Magnetic field assisted stabilization of circular double wall domain lattice in oxidized Fe3GeTe2 flakes

The family of 2D ferromagnets is in the center of research for novel spintronics applications. Among the various 2D ferromagnets, Fe3GeTe2 has drawn significant attention since it combines a high Curie temperature with a van der Waals structure, which allows easy exfoliation, and a high spin polarization/large spin–orbit coupling. The presence of interfacial DMI in 2D ferromagnets have a significant impact on the behavior of magnetic domain walls, which are fundamental in magnetic memory and logic devices. By controlling the interfacial DMI, it is possible to manipulate the motion of domain walls and the magnetic domain configuration, which is essential for the development of efficient and reliable magnetic devices. In this study, we investigate the effect of an, inversion symmetry breaking, oxidized layer on the magnetic domain structure of Fe3GeTe2 flakes due to the emergence of interfacial DMI. By combining magneto-optical Kerr effect microscopy images and micromagnetic simulations, we study the formation of a circular double wall (CDW) domain lattice in oxidized flakes under specific field cooling and magnetic field sweeping protocols. Their formation is attributed to a competition between the exchange interaction both symmetric and antisymmetric (associated to interfacial DMI), magnetocrystalline anisotropy and the external magnetic field. The CDW domains have a diameter of several microns, a magnetic structure resembling that of a skyrmionium and are arranged in regular lattice that survives thermal fluctuations close to T c. Our results suggest that these CDW domains transition to Néel type skyrmions after a magnetic field threshold. These findings could have important implications for the design and optimization of 2D ferromagnetic materials for spintronic applications.


Introduction
The two-dimensional (2D) materials with van der Waal (vdW) layered structures are one of the most promising class of materials for novel spintronic applications.Fe 3 GeTe 2 (FGT), a quasi-2D itinerant vdW ferromagnet, is a rare example with a high Curie temperature of T c = 220 K [1][2][3] which can be raised to room temperature via electrostatic gating [4].A long-range ferromagnetic order has been confirmed experimentally, ranging from bulk crystals down to monolayer [5].It presents strong perpendicular magnetic anisotropy [5], that makes it a good candidate for magnetic memory applications based on currentinduced switching, an energy efficiency method aiming to write information with limited energy consumption.Apart from magnetic anisotropy, FGT exhibits many other intriguing properties, such as large anomalous Hall current [6] and topologically protected spin textures called magnetic skyrmions.
There are reports revealing the formation of a wide range of topological spin textures as Bloch type [7][8][9][10][11] or Néel type [12][13][14][15][16][17][18] in the FGT with its nature remaining still a challenging issue.Néel type spin textures are unexpected in FGT since their formation is associated with the presence of a positive Dzyaloshinskii-Moriya interaction (DMI) and as FGT belongs to centrosymmetric space group, its crystal structure implies a zero DMI.Also, the stabilization of these textures cannot be attributed exclusively by DMI into the FGT crystal as it is not survived in the long-wavelength limit [19].So, the origin of DMI is still a key issue and mechanisms that could enable DMI in FGT crystal and its interfaces have been investigated recently [12].It has been proposed that the oxidized layer, formed spontaneously on the FGT surface, when it is left in ambient conditions, can induce an interfacial DMI and lead to the formation of Néel type skyrmions when current pulses are applied [12].Other key parameters are the thickness of the flake, as the configurations of domains vary significantly with the increasing FGT thickness [20,21] and the applied magnetic field protocol, leading to the formation of different domains structures, from labyrinth like pattern when a zero field cooling (ZFC) protocol is in place to circular and bubble configurations when cooling the sample under a magnetic field [18].In general, the stabilization and manipulation of topologically protected magnetic structures, such as skyrmions has been the main target of a series of recent research efforts.The potential application in novel spintronic devices and also the emerging physics in such systems has been the driving force behind a lot of experimental and theoretical works involving the generation and manipulation of skyrmions in composite magnetic systems [22], manipulation of spin-orbit interaction and DMI via the use of rare metals [23] for the generation of skyrmions, use of antiferromagnets as a source for topological spin textures [24] and current induced dynamics in antiferromagnetic skyrmions systems [25].Although there is a significant number of publications regarding the domain formation in FGT as we mention above, there are still many unclarified, or even contradicting points regarding the underlying physical mechanisms as well as the conditions under which non trivial domain patterns, such as circular skyrmion like domains emerge in FGT.In the present work, we aim to reexamine the domain formation in thick FGT flakes in different magnetic field-temperature range.Our magnetooptical Kerr effect microscope (MOKE) microscopy measurements revealed that appropriate field cooling (FC) schemes and the presence of an oxidized layer on the surface, change the preferred domain pattern from labyrinth like to dashed-line like.By sweeping the applied magnetic field near the saturation, circular double wall (CDW) domains emerge, forming also a regular lattice.These circular domains are stable in a range of magnetic fields just before magnetization saturation.By comparing our experimental results with micromagnetic simulations we examine the effect of magnetic field and interfacial DMI from an oxidized FGT epilayer to the observed domain patterns and conclude that the CDW collapse to skyrmions after a magnetic field value is exceeded.

