Introduction

The dynamics of domain walls play a pivotal role in the magnetic response of ferromagnetic materials. The noise signal within the ferromagnetic hysteresis curves, known as Barkhausen noise, is widely used in the steel industry for nondestructive testing (NDT)1,2,3,4, as it originates from the jerky motion of the domain walls, induced by the microstructural features and various chemical and physical imperfections in the material while exposed to external magnetic field5. Domain wall dynamics is also of great interest in the proposed concepts for domain wall-based memory and logic devices6,7,8,9,10. However, the physical details of the domain wall response, or the sensitivity of the spintronics devices to specific material imperfections is poorly known.

When it comes to the theoretical description and computational modeling of domain walls interacting with the disorder, one may mention at least two main lines of research. First, statistical physicists have built various minimal models, aimed at capturing and understanding the values of scaling exponents of the Barkhausen jump statistics, and hence the different universality classes11,12,13,14. In the simplest case where the material imposes a strong uniaxial anisotropy, the magnetic response can be modeled via the random field Ising model (RFIM) where the disorder (resulting from material imperfections) is modeled as a random (Gaussian) time-independent spatially uncorrelated local field interacting with the spins15,16. Another class of minimal models is given by the domain wall-based frameworks, including the Alessandro-Beatrice-Bertotti-Montorsi (ABBM) model17 as well as the model proposed by Cizeau, Zapperi, Durin, and Stanley (CZDS)18. There, the domain walls are treated as elastic interfaces, driven by an external magnetic field in a disorder field that is again sampled from a specific random distribution. A more accurate description of the magnetization dynamics (which is, therefore, more useful in the context of spintronics devices) can be obtained by means of micromagnetic simulations, taking into account the full vectorial nature of the magnetic moments19,20. In micromagnetic simulations, the disorder is often treated as a random spatial variation of the micromagnetic parameters, such as magnetocrystalline anisotropy, exchange stiffness, and saturation magnetization21,22,23, but there is a lack of physically justified representations for different types of single imperfections in the micromagnetic picture.

Dislocations are line defects in the lattice that, essentially, create an anisotropic internal stress field (i.e., its magnitude and sign depend on the direction from the dislocation core) in the material which decays as a function of the distance from the dislocation core24. In the literature, the effect of dislocations on magnetic domain wall dynamics is overlooked by arguing that the effect of individual dislocations is too weak to have a notable effect on the domain wall behavior25. The numerical research on domain wall–dislocation interaction is limited to in-plane magnetizations and the most simple configurations of dislocations, mostly appearing in the field of paleomagnetic applications26,27,28,29. In ferroics, dislocations have been shown to act as pinning sites for ferroelectric domain walls both experimentally30 and computationally through phase-field models31,32, but the implications in ferromagnets remain unclear. In this paper, we report on the implementation of dislocations in micromagnetic simulations via the conversion of the dislocation-induced stress field to an anisotropy field through the magnetoelastic constants. We show how the domain walls interact with the respective anisotropy fields for different surface orientations, film thicknesses, dislocation densities, and the relative orientations of the domain wall and the dislocation. Based on these numerical simulations, we argue that typical dislocation densities are likely to affect the domain wall dynamics in thin films with weak applied magnetic fields. We then illustrate a larger-scale use case of our approach by simulating the transient approach towards the dislocation-induced depinning transition of domain walls, and measuring the related scaling exponents characterizing the Barkhausen jumps. Our perspective here is in Barkhausen noise, and therefore we concentrate on Fe samples in body-centered cubic (bcc) phase, but our method should extend in a straightforward manner to modeling a large class of ferromagnetic systems with dislocations, including spintronics devices. To our knowledge, this is the first attempt to directly model the effect of dislocations in micromagnetic simulations.

Results

Dislocations in thin films: TEM imaging

We start by demonstrating the experimental motivation for the modeling approach presented in this paper; see methodological details in Methods. Transmission electron microscope (TEM) imaging can be used to not only image the microstructural features of the sample (e.g., dislocations, and other defects, see the top left panel of Fig. 1 showing an in-focus TEM image of a thin ferritic steel sample), but also to observe the magnetic domain walls by employing Lorenz microscopy (see the defocus image in the 2nd panel from the left in the top row of Fig. 1). Moreover, the objective lens of the TEM can be used to generate an out-of-plane magnetic field (the strength of which can be measured by a Hall-effect sensor holder33), which upon tilting the sample within the TEM can also result in an in-plane component capable of displacing the domain walls separating the in-plane magnetized domains (the latter is expected due to the shape anisotropy of the thin TEM samples, with the thickness less than 100 nm to ensure sufficient transmitted electron density). The remaining panels of Fig. 1 illustrate this by showing how the domain wall structure (visible as white and black lines) evolves when the sample is gradually tilted, resulting in a slowly increasing in-plane magnetic field driving the domain walls.

Fig. 1: Transmission electron microscope (TEM) images of thin ferritic steel sample.
figure 1

Top left panel: TEM in-focus image demonstrating a distribution of dislocations in the ferrite matrix. The other panels show Lorentz mode (defocus) images from the same area showing magnetic domain walls as white and black lines, with the different panels corresponding to different tilting angles as indicated in the figure. Non-zero tilting angles imply non-zero in-plane components of the small (a few mT) objective lens field, such that upon increasing the tilting angle (and hence the in-plane field), the domain walls get displaced from their initial positions. The sample preparation and TEM setup are described in Section Methods.

