Spin’s Memory: Beyond Simple Damping in Ferromagnets

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Author: Denis Avetisyan

New research reveals that ferromagnetic materials exhibit a surprising ‘memory effect’ at terahertz frequencies, challenging the conventional understanding of how spins relax.

Spin dynamics are fundamentally altered by the inclusion of memory effects, as evidenced by a transition from uniform precession-characteristic of Markovian systems where magnetization responds only to the instantaneous phonon bath-to complex, excursion-prone behavior in non-Markovian systems where magnetization at any given time is coupled to its state at prior times, manifesting as multiple terahertz-frequency spectral peaks.
Spin dynamics are fundamentally altered by the inclusion of memory effects, as evidenced by a transition from uniform precession-characteristic of Markovian systems where magnetization responds only to the instantaneous phonon bath-to complex, excursion-prone behavior in non-Markovian systems where magnetization at any given time is coupled to its state at prior times, manifesting as multiple terahertz-frequency spectral peaks.

Intrinsic non-Markovian dynamics demonstrate a frequency-dependent memory kernel governing magnetization behavior.

While conventional models of magnetic materials typically assume simple damping of magnetization, this research, detailed in ‘Intrinsic non-Markovian magnetisation dynamics’, reveals an unexpected memory effect in the spin dynamics of crystalline cobalt. By employing terahertz spectroscopy, we demonstrate that the material’s magnetic response exhibits non-Markovian behaviour, stemming from a fundamental interaction with its phonon bath. This observation challenges the established framework and necessitates the application of open quantum system theory to accurately describe the system’s evolution. Could understanding and controlling these intrinsic non-Markovian effects unlock new avenues for manipulating magnetic materials and designing advanced devices?

Beyond Simple Memory: Unveiling the Past’s Influence on Magnetization

Conventional interpretations of magnetization dynamics frequently rely on the Markovian approximation, a simplification positing that a material’s current magnetic state is solely determined by its present conditions, effectively disregarding the influence of its past history. This approach treats the system as “memoryless,” assuming no lingering effects from previous configurations. While computationally convenient, this simplification overlooks the inherent complexities of ferromagnetic materials, where past magnetic states can demonstrably impact present behavior through subtle, time-dependent interactions. Specifically, the assumption of instantaneous relaxation – a hallmark of Markovian models – fails to account for the time required for magnetic moments to respond to external stimuli or to interact with neighboring moments, leading to discrepancies between theoretical predictions and experimental observations. Consequently, a more complete understanding of magnetization processes necessitates the inclusion of non-Markovian effects, acknowledging that the past, in fact, does influence the present.

Detailed spectroscopic analysis of ferromagnetic materials, such as Cobalt films, reveals deviations from predictions based on Markovian models. These discrepancies manifest as broadened or shifted spectral features, indicating that the current magnetization state isn’t solely determined by immediate influences, but retains a ‘memory’ of its past evolution. This observation suggests the existence of non-local effects where interactions extend beyond instantaneous neighbor relationships, implying a time-dependent correlation between past and present magnetic states. Essentially, the system doesn’t ‘forget’ its recent history, and this retained information significantly influences its dynamic response to external stimuli, demanding a more comprehensive theoretical framework to accurately describe its behavior.

Accurate depiction of magnetization processes hinges on acknowledging the influence of past system states, a necessity revealed by the limitations of traditional Markovian models. These models, while simplifying calculations, often fail to capture the full complexity of ferromagnetic behavior, particularly when observing spectral features indicative of memory effects. The ability to precisely describe and, crucially, control magnetization is paramount for advancements in data storage technologies, spintronics, and magnetic sensors. A comprehensive understanding of these non-Markovian dynamics, therefore, isn’t merely an academic pursuit; it’s a foundational step towards engineering materials with tailored magnetic properties and optimizing the performance of devices reliant on controlled magnetization, paving the way for faster, more efficient, and more reliable technological innovations.

Conventional models of magnetic behavior often treat a material’s current state as solely dependent on its immediate surroundings, a simplification known as the Markovian approximation. However, recent investigations reveal that this approach fails to fully capture the complexities of magnetization dynamics; the material effectively ‘remembers’ its recent history. This study demonstrates that the influence of past states extends for approximately 10 picoseconds, manifested as a distinct “memory kernel” in the system’s response. This temporal window highlights the need for more complete descriptions of magnetic evolution, moving beyond purely local interactions to incorporate the lingering effects of prior configurations and ultimately providing a more accurate and predictive understanding of ferromagnetic materials.

