🔬 mesoscale physics

Anisotropic domain wall velocity profiles in the creep regime: the interplay of chiral damping, stiffness and Dzyaloshinskii-Moriya interaction

This paper presents an extended angular creep model that incorporates dispersive domain wall stiffness and chiral damping to accurately describe the anisotropic expansion of magnetic bubble domains, thereby enhancing the quantitative extraction of Dzyaloshinskii-Moriya interaction and chiral dynamical effects.

Original authors: Adriano Di Pietro, Alessandro Magni, Stefania Pizzini, Frowin Dörr, Yasser Shokr, Gianfranco Durin, Silvia Tacchi, Marco Madami, Giovanni Carlotti, Emily Darwin, Alexandra J. Huxtable, Christopher H.

Published 2026-07-29

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CC BY 4.0

Original authors: Adriano Di Pietro, Alessandro Magni, Stefania Pizzini, Frowin Dörr, Yasser Shokr, Gianfranco Durin, Silvia Tacchi, Marco Madami, Giovanni Carlotti, Emily Darwin, Alexandra J. Huxtable, Christopher H. Marrows, Bryan J. Hickey, Michaela Kuepferling

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the microscopic world of computer memory not as a grid of tiny switches, but as a bustling city of magnetic bubbles. In this city, information is stored in the shape and movement of these bubbles, which are essentially islands of magnetism floating in a sea of opposite magnetism. To make these bubbles move and carry data from one place to another, scientists use a special kind of magnetic "wind" called a Dzyaloshinskii–Moriya interaction (DMI). You can think of DMI as a subtle, invisible hand that twists the edge of the bubble, giving it a specific "handedness" or chirality, much like how a screw has a specific thread direction. This twist is crucial because it determines how fast and smoothly the bubble travels along its track. If we want to build faster, more reliable computers, we need to measure exactly how strong this invisible twisting hand is.

For a long time, scientists had a standard way to measure this twist. They would blow a magnetic bubble and watch how it expanded when pushed by a magnetic field. They noticed the bubble didn't grow in a perfect circle; it stretched out more in one direction than the other, like a balloon being squeezed. By measuring how fast the edge moved in different directions, they could calculate the strength of the DMI. However, this method relied on a simplified view of the bubble's edge, treating it like a stiff, unyielding rubber band. The researchers in this paper wondered: what if the edge isn't just a rubber band, but something more complex that can wiggle, relax, and even have a "friction" that depends on which way it's twisting? They set out to see if ignoring these extra details was leading them to the wrong answers about how strong the magnetic twist really is.

The authors of this study decided to upgrade the mathematical model used to describe these expanding bubbles. Instead of just looking at the energy of the bubble's edge, they added two new, sophisticated ingredients to their recipe: "domain-wall stiffness" and "chiral damping." Think of domain-wall stiffness as the edge's ability to bend and relax its shape as it moves, rather than staying rigid. It's the difference between a stiff wire and a flexible garden hose; the hose can twist and turn in ways the wire cannot, changing how much effort is needed to move it. "Chiral damping" is like a special kind of air resistance or friction that changes depending on the direction the bubble is spinning. If the bubble twists clockwise, it might feel a different kind of drag than if it twists counter-clockwise.

When the team applied this new, more complex model to their data, they found that the old, simple way of looking at things was missing the bigger picture. They showed that if you only look at the energy (the "rubber band" view), you get a distorted view of the bubble's speed. The new model revealed that the stiffness of the wall and the chiral damping play huge roles in how the bubble expands. In fact, these factors can make the bubble move much faster or slower in certain directions than the old models predicted. The researchers demonstrated that when they included these extra factors, the model could help address discrepancies in the data, but only if they used the correct, independently measured strength of the DMI.

Crucially, the paper argues against the idea that you can simply look at a single picture of a growing bubble and instantly know the exact strength of the DMI. The authors suggest that without knowing the specific details of the wall's stiffness and its unique friction, trying to guess the DMI strength from the bubble's shape alone is like trying to guess the weight of a car just by looking at its tires—you might get a rough idea, but you'll likely be wrong. They found that the old method tends to underestimate the true strength of the DMI. Furthermore, the study notes that while the full model helps, it introduces new uncertainties because the stiffness and damping values are hard to pin down just from the bubble's shape. The study concludes that to get a truly accurate measurement, scientists need to combine the bubble expansion pictures with other independent measurements of the material's properties. It's a reminder that in the tiny, twisted world of magnetic memory, the devil is in the details, and ignoring the "wiggle" and the "friction" can lead us astray.

Technical Summary: Anisotropic Domain Wall Velocity Profiles in the Creep Regime

Problem Statement
The reliable quantification of the interfacial Dzyaloshinskii–Moriya interaction (iDMI) is critical for the development of magnetic racetrack memories, where iDMI dictates domain wall (DW) chirality, stability, and motion efficiency. A standard method for determining iDMI involves analyzing the asymmetric expansion of magnetic bubble domains under an out-of-plane driving field and an in-plane magnetic field ( $H_x$ ). While conventional analyses rely on velocity measurements along specific directions, recent approaches utilize the full angular dependence of the velocity, $v(\theta)$ .

