We examine the substructures of magnetic domain walls (DWs) in [Pt/(Co/Ni)M/Ir]N multi-layers using a combination of micromagnetic theory and Lorentz transmission electron microscopy. Thermal stability calculations of Q=±1 substructures [2π vertical Bloch lines and DW skyrmions] were performed using a geodesic nudged elastic band model, which supports their metastability at room temperature. Experimental variation in strength of the interfacial Dzyaloshinskii–Moriya interaction and film thickness reveals conditions under which these substructures are present and enables the formation of a magnetic phase diagram. Reduced thickness is found to favor Q=±1 substructures likely due to the suppression of hybrid DWs. The results from this study provide an important framework for examining 1D DW substructures in chiral magnetic materials.

Discovery of the Dzyaloshinskii–Moriya Interaction (DMI) in bulk magnets1,2 and magnetic thin films3 has launched an intense research effort into its effects on the structure of magnetic bubbles and domain walls including the formation of topological excitations such as skyrmions4–7 and anti-skyrmions.8,9 The combination of topological protection, which offers improvement to thermal stability, and their susceptibility to manipulation by spin–orbit torques makes these features of great interest for future spintronic applications including non-volatile memory10–12 and neuromorphic computing.13–16 

While it is well established that the internal structure of a domain wall (DW) transitions from Bloch to Néel type with increasing interfacial DMI, less attention has been paid to its effect on the internal DW substructure. A feature known as a magnetic DW skyrmion has recently been theoretically predicted17–19 and is a 360° rotation of the internal magnetization of a chiral Néel DW (Fig. 1).19 These are the post-DMI analog of vertical Bloch lines (VBLs), which are 180° rotations in a Bloch DW and were once considered for universal computer memory.20 It is notable that DW skyrmions are predicted to be about one order of magnitude smaller than VBLs.19 Both of these substructures are confined to move within a magnetic DW and are, therefore, not subject to edge pinning (like a conventional DW) and are not able to drift in unwanted directions as with 2D skyrmions (via the skyrmion Hall effect). In addition to their potential use in non-volatile magnetic memory storage or neuromorphic computing applications, these DW substructures have been shown to affect DW motion21 and the formation of skyrmions via stripe pinching22 and nucleation.23,24

FIG. 1.

Representative schematics depicting the internal magnetization of a 2π vertical Bloch line along an achiral Bloch wall and a domain wall skyrmion along a chiral Néel wall. Note that topological charge, Q, is determined by following the magnetizations along the domain wall (L to R) rather than across it.19 

FIG. 1.

Representative schematics depicting the internal magnetization of a 2π vertical Bloch line along an achiral Bloch wall and a domain wall skyrmion along a chiral Néel wall. Note that topological charge, Q, is determined by following the magnetizations along the domain wall (L to R) rather than across it.19 

Close modal

Both DW skyrmions and VBLs can be described by their topological charge as calculated from 4πQ=dxdym(xm×ym), where m is the unit magnetization vector.25 A DW skyrmion has a topological charge of Q=±1, whereas a single VBL has a charge of Q=±1/2. In the case of a 2π VBL (i.e., a 360° rotation within a Bloch DW), the topological charge is equivalent to that of a DW skyrmion. Herein, DW skyrmions and 2π VBLs are collectively referred to as Q=±1 substructures.

Here, we examine both VBLs and DW skyrmions in perpendicularly magnetized thin films. Initial thermal stability calculations were performed via micromagnetic simulations to evaluate the metastability of VBLs and DW skyrmions. The presence of such substructures is expected to be strongly dependent on DMI strength as well as the thickness of the thin films, which is described in more detail in Sec. III B. As such, we leverage a highly tunable asymmetric multi-layer system based on (Pt/[Co/Ni]M/Ir)N, where a reduction in “M” leads to a greater interfacial DMI from the Pt/Co and Ni/Ir interfaces and “N” primarily modulates the total film thickness to identify the optimal conditions where VBLs and DW skyrmions may exist.26–29 These samples were imaged using Lorentz transmission electron microscopy (LTEM), which offers high resolution imaging of magnetic features in thin films. The results of this systematic study of M and N iterations were used to formulate a magnetic phase diagram describing substructures observed as a function of DMI strength and sample thickness.