Methods
In this study we used bulk FGT single crystals purchased from HQ Graphene.A first characterization of its magnetic properties was performed by acmagnetic susceptibility, χ ac , measurements using a home-made apparatus in the temperature range 80-350 K.The temperature dependence of χ ac was measured with an ac-field amplitude H 0 = 3 Oe and a frequency of f = 664 Hz.Our Kerr effect microscopy measurements were performed with a MOKE, in polar mode, from EvicoMagnetics, combined with an optical cryostat with continuous flow of LN 2 for low temperatures measurements (77-320 K).The magnetic domains at the surface of the FGT crystals and their evolution in relation to different magnetic fields applied perpendicular to sample surface (B z ) and temperatures were recorded with objective lenses of ×20, ×60 magnification.In order to acquire large, flat surfaces with good quality we exfoliated thick flakes (around 30 µm thickness) of the FGT crystals using scotch tape.The FGT flakes were left for a few hours to allow for the formation of a native oxide.Then, the sample was placed in a vacuum cryostat for the low temperature measurements.After the measurements the flake was stored in a vacuum sealed bag.To study the nature of magnetic domains in FGT crystals we performed a complete set of experiments with ZFC and FC protocols with several applied magnetic fields.In order to perform the FC protocol, we mounted small Nd magnets (nominal strength of 110 or 220 mT) inside the cryostat and below the sample and we combined them with our out of plane electromagnet capable of reaching 110 mT field.In this way we were able to explore higher magnetic fields than those accessible by our electromagnet, in order to reveal those new domain textures.In the experimental results section, we follow the convention that dark regions in MOKE image correspond to 'down domains' and bright ones to up, where 'up' and 'down' correspond to the direction towards to and away from the readers eye.The small Nd magnets were placed in such a way that their magnetic field is additive to the one of the electromagnet.In this way, the different magnetic domain configurations at 77 K were examined and also their evolution with temperature.We managed to form CDW domain by sweeping the applied magnetic field near the saturation field value.Interestingly, it is observed that those circular domains form an ordered lattice in the areas where they nucleate.We study the applied field, and temperature effect to the circular magnetic domain lattice.Further image processing with ImageJ software [26] and acquisition of domain size and shape characteristics was achieved for better understanding of their characteristics.Micromagnetic simulations were performed by the Mumax 3 package [27].The parameters used in our simulations were taken from the literature [28,29].More specifically we used saturation magnetization M sat = 376 × 10 3 A m −1 , exchange stiffness A = 10 −12 J m −1 , anisotropy constant K u = 4.46 × 10 5 J m −3 .We systematically examined the role of interfacial DMI and magnetic field application protocol on the domain formation in FGT and the nucleation of CDW domains.

Ac-magnetic susceptibility
In supplementary information (figure S1) we see the temperature dependence of the real part, χ ′ ac , of the ac-magnetic susceptibility.It indicates a Curie temperature close to 190 K, which is in accordance with previous reports [8,30].The observed drop at low temperature in χ ′ ac curve has been attributed to a transition from ferromagnetic to antiferromagnetic phase [30].Also, the extra peak at 150 K with the in plane magnetic field has been associated with the formation of spin spirals in thick FGT flakes [8].

