Nanoscale Skyrmions in a Nonchiral Metallic Multiferroic: Ni2MnGa

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Nanoscale skyrmions in a non-chiral metallic multiferroic: Ni2MnGa Charudatta Phatak, Olle Heinonen, Marc De Graef, and Amanda Petford-Long Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.6b01011 • Publication Date (Web): 17 May 2016 Downloaded from http://pubs.acs.org on May 27, 2016

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Nanoscale skyrmions in a non-chiral metallic multiferroic: Ni2MnGa Charudatta Phatak,∗,† Olle Heinonen,† Marc De Graef,‡ and Amanda Petford-Long†,¶ †Materials Science Division, Argonne National Laboratory, Lemont, IL 60439, USA. ‡Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA. ¶Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208, USA. E-mail: [email protected]

Abstract Magnetic skyrmions belong to a set of topologically non-trivial spin textures at the nanoscale that have received increased attention due to their emergent behavior and novel potential spintronic applications. Discovering materials systems that can host skyrmions at room temperature in the absence of external magnetic field is of crucial importance, not only from a fundamental aspect, but also from a technological point of view. So far, the observations of skyrmions in bulk metallic ferromagnets has been limited to low temperatures and to materials that exhibit strong chiral interactions. Here we show the formation of nanoscale skyrmions in a non-chiral multiferroic material which is ferromagnetic and ferroelastic, Ni2 MnGa at room temperature without the presence of external magnetic fields. By using Lorentz transmission electron microscopy (LTEM) in combination with micromagnetic simulations, we elucidate their

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formation, behavior and stability under applied magnetic fields at room temperature. The formation of skyrmions in a multiferroic material with no broken inversion symmetry presents new exciting opportunities for the exploration of the fundamental physics of topologically non-trivial spin textures. Keywords: skyrmions, multiferroic material, Lorentz transmission electron microscopy

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Topologically-protected systems are becoming a focal point for research in condensed matter physics. There is a fundamental interest in elucidating the underlying physical phenomena that control the behavior of these systems, but there is also interest in developing applications based on topological spin states, which will require the presence of robust states that are stable against external perturbations and defects. 1–3 In condensed matter systems with a directional order parameter (such as ferromagnets), the slowly varying texture of the order field (e.g. the magnetization in ferromagnets) and the requirement for local continuity of this field can lead to exciting and emergent phenomena. A prototypical example of such a phenomenon is the skyrmion, which is a topologically non-trivial magnetic spin texture observed in bulk materials as well as thin films. 4–7 Moreover, realization of such phenomena in multiferroic materials offers the advantage of being able to control the behavior through more than one single ferroic property, such as electric field and magnetic field or strain and magnetic field. The magnetic spin texture of a skyrmion can be characterized by the so called topological index defined as: 1 N= 4π

Z

n·

∂n ∂n × ∂x ∂y

dx dy,

(1)

where n is the unit magnetization vector. Topologically trivial spin textures have an index of N = 0, whereas non-trivial textures are characterized by a non-zero index: for instance, skyrmions are characterized by an integral skyrmion number, |NSk | ≥ 1. Based on the rotation of magnetization forming the skyrmion, they can be classified into two types: a Néel-type “hedgehog” skyrmion or a Bloch-type “spiral” skyrmion. 8 Although originally skyrmions were predicted in high energy physics by Skyrme, 9 theyhave been recently addressed in condensed matter systems and observed in a variety of chiral magnetic systems i.e. systems with broken inversion symmetry; and they are stabilized by the bulk or interfacial Dzyaloshinskii-Moriya (DM) interaction. Among the various bulk metallic systems in which the formation and stability of skyrmions have been explored, the most common are the B20-type alloys such as MnSi, Fe1−x Cox Si and FeGe. 10–13 The mag3