In order to utilize these DW substructures for possible spintronic applications, an understanding of their stability is necessary as thermal fluctuations can lead to their annihilation. Prior treatment of DW substructures considered the energy of the VBLs or DW skyrmions and their ground state configurations but did not consider the energy barrier associated with their formation or annihilation.19 Here, we have employed a geodesic nudged elastic band (GNEB) method30 in combination with a climbing image method31 implemented in the micromagnetic code M3, a MATLAB code based on finite-differences.32 The GNEB we use builds on the nudged elastic band model (NEB) but takes the constraint into account that the saturation magnetization of each cell in the simulation volume remains constant. For N magnetic moments, this method results in an unconstrained optimization within a 2N dimensional Riemannian manifold, as is discussed in detail in Ref. 30. For the evaluation of the geodesic distance between two images, we use Vincenty’s formula.33 In order to converge the images to the nearest minimum energy path, we use a steepest descent method34 with a Barzilai–Borwein step length selection method.35 To determine their stability, one has to find the activation barrier, which separates the skyrmion state from lower energy states. In the case of conventional skyrmions, this would be the skyrmion state and the simple ferromagnetic state. For the case of a DW skyrmion, the corresponding lower energy state is a skyrmion-free domain wall. Since thermally activated magnetic transitions are rare events, dynamical simulations using a stochastic Landau–Lifshitz–Gilbert equation are not practical. Therefore, the GNEB method is used to find the minimum energy path for the transition; this has been successfully applied to study the annihilation of conventional magnetic skyrmions.36,37

To stabilize the DW substructure, we consider an ultra-thin ferromagnetic film (2 nm) with an interfacial DMI interaction and a uniaxial perpendicular anisotropy.19 The total volume simulated was 128×256×2 nm3 where the cell size was 0.5×0.5×2 nm3. Magnetic parameters of MS=600 kA/m, Keff=2×105 J/m3, A=1.6×1011 J/m, and D=0.21.5 mJ/m2 were used, which are similar to previous measurements based on this Pt/Co/Ni/Ir system.28,38 The symmetric exchange and the dipole–dipole interaction are included in the micromagnetic simulations.39,40

To calculate the minimum energy path between those two states, 200 images were created to represent the transition path between the two fixed endpoint images. Figure 2(a) shows the domain wall with a DW skyrmion, Fig. 2(b) shows the domain wall just before DW skyrmion annihilation, and Fig. 2(c) shows the domain wall after the annihilation of the substructure. The transition between these states occurs through sharp narrowing of the DW skyrmion before the center spin flips direction concurrent with a change in the topological charge to Q=0.

FIG. 2.

Micromagnetic outputs of a Dzyaloshinskii domain wall (a) with a domain wall skyrmion (image distance = 0), (b) upon annihilation (image distance = 107), and (c) without one (image distance = 200). (d) and (e) Energy contributions as a function of the image distance between the domain wall skyrmion solution and the domain wall without a domain wall skyrmion. (f) Results from nudged elastic band calculations of creation and annihilation barriers for a domain wall skyrmion as a function of DMI strength.

FIG. 2.

Micromagnetic outputs of a Dzyaloshinskii domain wall (a) with a domain wall skyrmion (image distance = 0), (b) upon annihilation (image distance = 107), and (c) without one (image distance = 200). (d) and (e) Energy contributions as a function of the image distance between the domain wall skyrmion solution and the domain wall without a domain wall skyrmion. (f) Results from nudged elastic band calculations of creation and annihilation barriers for a domain wall skyrmion as a function of DMI strength.

