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Journal of Applied Physics / Volume 110 / Issue 12 / APPLIED PHYSICS REVIEWS
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J. Appl. Phys. 110, 121301 (2011); http://dx.doi.org/10.1063/1.3665219 (29 pages)

Gauge fields in spintronics

T. Fujita1, M. B. A. Jalil1,2, S. G. Tan1,3, and S. Murakami4

1Computational Nanoelectronics and Nano-device Laboratory, Electrical and Computer Engineering Department, National University of Singapore, 4 Engineering Drive 3, 117576, Singapore
2Information Storage Materials Laboratory, Electrical and Computer Engineering Department, National University of Singapore, 4 Engineering Drive 3, 117576, Singapore
3Data Storage Institute, A*STAR (Agency for Science, Technology and Research) DSI Building, 5 Engineering Drive 1, 117608, Singapore
4Department of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan

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(Received 22 August 2011; accepted 9 November 2011; published online 22 December 2011)

We present an overview of gauge fields in spintronics, focusing on their origin and physical consequences. Important topics, such as the Berry gauge field associated with adiabatic quantum evolution as well as gauge fields arising from other non-adiabatic considerations, are discussed. We examine the appearance and effects of gauge fields across three spaces, namely real-space, momentum-space, and time, taking on a largely semiclassical approach. We seize the opportunity to study other “spin-like” systems, including graphene, topological insulators, magnonics, and photonics, which emphasize the ubiquity and importance of gauge fields. We aim to provide an intuitive and pedagogical insight into the role played by gauge fields in spin transport.

© 2011 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. TERMINOLOGY
  3. SPIN- math SYSTEMS IN THE PRESENCE OF SPATIALLY VARYING MAGNETIC FIELD TEXTURES
    1. System
    2. Berry phase: Theoretical approaches
      1. Berry (1984)
      2. Path integral formalism
      3. Unitary transformation
    3. Physical consequences
      1. Spin-dependent forces: Chirality-driven spin-Hall effect
      2. Domain wall characterization
      3. Spin torque in domain walls
  4. REAL SPACE GAUGE FIELDS IN GRAPHENE
    1. A primer on graphene
    2. Modeling the effects of strain
    3. Physical consequences
      1. Valley filtering
      2. Valley-dependent forces
      3. Edge states
  5. SPIN-ORBIT COUPLING SYSTEMS: REAL SPACE ANALYSIS
    1. Spin-orbit coupling basics
    2. Non-Abelian gauge field representation
    3. Physical consequences
      1. Aharonov-Casher phase
      2. Spin-dependent transverse force
      3. Quantum spin-Hall effect
      4. Spin torque
    4. Spatially nonuniform spin-orbit coupling
  6. SPIN-ORBIT COUPLING SYSTEMS: math -SPACE ANALYSIS
    1. Derivation of math -space Berry curvature
    2. Physical consequences
      1. Spin-Hall effect
      2. Spin-Hall effect of light and optical Magnus effect
      3. Magnon-Hall effect in ferromagnetic insulators
      4. Valley-Hall effect in graphene
      5. Topological insulators
    3. Hall conductivity
  7. TIME-DEPENDENT MAGNETIC SYSTEMS
    1. Derivation
    2. Physical consequences
      1. Spin-Hall effect
      2. Pseudospin-Hall effect in graphene
      3. Rayleigh scattering of polaritons
      4. Spin motive force
    3. Semiclassical connection with k -space Berry curvature
  8. SUMMARY

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KEYWORDS and PACS

Keywords

Berry phase, graphene, magnetoelectronics, topological insulators

PACS

  • 85.75.-d

    Magnetoelectronics; spintronics: devices exploiting spin polarized transport or integrated magnetic fields

  • 72.25.-b

    Spin polarized transport

  • 73.20.-r

    Electron states at surfaces and interfaces

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3665219

PUBLICATION DATA

ISSN

0021-8979 (print)  
1089-7550 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

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Figures (11) Tables (1)

Figures (click on thumbnails to view enlargements)

FIG.1
(Color online) We illustrate the Berry phase for the problem of inhomogeneous magnetic fields math. (a) A Berry phase is acquired when quantum states undergo cyclic, adiabatic evolution C in magnetic field math-space. Adiabatic evolution corresponds to the limit T→∞. (b) The Berry curvature in Eq. ( 17 ) in math-space is a Dirac monopole of strength eg = math.