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netic structure in these materials undergoes a helical transition into a skyrmion phase with a transition temperature below room temperature: the highest reported transition temperature for FeGe is ∼ 278K. These systems show the formation of spiral skyrmions in a lattice, where the entire lattice can be controlled by external stimuli. Isolated skyrmions have been observed in ultra-thin magnetic nanowires at room temperature where the asymmetric interfaces provide the interfacial DM interactions necessary to stabilize the skyrmion texture. 14–19 These systems form hedgehog skyrmions which can be nucleated and transported along the nanowire by spin-transfer torque. However, they typically require an external magnetic field to be applied in order to form the skyrmions. In magnetic films, there exists another class of localized spin textures known as bubble domains, some of which can share the same topology as skyrmions observed in chiral materials. The latter are referred to as ‘skyrmion bubbles’, are typically stabilized by pure dipolar interactions and do not require broken inversion symmetry. 20 Such skyrmion bubbles have been observed in strongly uniaxial ferrites, such as Sc-doped barium ferrite; however, their size is typically larger than that of the skyrmions stabilized by DM interactions, and they do not show a fixed chirality. 21 Apart from these material systems, the only multiferroic material which has been shown to host a skyrmion phase is the insulating ferrimagnetic oxide, Cu2 OSeO3 , with the skyrmion phase observed only at temperatures well below room temperature. 22–24 Recent work has shown the formation of non-collinear magnetic spin structures in Mn-based Heusler alloys. 25 Similarly, skyrmions have been reported recently in β-Mn-type Co-Zn-Mn alloys which have a chiral space group symmetry. 26 But in both of these instances, the skyrmions are formed and stabilized by DM interactions and not by dipolar interactions. In this work, we present a novel class of skyrmions that form in a multiferroic (ferromagnetic and ferroelastic) material with inversion symmetry, in which they are stabilized by a competition between the intrinsic exchange, magnetocrystalline anisotropy and dipolar interaction. As a result, we observe the formation of a skyrmion lattice with a single chirality in the metallic Ni2 MnGa with ferromagnetic and ferroelastic properties at room temperature

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in the absence of any external magnetic field. Due to the presence of inversion symmetry however, different realizations of the lattice may result in different chirality. This is a unique observation in which we observe a skyrmion lattice formed due to geometric confinement of the magnetic structure by the microstructural twin boundaries, and since Ni2 MnGa is ferroelastic as well, it offers opportunity to control the skyrmions through externally applied strain. We have also studied the behavior of individual bubble skyrmions under application of an external magnetic field using in-situ Lorentz transmission electron microscopy (LTEM) in order to understand the stability of their spin texture. These results present an important step towards the fundamental understanding of the nature of skyrmions and how they can be stabilized in non-chiral metallic materials with uniaxial anisotropy, thus bringing us closer to their potential application in spintronic devices. 7 Ni2 MnGa is a classical example of a multiferroic material exhibiting a strong interplay between two order parameters: magnetization and strain. The alloy undergoes several temperature-dependent phase transformations, with one of the most interesting transitions occurring upon cooling from the cubic austenite phase to the room temperature martensite phase with the transition temperature of 316 K. The martensite typically has a modulated, nominally tetragonal, crystal structure that is conveniently described using a monoclinic unit cell (additional details in supplementary material). 27 During this phase transition, the magnetic properties of the alloy also undergo a dramatic change. The cubic phase has low magnetocrystalline anisotropy along the h111i cubic directions, whereas the martensite phase has high magnetocrystalline anisotropy along the [010] monoclinic axis. 28 The large tetragonal distortion is accommodated by {125} type-I twins, which are also responsible for the large magnetic field-induced strain and the resulting magnetic shape memory applications. The magnetic easy axis rotates by about 90◦ across each twin boundary. Additional details about the alloy composition, heat treatment, and sample preparation are given in the Methods section. A TEM sample was fabricated from a single crystal of Ni2 MnGa and heat treated to

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(a)

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(c) EA EA

X

250 nm

(b)