Close modal

The energy barrier to annihilation is determined by the difference in total energy between the first image and the maximum energy encountered during the annihilation process. This was done for a range of DMI strengths whereby the barrier to annihilation is observed to decrease with increased D and the barrier to creation increases initially but levels off with large D [Fig. 2(f)]. We note that a VBL with D=0 was not examined as the only pathway to annihilation for a 1π VBL is via the edge. However, as previously shown by Cheng et al., a critical DMI strength of 0.125 mJ/m2 is necessary to stabilize a full 360° winding with these parameters.19 More notably, even for the ultrathin film considered here, the energy barrier to annihilation is >60kBT for D<1.0 mJ/m2 at room temperature. This suggests a thermal stability lifetime >10 years.41 This energy barrier is directly rooted in the symmetric exchange (i.e., the exchange stiffness) as with 2D skyrmions. In the thin film approximation (i.e., uniform magnetization through the thickness), this value should scale linearly with thickness. However, it is reasonable to expect that dipolar interactions in substantially thicker films could lead to the formation of hybrid DWs characterized by magnetization that rotates through the film thickness.42–44 This would inherently create low-energy pathways for the DW skyrmion to annihilate as further discussed in Sec. III B.

[Pt(0.5 nm)/(Co(0.2 nm)/Ni(0.6 nm))M/ Ir(0.5 nm)]N multi-layers were prepared via rf (Ta layers) and dc (Pt, Co, Ni, Ir layers) magnetron sputtering on oxidized Si substrates for magnetic property measurements and 10 nm thick amorphous Si3N4 TEM membranes (Norcada) for LTEM imaging. The working pressure was fixed at 2.5×103 Torr of Ar. All samples had a Ta(3 nm)/Pt(3 nm) seed/adhesion layer and were capped with Ta(3 nm). The base pressure was maintained at less than 3×107 Torr. Magnetic properties were examined using alternating gradient field magnetometry (AGFM) and vibrating sample magnetometry (VSM), which confirms a strong perpendicular magnetic easy axis in all cases (Fig. S1 in the supplementary material).45 These films were imaged using LTEM using an aberration-corrected FEI Titan G2 80-300 at an accelerating voltage of 300 kV in a Lorentz mode (objective lens off). LTEM employs the inherent in-plane magnetic induction of the sample to deflect the electron beam and form magnetic contrast. The resulting contrast formed can give further information about the DW character. Fresnel-mode LTEM images shown here were taken with a defocus value of 1.0 to 7.0 mm depending on the thin film examined.

We have previously examined the magnetic structure of this Pt/Co/Ni/Ir multi-layer system and found the formation of labyrinthine domains in relatively thick (25 nm) samples for any combination of N and M.28 It was observed that the presence of Pt and Ir sandwiching Co/Ni layers induced an appreciable interfacial DMI manifested in the form of chiral Néel DWs for M=13 (Fig. 3). Symmetric samples did not have an appreciable DMI strength and instead displayed achiral Bloch DWs. In addition to these Bloch DWs, thick symmetric samples show sparsely distributed 1π VBLs and no 2π VBLs. This suggests an inherent instability of 2π VBLs, which would otherwise form when two 1π VBLs come into close proximity. It is worth noting, again, that such 2π VBLs are topologically equivalent to a DW skyrmion. It is reasonable to speculate that the relatively thick multi-layers examined here develop a non-uniform magnetization through the film thickness, which would create low-energy pathways to annihilation for both 2π VBLs and DW skyrmions.

FIG. 3.

Fresnel-mode Lorentz TEM micrographs of symmetric (a) M=10 and (b) M=100 samples. Arrows in (b) indicate locations of vertical Bloch lines denoted by contrast reversal from white/black to black/white (and vice versa) as seen more clearly in (a) where 2π vertical Bloch lines are present. (c) Fresnel-mode Lorentz TEM micrographs of an asymmetric M=2, N=20 sample in the tilted state. The red arrow in (c) designates that the direction samples were tilted. (d) Schematic of the asymmetric Pt/Co/Ni/Ir stack whereby M designates the number of Co/Ni repeats in each repetition of Pt through Ir, N.

FIG. 3.