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The imaging area of the TEM experiments like the one described above thus seems to be an interesting experimental system that could be modeled by micromagnetic simulations. This is because the reachable length scales in typical state-of-the-art micromagnetic simulations are limited to thin film samples with a side length of the order of a few micrometers, making it in principle possible to simulate systems the size of which approaches that of the images shown in Fig. 1. A key challenge that remains is the accurate description of the microstructure with which the magnetic degrees of freedom, most importantly the domain walls, are interacting. The aim of the present paper is to take a crucial step towards realistic modeling of microstructure and disorder in micromagnetic simulations, by developing the methodology to include the effect of dislocations in micromagnetic simulations. The challenges in studying the dislocations’ effect on the domain wall dynamics experimentally arise due to the large variation of areal dislocation densities (as in the top left panel of Fig. 1) and the many other factors that contribute to the domain wall dynamics simultaneously, such as grain boundaries, thickness variation, and embedded particles. Another issue with TEM is the challenging imaging of magnetic samples, forbidding the accurate determination of the dislocation line directions as the magnetized sample tends to bend the electron beam in indefinite directions. In this work, we simulate systems with dislocation lines parallel to the surface normal. The dislocation lines perpendicular to the normal experience an attractive force towards the surface, and consequently such line direction components are expected to move spontaneously to the surface and vanish24.

Coupling stress to anisotropy

Here, we describe our conversion of dislocation stress fields to local anisotropy.

Magnetic anisotropy causes the spin alignment being preferred in certain directions with respect to the atomic lattice (magnetocrystalline anisotropy) or to the shape and dimensions of the sample (shape anisotropy). As described in Section Methods, the anisotropy can be accounted for in the micromagnetic calculations straightforwardly. The first-order anisotropy energy term can be written as

$${E}_{{{{\rm{an}}}}}=-\int\,{d}^{3}{{{\bf{r}}}}{\sum}_{\alpha ,\beta }{K}_{\alpha \beta }({{{\bf{r}}}}){m}_{\alpha }({{{\bf{r}}}}){m}_{\beta }({{{\bf{r}}}})$$
(1)

where α and β denote the x, y, z components of the anisotropy tensor K and the magnetization m. In the following, we neglect the r dependence in the notations for readability. In addition to the magnetocrystalline anisotropy, which is consequent of the rotational symmetries of the atomic lattice, applied stress can also be responsible for anisotropic effects. In the first-order approximation, the stress is coupled to the magnetic anisotropy through material dependent magnetostriction constants λ100 and λ111 so that the local anisotropy energy can be written as34,35

$${E}_{{{{\rm{me}}}}}^{0}=-\frac{3}{2}{\lambda }_{100}{\sum}_{\alpha }\left({m}_{\alpha }^{2}-\frac{1}{3}\right){\sigma }_{\alpha \alpha }-3{\lambda }_{111}{\sum}_{\alpha ,\beta }{m}_{\alpha }{m}_{\beta }{\sigma }_{\alpha \beta },$$
(2)

where mi is the ith component of the local magnetization vector m and σ is the elastic stress tensor that is a function of r. The term with \(\frac{1}{3}\) leads to a constant term in the total energy expression that does not contribute to the magnetization dynamics, and therefore we neglect it and use the form of

$${E}_{{{{\rm{me}}}}}^{0}=-\frac{3}{2}{\lambda }_{100}{\sum}_{\alpha }{m}_{\alpha }^{2}{\sigma }_{\alpha \alpha }-3{\lambda }_{111}{\sum}_{\alpha ,\beta }{m}_{\alpha }{m}_{\beta }{\sigma }_{\alpha \beta }.$$
(3)

The total magnetoelastic anisotropy energy is thus obtained via

$${E}_{{{{\rm{me}}}}}=\int\,{d}^{3}{{{\bf{r}}}}\,{E}_{{{{\rm{me}}}}}^{0}({{{\bf{r}}}}).$$
(4)

In accordance to Eq. (1), the form of Eq. (3) allows us to write the local K matrix as

$$K=\lambda \circ \, \, \sigma$$
(5)

where the elementwise matrix product, denoted by ∘, is performed between the matrices

$$\lambda =\frac{3}{2}\left(\begin{array}{ccc}{\lambda }_{100}&{\lambda }_{111}&{\lambda }_{111}\\ {\lambda }_{111}&{\lambda }_{100}&{\lambda }_{111}\\ {\lambda }_{111}&{\lambda }_{111}&{\lambda }_{100}\end{array}\right)$$
(6)

and

$$\sigma =\left(\begin{array}{ccc}{\sigma }_{xx}&{\sigma }_{xy}&{\sigma }_{xz}\\ {\sigma }_{xy}&{\sigma }_{yy}&{\sigma }_{yz}\\ {\sigma }_{xz}&{\sigma }_{yz}&{\sigma }_{zz}\end{array}\right).$$
(7)

To describe the stress field as induced by dislocations, we compute the local stress tensor σ(r) in accordance to the Peierls-Nabarro model as represented in Methods.

The above relations are written in the primary cubic coordinate system so that [100] direction equals \(\hat{x}\) and [111] equals \(1/\sqrt{3}(\hat{x}+\hat{y}+\hat{z})\). Since λ is written in terms of the lattice directions, studying different surface lattice planes requires rotation of the full λ matrix as well; \(\hat{z}\) is always the surface normal. Coordinate transformations to different surface lattice planes are carried out using the transformed form of the K matrix as

$$K={R}^{T}\left(\lambda \circ \, \, (R\sigma {R}^{T})\right)R$$
(8)

where R is the ordinary rotation matrix between the basic (Eqs. (5)–(7)) and the desired orientation of the atomic unit cell.

The derived result (Eq. (5)) still does not allow us to use it directly in an application where the user provides the anisotropy as the directions and coefficients for the easy axes. Solving for the eigenvalues and -vectors (the principal vectors) of the anisotropy tensor K(r) we obtain the solution in the triaxial form

$${E}_{{{{\rm{me}}}}}= -\int\,{d}^{3}{{{\bf{r}}}}\left[{K}_{t1}({{{\bf{r}}}}){\left({{{{\bf{t}}}}}_{1}({{{\bf{r}}}})\cdot {{{\bf{m}}}}({{{\bf{r}}}})\right)}^{2}+{K}_{t2}({{{\bf{r}}}}){\left({{{{\bf{t}}}}}_{2}({{{\bf{r}}}})\cdot {{{\bf{m}}}}({{{\bf{r}}}})\right)}^{2}\right. \\ \left. +{K}_{t3}({{{\bf{r}}}}){\left({{{{\bf{t}}}}}_{3}({{{\bf{r}}}})\cdot {{{\bf{m}}}}({{{\bf{r}}}})\right)}^{2}\right],$$
(9)

where Kti and ti are the eigenvalues and -vectors of K(r) and, here, denote the magnitude and direction of the ith anisotropy axis, respectively. The vectors ti are mutually orthogonal as spanning the eigenspace of the K matrix. In MuMax3, the triaxial anisotropy of the form of Eq. (9) is not currently possible to set up but we use our custom code (see Code Availability).