100 nm Figure 1: (a) In-focus image from a finely twinned region of Ni2 MnGa with the dashed lines indicating the twin boundaries and the easy axis of magnetization is schematically shown for each twin. (b) shows the under-focused LTEM image showing the zig-zag domain walls (green arrow) and bright dots (red arrow). (c) shows a high magnification colored magnetization map with magnetization vectors overlaid to indicate the zig-zag domain walls. form the martensite phase. The geometric restrictions of a thin TEM sample influence the twins that are formed to accommodate the transformation from the austenite state to the multivariant martensite phase. The sample discussed here was not exposed to any external magnetic fields. An in-focus LTEM image of a finely twinned region is shown in Figure 1(a). The dashed lines indicate the twin boundaries and the easy axis in each twin variant is schematically shown as EA. The twin width varies from 100 nm near the left edge of the image to about 50 nm near the right edge. Figure 1(b) shows the corresponding under-focus LTEM image. Instead of a maze-like stripe domain contrast, we see an unexpected domain structure in which the domain walls (indicated by the green arrow) appear to zig-zag back and forth across the twin in variants where the easy axis is out-of-plane. Furthermore, as the width of the twin variants decreases, the nature of the domain wall changes, seen as a row of bright dots (indicated by the red arrow). Figure 1(c) shows a magnified colored magnetization map of the region outlined in Figure 1(b). The distinction between the twin

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variants with in-plane and out-of-plane magnetization can be seen more clearly. The twin variant with in-plane magnetization shows a uniform green color in the magnetization map. The zig-zag Bloch-type wall in the interleaving twins with out-of-plane magnetization is clearly visible and the origin of the row of bright spots can now be understood: as the width of the twin decreases, the zig-zag walls ‘fold up’ to form a row of individual skyrmion-like textures as seen near the bottom of the magnetization map.

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50 nm

x

(d) X

Figure 2: (a) High resolution under-focused LTEM image showing the bright spots corresponding to skyrmion cores in alternate twins, and (b) shows the corresponding colored magnetization map. The inset in (a) shows the Fourier transform of the under-focused LTEM image. (c) A 3D schematic showing the hexagonal skyrmion lattice in the narrow twinned region of Ni2 MnGa indicating the presence of a skyrmion lattice. (d) A schematic illustrating how the magnetization pattern is formed within the narrow twin. The red and the blue colors indicate the out-of-plane magnetization direction and the yellow arrows indicate the in-plane magnetization along the Bloch walls. Figure 2(a) shows an under-focus LTEM image from the region displaying the rows of bright-dot contrast in alternating twins, indicated by the red arrow in Figure 1(b). Figure 2(b) shows the corresponding magnetization color map with the magnetization vectors overlaid. A complete through-focus series of images is available in the supplementary material. From the magnetization map, it can be seen that each bright dot location corresponds 7


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to an individual skyrmion. Correlating the skyrmion positions with the under-focus image shows that they are present only in alternate twins, in which the easy-axis is out of the sample plane. The inset in Figure 2(a) shows the Fourier transform revealing the near 6fold coordination of the skyrmion lattice. The distance between the skyrmion cores along a single twin is 80 nm, while the distance between the out-of-plane twins is 100 nm, thereby forming a slightly distorted hexagonal lattice. We also note that all the skyrmions are spiral skyrmions (Bloch-type) and have a fixed helicity, γ. Figure 2(c) shows schematically the magnetic spin texture of the observed skyrmion lattice, where the skyrmions are seen in alternate twins and extend throughout the thickness of the sample. Other regions of the sample with similar twin variant sizes also showed similar skyrmion spin texture. Additional images from these regions are included in the supplementary material. Thus, we observe a spontaneous formation of a skyrmion lattice without the presence of any chiral interactions such as DM interaction at room temperature without application of an external magnetic field. We can now try to understand the formation of the skyrmion lattice in the region with narrow twin variants. The relevant length scales here are the domain wall width p √ δ = π A/K ≈ 30 nm and the minimum domain width Dm ≈ 2.35 AK/MS2 ≈ 120 nm.