FIG.1 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.2
(Color online) A spatially varying magnetic field system (left), characterized by math(math), can be transformed via local rotation to a locally uniform system (right). The effect of this is to modify the momentum of the carriers math→math+math, where math is a gauge field (see Eq. ( 28 )). math is a true gauge field, in the sense that it obeys the transformation rule in Eq. ( 30 ) with respect to unitary transformations.

FIG.2 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.3
(Color online) (a) Monolayer graphene comprised of two triangular sublattices A and B. math1 and math2 are unit lattice vectors, and mathi (i = 1,2,3) are the three nearest neighbor vectors. (b) The Brillouin zone of monolayer graphene is hexagonal, with two inequivalent corners K and K', known as valleys. The energy spectrum is degenerate at the corners, and the low energy Hamiltonian is centered about them in a Dirac cone configuration with slope υF = 1×106 ms-1.

FIG.3 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.4
(Color online) Valley filter device comprised of (a) a region of uniform uniaxial strain along the armchair direction (strained bonds are highlighted) followed by (b) a magnetic barrier region.

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FIG.5
(Color online) Transmission probability through the strained region of Fig. 4a as a function of φ (the direction of the incident wavevector). Valley K (K') is denoted by solid (dashed) lines. Filtering in φ-space, facilitated by magnetic field barriers shown in Fig. 4b, therefore, results in valley filtering. The shaded region indicates the action of a φ-filter, which allows for valley K to be transmitted freely, while blocking valley K'.

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FIG.6
(Color online) (a) The quantum Hall state, induced by strong perpendicular magnetic fields in two-dimensional systems, comprised of an insulating bulk and metallic edges. The edge states are chiral (uni-directional), owing to the broken TR symmetry, and consequently resist backscattering by impurities. (b) The quantum valley-Hall effect arising in strained graphene is a TR symmetric cousin of the quantum Hall effect. Here, electrons in opposite valleys (K and K') experience opposite vertical magnetic fields, which form two valley-resolved (chiral and anti-chiral) quantum Hall states. Backscattering of edge states is only permitted with a corresponding switching of valleys. Thus, in the absence of TR-breaking impurities, the edge states will remain robust.

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FIG.7
(Color online) Forming graphene edges by severing bonds as shown for (a) zig-zag and (b) armchair edges. This induces local gauge fields, whose curvatures represent effective magnetic fields capable of polarizing the pseudospin. Zig-zag edges are associated with a finite pseudospin polarization and, thus, are able to support edge states. Armchair edges, on the other hand, do not polarize the pseudospin and, thus, cannot support edge states.

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FIG.8
(Color online) Transverse spin separation mediated by spin-orbit coupling. Up and down spins experience opposing Lorentz forces, resulting in a spin-Hall effect.

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FIG.9
(Color online) (a) Illustration of proposed device, in which a transverse separation of spins occurs in response to a longitudinal charge current J. The separation occurs heuristically as a result of spin-dependent forces due to (i) Rashba SOC, which is characterized by the perpendicular electric field, mathSO (vertical, dark arrow), and (ii) a spatially nonuniform magnetic field, math(math). The directions of the spin-dependent force arising from the Rashba SOC, mathSO, and from the Berry curvature, mathBerry, are indicated by arrows. We note that the forces from the two contributions act in opposite directions. The degree of cancellation between the two forces can be modulated via a gate bias. This leads to the potential modulation of the transverse spin-current by purely electric means. (b) The configuration of the spatially nonuniform magnetic field characterized by chirality θ.

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FIG.10
(Color online) (a) Trilayer structure in which the 2DEG channel has Rashba spin-orbit coupling, while the two contact regions do not. (b) The spatial discontinuity of the Rashba spin-orbit coupling induces spatially narrow effective magnetic fields at the interfaces, which are spin-dependent, σx = ±1.

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FIG.11
(Color online) In the presence of a time-dependent magnetic field, math(t) = |math(t)|math(t), an additional magnetic field math⊥ = math×math (vertical arrow) is seen by spins. The net instantaneous magnetic field felt by spins is the vector sum of math(t) and math⊥, denoted by the dashed arrow.

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Tables

Table I. A summary of gauge fields, the context in which they occur, and their physical (measurable) consequences.

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