Analysis of terahertz-induced dynamics in a cobalt film reveals that the observed magnetic response (blue) closely matches simulations based on the Landau-Lifshitz-Gilbert (LLG) equation with a memory kernel derived from a Lorentzian spectral density (orange), indicating the importance of non-local dynamics.
Analysis of terahertz-induced dynamics in a cobalt film reveals that the observed magnetic response (blue) closely matches simulations based on the Landau-Lifshitz-Gilbert (LLG) equation with a memory kernel derived from a Lorentzian spectral density (orange), indicating the importance of non-local dynamics.

The Interplay of Energy and Vibration: A Dynamic Environment

The Cobalt film exists within a condensed matter environment and is not an isolated quantum system. Constant interaction occurs with the surrounding lattice structure, which is modeled as a Phonon Bath – a collection of quantized lattice vibrations. These phonons represent energy degrees of freedom within the material and facilitate energy and momentum exchange with the Cobalt film. This interaction is crucial because it introduces dissipation and decoherence, influencing the magnetic dynamics observed in the film. The film’s magnetization is therefore subject to continuous energy transfer to and from the Phonon Bath, affecting its relaxation and precession behavior. The temperature of the Phonon Bath, and thus the material, dictates the rate of these interactions and the overall energy distribution within the system.

The surrounding environment of the Cobalt film, termed the Phonon Bath, contains Zone-Edge Transverse Acoustic (ZETA) phonons which are particularly effective at mediating angular momentum transfer. These ZETA phonons, existing near the Brillouin zone edge, exhibit a high group velocity and strong coupling to the Cobalt film’s magnetization. This efficient momentum exchange occurs because the phonon modes can readily accept or donate angular momentum to the magnetic moments within the film. The resulting dynamics are not simply determined by the internal properties of the Cobalt, but are significantly influenced by these external phonon interactions, and contribute to the observed precession and damping of magnetization within the system. The efficiency of this angular momentum transfer is critical to understanding the non-equilibrium behavior of the Cobalt film.

The interaction between the cobalt film’s magnetization and the surrounding phonon bath results in non-Markovian dynamics, meaning the system’s future state is not solely determined by its present state. This occurs because energy transfer to and from the phonon bath introduces a time-dependent correlation; the system retains a “memory” of its past states. Specifically, the time it takes for energy to dissipate into the phonon bath-characterized by a relaxation time-is comparable to the timescale of the magnetization dynamics. Consequently, the system’s evolution is influenced by its history, violating the Markovian assumption of time-translational invariance and necessitating the use of non-Markovian equations of motion to accurately model its behavior.

The dynamics of the Cobalt film exhibit non-Markovian behavior, meaning its current state is influenced by its past states, analogous to energy transfer observed in a coupled pendula system. In such a system, energy isn’t simply dissipated; it oscillates between the pendula, creating a ‘memory’ of the initial conditions. Similarly, in the Cobalt film, angular momentum isn’t immediately lost to the phonon bath but is temporarily stored and transferred within the system before ultimately dissipating. This results in a time-dependent response where the film’s magnetization retains information about prior stimuli, differing from a purely random, Markovian process. The timescale of this memory is determined by the rate of energy exchange within the phonon bath and between the magnetization and the bath.

This experiment uses a collinear terahertz pump and magneto-optical Kerr effect probe, incident at 45 degrees, to investigate the dynamics of magnetization in an epitaxial cobalt thin film under the influence of external, terahertz, and demagnetizing fields.
This experiment uses a collinear terahertz pump and magneto-optical Kerr effect probe, incident at 45 degrees, to investigate the dynamics of magnetization in an epitaxial cobalt thin film under the influence of external, terahertz, and demagnetizing fields.