Here A is the micromagnetic exchange constant, A ≈ 12 pJ/m and K the uniaxial anisotropy

energy density, K ≈ 0.19MJ/m−3 . We note that the thickness of the film is smaller than Dm so that the system tries to lower its magnetostatic energy by creating stripes in the perpendicular variant. On the other hand, because of the alternating in-plane and out-of-plane easy axis variants, the system tries to lower its energy by creating a texture in the direction perpendicular to the twins by forming consecutive in-plane and out-of-plane magnetization separated by domain walls. When the width of the variants becomes comparable to the domain wall width δ this texture becomes smoothly varying along the direction perpendicular to the twins The result is that the entire magnetization texture deforms smoothly to form a skyrmion lattice in order to minimize demagnetizing energy, anisotropy energy, as

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well as domain wall energy. Figure 2(d) shows a cartoon of the twin with an out-of-plane easy axis of magnetization. Within such a twin variant the magnetization can minimize the demagnetization energy by forming stripe domains; however, due to geometric confinement, the domain wall becomes constricted to a zig-zag orientation as seen in Figure 1(b). As the width of the twin further decreases, the zig-zag walls ‘fold up’, ultimately forming a row of individual skyrmions. This skyrmion formation is a direct result of the balance between the magnetic anisotropy energy, exchange energy and demagnetization energy, at a length scale of a few tens of nanometers. The magnetic anisotropy favors the magnetization to lie along the out-of-plane direction, and the exchange energy favors a continuous transition of magnetization directions. However, since the demagnetization energy (dipolar interaction) prefers to minimize the stray field, a smooth zig-zag formation of domain walls is observed. As these domain walls get constricted further, they disintegrate to form individual skyrmions, thereby further reducing the domain wall and exchange energies. The domain wall width in p the martensite state δ = π A/K, is about 30 nm which is close to the width of the twin

(∼ 50 nm). Additionally, due to the dipolar interactions with the neighboring twins, which have in-plane magnetization, all the skyrmions are found to have the same chirality. Based on these observations, we can also anticipate the formation of skyrmions in wider twin variants (> 1 µm) with out-of-plane magnetic anisotropy under application of an external magnetic field. The skyrmions would then be formed simply by a balance between the Zeeman energy due to the applied field and the dipolar interaction energy. Similar observations in regions with out-of-plane magnetic anisotropy have been reported in Sc-doped barium ferrite as well as bilayer manganite La2−2x Sr1+2x Mn2 O7 with x = 0.315. 21,29 In the

latter case, the application of a magnetic field resulted in the formation of biskyrmions with a skyrmion number of NSk = 2. In the present work, we analyzed the effect of applying a magnetic field to a focused ion beam (FIB) lift-out sample of Ni2 MnGa with uniform thickness, containing a twin variant with out-of-plane easy axis of magnetization and a width of ∼ 2 µm. Figure 3 shows a series of under-focus LTEM images as a function of magnetic field 9


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(indicated in the top right corner), applied perpendicular to the sample plane; the field was applied by exciting the objective lens of the TEM. The field-free domain structure shows a mixture of maze-like stripe domains and bubble domains (indicated by B1 in Figure 3). The domain walls between the stripe domains can be identified as Bloch walls with an in-plane component of magnetization leading to double contrast (black-white) at the location of each wall. Analysis of this contrast shows that the Bloch walls do not maintain a specific chirality, due to the lack of any additional anisotropy, such as the DM interaction, in the sample. There are two interesting features to note, indicated by the red and white arrows at 0 mT. The red arrow points towards a Bloch line present in a domain wall at which the domain wall chirality switches. The contrast of the domain wall below the Bloch line is black on the left and white on the right side which reverses above the Bloch line. Observation of such features is key to understanding domain behavior, as they act as sites that can lead to twisted domain walls. The white arrow points towards a ‘U-shaped’ domain wall that only has a faint single white contrast associated with it; such contrast has previously been reported in Ni2 MnGa alloys and corresponds to an antiphase boundary in the martensitic state. 30 As the applied perpendicular field is increased, the width of the stripe domains decreases and the bubble domains start to shrink. At an applied field of 180 mT, the Bloch wall crosses the antiphase boundary, resulting in a contrast inversion from white to black for the antiphase boundary (black arrow), as has been previously reported. 31 Another key feature to note is that the change in contrast of the wall surrounding the bubble domain B1 . From being uniformly white inside and black outside, the contrast now changes to half white and half black akin to the ‘yin-yang’ symbol. Further increasing the applied field reduces the width of the stripe domains significantly, with some of them collapsing to form more bubble domains as seen at an applied field of 340 mT. The Bloch walls surrounding the two bubble domains remain stable until a very high field is applied, and finally disappear at an applied field of 440 mT, likely as a result of being pinned at the antiphase boundary. It is interesting

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0 mT

100 mT

180 mT

280 mT

340 mT

440 mT

B1

300 nm

Figure 3: A series of under-focus LTEM images during the in-situ magnetizing process where the field was applied along a direction perpendicular to the sample going into the plane of the paper. The field value is indicated in the top right corner.

