Inertia is a fundamental property of a particle that can be understood from Newtons laws of motion. In a similar way, any magnetized body must possess magnetic inertia by the virtue of its magnetization. The influence of possible magnetic inertia effects has recently drawn attention in ultrafast magnetization dynamics and switching. Magnetization dynamics at the inertial regime has been investigated using thermodynamic theories that predicted the magnetic inertia can become impactful at shorter timescales. However, at the fundamental level, the origin of magnetic inertial dynamics is still unknown. <p> </p>Here, we derive rigorously a description of magnetic inertia in the extended Landau-Lifshitz-Gilbert (LLG) equation starting from a fundamental and relativistic Dirac-Kohn-Sham framework. We use a unitary transformation, the so called called Foldy-Wouthuysen transformation, up to the order of 1/c<sup>4</sup>. In this way, the particle and anti-particle in fully relativistic description become decoupled and a Hamiltonian describing only the particles is derived. This Hamiltonian involves the nonrelativistic Schrodinger-Pauli Hamiltonian together with the relativistic corrections of the order 1/c<sup>2</sup> and 1/c<sup>4</sup>. <p> </p>With the thus-derived Hamiltonian, we calculate the corresponding spin dynamics leading to the LLG equation of motion. Our result exemplify that the relativistic correction terms of 1/c<sup>2</sup> are responsible for the Gilbert damping, however, the relativistic correction terms of 1/c<sup>4</sup> are responsible for magnetic inertial dynamics. Therefore, we predict that the intrinsic magnetic inertia is a higher-order relativistic spin-orbit coupling effect and is expected to be prominent only on ultrashort timescales (subpicoseconds). We also show that the corresponding Gilbert damping and magnetic inertia parameters are related to one another through the imaginary and real parts of the magnetic susceptibility tensor, respectively.
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
<p> </p>ħ/4m<sup>2</sup>c<sup>2</sup> [<i>E×p</i>] • σ (1)<p> </p>
where <i>p </i>is the momentum operator, <i>E</i> 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:
<p> </p>ξ/2 [r × (E × B)] M, (2)<p> </p>
with <i>M</i> the magnetization, <i>r</i> 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. <p> </p>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:
<p> </p>ξ [E × A] • σ (3) <p> </p>
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.