In our micromagnetic simulations, the total anisotropy term has

$${E}_{{{{\rm{an}}}}}={E}_{{{{\rm{c}}}}}^{{{{\rm{Fe}}}}}+{E}_{{{{\rm{me}}}}}$$
(10)

where \({E}_{{{{\rm{c}}}}}^{{{{\rm{Fe}}}}}\) is the cubic magnetocrystalline anisotropy energy of Fe. We note that summing up the anisotropy energies as such is not completely accurate, since the cubic anisotropy term \({E}_{{{{\rm{c}}}}}^{{{{\rm{Fe}}}}}\) originates from the cubic symmetry of the lattice which is broken in the vicinity of a dislocation. However, a comparison of the results excluding and including the cubic anisotropy term in the following sections indicates that this coarse approximation is valid for the means of domain wall dynamics. We should also mention an additional approximation: Even if single-crystalline iron is elastically somewhat anisotropic, the anisotropy field due to the dislocation stress fields is here computed using expressions for the latter derived under the assumption of isotropic elasticity.

In addition, we note that the magneto-elastic effect responsible for the dislocation-induced anisotropy is in principle a two-way phenomenon, such that in addition to mechanical stresses (here, stresses due to the dislocations) affecting the magnetization, also the magnetization will in principle affect the mechanical degrees of freedom, i.e., there would be a non-zero magnetostrictive strain. We do not describe the latter explicitly in our model as we are primarily interested in the leading order effect of dislocations on domain wall dynamics. This can be rationalized as follows: As a line-like domain wall moves within the magnet, the magnetization will locally change along the domain wall, and as a result, one would get a local change in the magnetostrictive strain. This effect is however not dependent on the position of the domain wall, and therefore the stress field (and hence the anisotropy field) emerging via Hooke’s law from the magnetostrictive strain due to the domain wall motion does not result in domain wall pinning at any specific location. On the other hand, the anisotropy field due to dislocations that are fixed in space is spatially localized around the dislocations, and hence the moving domain wall will experience a spatially varying anisotropy field which might pin the domain wall.

Dislocation-induced anisotropy on different surface planes

In Fig. 2, we illustrate the shape of the edge dislocation-induced anisotropy fields by showing the energy densities for various Fe surfaces and directions of the uniform magnetization; the cubic anisotropy is neglected in this visualization. We notice that the shape of the energy landscape is dependent on the surface lattice plane, which is a consequence of the anisotropic magnetostriction of the lattice. The quadrupole shape of the edge dislocation stress field is clearly reproduced in certain energy landscape shapes. In each case, the energy density is naturally highest close to the dislocation core. The values are only shown up to  ± 50 kJ m−3, beyond which the color is the same as that of the maximum/minimum of the color scale. The value of 50 kJ m−3 is chosen as being close to the cubic anisotropy constant of Fe (=48 kJ m−3). Close to the dislocation core, we see that the stress-induced anisotropy value exceeds that of the cubic anisotropy, indicating that the dislocation is able to affect the local magnetic state of the Fe sample through its stress field. The pixel size in Fig. 2 is 3 nm, which is the same value that we use as a grid spacing in the micromagnetic calculations for Fe. Close to the dislocation core, the energy landscape varies rapidly, and therefore one has to be careful when drawing quantitative conclusions of the results. The domain wall width in Fe is relatively large so that the finest details of the dislocation-induced potential have not much effect on the depinning fields, as we demonstrate in the Supplementary Information.

Fig. 2: Energy densities of uniform magnetizations in the edge dislocation-induced triaxial anisotropy fields.
figure 2

The anisotropy fields are generated according to Eq. (3). To better visualize and compare the shapes of the energy landscapes, the solid and dashed lines show the energy density isolevels of 5 kJ m−3 and  − 5 kJ m−3 respectively. Top row: (100), middle row: (110), bottom row: (111) surface. The Burger’s vector has \(\hat{b}=\hat{x}\) on each surface.

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The corresponding anisotropy field profiles for screw dislocations are shown in Supplementary Fig. 1 (Fig. S1). For screw dislocations, the magnetization pointing towards each of the x, y, and z directions leads to constant energy landscapes at (100) and (110) surfaces. We note that this result is reproduced for each in-plane magnetization direction at (100) and (110) surfaces. Instead, the introduction of a finite out-of-plane magnetization component creates nontrivial energy landscapes, as demonstrated in Supplementary Fig. 2 (Fig. S2). This indicates that in-plane Neél type domain walls are not affected at all by vertical screw dislocations.

In the rest of the paper, we therefore concentrate on the effect of edge dislocations on the domain wall dynamics.

Individual dislocation effect on domain walls

In the present paper, we are interested in how the dislocation-induced anisotropy field affects the shape and dynamics of magnetic domain walls.

Let us first describe the possible relative orientations of the domain wall and Burger’s vectors, respective to the surface lattice plane. 180∘ domain walls are most stable in the directions of the magnetocrystalline anisotropy easy axes. For (100) and (110) surface lattice planes, stable domain walls exist in the direction of the cubic in-plane easy axes, that is 〈100〉 directions for Fe. The (111) surface has no in-plane anisotropy, and therefore infinitely long, purely in-plane domain walls (Neél walls) are not stable on (111) surfaces unless there are other sources for anisotropy, such as residual stresses or shape anisotropy. The dislocation orientation is determined by its Burger’s vector. On (100) surface, we regard dislocations with Burger’s vector in 〈100〉 directions. On (110) surface, we regard dislocations with Burger’s vectors in 〈111〉 directions as manifesting the most energetically favorable configuration24. In both cases, there are four distinct Burger’s vector directions as visualized in Fig. 3a.