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to note however, that the bubble domains do not form in any regular lattice arrangement and there is always a mixture of stripe domains and bubble domains under an externally applied magnetic field. Next, we consider the topology of the bubble domains in more detail. Figure 4(a) and (d) show high resolution under-focus images of the bubble B1 from Figure 3 at applied fields of 0 mT and 180 mT, respectively. Figure 4(b) and (e) show the corresponding magnetization color maps from which we can determine the skyrmion number, NSk , for the bubble domain to be NSk = 1, identifying it as a skyrmion bubble. At an applied field of 180 mT, the skyrmion bubble contrast changes significantly. The corresponding magnetization map shows that the magnetization in the domain wall is no longer circular but, instead, points upwards in both branches of the domain wall. The skyrmion number has now changed to NSk = 0, and the skyrmion bubble has evolved to a topologically trivial bubble domain. Figure 4(c) and (f) show simulated under-focus TEM images obtained from micromagnetic simulations for the two cases of a skyrmion bubble with NSk = 1 and NSk = 0, respectively. Details of the simulations are given in the Methods section. There is excellent agreement between the simulated and experimental images, thus indicating that we observe a change in the topology of the skyrmion as a function of applied field, from non-trivial to trivial. Figure 4(g) and (h) show 3D representations of the magnetization (color indicates Mz ) for the NSk = 1 and NSk = 0 bubble domains respectively, along with the magnetization vectors along the domain wall. The 3D visualization shows that the domain wall magnetization spirals as a function of thickness for the skyrmion bubble, whereas for the trivial bubble domain (NSk = 0) the magnetization direction remains constant through the thickness. This change can be used to assess the skyrmion’s topological stability. Typically for chiral materials, the DM interaction that stabilizes the skyrmion must be overcome by the externally applied field. However, since Ni2 MnGa is not a chiral material and does not exhibit a DM interaction, the likely reason for the topological change is a misalignment between the easy axis direction of the magnetization and the applied field direction. As a

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Experimental Underfocus Image (a)

Experimental Mag. color map (b)

Simulated Underfocus Image (c)

(e)

(f)

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Bloch Skyrmion

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(g)

(h)

Figure 4: High resolution under-focus image of a bubble domain at 0 mT (a) and 180 mT (d). The corresponding colored magnetization map of the bubble domains, (b) and (e) respectively. The last column shows the simulated under-focus LTEM images for the two cases, (c) and (f) respectively. (g) and (h) show the 3D representation of the simulated magnetic spin texture of the two bubble configurations, (c) and (f) respectively, to elucidate the difference in topology.

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result, when the applied field is increased, the skyrmion reduces its domain wall energy by forming an ‘onion’ state i.e., changing its skyrmion number to NSk = 0.

(a)

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100 mT

140 mT

160 mT

240 mT

Figure 5: Simulated under-focus LTEM images from micromagnetic simulations for NMG under an applied perpendicular field. The simulations are performed for a misalignment between the applied field and easy axis of (a) 0◦ and (b) 10◦ . The field values are indicated on the right side. We performed micromagnetic simulations to understand the details of the observed domain behavior of Ni2 MnGa. Figure 5 shows under-focus images computed from the magnetization profiles obtained from the micromagnetic simulations. The details of both simulations are provided in the Methods section. Two cases for the in-situ applied field were considered; in Figure 5(a), the field lies along the easy-axis of magnetization (perpendicular to the plane of the figure), and in Figure 5(b) there is a misalignment of θ = 10◦ between the applied field and the easy axis. The initial domain configuration at 0 mT is similar in both the cases, revealing stripe domains with Bloch lines, and skyrmions. As the applied field is increased, the behavior of the stripe domains is similar to that observed experimentally. However, for an applied field at θ = 0◦ , one only observes a reduction of the skyrmion size with no change in their vorticity. For a field inclined at θ = 10◦ , we observe the formation of skyrmions with winding number N = 0 at an applied field of 240 mT. This supports our hypothesis that change in vorticity results from a misalignment between the applied field direction and the 14


