Ultrafast dynamics of moments in bulk ferromagnets

TL;DR

The paper tackles ultrafast magnetization dynamics in bulk ferromagnets subject to laser pulses by developing a unified, thermodynamically consistent framework that couples a dynamical Landau-Lifshitz-Bloch (dLLB) description to a quantum fluctuation-dissipation theorem and to spin, electron, and phonon reservoirs via a three-temperature model (3TM) and a two-temperature model (2TM). It derives closed equations for the first and second moments of the spin—$\boldsymbol{s}_i$ and $\Sigma_i$, with the second cumulant $\chi_i=\Sigma_i-\Gamma_i$—under a mean-field exchange $J_{ij}$ and includes a noise amplitude $D$ tied to the spin temperature, enabling consistent thermodynamic predictions. A quantum heat bath approach, incorporating a magnon-density-of-states $g_m(\omega,T)$ and a Callen-Callen temperature dependence $A(T)=A_0\langle S\rangle^2$, with a spin-wave correction $\gamma_{SW}$ to $J_{ij}$, yields excellent agreement with experimental Curie temperatures and magnetization curves for Ni, Fe, and Co, and accurately captures ultrafast demagnetization when coupled to the 2TM with laser driving $P(t)$. The framework reproduces fluence-dependent demagnetization dynamics and reveals how dynamic susceptibility $\chi$ evolves under isotropy and under an applied DC field, highlighting off-diagonal features that reflect spin correlations. Overall, the approach offers a fast, predictive, and extensible tool for modeling ultrafast magnetism in bulk and layered ferromagnets, with potential extensions to ferrimagnets, antiferromagnets, and multilayer systems.

Abstract

A robust and efficient model for investigating the ultrafast dynamics of magnetic materials excited by laser pulses has been created, integrating dynamic Landau-Lifshitz-Bloch equations with a quantum thermostat and a two-temperature model. The model has been successfully applied to three archetypal materials in the literature: nickel, cobalt, and iron. Additionally, analysis of the ultrafast dynamic susceptibility tensor indicates that off-diagonal components display specific features depending on whether a continuous external magnetic field is present.

Figures (3)

• Figure 1: Norm of the volume average of the first moment as a function of an external temperature for a) Nickel, b) Iron, c) Cobalt, computed by atomistic spin dynamics of dLLB equations with both classical (CFDR) and quantum-corrected (QFDR) thermal baths. The experiments are taken from Ref. crangleMagnetizationPureIron1971kuzminShapeTemperatureDependence2005 (see text).
• Figure 2: Norm of the volume average of the first moment as a function of time for a) Nickel, b) Iron, c) Cobalt, computed by atomistic spin dynamics of dLLB equations coupled to the 2TM with the parameters highlighted in Table.\ref{['table:2TM_parameters']}. The experimental data for Ni and Co are adapted from koopmansExplainingParadoxicalDiversity2010, while for Fe, the data are adapted from carpeneDynamicsElectronmagnonInteraction2008. The colors demonstrating an increase in experimental fluence are black, purple, and red, respectively.
• Figure 3: a) and d) Volume averaged first moment component as a function of time. b) and e) Diagonal elements of the second moment tensor of dLLB equation as a function of time. c) and f) Off-diagonal elements of second moment tensor of dLLB equation as a function of time. For a),b) and c) ${\bm B}={\bm 0}$. For d), e) and f) ${\bm B}=B_0{\bm e}_z$, with $B_0=1.5\,\mathrm{T}$ being the DC magnetic field.