Fig. 3: A domain wall interacting with individual dislocations for different lattice and Burger’s vector orientations.
figure 3

a Directions of the lattices and respective symbols for Burger’s vectors treated in this paper. b Illustrative magnetic configuration of a 180∘ domain wall with a T shaped symbols at (0, 0) and (−768 nm, 0) indicating the positions of the dislocations and the directions of their Burger’s vectors. The arrows denote the magnetization direction at every 32nd grid point. The color map indicates the magnetization directions. c Average energy density of the system as a function of the domain wall distance from the dislocation. The distance dependence is shown for surface lattice planes (100) and (110) as well as for multiple Burger’s vector directions as indicated in the legend. The energy curves are shifted along y axis by their mean values, as the incomplete description of periodicity shifts the zero level of the energy (see original data in Supplementary Fig. 3 (Fig. S3)). d Depinning fields of dislocations in micromagnetic simulations for various sample thicknesses at (100) surface. e Similar data for (110) surface. The legend indicates the thickness of the system. The dashed line for a rigid wall shows the toy model predictions for a system of 16 nm thickness. f Depinning fields of dislocations in micromagnetic simulations for various linear dislocation densities in (110) surface. The dislocation density is varied by changing the size of the periodic simulation cell accordingly in the direction of the periodicity (that is the direction of the domain wall). Linear dependence of the depinning fields and the dislocation density is observed.

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To demonstrate the effect of dislocations on a magnetic domain wall, we first consider a system with a straight, rigid (i.e., a domain wall that maintains its structure during motion and, e.g., does not bend as a result of interaction with the dislocation) 180∘ domain wall in the vicinity of a dislocation as shown in Fig. 3b; we call it the toy model. The second dislocation at the left-hand edge of the system ensures a valid description of the anisotropy field periodicity (see details in Section Methods). The domain wall shape is given by

$${m}_{x}(x,y,z) = \sin \left(2\arctan {e}^{(x-{x}_{0})/\delta }\right),\\ {m}_{y}(x,y,z) = \cos \left(2\arctan {e}^{(x-{x}_{0})/\delta }\right),\\ {m}_{z}(x,y,z) =0$$
(11)

where x0 is the domain wall position and δ is a width parameter36. Here, we use the value of δ = 30 nm that is of the order of magnitude of the domain wall width in Fe thin films.

In Fig. 3c, we show the total anisotropy energy of the system as a function of the domain wall position with respect to the dislocation core; here we disregard the cubic anisotropy of Fe that would only lead to a constant term in the energy curves. We see that, for some surfaces and dislocation orientations, the dislocation is able to pin the domain wall movement (towards the positive x axis) by clearly producing an energy barrier to the domain wall energy landscape. For 0∘ and 180∘ orientations on (100) surface, the energy profiles indicate that the domain wall does not feel the dislocation: the energy landscape is flat and the barrier is absent. The 90∘ orientation on (100) surface shows a high energy barrier at the position of the dislocation, whereas the 270∘ orientation shows an equally high energy drop at the dislocation; this is expected due to the symmetry of the dislocation stress fields. Also, if the x component of the magnetization inside the domain wall is inverse, the energy profiles of 0∘ and 180∘, and 90∘ and 270∘ orientations would be interchanged by the same symmetry. We note that the convergence of the energy curves to nonzero energies far from the dislocation is a simulation effect that origins together from the second dislocation in the unit cell and the small size of the unit cell with periodic boundary conditions. With larger Lx, the expected 1/r decay towards zero would be apparent. However, we are interested in the maximum slopes of the energy curves as described in the next section and therefore the zero level of the energy is not important.

For 144∘ and 215∘ orientations on (110) surface in Fig. 3c, the domain wall experiences an energy minimum close to but not exactly at the location of the dislocation core. That is, those dislocations have an attractive nature towards the domain wall. For 35∘ and 324∘ orientations, the dislocation creates a clear energy barrier for the domain wall and effectively repels the domain wall.

To summarize, the simple energy examination indicates that domain walls react to dislocations in various ways. They are pinned or pushed spontaneously forward, or they do not feel the dislocations at all, depending on the relative orientations of the domain wall and the dislocation Burger’s vector as well as the surface lattice plane.

The above treatment neglects the real, dynamical effects introduced by the exchange interaction and the magnetostatic interaction between the magnetic moments, including the possibility for domain wall bending under the influence of the dislocation. We run micromagnetic simulations to treat these factors and study their effect on the domain wall reaction to dislocation stress fields.

In Fig. 3d and e, we show the depinning fields of edge dislocation arrays with fixed density. The dimension of the simulated system is 768 nm in the direction of the domain wall. Due to the periodic boundary condition in the same direction, the linear dislocation density is 1/768 nm−1 along the domain wall direction. The depinning fields are related to the maximum values of the slopes in the energy curves shown in Fig. 3c that are converted to field values by \(B={F}_{\max }/(2{M}_{s}{L}_{y}{L}_{z})\) where Ms is the saturation magnetization value of Fe, Ly and Lz are the sizes of the system in y and z directions, respectively (Lz is thus the sample thickness), and \({F}_{\max }={({{{\rm{d}}}}E/{{{\rm{d}}}}x)}_{\max }\) is the maximum slope of the energy curve. For the 16 nm thick systems, the predicted depinning fields by the toy model correspond very well to the ones predicted by the micromagnetic framework.

The thickness of the sample plays an interesting role for the depinning fields on (100) surface. The depinning strength increases as the thickness increases from 16 nm to 32 nm and decreases again while increasing the thickness further to 64 nm. The reduced depinning field for 64 nm is attributed to the spontaneous formation of asymmetric Bloch walls37 above the critical thickness that, according to our simulations, is between 32 and 64 nm for (100) surface lattice plane. The appearance of the asymmetric Bloch walls offers more flexibility for the domain wall to pass the pinning center, and therefore the depinning field is decreased in comparison to the pure Neél wall. The spontaneous birth of asymmetric Bloch walls originates from the minimized magnetostatic energy, and thus the toy model would not be able to reveal the thickness effect as lacking the demagnetizing interactions. Therefore, the micromagnetic formulation is required to catch the effect.