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easy-axis direction. A similar effect of misalignment of applied field direction and formation of bubble domains with N = 0 has been reported previously, 21 where even a misalignment of 2◦ is sufficient to form these bubble domains. This shows that there is a delicate balance between the energies that stabilize the skyrmion lattice in a non-chiral material. Within the same material, we have shown the existence of two types of skyrmions; one type that forms in a “lattice” state in the martensitic twins and a second type that forms in an isolated state under an externally applied field. There is a difference in size between the two types of skyrmion that are observed. The skyrmions in the “lattice” state are much smaller than the isolated ones. This is due to the fact that there is physical confinement (the twin boundaries) that restricts the magnetization to lie within one twin. Hence the size of the skyrmions in this case is dictated by the physical size of the twin variant. However, in the case of an isolated skyrmion, no such physical boundary is present, and the skyrmion size is solely dependent on the exchange and magnetostatic energies under an applied magnetic field. The observation of a skyrmion lattice presented here demonstrates that chiral interactions are not a necessity for a material to host the skyrmion phase. The rich variety of magnetic interactions in thin films, such as exchange energy and magnetocrystalline anisotropy, can be tailored to obtain a much wider spectrum of complex magnetic spin textures. Here the skyrmion lattice is localized to twin variants that have an out-of-plane easy axis of magnetization interleaved with variants with an in-plane easy axis. Such an arrangement naturally leads to channels that can become conduits for transport of skyrmions. Since Ni2 MnGa is metallic at room temperature and does not require an external magnetic field to stabilize the skyrmion lattice, they can be expected to be mobile and stable at lower current density as observed by others in MnSi. 11 Furthermore, on a broader perspective, the skyrmion lattice was stabilized without any special requirements apart from the geometric confinement, which was attained in Ni2 MnGa through control of its microstructure, and there have been reports on the stabilization of martensitic phase in Ni2 MnGa thin films, with or without a

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substrate, at room temperature. 32,33 The geometric confinement gives rise to competition between exchange, magnetic anisotropy and demagnetization energy. However, this can be easily extended to artificially patterned structures which would allow for more freedom and precise control over the widths of the constrictions and thus the magnetic behavior of the system. For example, using templated self-assembly, two dissimilar magnetic materials could be patterned in a geometric arrangement similar to the microstructure of Ni2 MnGa where the easy axis of magnetization can vary by 90◦ between adjacent constrictions. This opens an entire new area of artificially-designed magnetic systems that can host skyrmion lattices. Furthermore, since Ni2 MnGa is a multiferroic with strong coupling between strain and magnetization through magnetic field-induced strain, an additional parameter, namely application of external strain, could be used to control the skyrmion behavior. In conclusion, we have observed the formation of a skyrmion lattice in a non-chiral metallic multiferroic material at room temperature and in the absence of an external magnetic field. The magnetic structure was confined within narrow martensitic twins with easy axis of magnetization alternating by 90◦ between the adjacent variants. This additional confinement, in conjunction with dipolar interactions and exchange interactions, result in folding of domain walls to form the skyrmions. The role of the geometric confinement was elucidated by exploring the behavior of stripe domains and skyrmion bubbles in much wider twins with out-of-plane easy axis of magnetization. We showed that an applied field along the easy axis can result in shrinking of the skyrmion bubbles and ultimately a change of their topology to a trivial bubble domain. The results presented here bring us closer to understanding the basic physics of skyrmion spin textures, in which long range forces (such as dipolar interactions) can be tailored to stabilize them. Additionally, due to the requirement of geometric confinement, the resulting conduits could be used for skyrmion transport, bringing us closer to ushering in the era of “skyrmionics”.

