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# Electro and magneto statics of topological insulators as modeled by planar, spherical and cylindrical $$\theta$$ boundaries: Green function approach

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### Abstract

The Green function (GF) method is used to analyze the boundary effects produced by a Chern Simons (CS) extension to electrodynamics. We consider the electromagnetic field coupled to a $$\theta$$ term that is piecewise constant in different regions of space, separated by a common interface $$\Sigma$$, the $$\theta$$ boundary, model which we will refer to as $$\theta$$ electrodynamics ($$\theta$$ ED). This model provides a correct low energy effective action for describing topological insulators (TI). In this work we construct the static GF in $$\theta$$ ED for different geometrical configurations of the $$\theta$$ boundary, namely: planar, spherical and cylindrical $$\theta$$ interfaces. Also we adapt the standard Green theorem to include the effects of the $$\theta$$ boundary. These are the most important results of our work, since they allow to obtain the corresponding static electric and magnetic fields for arbitrary sources and arbitrary boundary conditions in the given geometries. Also, the method provides a well defined starting point for either analytical or numerical approximations in the cases where the exact analytical calculations are not possible. Explicit solutions for simple cases in each of the aforementioned geometries for $$\theta$$ boundaries are provided. The adapted Green theorem is illustrated by studying the problem of a point like electric charge interacting with a planar TI with prescribed boundary conditions. Our generalization, when particularized to specific cases, is successfully compared with previously reported results, most of which have been obtained by using the methods of images.

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### Spin Transfer Torque Generated by the Topological Insulator Bi_2Se_3

(2014)
Magnetic devices are a leading contender for implementing memory and logic technologies that are nonvolatile, that can scale to high density and high speed, and that do not suffer wear-out. However, widespread applications of magnetic memory and logic devices will require the development of efficient mechanisms for reorienting their magnetization using the least possible current and power. There has been considerable recent progress in this effort, in particular discoveries that spin-orbit interactions in heavy metal/ferromagnet bilayers can yield strong current-driven torques on the magnetic layer, via the spin Hall effect in the heavy metal or the Rashba-Edelstein effect in the ferromagnet. As part of the search for materials to provide even more efficient spin-orbit-induced torques, some proposals have suggested topological insulators (TIs), which possess a surface state in which the effects of spin-orbit coupling are maximal in the sense that an electron's spin orientation is locked relative to its propagation direction. Here we report experiments showing that charge current flowing in-plane in a thin film of the TI Bi_2Se_3 at room temperature can indeed apply a strong spin-transfer torque to an adjacent ferromagnetic permalloy (Py = Ni81Fe19) thin film, with a direction consistent with that expected from the topological surface state. We find that the strength of the torque per unit charge current density in the Bi_2Se_3 is greater than for any other spin-torque source material measured to date, even for non-ideal TI films wherein the surface states coexist with bulk conduction. Our data suggest that TIs have potential to enable very efficient electrical manipulation of magnetic materials at room temperature for memory and logic applications.
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### Nickel-based cocatalysts for photocatalytic hydrogen production

(2015)
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### Author and article information

###### Journal
1601.06196
10.1103/PhysRevD.93.045022