In 32 nm runs, asymmetric Bloch walls are not formed but, instead, multiple Bloch lines are formed along the domain wall when the external field is set on. This seems to increase the depinning field in comparison to pure Neél walls in the 16 nm thick sample. To verify this, we run the same depinning field analysis for 24 nm thick sample and observe (not shown in the figure) that no Bloch lines are formed and that the computed depinning fields are exactly the same as those for the 16 nm case.

On (110) surface, the thinnest samples have the largest depinning fields. Bloch lines are formed again in 32 nm thick sample, which in this case decreases the depinning field compared to the 16 nm thickness and pure Neél walls. This is a consequence of the existence of two off-plane anisotropy easy axes on (110) surface as a result of the orientation of the cubic unit cell of bcc Fe, whereas on (100) surface two of the easy axes are in plane. 24 nm thickness again shows the same depinning fields as 16 nm. The depinning fields for systems of thickness 64 nm are again smaller than for 32 nm as a result of the formation of asymmetric Bloch walls. As a small final observation, we note that the maximum depinning field at (100) surface is surprisingly much larger than the values at (110) surface.

The depinning field values in Fig. 3d and e are computed for very low dislocation densities in comparison to experimental values that are of the order of 1014 m−2. In linear density, this corresponds to an average of 1/100 nm−1. In Fig. 3f, we show that the predicted depinning field on (110) surface evolves linearly as a function of the linear, regular dislocation density along the domain wall up to the dislocation density of 1/48 nm−1. We thus conclude that, in this density range, bending and other changes in the internal structure of the domain wall are not remarkable in terms of the depinning effect.

Barkhausen noise from domain wall–dislocation interactions

To demonstrate the effect of dislocations on Barkhausen noise, we simulate the transient approach towards the domain wall depinning transition due to a slowly increasing external magnetic field in 2D thin films of linear size (i.e., the side length of the simulated thin film sample) of 2.99 μm and thickness of 16 nm with a disorder field due to multiple, randomly positioned dislocations. In such a system, the domain wall dynamics is characterized by a sequence of propagation bursts, or Barkhausen jumps, and we focus here on the numerical characterization of their statistical properties. The details of the driving scheme used are given in Section Methods.

The burst statistics are collected from multiple simulation runs, considering also different disorder strengths. The latter is here determined by the dislocation density of the system. We examine dislocation densities of four different values, that are of the typical order of magnitude in ferritic steels: D0.5 = 0.5 × 1014 m−2, D1.0 = 1.0 × 1014 m−2, D2.0 = 2.0 × 1014 m−2, and D4.0 = 4.0 × 1014 m−2, corresponding to 447, 894, 1788, and 3576 dislocations placed randomly in the simulation box, respectively. We run 200 simulations for the two smallest disorder strengths and 300 simulations for the two largest to obtain reasonable statistical ensemble sizes. A burst signal comprises the spatially averaged domain wall velocity vDW(t), out of which we determine the propagation distance si of the ith burst as

$${s}_{i}=\int_{{t}_{i}^{{{{\rm{start}}}}}}^{{t}_{i}^{{{{\rm{end}}}}}}{v}_{{{{\rm{DW}}}}}(t)\,{{{\rm{dt}}}}$$
(12)

where \({t}_{i}^{{{{\rm{start}}}}}\) and \({t}_{i}^{{{{\rm{end}}}}}\) are the starting and ending times of the burst, respectively. The burst duration is given by \({T}_{i}={t}_{i}^{{{{\rm{end}}}}}-{t}_{i}^{{{{\rm{start}}}}}-{t}_{w}^{{{{\rm{burst}}}}}\), where the waiting time \({t}_{w}^{{{{\rm{burst}}}}}\) (see Fig. 4a) is subtracted to consider only the most active part of the burst.

Fig. 4: Barkhausen noise from motion of domain walls interacting with dislocations.
figure 4

a Typical burst where the domain wall does not show Bloch lines. The propagation distance of the domain wall during the burst is 416 nm. Dislocation density is D1.0 = 1.0 × 1014 m−2. The green and red fillings show the burst duration T and the waiting time \({t}_{w}^{{{{\rm{burst}}}}}=5\) ns, respectively. The dashed horizontal lines show the threshold speed vt = ± 5 m/s. The damping oscillation immediately after tend is caused by relaxation of the magnetization after shifting the simulation window (see text for details). b Magnetic configuration at the end of the burst shown in a. The arrows denote the magnetization direction at every 96th grid point. c Typical burst where Bloch lines appear along the domain wall. The propagation distance of the domain wall during the burst is 507 nm. The dislocation density is D4.0 = 4.0 × 1014 m−2. d Magnetic configuration at the end of the burst shown in (c). A pair of Bloch lines gets nucleated within the domain wall. e Burst size distribution. f Burst duration distribution. g Burst size scaling with duration. The labels indicate the disorder strength with dislocation densities of D0.5 = 0.5 × 1014 m−2, D1.0 = 1.0 × 1014 m−2, D2.0 = 2.0 × 1014 m−2, and D4.0 = 4.0 × 1014 m−2, respectively, as well as the fit exponents for each data set; the number in parenthesis gives the statistical error of the fit. To enhance readability, the data points of the P(s) and P(T) distributions as well as for the 〈s(T)〉 data for D0.5, D1.0, and D2.0 have been shifted along the y axis by multiplying the y-coordinates of the data points by appropriate factors such that the different distributions do not overlap.