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Materials and Methods Sample preparation Bulk single crystals of Ni2 MnGa with a nominal composition of Ni49.9 Mn28.3 Ga21.8 were used in this work. This alloy has a Curie temperature of TC = 372 K, and a martensite start temperature of MS = 316 K. A TEM sample was prepared using twin jet electropolishing without mechanical polishing in an electrolyte containing 95% ethanol and 5% perchloric acid. The voltage and temperature of the bath were maintained at 11 V and 233 K. The samples were heated above the martensitic transition temperature to 350 K and then cooled to form a multivariant martensitic state with multiple twins at room temperature. A FIB lift-out sample was prepared from the same TEM sample followed by low energy Ar+ ion milling to produce a sample with uniform thickness for the in-situ magnetizing study. At room temperature, the microstructure consists of a mixture of mostly martensite with retained austenite pockets. The martensite has a 5M modulated structure. Lorentz transmission electron microscopy The magnetic domain structures were observed using Lorentz transmission electron microscopy on a dedicated JEOL 2100F TEM operating at 200 kV and equipped with a spherical aberration corrector, which enables imaging of magnetic structure at a spatial resolution of a few nanometers. The out-of-focus Fresnel images were obtained using a nominal defocus value of 360 µm. Magnetization maps were then reconstructed from through-focus series of images using a phase-retrieval method based on the transport-of-intensity equation (TIE). 34 In-situ magnetizing experiments were performed on the FEI Tecnai F20ST TEM operating at 200 kV in Lorentz mode. The perpendicular magnetic field was applied by exciting the objective lens of the microscope which was previously calibrated using a Hall probe. Micromagnetic simulations Micromagnetic simulations were performed on slab geometries of dimensions 2000 nm×500 nm×50 nm. The system was divided into a regular rectangular mesh of size 5 nm×5 nm×5 nm with periodic boundary conditions in the xand y-directions. The magnetization structures were achieved by integrating the Landau-

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Lifshitz-Gilbert (LLG) equation for the magnetization director m: ˆ dm ˆ |γe | |γe |α =− m ˆ × H − m ˆ × [m ˆ × Heff ] , eff dt 1 + α2 1 + α2

(2)

where γe is the electron gyromagnetic constant and the effective field Heff includes exchange field, magnetostatic fields, anisotropic effective field, and external field; α is the dimensionless damping and we used a value of α = 0.2 to dissipate energy efficiently and bring the system to a local equilibrium state. The magnetostatic fields were calculated using fast Fourier transforms with appropriate zero-padding along the z-direction, and the time integration performed using a Bulirsch-Stoer semi-implicit integrator adapted specifically for the LLG equation. We started each set of simulations with zero applied field and a random magnetization configuration, and then increased the field in steps of 100 Oe using the previously converged magnetization structure as input. For materials parameters, we used a saturation magnetization of MS = 537 emu/cm3 and a micromagnetic exchange coupling of A = 1.2 µerg/cm. There is some uncertainty regarding the anisotropy field, and we performed simulations with anisotropy fields of 7 kOe, 8 kOe, 9.3 kOe, and 10 kOe. The results closest in agreement with the LTEM images were obtained using an anisotropy field of 7 kOe. The LTEM images were simulated by using the output magnetization from the micromagnetic simulations and computing the phase shift of electrons using the Mansuripur method. 35 The phase shift along with the transfer function of the microscope were used to simulate the image. The microscope parameters that were used for the simulations were: E = 200 kV, Cs = 500 µm, defocus = 10 µm.

Acknowledgement Work by C.P, A.P.L, O.H was supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. Use of the Center for Nanoscale Materials, an Office of Science user facility, was supported by the U.S. 18


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Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02- 06CH11357. M.DG would like to acknowledge support from the National Science Foundation, grant # DMR-1306296. We gratefully acknowledge the computing resources provided on Blues and Fusion, high-performance computing clusters operated by the Laboratory Computing Resource Center at Argonne National Laboratory.

Supporting Information Available Additional information of crystal structure and electron diffraction of Ni2 MnGa , and additional through-focus series images

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