Hysteresis in layered spring magnets

TL;DR

The paper addresses magnetization reversal in exchange-spring layered magnets using a one-dimensional hard/soft bilayer model with uniform per-layer magnetization governed by the Landau–Lifshitz–Gilbert equation. It introduces a magnitude-preserving, unconditionally stable integration scheme and an algorithm to compute equilibrium spin configurations, enabling robust simulations under rotating in-plane fields. The results reveal two rotational hysteresis regimes: a $2\pi$-period loop at moderate fields due to partial-length spin-chain transitions, and a $\pi$-period (or multiples thereof) loop at strong fields from full-length transitions, with a clear role for chirality and chain stiffness. While the qualitative behavior agrees with torque and magneto-optical experiments, the one-dimensional model overestimates demagnetization energy and misses nanodomain dynamics, underscoring the need for multidimensional modeling for quantitative accuracy.

Abstract

This article addresses a problem of micromagnetics: the reversal of magnetic moments in layered spring magnets. A one-dimensional model is used of a film consisting of several atomic layers of a soft material on top of several atomic layers of a hard material. Each atomic layer is taken to be uniformly magnetized, and spatial inhomogeneities within an atomic layer are neglected. The state of such a system is described by a chain of magnetic spin vectors. Each spin vector behaves like a spinning top driven locally by the effective magnetic field and subject to damping (Landau-Lifshitz-Gilbert equation). A numerical integration scheme for the LLG equation is presented that is unconditionally stable and preserves the magnitude of the magnetization vector at all times. The results of numerical investigations for a bilayer in a rotating in-plane magnetic field show hysteresis with a basic period of $2π$ at moderate fields and hysteresis with a basic period of $π$ at strong fields.

Figures (10)

• Figure 1: Equilibrium spin configurations; $H_a = 4800$ oersteds. Left: $\theta_a$ increasing, right: $\theta_a$ decreasing.
• Figure 2: In-plane angle $\theta_i$ vs. $i$; $H_a = 4800$ oersteds; (a) $\theta_a = 45$, (b) $\theta_a = 90$, (c) $\theta_a = 135$, (d) $\theta_a = 180$, (e) $\theta_a = 225$, (f) $\theta_a = 270$, (g) $\theta_a = 301.5$, (h) $\theta_a = 301.6$, (i) $\theta_a = 315$ degrees.
• Figure 3: Rotational hysteresis: in-plane angle $\theta_i$ vs. $\theta_a$; $H_a = 4800$ oersteds; $i = 95, 115, 135, 155, 175, 195, 215$.
• Figure 4: The critical angle $\theta_c$ as a function of $H_a$.
• Figure 5: In-plane angle $\theta_i$ vs. $i$; (a) $H_a = 1000$, (b) $H_a = 2000$, (c) $H_a = 3000$, (d) $H_a = 5000$, (e) $H_a = 6000$, (f) $H_a = 7000$, (g) $H_a = 8000$, (h) $H_a = 9000$, (i) $H_a = 10,000$ oersteds. Right branches: $\theta_a$ just below $\theta_c$, left branches: $\theta_a$ just above $\theta_c$.