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In Fig. 4a and c, we show two examples of the burst shapes and the corresponding shapes of the domain wall at the end of the bursts from systems with two different dislocation densities/disorder strengths. The typical behavior of a burst is different depending on the appearance of Bloch lines during the burst, which in turn is controlled by the disorder strength. Figure 4a shows a typical burst for weak disorder in the absence of Bloch lines: The domain wall propagation velocity exhibits irregular time-dependence until the velocity drops to the negative side and settles close to zero. The temporary negative values of the velocity are due to the precession of the magnetization. The two bursts in Fig. 4a and c (with the latter for stronger disorder, exhibiting a more irregular velocity profile) have the propagation distance of the same order of magnitude, but the durations are very different (7 ns and 38 ns). The major part of the difference is due to the strong ringing of the signal in Fig. 4c at the latter half of the burst. During this time, the magnetization fluctuates so that the domain wall is not propagating but the Bloch lines are meandering along the domain wall causing the slight oscillation in the total magnetization above the threshold vt. The appearance of the Bloch lines is visualized in Fig. 4d. Below, we describe how the appearance of Bloch lines affects the burst statistics.

The burst statistics are illustrated in Fig. 4e–g. The distributions P(s), P(T) and the function 〈s(T)〉 show (partial) power law behavior, as expected for domain wall burst dynamics18. In the scaling regime, they scale with power law exponents τ, α and γ, respectively, according to

$$P(s) \sim {s}^{-\tau },$$
(13)
$$P(T) \sim {T}^{-\alpha },$$
(14)

and

$$\langle s(T)\rangle \sim {T}^{\gamma }.$$
(15)

We observe that the exponent values seem to indicate two apparently different scaling behaviors: For weak disorder, i.e., disorder strengths D0.5 and D1.0 the exponent values are close to each other and roughly equal to τ ≈ 1.1, α between 1.0 and 1.1, and γ ≈ 1.8. For stronger disorder (D2.0 and D4.0), the exponents have slightly different values, i.e., τ ≈ 1.3, α ≈ 1.1, and γ in the range between approximately 1.4 and 1.7. Out of these quantities we expect the avalanche/burst size s to be the most robust, while the event duration generally exhibits a shorter scaling regime (and hence less precise exponent value). In our case, an additional complication arises due to inertia in the domain wall dynamics, which is visible as ringing of the domain wall velocity at end of each event (see the discussion above and Fig. 4c). This implies that the end time of the burst (\({t}_{i}^{{{{\rm{end}}}}}\)) is challenging to determine accurately and that its ambiguity can potentially distort the scaling behavior of duration dependent quantities (P(T) and 〈s(T)〉) in the small system at hand. Hence, when interpreting the results, we focus here on the avalanche size distribution P(s). We emphasize that given our setup where the depinning field is approached from below by slowly ramping up the external field, giving rise to a non-stationary avalanche sequence, the τ (and α) exponents we measure should be interpreted as the integrated exponents τint = τs + σ where τs is the stationary exponent38. The value of σ is given by σ = (ν(1+ζ))−1 with the roughness exponent ζ and correlation length exponent ν for 2 dimensional system18. For 1+1 dimensional quenched Edwards-Wilkinson (qEW) equation, the exponents have known values of \({\tau }_{{{{\rm{s}}}}}^{{{{\rm{qEW}}}}}\approx 1.1\), ζqEW = 1.25, and νqEW = 1.36139. Therefore \({\tau }_{{{{\rm{int}}}}}^{{{{\rm{qEW}}}}}\approx 1.4\) for the qEW equation, in reasonable agreement with our strong disorder estimate. Thus, a possible interpretation is that for weak disorder (small dislocation densities), the Larkin length is of the order of the system size (or even larger), and hence the effective avalanche exponents for weak disorder are affected by finite-size effects. For stronger disorder (high dislocation density), the Larkin length is smaller, and hence the true critical avalanches start to become visible in the small system we study. This then leads to scaling of P(s) which is within errorbars consistent with that of the qEW equation. This is in agreement with our observations above that individual dislocations are relatively weak pinning centers for the domain wall and, hence, a rather high density is needed to significantly perturb the domain wall from a straight configuration. We illustrate the difference of the disorder fields for different dislocation densities in Supplementary Fig. 4 (Fig. S4). While strong disorder is here seen to sometimes result in nucleation of Bloch lines within the domain wall, there are typically only a small number of them, and hence it is unlikely that they would change the scaling behavior as observed in ref. 40.

Similar conclusions can be established qualitatively by observing the domain wall morphology for different dislocation densities. A high dislocation density results in a somewhat rough domain wall, as demonstrated in Supplementary Fig. 5 (Fig. S5). For a small dislocation density, the domain walls are relatively straight, suggesting a large Larkin length. For high density, the exact position of the domain wall can also be somewhat ambiguous due to the strong disorder: the domain wall structure is in some cases found to be more spread-out and therefore less line-like. We note that the latter observation does not affect our statistical distributions since the propagation distances are determined based on the changes of the total magnetization only (see Methods).

As discussed above, due to inertial effects, the burst duration cannot be determined unambiguously, as the domain wall velocity tends to exhibit damped oscillations (visible as ringing of vDW(t)) after the domain wall has found a new metastable state. This ringing effect seems to affect especially large/long events, and is stronger in systems with a higher dislocation density (see again Fig. 4c). We attribute this behavior to the appearance of Bloch lines along the domain wall as the disorder strength is increased. Higher disorder strength leads to higher gradients of the local disorder, enabling nucleation of Bloch lines while the domain wall is driven forward. The motion of Bloch lines along the domain wall mediates precession of the wall, and manifests itself as an increased inertia in the domain wall dynamics. Specifically, when the domain wall finds a new metastable state at the end of a burst, the Bloch lines still tend to move along the domain wall, resulting in oscillations of the average domain wall position around the local energy minimum. The resulting ringing effect is more pronounced for stronger disorder, and hence for stronger disorder an increasing contribution of the burst duration consists of the ringing of the burst. This is visible as distorted scaling behavior of the duration-related quantities in Fig. 4f: For instance, in the P(T) distribution of D4.0 (the strongest disorder case considered), clearly some events appear to be missing for T around 10 ns, while there is a rather pronounced bump corresponding to an excess of events around T = 25 ns. It is likely that events whose duration should be considered shorter have an excess contribution in their duration due to the ringing effect. A similar effect is visible also in the strong-disorder data for 〈s(T)〉. The problem in mitigating this issue is in selecting the value of the threshold vt: It should be large enough to recognize where the burst is being stopped and small enough to distinguish between the smallest bursts.