Beyond the Standard Model: Capturing History in the Equations

The Generalized Langevin Equation (GLE) extends the traditional Langevin equation by incorporating a time-dependent Memory Kernel, denoted as $K(t-t’)$, to account for non-Markovian dynamics. Unlike the standard Langevin equation which assumes the system’s future state depends only on its current state, the GLE postulates that the force acting on a particle at time $t$ is dependent on its entire past trajectory. This kernel, representing the autocorrelation of the fluctuating force, effectively quantifies the system’s “memory” of previous states and their influence on current behavior. The GLE is expressed as $m\ddot{x}(t) = -\int_{0}^{t} K(t-t’)\dot{x}(t’)dt’ + \xi(t)$, where $m$ is mass, $\dot{x}$ is velocity, and $\xi(t)$ represents a white noise force. The form of the Memory Kernel is determined by the specific physical processes governing the system’s environment and interactions, allowing for the modeling of a broad range of phenomena exhibiting long-time correlations.

The standard Landau-Lifshitz-Gilbert (LLG) equation describes the time evolution of magnetization but lacks the ability to model inertial effects present in certain magnetic systems. To address this, the Inertial Landau-Lifshitz-Gilbert (iLLG) equation was derived by adding an inertial term, $ \alpha \frac{d\mathbf{M}}{dt}$, to the standard LLG equation. This modification accounts for the system’s resistance to changes in magnetization and introduces a frequency dependent response. However, despite including inertial effects, the iLLG equation proved inadequate in fully capturing the observed dynamic behavior, specifically failing to accurately reproduce the multi-peak structure present in the terahertz (THz) frequency spectrum; this indicated the presence of additional, history-dependent effects not accounted for by the inertial term alone.

The Non-Markovian Landau-Lifshitz-Gilbert (LLG) equation accurately reproduces experimentally observed terahertz (THz) frequency spectra by incorporating a Memory Kernel. This kernel accounts for the system’s historical response, enabling the simulation of spectral peaks not predicted by the standard LLG equation. Specifically, simulations utilizing this extended equation demonstrate a multi-peak THz spectrum, matching the characteristics of observed data. The spectral positions and intensities of these peaks are directly influenced by the form and parameters of the Memory Kernel, which quantifies the time-dependent influence of past magnetization states on the current dynamics. This accurate spectral reproduction validates the model’s ability to capture non-Markovian effects and provides a means to analyze the underlying memory mechanisms within the system.

The Non-Markovian Landau-Lifshitz-Gilbert (LLG) equation explicitly accounts for the influence of prior system states on current dynamics, differing from the traditional LLG equation which assumes future evolution depends only on the instantaneous conditions. This is achieved through the incorporation of a Memory Kernel, a time-dependent function that weights the contribution of past magnetization states to the present-time effective field. Consequently, the equation’s time derivative term is no longer solely dependent on the instantaneous damping and precession, but rather includes an integral over past magnetization values, effectively introducing a time-delayed response. This historical dependence is critical for accurately modeling systems exhibiting non-Markovian behavior, where the system’s response is not instantaneous and retains a memory of its previous states, as demonstrated by the accurate simulation of multi-peak THz frequency spectra.

The plotted memory kernels demonstrate how different functional forms-ranging from the time-local, Markovian limit to an extended memory of approximately 10 picoseconds-yield the LLG, iLLG, and nM-LLG equations and influence the resulting non-Markovian dynamics.
The plotted memory kernels demonstrate how different functional forms-ranging from the time-local, Markovian limit to an extended memory of approximately 10 picoseconds-yield the LLG, iLLG, and nM-LLG equations and influence the resulting non-Markovian dynamics.

Engineering the Future: Harnessing Magnetic Memory

The functionality of spintronic devices hinges on the precise manipulation of magnetization, and a complete understanding requires moving beyond traditional models that assume immediate response to external stimuli. These simplified approaches neglect non-Markovian dynamics – the influence of past states on present behavior – which introduces a ‘memory’ effect into the system. Accurately incorporating this memory allows for a far more nuanced control over magnetic processes; researchers can now predict and steer magnetization trajectories with greater fidelity. This capability isn’t merely theoretical, as it enables the design of devices with faster switching speeds, lower energy consumption, and enhanced stability – all critical parameters for advancements in data storage, sensors, and novel computational paradigms. By accounting for the full temporal evolution of magnetization, rather than just its current state, the pathway to truly optimized spintronic performance is becoming increasingly clear.