Fresnel-mode Lorentz TEM micrographs of symmetric (a) M=10 and (b) M=100 samples. Arrows in (b) indicate locations of vertical Bloch lines denoted by contrast reversal from white/black to black/white (and vice versa) as seen more clearly in (a) where 2π vertical Bloch lines are present. (c) Fresnel-mode Lorentz TEM micrographs of an asymmetric M=2, N=20 sample in the tilted state. The red arrow in (c) designates that the direction samples were tilted. (d) Schematic of the asymmetric Pt/Co/Ni/Ir stack whereby M designates the number of Co/Ni repeats in each repetition of Pt through Ir, N.

Close modal

Noting that neither DW skyrmions nor 2π VBLs are stable in thicker multi-layer samples, we turn to thinner films where it is more reasonable to expect uniform magnetization through the film thickness, which may support them. In thinner symmetric samples (M=10, N=1), we found a larger presence of 1π VBLs than in thicker ones (M=100, N=1) as seen in Fig. 3. More importantly, we note significant VBL pileups, which include many 2π VBLs. This suggests that any low-energy paths to annihilation of 2π VBLs associated with the larger thickness have been suppressed. Therefore, we postulate that reducing the total film thickness of similar asymmetric films while maintaining a sufficiently large DMI to support chiral Néel DW formation would lead to DW skyrmion formation. Having established broadly the importance of sample thickness and DMI on the stability of these excitations, we now expand on our experimental characterization of this material system through systematic variation of both these critical properties toward the development of a DW substructure phase diagram.

We began by examining thin (N=12) asymmetric films of M=310 to minimize overall film thickness. Although DMI strength increases with decreasing M, we observe some Bloch component to the DWs in each of these films [Figs. 4(a)4(d)]. Additionally, we note an overall reduction in contrast at the DW with decreasing M, which likely stems from both an increase in the Néel character of the DW as well as reduced overall film magnetization. Even though contrast consistent with 2π VBL pileups is observed in each of these films, it is difficult to ascertain the true magnetic character of these substructures because the DWs are likely of mixed Bloch–Néel character. For M=3, N=2, exclusively chiral Néel DWs are observed. The Fresnel-mode contrast locally matches that predicted for DW skyrmions from micromagnetic simulations including an asymmetric distortion of the DW itself [Fig. 4(e)]. We note that an inherent sample tilt is present due to bends in the underlying TEM membrane, which leads to the appearance of DW contrast parallel to the tilt axis.29,46 Details can be found in the supplementary material. Moreover, we examine the response of multiple similar sites in situ as a perpendicular magnetic field is applied [Figs. 4(e)4(h)]. It is reasonable to expect (as previously understood for VBLs)47–49 that any substructure should have a pinning effect on the overall DW motion. Indeed, it is seen in all cases that the defect-free portions of the wall bow around the point of interest. Eventually, the wall does break past and the local dipole like contrast vanishes. We also note the absence of contrast where the DW was initially pinned confirming the absence of any larger microstructural defect (Fig. S5 in the supplementary material). These observations all point toward the existence of DW skyrmions. From the systematic examination of these different iterations, we now propose a phase diagram of DW substructures with respect to sample thickness (M×N) and DMI strength (Fig. 5).

FIG. 4.

Fresnel-mode Lorentz TEM micrographs of M=410, N=1 samples where M= (a) 10, (b) 8, (c) 6, and (d) 4 depict Bloch domain walls suggesting that relatively small DMI strength is displayed. Contrast reversal from white/black to black/white (and vice versa) along domain walls indicates locations of vertical Bloch lines and 2π vertical Bloch lines. (e)–(h) Fresnel-mode Lorentz TEM micrographs of an M=3, N=2 sample in the presence of increasing perpendicular magnetic field. White arrows indicate locations of domain wall skyrmions that are seen to pin domain wall motion. The red arrows in (e)–(h) designate that the direction samples were tilted.

FIG. 4.