However, existing angular creep models suffer from limitations. In the thermally activated, low-velocity creep regime, the DW velocity depends exponentially on the elastic energy scale. Current models primarily focus on the orientation-dependent wall energy but neglect two critical factors:

Dispersive Domain Wall Stiffness: The relevant elastic quantity is not just the equilibrium energy density, but the relaxed stiffness, which includes the angular curvature of the energy landscape and the relaxation of the internal magnetization angle. Neglecting this can lead to stiffness-induced velocity changes being misinterpreted as changes in the iDMI field.
Chiral Damping: Dynamical prefactors, such as chirality-dependent damping, introduce additional asymmetries that are often overlooked.

Consequently, fitting experimental data with energy-only models may yield iDMI values that are inconsistent with those obtained from independent, more reliable methods (such as flow-regime measurements or Brillouin light scattering).

Methodology
The authors develop an extended angular creep model that generalizes the description of domain wall motion around a bubble domain. The methodology involves:

Theoretical Framework: The model defines the domain wall energy density $\sigma$ considering the interplay between exchange stiffness, effective perpendicular anisotropy, and the DMI field ( $H_{DMI}$ ).
Incorporation of Stiffness: The authors introduce a relaxed domain-wall stiffness term, $\tilde{\sigma}(\theta, H_x)$ , which accounts for the energy landscape and the relaxation of the internal magnetization angle $\phi$ along a finite wall segment of length $L$ . This contrasts with the "stiffness off" limit where $\tilde{\sigma} = \sigma$ .
Incorporation of Chiral Damping: An orientation-dependent effective damping, $\alpha_{eff}(\theta, H_x)$ , is introduced to account for chiral damping effects, parameterized by a strength factor $\alpha_{cd}$ .
Velocity Formulation: The creep velocity is expressed in a reference-normalized form (Eq. 5), which, when logarithmically decomposed (Eq. 7), separates three distinct contributions:
1. The chiral damping term (algebraic prefactor).
2. The domain wall energy dependent prefactor (associated with the $\beta$ exponent).
3. The stiffness-dependent creep activation barrier (the exponential term).
Experimental Validation: The model is tested against experimental data from two Pt/Co-based ultrathin samples (labeled a4 and a5). The DMI fields for these samples were independently determined using asymmetric bubble expansion in the flow regime and Brillouin light scattering (BLS). These independent values were then fixed in the angular creep model to see if the full model could reproduce the measured angular velocity profiles $v(\theta)$ .

Key Results

Stiffness Dominance: Under typical creep conditions (where the scaling constant $\chi_0$ is large), the stiffness contribution dominates the angular dependence by modifying the activation barrier in the exponential term. Including the relaxed stiffness $\tilde{\sigma}(\theta)$ produces significant quantitative modifications to the velocity profile $v(\theta)$ .
Chiral Damping Role: Chiral damping primarily acts as a prefactor correction. While it modifies the curve, it does not generally produce a unique qualitative signature distinguishable from stiffness effects in the velocity profile alone.
Resolution of Discrepancies: When using the independent, reliable DMI values (from flow regime and BLS) as fixed inputs, the conventional "energy-only" model fails to reproduce the experimental angular profiles, often overestimating the velocity asymmetry. In contrast, the extended model (incorporating both stiffness and chiral damping) successfully reproduces the experimental data while maintaining consistency with the independently measured DMI values.
Model Artifacts: The inclusion of relaxed stiffness can introduce sharp kinks or cusps in the calculated angular velocity near the crossover between mixed Bloch-Néel and saturated Néel configurations. The authors note that these features are likely rounded out in experiments due to disorder, thermal averaging, and finite imaging resolution, and their absence in data should not be taken as evidence against the stiffness contribution.

Significance and Claims
The paper establishes that a rigorous determination of the DMI field from creep-regime velocity profiles cannot rely on energy-only models. The authors claim that:

Necessity of Multi-Parameter Analysis: Reliable extraction of iDMI requires separating energetic, elastic (stiffness), and dissipative (chiral damping) contributions. Assigning measured asymmetry solely to DMI leads to systematic errors (specifically, an underestimation of DMI in the creep regime).
Limitations of Single-Image Metrology: While single-image angular bubble expansion measurements reduce experimental effort, the resulting velocity profile $v(\theta)$ is not a standalone route to determine $H_{DMI}$ without prior or complementary information on chiral damping and domain-wall stiffness.
Framework for Interpretation: The extended model provides a more complete framework for interpreting chiral domain wall dynamics. It highlights that elastic effects and chiral damping significantly reshape the apparent DMI-induced velocity asymmetry, which is crucial for the design of velocity-selective DW logic and memory devices.

The authors modestly conclude that while their model resolves the discrepancy between creep and flow-regime DMI measurements, the parameters for chiral damping strength ( $\alpha_{CD}$ ) and stiffness relaxation length ( $L$ ) remain phenomenological in the absence of independent experimental constraints for these specific quantities. Future work is identified as necessary to independently measure domain-wall stiffness relevant to creep dynamics.

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