Noncommutative geometry and non-Abelian Berry phase in the wave-packet dynamics of Bloch electrons

Shindou, Ryuichi; Imura, Ken‐Ichiro

doi:10.1016/j.nuclphysb.2005.05.019

Cited by 78 publications

(113 citation statements)

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“…In a series of papers [9] (see also [10]), a new set of semiclassical equations with a Berry phase correction was proposed to account for the semiclassical dynamics of electrons in magnetic Bloch bands (in the usual one band approximation). These equations were derived by considering a wave packet in a band and using a time-dependent variational principle in a Lagrangian formulation.…”

Section: Introductionmentioning

confidence: 99%

Semiclassical diagonalization of quantum Hamiltonian and equations of motion with Berry phase corrections

2007

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It has been recently found that the equations of motion of several semiclassical systems must take into account terms arising from Berry phases contributions. Those terms are responsible for the spin Hall effect in semiconductor as well as the Magnus effect of light propagating in inhomogeneous media. Intensive ongoing research on this subject seems to indicate that a broad class of quantum systems may be affected by Berry phase terms. It is therefore important to find a general procedure allowing for the determination of semiclassical Hamiltonian with Berry Phase corrections. This article presents a general diagonalization method at orderh for a large class of quantum Hamiltonians directly inducing Berry phase corrections. As a consequence, Berry phase terms on both coordinates and momentum operators naturally arise during the diagonalization procedure. This leads to new equations of motion for a wide class of semiclassical system. As physical applications we consider here a Dirac particle in an electromagnetic or static gravitational field, and the propagation of a Bloch electrons in an external electromagnetic field.

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Section: Introductionmentioning

confidence: 99%

Semiclassical diagonalization of quantum Hamiltonian and equations of motion with Berry phase corrections

2007

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“…In a series of papers [8,9] (see also [10]), the following new set of semiclassical equations with a Berry-phase correction was proposed to account for the semiclassical dynamic of electrons in magnetic Bloch bands (in the usual one-band approximation)ṙ…”

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confidence: 99%

Semiclassical dynamics of electrons in magnetic Bloch bands: A Hamiltonian approach

et al. 2006

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By formally diagonalizing with accuracy the Hamiltonian of electrons in a crystal subject to electromagnetic perturbations, we resolve the debate on the Hamiltonian nature of semiclassical equations of motion with Berry-phase corrections, and therefore confirm the validity of the Liouville theorem. We show that both the position and momentum operators acquire a Berry-phase dependence, leading to a non-canonical Hamiltonian dynamics. The equations of motion turn out to be identical to the ones previously derived in the context of electron wave-packets dynamics. PACS numbers:The notion of Berry phase has found many applications in several branches of quantum physics, such as atomic and molecular physics, optic and gauge theories, and more recently, in spintronics, to cite just a few. Most studies focused on the geometric phase a wave function acquires when a quantum mechanical system has an adiabatic evolution. It is only recently that a possible influence of the Berry phase on semiclassical dynamics of several physical systems has been investigated. It was then shown that Berry phases modify semiclassical dynamics of spinning particles in electric [1] and magnetic fields [2], as well as in semiconductors [3]. In the above cited examples, a noncommutative geometry, originating from the presence of a Berry phase, which turns out to be a spin-orbit coupling, underlies the semiclassical dynamics. Also, spin-orbit contributions to the propagation of light have been the focus of several other works [1,4,5], and have led to a generalization of geometric optics called geometric spinoptics [6].Semiclassical methods in solid-state physics have also played an important role in studying the dynamics of electrons to account for the various properties of metals, semiconductors, and insulators [7]. In a series of papers [8,9] (see also [10]), the following new set of semiclassical equations with a Berry-phase correction was proposed to account for the semiclassical dynamic of electrons in magnetic Bloch bands (in the usual one-band approximation)ṙwhere E and B are the electric and magnetic fields respectively and E(k) = E 0 (k)−m(k).B is the energy of the nth band with a correction due to the orbital magnetic moment [9]. The correction term to the velocity −k × Θ with Θ(k) the Berry curvature of electronic Bloch state in the nth band is known as the anomalous velocity predicted to give rise to a spontaneous Hall conductivity in ferromagnets [11]. For crystals with broken time-reversal symmetry or spatial inversion symmetry, the Berry curvature is nonzero [9]. Eqs.1 were derived by considering a wave packet in a band and using a time-dependent variational principle in a Lagrangian formulation. The derivation of a semiclassical Hamiltonian was shown to lead to difficulties in the presence of Berry-phase terms [9]. The apparent non-Hamiltonian character of Eqs.1 led the authors of [12] to conclude that the naive phase space volume is not conserved in the presence of a Berry phase, thus violating Liouville's theorem. To remedy this...

