Formalisms of Band Theory

Blount, E. I.

doi:10.1016/s0081-1947(08)60459-2

Cited by 419 publications

(207 citation statements)

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“…There are, however, other contributions to the Hall effect that are ignored in such a theory including anomalous velocity 12,17,18,26,27 and the side-jump 6,[28][29][30][31][32][33] effects. The presence of these additional contributions is directly linked to induced nonzero interband elements of the density matrix, in other words, the interband coherence, as recognized since the work of Luttinger and Kohn.…”

Section: Introductionmentioning

confidence: 99%

Anomalous Hall effect in a two-dimensional Dirac band: The link between the Kubo-Streda formula and the semiclassical Boltzmann equation approach

et al. 2007

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The anomalous Hall effect ͑AHE͒ is a consequence of spin-orbit coupling in a ferromagnetic metal and related primarily to density-matrix response to an electric field that is off-diagonal in band index. For this reason disorder contributions to the AHE are difficult to treat systematically using a semiclassical Boltzmann equation approach, even when weak localization corrections are disregarded. In this article we explicitly demonstrate the equivalence of an appropriately modified semiclassical transport theory which includes anomalous velocity and side-jump contributions and microscopic Kubo-Streda perturbation theory, with particular unconventional contributions in the semiclassical theory identified with particular Feynman diagrams when calculations are carried out in a band-eigenstate representation. The equivalence we establish is verified by explicit calculations for the case of the two-dimensional Dirac model Hamiltonian relevant to graphene.

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Section: Introductionmentioning

confidence: 99%

Anomalous Hall effect in a two-dimensional Dirac band: The link between the Kubo-Streda formula and the semiclassical Boltzmann equation approach

et al. 2007

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“…This is regarded as the χ-representation for the operator T µ , and analogous to the crystal momentum representation of the spatial coordinate x [16]. The second term of (2) is the Berry connection.…”

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Hall Effect of Spin Waves in Frustrated Magnets

Fujimoto

2009

Phys. Rev. Lett.

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We examine a possible spin Hall effect for localized spin systems with no charge degrees of freedom. In this scenario, a longitudinal magnetic field gradient induces a transverse spin current carried by spin wave excitations with an anomalous velocity which is caused by a topological Berry-phase effect associated with spin chirality, in analogy with anomalous Hall effects in itinerant electron systems. Our argument is based on a semiclassical equations of motion applicable to general spin systems. Also, a microscopic model of frustrated magnets which exhibits the anomalous spin Hall effect is presented. PACS numbers:Introduction -Spin transport phenomena in condensed matter systems have been attracting much interest because of potential applications to spintronics [1,2], and also because of their fundamental relation with the notion of topologically-induced spin currents [3,4,5]. For semiconductors and metals with spin-orbit interactions, the spin Hall effect (SHE) in the absence of magnetic fields was predicted theoretically [1], and later experimentally realized [2]. This effect allows for topological interpretation in terms of the Berry phase; the spinorbit interaction gives rise to the Berry curvature associated with the Bloch wave function, which results in the nonzero anomalous velocity producing a transverse force even in the absence of the Lorentz force [3]. This leads to the anomalous Hall effect (AHE) for the case without time-reversal symmetry [3,4,5,6,7], and the SHE for the case with time-reversal symmetry [1]. On the other hand, it was shown by several authors that the anomalous velocity is also generated by spin textures accompanying spin chirality, and the AHE occurs in itinerant electron systems coupled with localized spins, the orientation of which possesses spin chirality order [8]. In this letter, we propose a possible analogue of this phenomenon for localized spin systems with no charge degrees of freedom. We demonstrate that, in analogy with metallic systems, the topological Berry-phase effect associated with spin chirality raises the dissipationless spin Hall current induced by a longitudinal magnetic field gradient: i.e. Hall effect of spin waves.

