26 September 2016 New relativistic Hamiltonian: the angular magnetoelectric coupling
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Abstract
Spin-Orbit Coupling (SOC) is a ubiquitous phenomenon in the spintronics area, as it plays a major role in allowing for enhancing many well-known phenomena, such as the Dzyaloshinskii-Moriya interaction, magnetocrystalline anisotropy, the Rashba effect, etc. However, the usual expression of the SOC interaction

ħ/4m2c2 [E×p] • σ (1)

where p is the momentum operator, E the electric field, σ the vector of Pauli matrices, breaks the gauge invariance required by the electronic Hamiltonian. On the other hand, very recently, a new phenomenological interaction, coupling the angular momentum of light and magnetic moments, has been proposed based on symmetry arguments:

ξ/2 [r × (E × B)] M, (2)

with M the magnetization, r the position, and ξ the interaction strength constant. This interaction has been demonstrated to contribute and/or give rise, in a straightforward way, to various magnetoelectric phenomena,such as the anomalous Hall effect (AHE), the anisotropic magnetoresistance (AMR), the planar Hall effect and Rashba-like effects, or the spin-current model in multiferroics. This last model is known to be the origin of the cycloidal spin arrangement in bismuth ferrite for instance. However, the coupling of the angular momentum of light with magnetic moments lacked a fundamental theoretical basis.

Starting from the Dirac equation, we derive a relativistic interaction Hamiltonian which linearly couples the angular momentum density of the electromagnetic (EM) field and the electrons spin. We name this coupling the Angular MagnetoElectric (AME) coupling. We show that in the limit of uniform magnetic field, the AME coupling yields an interaction exactly of the form of Eq. (2), thereby giving a firm theoretical basis to earlier works. The AME coupling can be expressed as:

ξ [E × A] • σ (3)

with A being the vector potential. Interestingly, the AME coupling was shown to be complementary to the traditional SOC, and together they restore the gauge invariance of the Hamiltonian. As an illustration of the AME coupling, we straightforwardly derived a relativistic correction to the so-called Inverse Faraday Effect (IFE), which is the emergence of an effective magnetic field under illumination by a circularly polarized light.
© (2016) COPYRIGHT Society of Photo-Optical Instrumentation Engineers (SPIE). Downloading of the abstract is permitted for personal use only.
Charles Paillard, Charles Paillard, Ritwik Mondal, Ritwik Mondal, Marco Berritta, Marco Berritta, Brahim Dkhil, Brahim Dkhil, Surendra Singh, Surendra Singh, Peter M. Oppeneer, Peter M. Oppeneer, Laurent Bellaiche, Laurent Bellaiche, } "New relativistic Hamiltonian: the angular magnetoelectric coupling", Proc. SPIE 9931, Spintronics IX, 99312E (26 September 2016); doi: 10.1117/12.2238196; https://doi.org/10.1117/12.2238196
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