Elsevier

Annals of Physics

Volume 325, Issue 8, August 2010, Pages 1537-1549
Annals of Physics

Monopole and topological electron dynamics in adiabatic spintronic and graphene systems

https://doi.org/10.1016/j.aop.2010.04.007Get rights and content

Abstract

A unified theoretical treatment is presented to describe the physics of electron dynamics in semiconductor and graphene systems. Electron spin's fast alignment with the Zeeman magnetic field (physical or effective) is treated as a form of adiabatic spin evolution which necessarily generates a monopole in magnetic space. One could transform this monopole into the physical and intuitive topological magnetic fields in the useful momentum (K) or real spaces (R). The physics of electron dynamics related to spin Hall, torque, oscillations and other technologically useful spinor effects can be inferred from the topological magnetic fields in spintronic, graphene and other SU(2) systems.

Introduction

Spintronics [1], [2] is an interesting area of research that straddles the border between fundamental physics and technology, offering an almost unique opportunity to translate physics into real applications. One example is the giant magnetoresistance [3], [4] (GMR) effects which have found applications in the form of multilayer spin valve [5], [6] recording heads for reading magnetic data stored on the magnetic media. Another phenomenon, the spin transfer torque [7], [8], [9], is currently under intense investigation for current-induced magnetization switching, noise control in spin valve, and sustained spin torque oscillations [10]. Micromagnetic studies of magnetization configurations have improved the design of magnetic media and read-heads for recording purposes. Modern interest in micromagnetics consider the dynamics of both itinerant and local electron spin, providing new insights into anomalous Hall, as well as spin transfer with respect to the itinerant, local spin dynamics, respectively. Of interest recently is the spin orbital (SO) effect [11], [12], [13], [14], particularly the Rashba and the Dresselhaus in semiconductor materials. The SO effect has direct implication to the spin and the momentum dynamics of electrons, leading to recent interest that spans fundamental, device and engineering physics, and subjects that range from spin Hall [15], [16], [17] to spin current [18] transistor. In fact, SO effect is highly relevant to spintronics, ranging from the well-known anisotropic magnetoresistance, the anisotropy energy of local moment density, the keenly studied spin Hall effect and spin current in semiconductor spintronics, to more subtle implications like spin torque, spin dynamics, spin oscillations, and Zitterbewegung.

A SO system can be viewed as one which provides an effective Zeeman magnetic (b) field which varies in the momentum (K) space. As such, one can draw an analogy between a SO system and a locally varying b field system, in which a conduction electron experiences the varying b field in real space (R). In the event that the electron spin evolves and aligns adiabatically to the Zeeman b field in their respective K or R spaces, under the theoretical framework of gauge and symmetry [19], [20], the two systems will be analogous in that electron spin evolving adiabatically in both systems “see” a Dirac monopole in the Zeeman field space (B). Since monopole in B space has no direct bearing on the spin or orbital dynamics of electrons, it would be essential to transform the B space monopole to some topological magnetic fields (curvature) in the more useful space of K or R under which the equation of motion [15], [21] can be constructed to describe the electron's orbital dynamics. A similar SU(2) system which resembles the SO is the special carbon system of monolayer and bilayer graphene. But the spinor of these systems does not represent the spin state of conducting particle in the carbon system. Instead, the spinor describes the pseudo-spin which consists of a linear combination of waves due to different sub-lattice sites. This article would be devoted to discussing the physics of monopole fields [22] originating from spinor dynamics in spintronics and graphene. The monopole field in these so-called SU(2) systems can be viewed as a mathematical object which can lead to instructive description of the electron's orbital dynamics [23], [24] or motion. Here we will present a thorough description of the Dirac gauge potential arising from spinor dynamics (fast alignment with b fields) in the strong Zeeman field (adiabatic) limit. The strong Zeeman effect has direct relevance to both the SO or graphene systems and the local magnetic system; the former would relate to the transformed topological magnetic fields in K space, the latter to R space. In local micromagnetic systems which have been studied intensively in the magnetic media for hard disk drives, or domain wall spintronics, one needs to investigate the topological magnetic field in real spaces which will not be discussed in this article.

Section snippets

Theory

When an electron propagates in the SU(2) system, its spin precesses about the effective b field. This mechanism has been studied in great details in spintronics where precession of spin due to the Rashba or the Dresselhaus effects leads to spin current [25], [26] when a finite dimension (boundary condition) is imposed on the system. But here we will consider a system where the effective Zeeman b field is infinitely strong, such that in this limit electron spin relaxes to the field. The

Acknowledgments

The authors would like to thank the Agency for Science, Technology and Research (A*STAR) of Singapore, SERC Grant No. 092 101 0060 (R-398-000-061-305) and the National University of Singapore Nanoscience and Nanotechnology Initiative for financially supporting their work.

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    Division of Spintronic, Media and Interface, Data Storage Institute (Agency for Science and Technology Research), DSI Building 5 Engineering Drive 1 (Off Kent Ridge Crescent, NUS), Singapore 117608, Singapore.

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