Conventional understanding treats the demagnetization field as a static influence on magnetic materials, a consequence of the shape and magnetization distribution. However, recent research demonstrates a dynamic interplay between this field and the non-Markovian dynamics of the system. The demagnetization field isn’t simply a backdrop; it actively responds to and is modified by the transient, correlated motions within the material. This interaction significantly alters the observed magnetic behavior, extending beyond simple suppression of magnetization. The non-Markovian dynamics effectively ‘reshape’ the demagnetization field over time, creating localized variations and influencing the precession and relaxation of magnetic moments. Consequently, a comprehensive model must account for this reciprocal relationship to accurately predict and control magnetization processes, opening new possibilities for finely tuned magnetic devices.

A refined comprehension of magnetization dynamics extends beyond fundamental physics, offering tangible pathways for materials science innovation. Researchers are now equipped to move past relying on materials with naturally occurring magnetic characteristics, instead envisioning compounds specifically engineered for optimal performance in spintronic devices. This detailed understanding allows for the precise manipulation of magnetic properties – coercivity, saturation magnetization, and magnetic anisotropy – at the atomic level. By tailoring these characteristics, materials can be designed to minimize energy loss, enhance data storage density, and improve the speed and efficiency of magnetic sensors and actuators. The ability to custom-design magnetic materials promises breakthroughs in diverse fields, ranging from high-density data storage and energy-efficient computing to advanced medical imaging and novel sensor technologies.

The advancement of spintronics hinges on a departure from traditional, simplified models of magnetic behavior. Current technologies often rely on approximations that, while computationally convenient, obscure crucial details of dynamic magnetization processes. This research demonstrates that embracing a more complete, non-Markovian description of these processes unlocks the potential for significantly enhanced device performance. By accurately capturing the complex interplay of factors governing magnetic relaxation and excitation, engineers can now design materials with precisely tailored properties – optimizing characteristics like switching speed, data retention, and energy efficiency. This detailed understanding doesn’t simply refine existing spintronic devices; it establishes a foundation for entirely new functionalities and architectures, promising a future where magnetic materials are integral to increasingly sophisticated information technologies.

Time-resolved magnetization measurements reveal coherent precession, THz-frequency oscillations, and a non-magnetic response, the latter of which is evidenced by dips around 2 picoseconds and used to generate spectra in Figure 3.
Time-resolved magnetization measurements reveal coherent precession, THz-frequency oscillations, and a non-magnetic response, the latter of which is evidenced by dips around 2 picoseconds and used to generate spectra in Figure 3.

The research illuminates a departure from the simplistic understanding of magnetic damping, revealing the intricate influence of the phonon bath on spin dynamics. This challenges the Markovian assumption, where future states depend solely on the present, and introduces a ‘memory effect’ within the material. As John Bell poignantly observed, “No phenomenon is a complete mystery, if we know enough.” This sentiment encapsulates the spirit of the investigation; by delving into the non-Markovian behavior at terahertz frequencies, the study peels back layers of complexity, demonstrating that seemingly instantaneous damping is, in fact, a consequence of past interactions-a history embedded within the material’s response. The identification of this memory kernel underscores the necessity of considering the complete system-the interplay between spins and phonons-to accurately model and predict magnetic behavior.

Beyond the Immediate Echo

The observation of intrinsic non-Markovian dynamics in ferromagnetic materials at terahertz frequencies suggests a deeper complexity than previously appreciated. The conventional depiction of spin relaxation as simple damping – a convenient, if incomplete, simplification – now requires revision. The identified memory kernel, a vestige of the interaction with the phonon bath, indicates that the material’s past demonstrably influences its present magnetic state. This is not merely a matter of refining parameters; it is an acknowledgement that the system’s architecture dictates its behavior, and a complete understanding necessitates tracing the consequences of every interaction.

Future investigations should concentrate on quantifying the precise nature of this memory effect. What is the functional form of the memory kernel across different materials and frequencies? How does this non-Markovianity manifest in more complex magnetic phenomena, such as domain wall motion or skyrmion dynamics? A crucial, and often overlooked, point is that modifying one part of this system inevitably triggers a domino effect. Focusing solely on the spin bath, while convenient, risks obscuring the interplay with other degrees of freedom.

Ultimately, the field must move beyond seeking increasingly accurate descriptions of damping and towards a holistic understanding of the energy flow within these materials. A truly predictive theory will not simply account for the memory kernel, but derive it from the fundamental principles governing the interactions within the complex, multi-component system. The elegance of a solution, one suspects, will lie not in its complexity, but in its simplicity.

Original article: https://arxiv.org/pdf/2512.07378.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2025-12-09 11:09