We used a deformed Fe85.2Si0.5B9.5P4Cu0.8 (at.%) amorphous metallic alloy (metallic glass) with saturation magnetostriction of ls ~ + 40×10− 6 34, 35 as an example in this study. 26, 36, 37 The alloy was deformed by scratch testing under ambient conditions using a diamond tip. As a result, shear offsets occurred on the scratched surface. Figure 2a shows an ADF-STEM image of the TEM lamella, where the location of shear bands can be estimated based on the shear offset at the surface. We acquired a LA-Ltz-4D-STEM map in the area (green rectangle) including the shear bands indicated by the red arrows, with a step size of 10 nm for balancing the field of view and data size.
Figure 2b-d shows the LA-Ltz-4D-STEM results: maps of the magnetic field (\(\overrightarrow{B}\)), compressive strain (\({\overrightarrow{\epsilon }}_{\text{c}\text{o}\text{m}}\)), and relative density (\(\varDelta {\rho }\)). The brightness in the \(\overrightarrow{B}\) and the \({\overrightarrow{\epsilon }}_{\text{c}\text{o}\text{m}}\) images represent the strength of the fields, and the colors represent their orientation. Complicated magnetic nanostructures are observed in the heavily deformed zones, i.e., the vicinity of the shear bands. They are clearly different from the magnetic domain structure of the undeformed sample (Figure S4), which exhibits larger homogeneous domains (\(>3 {\mu }\text{m}\times 3 {\mu }\text{m}\)).
Figure 2c visualizes that the strain concentrates near the shear bands with an orientation difference of ~ 90° at each side of the shear bands. The asymmetrical strain fields with a sharp transition across shear bands are in line with the previous strain observations in deformed amorphous metallic alloys. 23, 38, 39 The strain also induces the variation of the relative atomic density \(\varDelta {\rho }\) which reflects the net volume change due to the hydrostatic stress. \(\varDelta {\rho }\) suffers a sudden change from positive to negative across the shear plane (Fig. 2d), namely the pop-in side (where the surface of the material was pressed down) suffers a mainly compressive strain, while on the opposite, the pop-out side (the material surface was pushed out) suffers tractive force that induces material dilatation. The strain field gradually fades out away from the shear bands in both compressed and tensile regions.
To rule out that density variations can introduce a phase shift of the exit electron wave (hence inducing a beam tilt), resulting in artifacts in the magnetic image, we conducted a conventional 4D-STEM measurement at the same sample position with the objective lens on (in a fully out-of-plane magnetized state due to the strong magnetic field generated from the objective lens). None of the contrast variations across the shear bands was visible in the DPC image (Figure S5). The local density gradients are too small to contribute any disturbance to the magnetic image and the observed features in Fig. 1b reflect the pure magnetic field of the sample.
For materials with isotropic magnetostriction properties, the magnetoelastic energy density can be written as \({e}_{ME}=-\frac{3}{2}{\lambda }_{s}\sum _{i=1}^{3}{\sigma }_{i}{\gamma }_{i}^{2}\), 40 where \({\lambda }_{s}\) is saturation magnetostriction, \({\sigma }_{i}\) is the deviatoric strain and its in-plane component can be quantified through the strain measurement (Figure S6a), \({\gamma }_{i}\) is the sine of the misorientation angle between \(\overrightarrow{B}\) and \({\overrightarrow{\epsilon }}_{\text{t}\text{e}\text{n}}\). Taking advantage of the correlative imaging, a map of \({e}_{ME}\) can be obtained if the strain and magnetic vectors were measured in 3D by involving the tomography strategy for vector field 41, 42 in our new approach. As a simplified illustration, Figure S6b shows the map of the in-plane contribution of \({e}_{ME}\). As the positive saturation magnetostriction (\({\lambda }_{\text{s}}\)~40 ppm), the \({e}_{ME}\) tends to align magnetic moments parallel to the tensile strain direction and perpendicular to the compressive strain direction. 43, 44 For magnetically soft ferromagnetic materials, the physical size of the domains is usually large to minimize the density of domain walls. 7 However, a high (magnetostatic) energy configuration is observed around the shear bands: small and periodically arranged magnetic domains are present on the pop-in side of the shear band. They are confined by the strain-induced magnetic anisotropy perpendicularly aligned to the shear band orientation due to strong compressive strain parallel to the shear band (strong tensile strain perpendicular to the shear band due to Poisson’s effect). In contrast, on the pop-out side of the shear bands, the magnetic structure is simpler and aligned parallel to the shear band direction due to the tensile strain parallel to the shear bands. The closure domains extend over both regions to reduce the magnetic anisotropy contributions of the magnetic domain state. Similar magnetic domain structures have been discussed in samples with orthogonal magnetic anisotropies in soft magnetic amorphous thin films induced by ion irradiation 45 and locally induced stress variation 46.
LA-Ltz-4D-STEM enables pixel-to-pixel correlation of the magnetic and atomic structure. We plot both information together in Fig. 2e. The white arrows represent the local \(\overrightarrow{B}\) vectors. The red sticks representing \({\overrightarrow{\epsilon }}_{\text{c}\text{o}\text{m}}\)visualize the strain field (\({\overrightarrow{\epsilon }}_{\text{t}\text{e}\text{n}}\) and \({\overrightarrow{\epsilon }}_{\text{c}\text{o}\text{m}}\) are perpendicular to each other). The arrows and sticks are overlay on the magnetic field image. Figure 2e shows that the orientation of \(\overrightarrow{B}\) and \({\overrightarrow{\epsilon }}_{\text{c}\text{o}\text{m}}\) are well correlated (namely, white arrows are close to being perpendicular to the red sticks) in the strongly strained area, except at domain walls and vortexes, showing that the magnetization is dominated by the local strain.
