Voltage dependence of Landau-Lifshitz-Gilbert damping of spin in a current-driven tunnel junction

Abstract

We present a theory of Landau-Lifshitz-Gilbert damping α for a localized spin S in the junction coupled to the conduction electrons in both leads under an applied volatege V . We find the voltage dependence of the damping term reflecting the energy dependence of the density of states. We find the effect is linear in the voltage and cotrolled by particle-hole asymmetry of the leads. PACS numbers: 75.80.+q, 71.70.Ej, INTRODUCTIONSpintronics is an emerging subfield that holds the potential to replace conventional electronic devices with spintronic analogues where the manipulation, control, and readout of spins will enable novel functionality with no or little electronic charge dynamics [1]. In order to realize this promise, the spin dynamics of the small scale devices needs to be well controlled. One of the most pressing questions concerns a set up which would preserve coherence and allow a manipulation of spins. In most systems, the relevant spin degrees of freedom are coupled to some bath, such as a fermionic bath of electrons. The detailed dynamics of single spins when in contact with such a bath plays a pivotal role in addressing decoherence in potential spintronic systems.The conventional way to treat this problem is via a Caldeira-Leggett approach where the external bath is modeled by collective excitations which are capable of destroying coherent spin dynamics. Often, spin dynamics is described by a Landau-Lifshitz-Gilbert equation [2,3]:where h is, up to constant prefactors, the external magnetic field and the coefficient α captures the damping due to the external bath. A caricature of the solution of this equation [4] is provided in Fig.1. There are standard methods to calculate α in an equilibrium situation when, say, one considers a spin in a Fermi liquid [5,6].In the current publication, we address a related novel question concerning the effect of an applied voltage bias on the Gilbert coefficient α. Our work complements the recent results of [7] wherein the effects of the "retarded" electronic contributions in the equations of motion for a system of spins were studied. Both such retarded correlations [8] as well as additional "Keldysh" correlations generally manifest themselves in the single spin equations of motion, see e.g. [9] for general spin equations of motion entailing the effects of both correlations. In the current work, we examine the voltage dependence of Gilbert damping. For the sake of clarity, we depart from the Keldysh contour formalism of [7,8,9], and use a Caldeira-Leggett approach.In what follows, we consider the case of a junction between two electrodes that contains one spin S, see Fig.2. This spin S may be the spin of a single magnetic impurity or it may portray the spin of a cluster at low temperature when the spins in the cluster are locked. Upon applying a finite bias between the electrodes of Fig.1, a current flow is generated. Thereafter, at long times, the system is at a steady but non-equilibrium state so long as the voltage bias V is applied. We will focus on the voltage d...

Influence of a periodic magnetic field and spin-polarized current on the magnetic dynamics of a monodomain ferromagnet

“…Several possible cases of such dependences were already discussed in the literature. In particular, Katsura et al 40 showed that ␣ changes linearly with the applied voltage in a magnetic junction. Accounting for the angular dependence of Gilbert damping coefficient on the vertical angle , Kim et al 41 demonstrated that the damping coefficient can be twice as large as ␣ 0 introduced by Gilbert for Py/Cu and Co/Cu interfaces.…”

Geometric Quantum Noise of Spin

“…Our approach can be viewed as a generalization of the Landau-Lifschitz-Gilbert (LLG)-Langevin equation [7,8], central to the field of spintronics [9], to a regime where quantum dynamics dominates. Stochastic LLG equations have been derived in numerous publications for both a localized spin in an electronic environment (a situation of the Caldeira-Leggett type) [10,11] and for a magnetization formed by itinerant electrons [12,13]. In all these papers the precession frequency was assumed to be lower than the temperature or the voltage, thus justifying the semi-classical treatment of the problem.…”

Transient spin dynamics in a single-molecule magnet

“…Methods using quantum master equations [49][50][51][52] and stochastic LLG equation [53,54] have been thoroughly investigated. Another technique, which will be used in this paper, is to derive a spin equation of motion (SEOM) from the spin action defined on the Keldysh contour, considering the nonequilibrium properties of the effective spin moment [55][56][57][58][59]. This provides a general description of the spin dynamics and exchange interactions in the nonequilibrium regime [60][61][62][63][64].…”