Discussion

We implement the dislocation-induced stress field and the resulting magnetoelastic anisotropy field in micromagnetic simulations. In contrast to the general understanding, we observe that the presence of dislocations is able to affect the domain wall dynamics in ferromagnetic thin films. Dislocations bend the domain walls and pin their movement under the applied magnetic field in the 2-dimensional thin film system. We demonstrate that the magnitudes of the pinning effects depend on the surface orientation, film thickness, relative orientation of the domain wall and the dislocation, and the dislocation density. The statistical inspection of domain wall dynamics under the influence of random dislocation fields reveals the power law behavior of the domain wall burst statistics in a way that clearly resembles the Barkhausen effect. We acknowledge that the random dislocation configuration is not a realistic setup in which the dislocations would tend to move spontaneously to relaxed positions. It would be interesting to exploit discrete dislocation dynamics simulations to obtain relaxed dislocation configurations or ones obtained after straining the system to a finite strain to create correlated disorder fields in the micromagnetic simulations, something that could affect the burst statistics. Another interesting possibility for future work is to consider effects of dislocations in atomistic spin dynamics (ASD) simulations41, where dislocations can be built directly into the lattice. That approach would however be quite limited in terms of the reachable length scales, but in addition to simulating nanoscale systems with a small number of dislocations ASD simulations might give rise to interesting multiscale modeling possibilities of structural defects in magnets when combined with continuum micromagnetic simulations.

As a side result, we observe that the appearance of asymmetric Bloch walls in thick enough samples affects the pinning effect of dislocations. This might change the statistical features of dynamics of the domain walls, possibly leading to different power law exponents from those we obtained for the 2-dimensional thin film case; the dimensionality is expected to play an essential role in the universal power law exponents of jerky dynamics18. This effect is most probably reproduced by other types of defects as well.

We note that the presence of dislocations affects the local micromagnetic parameters in various ways, but our model only takes into account the anisotropy field, induced by the local stress field of the dislocation. For example, the saturation magnetization is probably locally decreased close to the dislocation core and the addition of this factor would presumably change the pinning effect of dislocations. Studying the variance of other micromagnetic parameters in the vicinity of the dislocation core would require challenging atomistic calculations with magnetic moments included, something that should be worth future research efforts.

Finally, we point out that it has recently become possible to image the jerky dynamics of domain walls in situ in thin films with a transmission electron microscope in the Lorentz mode42,43. It is also possible to resolve the local densities of geometrically necessary dislocations and average Burger’s vector directions via the method of transmission Kikuchi diffraction. This combination of the experimental techniques enables the experimental verification of our findings in the future.

Methods

Transmission electron microscopy

Dislocations in the ferritic sample were imaged with a (scanning) transmission electron microscope (S/TEM, JEM-F200, JEOL). Domain walls in the sample were studied by Lorentz microscopy using TEM. Domain wall dynamics was studied by increasing gradually an in-plane magnetization component by tilting the sample in TEM. The strength of the out-of-plane magnetic field B⊥ in the sample area was few mT measured with a Hall-effect sensor holder33. The in-plane field B∣∣ responsible for displacing the domain walls within the sample is then given by \({B}_{| | }={B}_{\perp }\sin \theta\), where θ is the tilting angle. A thin sample for S/TEM and Lorentz microscopy studies was prepared by machining a 3 mm diameter disc (thickness  < 100 μm) followed by electropolishing (TenuPol, Struers) until perforation using a solution (−25 ∘C) of nitric acid in methanol (1:3).

Micromagnetic simulations

The total energy of a ferromagnetic system in the micromagnetic formulation is given by

$${E}_{{{{\rm{tot}}}}}={E}_{{{{\rm{applied}}}}}+{E}_{{{{\rm{ex}}}}}+{E}_{{{{\rm{demag}}}}}+{E}_{{{{\rm{an}}}}},$$
(16)

where Eapplied is the energy contribution from the external field, Eex from the electron exchange interaction, Edemag from the long-range magnetic dipolar interaction (or the resulting demagnetizing field), and essentially Ean is the energy contribution from the magnetic anisotropy of the material.

In this work, we use the GPU-accelerated code package MuMax344 and implement the resulting stress field-induced triaxial anisotropy therein (see Code Availability). The material parameters used in the work are given in Table 1. The dynamics is run using the RK45 algorithm as implemented in MuMax344,45. For additional details on the validity of the micromagnetic model used here we refer to Supplementary Note 1 and Supplementary Fig. 6 (Fig. S6).

Table 1 Simulation parameters for Fe
Full size table

Dislocation stress fields

Here, we describe how the dislocation stress fields are computed in the vicinity of the dislocation cores. We will concentrate on the cases of pure screw and edge dislocations with line directions parallel to the surface normal.

Edge dislocation

Let’s assume an edge dislocation in cubic lattice with Burger’s vector parallel to the x-axis i.e. the [100] direction, and the dislocation line parallel to the z-axis ([001]). We model the dislocation in accordance to the Peierls-Nabarro model which is divergence-free close to the dislocation core24,46. The symmetric stress tensor is given by the components

$${\sigma }_{xx}=-{C}_{e}\left(\frac{3y\pm 2\zeta }{{D}_{e}}-\frac{2y{(y\pm \zeta )}^{2}}{{D}_{e}^{2}}\right),$$
(17)
$${\sigma }_{xy}={C}_{e}\left(\frac{x}{{D}_{e}}-\frac{2xy(y\pm \zeta )}{{D}_{e}^{2}}\right),$$
(18)
$${\sigma }_{yy}=-{C}_{e}\left(\frac{y}{{D}_{e}}-\frac{2{x}^{2}y}{{D}_{e}^{2}}\right),$$
(19)

and

$${\sigma }_{zz}=\nu ({\sigma }_{xx}+{\sigma }_{yy}),$$
(20)

where  ± relates to the sign of the y coordinate, and ζ is a dislocation width parameter with the shape of

$$\zeta =\frac{d}{2(1-\nu )}$$
(21)

where d is the lattice spacing and ν the Poisson ratio. De is a distance function (with units of length squared) given by

$${D}_{e}={x}^{2}+{(y\pm \zeta )}^{2}.$$

The constant Ce in the stress tensor is given by

$${C}_{e}=\frac{\mu b}{2\pi (1-\nu )}$$
(22)

where μ is the Young’s modulus and b the magnitude of Burger’s vector.