We statistically studied the magnetoelastic coupling in the highly strained regions around the shear band (Fig. 3a). Figure 3b and d show 2D histograms of \({\overrightarrow{\epsilon }}_{\text{t}\text{e}\text{n}}\) and \(\overrightarrow{B}\) respectively, where the axes correspond to the horizontal and vertical components of \({\overrightarrow{\epsilon }}_{\text{t}\text{e}\text{n}}\) and \(\overrightarrow{B}\) in the multi-information map (Fig. 3a) and the color corresponds to the number of pixels counted for each map. Figure 3b reveals a two-fold symmetry of the strain field with the principal orientations along (black arrows) and orthogonal (white arrows) to the shear band. Figure 3c shows the spatial distribution of the tensile strains binarized according to their orientation following the principal orientations shown in Fig. 3b (blue is for perpendicular and light gray for parallel to the shear band). It reveals a 90° rotation of the strain field between the pop-in and pop-out sides across the shear plane. Figure 3d shows a clustered distribution of \(\overrightarrow{B}\), where each cluster corresponds to a magnetic domain (Figure S7 shows the result from the whole map). The magnetic moments in the highly strained areas are preferentially orientated along the two principal strain directions, which are perpendicular to each other. Figure 3e shows the spatial distribution of the clusters (shown in Fig. 3b) according to their orientations following the principal orientations. The magnetic domains are highly coherent with the strain field.
As shown above, LA-Ltz-4D-STEM maps enable the analysis of multiple physical quantities. In the current study, these are the in-plane \(\overrightarrow{B}\) field with two degrees of freedom and in-plane strain tensor with 3 degrees of freedom (symmetric 2×2 matrix, in the above results represented in the form of two perpendicular principal strains) which can also be interpreted in different aspects such as deviatoric and volumetric strain. Owing to the pixel-to-pixel correlation between the different properties, there are many possibilities to correlatively analyze these quantities. Figure 3f-h shows an example. Figure 3f is a 2D distribution plot (2D histogram) of \(\left|\overrightarrow{B}\right|\) and the misorientation angle between \({\overrightarrow{\epsilon }}_{\text{t}\text{e}\text{n}}\) and \(\overrightarrow{B}\). The major population of the pixels concentrates at small misorientation angles (< 30˚), forming cluster ①. This reveals a preference that the in-plane component of \(\overrightarrow{B}\) increases linearly with decreasing misorientation between \({\overrightarrow{\epsilon }}_{\text{t}\text{e}\text{n}}\) and \(\overrightarrow{B}\). This suggests that in-plane tensile strain tilts the magnetization in-plane to reduce the magnetoelastic energy, resulting in a stronger strength of the projected magnetic field. In cluster ②, \(\overrightarrow{B}\) is independent of the strain orientation. Figure 3g and h visualize the spatial distribution corresponding to cluster ① and ②. The region for ① coincides with the well-defined primary domains with a low angular mismatch between the magnetic and strain fields, namely the magnetization and strain are well correlated. The region for ② contains the closure domains and vortexes with more complex magnetic structures, where the magnetoelastic coupling is weak. It is because of that the magnetic configuration is governed by both the magnetic anisotropy (magnetoelastic) and the dipole-dipole interaction (magnetostatics). Due to the sharp 90° transition of the strain states crossing the shear bands, the magnetic moments close to shear bands suffer strong magnetoelastic and magnetostatic competition, which results in the complex magnetic structures.
Interestingly, the amplitude of the in-plane magnetic components exhibits significant variations in different domains (Figure S5a). For example, the in-plane \(\left|\overrightarrow{B}\right|\) in Region ② is weaker than that in Region ①. The exact reasoning is not clear from our images, but this observation implies additional out-of-plane magnetization contributions to minimize the magnetization energy within the small bar-shaped sample. Especially, the complex patterns observed in the closure domain regions suggest a possible supplementary magnetic structure that varies in the sample thickness direction due to additional flux closure between the submicron domains. This could contribute to the observed decrease in the net in-plane magnetic flux density in some domains. Of course, we cannot discard the possibility of out-of-plane strain, which could also play a role in the presence of out-of-plane magnetization components.
Additional examples of correlatively analyzing the multiple physical quantities are demostrated in Figures S8 and S9. A direct correlative invesitgation of a domain wall properties is illustrated in Figure S10.
Figure 4 shows an in-situ magnetization test for the same TEM sample (supplementary video 1). The lamella was tilted to 10 degrees to apply a true in-plane magnetic field to the sample. The objective lens is gradually excited from 0–9%, corresponding strength of the magnetic field in the sample plane is from 0 to 30 mT (estimated by the fact that the 100% objective lens produces a 2 T magnetic field). While the major magnetization in the sample is oriented along the external magnetic field (\({\overrightarrow{B}}_{\text{e}\text{x}}\)), the areas exhibiting complex domain patterns are strongly resistant to change, especially when compared to the undeformed sample (supplementary video 2). We note that the magnetic domain structure almost recovered back to the original magnetization state after the objective lens current was set back to zero at the end of the test (Fig. 4g and h), except for only slight variations in some domain walls and vortexes. This indicates very low magnetic hysteresis and high reproducibility of the magnetization state in the material. The magnetic domain patterns are determined by the strain-induced magnetoelastic anisotropy. This may lead to potential applications for nanomagnetic devices.