Fresnel-mode Lorentz TEM micrographs of M=410, N=1 samples where M= (a) 10, (b) 8, (c) 6, and (d) 4 depict Bloch domain walls suggesting that relatively small DMI strength is displayed. Contrast reversal from white/black to black/white (and vice versa) along domain walls indicates locations of vertical Bloch lines and 2π vertical Bloch lines. (e)–(h) Fresnel-mode Lorentz TEM micrographs of an M=3, N=2 sample in the presence of increasing perpendicular magnetic field. White arrows indicate locations of domain wall skyrmions that are seen to pin domain wall motion. The red arrows in (e)–(h) designate that the direction samples were tilted.

Close modal
FIG. 5.

(a) Qualitatively formulated phase diagram depicting conditions where the domain wall substructures depicted in (a)–(d) can be expected to be observed with respect to DMI strength and thickness (M×N). Stars on the phase diagram demark 1 of 15 iterations of the Pt/Co/Ni/Ir multi-layer examined. (b)–(e) Representative schematics, (f)–(i) predicted Lorentz TEM micrographs, and (j)–(m) representative experimental LTEM images of the corresponding regions are shown on the right. The red arrow in (g), (i), and (k) designates that the direction samples were tilted.

FIG. 5.

(a) Qualitatively formulated phase diagram depicting conditions where the domain wall substructures depicted in (a)–(d) can be expected to be observed with respect to DMI strength and thickness (M×N). Stars on the phase diagram demark 1 of 15 iterations of the Pt/Co/Ni/Ir multi-layer examined. (b)–(e) Representative schematics, (f)–(i) predicted Lorentz TEM micrographs, and (j)–(m) representative experimental LTEM images of the corresponding regions are shown on the right. The red arrow in (g), (i), and (k) designates that the direction samples were tilted.

Close modal

We divide the diagram into the four regimes described in Fig. 5 based on our observations. Region I ( DMI, thickness) contains magnetic domain walls that have a non-zero Bloch component, which include 1π VBLs only. Region II ( DMI, thickness) contains only chiral Néel DWs with no substructure. Region III ( DMI, thickness) contains magnetic domain walls that have a non-zero Bloch component characterized by highly variant DW substructures including 2π VBLs and VBL pileups. Region IV ( DMI, thickness) contains chiral Néel DWs with isolated DW skyrmions. While the points correspond to experimentally investigated samples, the lines denote qualitative transitions between regions. The transitions between I and II as well as III and IV correspond simply to the critical DMI strength required to overcome the DW anisotropy stabilizing Bloch DWs. The transitions between I and III as well as II and IV correspond to the approximate thicknesses where hybrid DWs exist and create low-energy paths to annihilation of DW skyrmions or 2π VBLs.

In summary, we modeled the thermal stability of DW substructures in Pt/Co/Ni/Ir multi-layers and experimentally characterized the parameter space (DMI and thickness) where they may exist. Thermal stability calculations revealed high energy barriers to annihilation for both DW skyrmions and VBLs, most notably >60 kBT for D<1.0 mJ/m3 within the approximation that magnetization is uniform through the film thickness. Lorentz TEM examination of different iterations of our quaternary system revealed four regions characterized by the DW type and substructure. Increased thickness, which is known to support the formation of hybrid DWs, inhibits stabilization of the topologically equivalent 2π VBLs and DW skyrmions in low and high DMI samples, respectively. Upon reducing thickness, low-energy paths to annihilate these features are suppressed, and, in the high DMI case, DW skyrmions are observed. DW skyrmions are observed to locally pin the DW as expected due to their associated increase in elastic energy. Although our observations were made in Pt/Co/Ni/Ir multi-layers, which serves as a suitable test-bed, we expect that these results are not unique to this material system.

See the supplementary material that contains M–H hysteresis loops, calculated MS and Keff, and additional Lorentz TEM images.

This research was supported by the Defense Advanced Research Projects Agency (DARPA) program on Topological Excitations in Electronics (TEE) under Grant No. D18AP00011. The authors also acknowledge use of the Materials Characterization Facility at Carnegie Mellon University supported by Grant No. MCF-677785.

The data that support the findings of this study are available within the article and its supplementary material.

The authors have no conflicts to disclose.

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Supplementary Material