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Dynamic Landau-Lifshitz-Bloch-Slonczewski equations for spintronics

Pascal Thibaudeau, Mouad Fattouhi, Liliana D. Buda-Prejbeanu

TL;DR

This work tackles the inadequacy of constant-magnitude LLG under significant Joule heating by deriving dynamic Landau-Lifshitz-Bloch-Slonczewski (dLLBS) equations from a statistical-bath framework. It introduces ensemble-averaged magnetization $S$ and its covariance $\Sigma$, yielding coupled equations with a temperature-dependent noise amplitude $D$ and an effective torque $T_S^{eff}$, while clarifying the role of the coupling matrix $\mathfrak{M}$. The results reproduce stochastic-LLG behavior while offering faster predictions and explicit tracking of fluctuations, improving estimates of critical currents and switching times in MTJs under thermal stress. Overall, the approach provides a practical, physics-grounded method for modeling heating effects in high-current spintronic devices and opens avenues for probabilistic spintronics and noise-enabled device concepts.

Abstract

The atomistic Landau-Lifshitz-Gilbert equation is challenged when modeling spintronic devices where Joule heating is significant, due to its core assumption of a constant magnetization magnitude. Based on a statistical framework that treats the magnetization magnitude as a dynamic variable coupled to a thermal bath, we derive a dynamic Landau-Lifshitz-Bloch-Slonczewski set of equations for torques, that captures the transient, heating-induced demagnetization that occurs during high-current operation. Integrating these dynamic equations and comparing them to their stochastic equivalents reveals that both the energy landscape and switching dynamics in high-anisotropy systems are similarly modified. This approach yields accurate and accelerated predictions of critical currents and switching times.

Dynamic Landau-Lifshitz-Bloch-Slonczewski equations for spintronics

TL;DR

This work tackles the inadequacy of constant-magnitude LLG under significant Joule heating by deriving dynamic Landau-Lifshitz-Bloch-Slonczewski (dLLBS) equations from a statistical-bath framework. It introduces ensemble-averaged magnetization and its covariance , yielding coupled equations with a temperature-dependent noise amplitude and an effective torque , while clarifying the role of the coupling matrix . The results reproduce stochastic-LLG behavior while offering faster predictions and explicit tracking of fluctuations, improving estimates of critical currents and switching times in MTJs under thermal stress. Overall, the approach provides a practical, physics-grounded method for modeling heating effects in high-current spintronic devices and opens avenues for probabilistic spintronics and noise-enabled device concepts.

Abstract

The atomistic Landau-Lifshitz-Gilbert equation is challenged when modeling spintronic devices where Joule heating is significant, due to its core assumption of a constant magnetization magnitude. Based on a statistical framework that treats the magnetization magnitude as a dynamic variable coupled to a thermal bath, we derive a dynamic Landau-Lifshitz-Bloch-Slonczewski set of equations for torques, that captures the transient, heating-induced demagnetization that occurs during high-current operation. Integrating these dynamic equations and comparing them to their stochastic equivalents reveals that both the energy landscape and switching dynamics in high-anisotropy systems are similarly modified. This approach yields accurate and accelerated predictions of critical currents and switching times.

Paper Structure

This paper contains 5 sections, 18 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Dynamic Landau-Lifshitz-Bloch-Slonczewski equations
  3. sLLG versus dLLBS analysis
  4. Conclusion
  5. Derivation of the noise-averaged sLLG equations

Figures (2)

  • Figure 1: Dynamics of the average magnetization switching of a free MTJ at different temperatures. The average is performed on 100 stochastic realizations of the sLLB equation (Eq.\ref{['sLLG']}) (lines) and compared to a single simulation using the dLLBS equations (Eqs.\ref{['dLLB1']}, \ref{['dLLB2']}) (dots) (see text).
  • Figure 2: Switching of an MTJ induced by a (damping-like) spin-transfer torque, simulated via the dLLBS approach for increasing operating temperatures, in the presence of a magnetic anisotropy axis in the free-layer. (see text).