomb potential, with particular emphasis on the Haldane model, its phase diagram, and the breakdown of the Peierls substitution. Tunable honeycomb lattices, characterized by movable Dirac points, are also considered. Finally, general considerations for dealing with the interaction terms are presented.
Wannier function
Tight binding
Ultracold atom
Lattice (music)
Maxima and minima
Hamiltonian (control theory)
Honeycomb
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A simple tight-binding Hamiltonian is used in the band model to calculate binding energies of crystal structures in silicon using the recursion method. The use of matrix orthogonal polynomials is described for efficient computation of the symmetry-preserving matrix Green functions. This provides the basis for a proposed rotationally invariant interatomic force algorithm.
Tight binding
Hamiltonian (control theory)
Matrix (chemical analysis)
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Tight binding
Wannier function
Ultracold atom
Lattice (music)
Hamiltonian (control theory)
Maxima and minima
Honeycomb
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Twisted multi-layer heterostructures have been considered a platform for studying highly correlated many-particle systems, hosting the emerging phenomena of correlation physics. Electronic structure calculations of such materials which mainly focused on Twisted Bilayer Graphene (TBG) in the framework of the independent-electron approximation, despite the complexity, accurately match the experimental results around the Fermi level. Here, we present a convenient ab-initio π-bands tight-binding model to calculate the band structure of TBG based on the Wannier interlayer hopping parameters obtained from non-twisted bilayer graphene. Our approach is based on mapping hopping parameters onto the real-space TBG interlayer couplings, using Radial Basis Function (RBF) interpolation method. This accurate mapping, naturally, transfers all the symmetries and orbital shape features of the bilayer graphene lattice to the TBG lattice and preserves coupling orientation dependencies in addition to distance dependencies. The importance of this fact is explained by showing the effect of the threefold rotation symmetry of AB interlayer coupling on the flat band of the TBG band structure. In addition, due to real-space study, the model gives us a comprehensive and intuitive view of the role of each interlayer orbital coupling in the TBG band structure and the structural variation relative to twisting. Using this model, it is not only possible to perform calculations for any combination of heterostructure materials, but the Hamiltonian output can be used in other calculations.
Tight binding
Wannier function
Bilayer graphene
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Here, the electronic behavior of half-Heusler NaAuS is studied using PBEsol exchange correlation functional by plotting the band structure curve. These bands are reproduced using maximally localized Wannier function using WANNIER90. Tight-binding bands are nicely matched with density functional theory bands. By fitting the tight-binding model, hopping parameter for NaAuS is obtained by including Na 2s, 2p, Au 6s, 5p, 5d and S 3s, 3p orbitals within the energy interval of -5 to 16 eV around the Fermi level. In present study, hopping integrals for NaAuS are computed for the first primitive unit cell atoms as well as the first nearest neighbor primitive unit cell. The most dominating hopping integrals are found for Na (3s) – S (3s), Na (2px) – S (2px), Au (6s) – S (3px), Au (6s) – S (3py) and Au (6s) – S (3pz) orbitals. The hopping integrals for the first nearest neighbor primitive unit cell are also discussed in this manuscript. In future, these hopping integrals are very important to find the topological invariant for NaAuS compound.
Wannier function
Tight binding
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Tight-binding models for ultracold atoms in optical lattices can be properly defined by using the concept of maximally localized Wannier functions for composite bands. The basic principles of this approach are reviewed here, along with different applications to lattice potentials with two minima per unit cell, in one and two spatial dimensions. Two independent methods for computing the tight-binding coefficients - one ab initio, based on the maximally localized Wannier functions, the other through analytic expressions in terms of the energy spectrum - are considered. In the one dimensional case, where the tight-binding coefficients can be obtained by designing a specific gauge transformation, we consider both the case of quasi resonance between the two lowest bands, and that between s and p orbitals. In the latter case, the role of the Wannier functions in the derivation of an effective Dirac equation is also reviewed. Then, we consider the case of a two dimensional honeycomb potential, with particular emphasis on the Haldane model, its phase diagram, and the breakdown of the Peierls substitution. Tunable honeycomb lattices, characterized by movable Dirac points, are also considered. Finally, general considerations for dealing with the interaction terms are presented.
Tight binding
Wannier function
Ultracold atom
Lattice (music)
Maxima and minima
Hamiltonian (control theory)
Honeycomb
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Wannier function
Tight binding
Formalism (music)
Dielectric function
Basis function
Basis (linear algebra)
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Wannier tight-binding models are effective models constructed from first-principles calculations. As such, they bridge a gap between the accuracy of first-principles calculations and the computational simplicity of effective models. In this work, we extend the existing methodology of creating Wannier tight-binding models from first-principles calculations by introducing the symmetrization post-processing step, which enables the production of Wannier-like models that respect the symmetries of the considered crystal. Furthermore, we implement automatic workflows, which allow for producing a large number of tight-binding models for large classes of chemically and structurally similar compounds or materials subject to external influence such as strain. As a particular illustration, these workflows are applied to strained III-V semiconductor materials. These results can be used for further study of topological phase transitions in III-V quantum wells.
Wannier function
Tight binding
Symmetrization
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Citations (54)