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“…Ap, A growing interest in localized real-space descriptions of the electronic structure of solids has been motivated by the development of computationally efficient "linearscaling" algorithms [1,2] and by the desirability of a local real-space mapping of chemical [3,4] and dielectric [5,6] properties. A primary avenue to such a description is the use of Wannier functions [7][8][9] (WFs), i.e., a set of localized wavefunctions w R (r) obtained from the Bloch functions ψ k (r) by a Fourier-like unitary transformation. A closely related approach is to represent the electronic structure in terms of the density matrix n(r, r ′ ).…”

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confidence: 99%

“…[8]. The exponential decay of the WFs is then governed by the positions of branch points in the "complex band structure" E n (k) constructed by regarding complex E n to be a function of complex k via analytic continuation from the real axis [8,9]. Specifically, there is a Riemann sheet E n (k) for each band n, and the branch points k n are the points at which the sheets are connected, E n (k n ) = E n+1 (k n ).…”

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Exponential Decay Properties of Wannier Functions and Related Quantities

2001

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The spatial decay properties of Wannier functions and related quantities have been investigated using analytical and numerical methods. We find that the form of the decay is a power law times an exponential, with a particular power-law exponent that is universal for each kind of quantity. In one dimension we find an exponent of −3/4 for Wannier functions, −1/2 for the density matrix and for energy matrix elements, and −1/2 or −3/2 for different constructions of non-orthonormal Wannier-like functions.PACS: 71.15. Ap, A growing interest in localized real-space descriptions of the electronic structure of solids has been motivated by the development of computationally efficient "linearscaling" algorithms [1,2] and by the desirability of a local real-space mapping of chemical [3,4] and dielectric [5,6] properties. A primary avenue to such a description is the use of Wannier functions [7][8][9] (WFs), i.e., a set of localized wavefunctions w R (r) obtained from the Bloch functions ψ k (r) by a Fourier-like unitary transformation. A closely related approach is to represent the electronic structure in terms of the density matrix n(r, r ′ ). It is thus not surprising to find considerable recent interest in the localization properties of the WFs [3] and of the density matrix [10,11].In a classic 1959 paper, Kohn proved, for the case of a centrosymmetric crystal in one dimension (1D), that the WFs have an "exponential decay" w(x) ≈ e −hx , where h is the distance of a branch point from the real axis in the complex-k plane [8]. More precisely, lim x→∞ w(x) e qx = 0,The density matrix has a similar decay n(x, x ′ ) ≈ e −h|x−x ′ | . The exponential decay of the WFs has since been proven for the general 1D [12] and single-band 3D[13] cases, and that of the density matrix (more precisely, of the band projection operator) has been proven in general [12]. The energy matrix elements E(R) = w R |H|w 0 , with w R (x) = w(x − R) and R = la a lattice vector, are also expected to have a similar decay,The purpose of this Letter is to address two questions. First, Eq. (1) allows considerable freedom; in fact, it is consistent withfor any exponent α, i.e., a decay which could be faster (α > 0) or slower (α < 0) than pure exponential. Does such a power-law prefactor exist, and if so, what is the exponent α? Second, it has long been understood that relaxation of the orthogonality constraint w 0 |w R = δ 0,R can give "more localized" Wannier-like functions [14][15][16].In what sense are these more localized -a larger h, or a larger α for the same h, or only a smaller prefactor of the tail? We show that the power-law prefactors of Eq. (2) do exist, and that the various quantities have a common inverse decay length h but different exponents α. In 1D we find that α = 3/4 for usual (orthonormal) WFs, α = 1/2 for n(x, x ′ ) and E(R), and α = 1/2 or α = 3/2 for two different constructions of nonorthonormal Wannier-like functions (NWFs). The NWFs of superior decay (∼ x −3/2 e −hx ) can be constructed by a projection method as duals to a set of tria...

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Formalisms of Band Theory

Cited by 419 publications

References 28 publications

Anomalous Hall effect in a two-dimensional Dirac band: The link between the Kubo-Streda formula and the semiclassical Boltzmann equation approach

Anomalous Hall effect in a two-dimensional Dirac band: The link between the Kubo-Streda formula and the semiclassical Boltzmann equation approach

Hall Effect of Spin Waves in Frustrated Magnets

Exponential Decay Properties of Wannier Functions and Related Quantities

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