Screw dislocation

For a screw dislocation, the Burger’s vector is parallel to the dislocation line. Given that the Burger’s vector points to z and the glide plane has y = 0, we have the stress tensor components

$${\sigma }_{xz}=-{C}_{s}\frac{y\pm \eta }{{D}_{s}}$$
(23)

and

$${\sigma }_{yz}={C}_{s}\frac{x}{{D}_{s}},$$
(24)

where

$${C}_{s}=\frac{\mu b}{2\pi }$$
(25)

and

$${D}_{s}={x}^{2}+{(y\pm \eta )}^{2}.$$
(26)

Here the dislocation width parameter η has

$$\eta =(1-\nu )\zeta =\frac{d}{2}.$$
(27)

Periodic dislocation stress field

Since we run the micromagnetic simulations with periodic boundary conditions in the direction of the domain wall, the sum of all Burger’s vectors has to be zero. Otherwise, the desired periodic symmetry would be broken. Therefore, Burger’s vectors are selected so that their sum is zero in each calculation presented in this paper. The periodicity of the stress field is achieved directly by copying the mirror image of each dislocation in the periodic directions x and y and calculating its contribution to the stress field in the primary unit cell. For the data in Fig. 3d and e, this procedure is done with 4 mirror images along both positive and negative x and y axes around the unit cell. For data in Fig. 3f, the size of the unit cell is varied, and therefore the number of mirror images is changed between the different dislocation densities: There are 4 images along  ± x directions, and the system size in y direction times the number of images along  ± y directions is kept constant (the same as for Fig. 3d and e). In the data for Fig. 4, we use 4 mirror images along both positive and negative x and y axes around the unit cell.

Computing depinning fields

In Fig. 3d–f, the depinning fields for a domain wall are computed with micromagnetic simulations. The grid spacing is 3 nm, 3 nm, and 4 nm in x, y, and z directions, respectively. In data of Fig. 3d and e, the grid size is 512 × 256 along x and y axes, and the grid size along z is determined by the desired thickness of the sample. In Fig. 3f, the grid size in y direction is halved to double the dislocation density. A dislocation is placed at the center of the system by precomputing the stress field and the derived anisotropy field and then using the anisotropy field in the MuMax3 run. In the MuMax3 run, a domain wall is initialized by positioning it at 50 nm left from the dislocation core (similar to Fig. 3a). The magnetic configuration is then relaxed. The external field is increased in steps of 0.02 mT, and, at every step, dynamics is run for 50 ns with the constant field. The depinning field is reached when my (the mean of the y component of the magnetization) increases above the value of 0.1.

Domain wall burst dynamics

In the simulations for Fig. 4, the simulated cell has a grid size of 1024 × 1024 × 1 with grid spacings 2.92 nm, 2.92 nm, and 16 nm in x, y and z directions, respectively. Here z axis is normal to the surface. The dislocation coordinates are randomly distributed in the system area, and the sum of the Burger’s vectors is enforced to zero in order to obtain the reasonable periodic stress field. The Burger’s vectors have 4 possible directions as illustrated in Fig. 3a. The magnetic configuration is initialized by setting \(\hat{m}=\hat{y}\) at x < 0, and \(\hat{m}=-\hat{y}\) at x > 0, and \(\hat{m}=\hat{x}\) at x = 0 so that there is a domain wall at the center of the system (x = 0) parallel to y axis. This initial configuration is relaxed. To move the domain wall towards the positive x axis, the applied field is directed to the positive y axis. In the beginning of the run, the field is increased linearly as Bext = kt with the rate of k = 2 mT ns−1. Once the speed of the domain wall exceeds a threshold value vt = 5 m s−1, we start recording a burst. During a burst, the field is kept constant with the magnitude the same as that at the start of the burst. The burst is stopped once the absolute value of the speed has stayed below vt for \({t}_{w}^{{{{\rm{burst}}}}}=5\) ns. The domain wall is then shifted back to the center of the cell with the built-in MuMax3 function ext_centerwall, padding the empty region with the magnetization vectors parallel to the negative y axis. Similarly, the underlying dislocation-induced anisotropy field is shifted by padding with the values from the opposite side (”rolling”). After the shift, the applied field is kept constant for \({t}_{w}^{{{{\rm{relax}}}}}=20\) ns to damp the precession caused by shifting the system and padding the magnetization field with slightly non-optimal values. The run is continued until the domain wall position reaches 1.4 μm, calculated from the center of the computational cell, which is very close to the edge of the system. Typically, if the run is continued after that, the domain wall starts moving immediately after shifting the system due to the absence of the restoring force that is exerted to the domain wall by the demagnetizing effect close to the edge of the system. This would lead to burst statistics that have large error due to the restrictions in the simulation setup, in particular the restriction on the system size due to memory issues.

The speed of the domain wall is defined as a finite difference value

$${v}_{{{{\rm{DW}}}}}(t)=\frac{{x}_{{{{\rm{DW}}}}}(t)-{x}_{{{{\rm{DW}}}}}(t-\Delta t)}{\Delta t}$$
(28)

where xDW(t) is the position of the domain wall at time t and Δt is the increment of time between two-time steps during the dynamics. The position of the domain wall is defined via the total magnetization as

$${x}_{{{{\rm{DW}}}}}(t)={m}_{y}(t)\frac{{L}_{x}}{2}$$
(29)

where my is the mean of the y component of the full magnetization of the system, and Lx = 2.99 μm is the dimension of the simulation box in x direction.