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“…[9,10] Our simulations show that the uniform spreading of the wave packets is limited along the transverse channel direction due to both interference effects and Coulomb repulsion. As a result, the electron density is strongly modulated along the y-direction, exhibiting a stationary statelike distribution.…”

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confidence: 85%

“…[2,4,5,7,8] Here, we address the charge transport from an intermediate, semiclassical perspective, [9,10] by solving the Schrödinger equation numerically for a pair of interacting electron wave packets that propagate in a planar nanochannel. This approach interpolates between the classical, particle-like and the quantum, wavelike nature of electrons ( Fig.…”

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Interacting Electron Wave Packet Dynamics in a Two-Dimensional Nanochannel

Puetter

Konabe

Hatsugai

et al. 2013

Appl. Phys. Express

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Classical and quantum dynamics are important limits for the understanding of the transport characteristics of interacting electrons in nanodevices. Here, we apply an intermediate semiclassical approach to investigate the dynamics of two interacting electrons in a planar nanochannel as a function of Coulomb repulsion and electric field. We find that charge is mostly redistributed to the channel edges and that an electric field enhances the particle-like character of electrons. These results may have significant implications for the design and study of future nanodevices.Ideal ultrasmall logic nanodevices feature high switching speeds, low power consumption, excellent on-off current ratios and high scalability and integrability. Electron transport in ultrasmall nanodevices approaching channel lengths of approximately 10 nm, [1] however, turns out to be quasi-ballistic and intricate [2,3] due to Brownian motion (thermal noise caused by scattering and diffusion) and the discreteness of the electric charge (leading to shot noise enhanced by unscreened trapped charges). These effects give rise to significant current fluctuations at high clock speeds and low voltages, [4,5] which are detrimental to efficient device operation. This indicates that high-speed electrons in nanoscale regions cannot be described by conventional statistical frameworks such as Maxwell-Boltzmann approaches. Highly doped drain and source regions can further impact channel electrons, e.g., due to the build-up of mirror charges, suggesting that Coulomb interaction is a paramount ingredient in describing transport, dissipation, and equilibration in nanostructures. [2,3,6] A comprehensive understanding of electron dynamics on small time and length scales therefore is desirable to improve the device performance.Various approaches to electron transport in nanodevices have been taken so far, ranging from classical Monte Carlo and molecular dynamics methods to quantum nonequilibrium Green function calculations. [2,4,5,7,8] Here, we address the charge transport from an intermediate, semiclassical perspective, [9,10] by solving the Schrödinger equation numerically for a pair of interacting electron wave packets that propagate in a planar nanochannel. This approach interpolates between the classical, particle-like and the quantum, wavelike nature of electrons (Fig. 1) and approximately preserves the discreteness of the transported electron charge over appropriate (not too short) time scales. Wave packet approaches have been considered previously, but relied on strictly one-dimensional structures and/or on noninteracting electrons, or were based on approximate schemes such as the time-dependent Hartree-Fock theory. [11][12][13][14][15][16][17] Below, we study in detail the effect of Coulomb repulsion and an external electric field on wave packet propagation in a two-dimensional nanochannel. A main finding is that Coulomb repulsion redistributes the charge density to the channel walls, which makes electron transport more sensitive to perturbations at the in...

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Noncommutative geometry and non-Abelian Berry phase in the wave-packet dynamics of Bloch electrons

Cited by 78 publications

References 53 publications

Semiclassical diagonalization of quantum Hamiltonian and equations of motion with Berry phase corrections

Semiclassical diagonalization of quantum Hamiltonian and equations of motion with Berry phase corrections

Semiclassical dynamics of electrons in magnetic Bloch bands: A Hamiltonian approach

Interacting Electron Wave Packet Dynamics in a Two-Dimensional Nanochannel

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