Oxidization of FGT flakes
In order to access the degree of oxidization, we placed the oxidized flake in a XPS chamber along with a FGT flake obtained immediately after its exfoliation, as a reference sample.XPS spectra were taken from the two flakes in order to monitor the effect of oxidization after air exposure compared to an almost pristine, unoxidized film.Due to its high sensitivity factor, we monitored the Te-3d XPS doublet for the oxidized and as-exfoliated flake (figure 2).We can clearly see that, for the oxidized flake there is a clear enhancement of the XPS peak associated to oxidized FGT compared to the as-exfoliated flake but still for the oxidized flake the XPS peak associated with the unoxidized FGT is still visible.Keeping in mind that XPS is a surface sensitive technique, probing material about a few (∼5 nm) nanometers from the surface, we can conclude that a native oxide layer has been formed on FGT with a thickness of few nanometers.It should be noted, that an inhomogeneous oxidation over the depth cannot be excluded.

Magnetic textures
In figure 1 we see the usual domain pattern of thick FGT flakes is a labyrinth/wormlike pattern which appear spontaneously below T c .The domains width is of the order of 2 µm.
On the other hand, ZFC for the oxidized flake yielded a stripe-chain like pattern with intermediate bubble domains of opposite polarity at 77 K (figure 2).The width of the basic domain (stripelike) is of the order of 2-3 µm whereas the bubblelike domains are 0.5-1.5 µm.This domain pattern is usual in ferromagnets with high uniaxial anisotropy whereas a stripe like pattern is dictated.In the case of specimens with reduced thickness in the same direction as the easy axis, the demagnetizing field further invokes the appearance of antiparallel domains (in this case bubble like) with opposite direction of magnetization.
By following a series of FC protocols, a plethora of different domain patterns emerged.FC with 40 mT yields a fractal like pattern at 77 K (figure S3).Due to the FC the 'dark' ('down') domains are favored (table 1) but the anisotropy and Zeeman energy does not dominate over the depolarizing term completely.Thus, the system is forced to increase the domain density in order to minimize the depolarizing field yielding this fractal like pattern.
By performing FC with 110 mT magnetic field, the 'dark' domains are favored even more (see table 1) as the Zeeman energy term combined with the anisotropy term start to dominate over the depolarizing field energy.Thus, a branching domains configuration emerge with some circular domains also that are close to each other forming a chain at 77 K (figure S4).Domain width is again of the order of 2 µm.
Going even further with the FC field (210 mT) we see these linear domains emerging at 77 K (figure S5).Linear segments have approximately width of the order of 2 µm and length ∼10 µm.Following the same arguments as above, we see that the 'dark' domains dominate, limiting the presence of the antiparallel ones to those stripes.In the line domains regime further increase of the applied field drives the system into a discontinuous branched line pattern (figure S6).
In order to access the percentage of up and down domains we post process our MOKE images with ImageJ software.We choose a ROI of 26 × 26 µm and by acquiring the histograms of the images we calculated the relative percentage of 'bright' and 'dark' domains.The results are presented in table 1 below.
The relative percentage of 'bright' and 'dark' domains for the different FC protocols are in line with the assumptions made above about the effect of interplay between the demagnetizing filed, Zeeman energy and anisotropy for the different FC protocols for high out of plane anisotropy ferromagnets [31].

Generation of CDW domains and lattice formation
By sweeping the applied magnetic field just before saturation, we noticed a history dependent magnetic field cycle of the domain pattern (figure 3(a)).At first, by decreasing (256 mT → 242 mT) (figure 3(a)     The behavior and properties of this new CDW lattice pattern was examined thoroughly as a function of the magnetic field.It seems that the maximum range of magnetic field that the lattice remains intact is around 100 mT from B z ≈ 180 mT to B z ≈ 280 mT (figure S7).It is important to notice that as it is also revealed in other publications [16], the magnetic history of the sample plays a role in the domain evolution.We have also noticed that based on the previous magnetic field cycling, the above mentioned values of the magnetic field range in which the lattice survive might differ.Inside this range of the applied magnetic field the CDW domains retain their closed circle form without any branches coming out from their perimeter.One can observe that the distances between the domains remain the same and their size changes slightly.

Temperature dependence
In addition to the magnetic field influence we study the evolution of the formed lattice with temperature change.As figure 4 demonstrates, the CDW domain lattice, exists up until 170 K.The thermal fluctuations affect the CDW lattice, thus making the CDW less bright and the whole image contrast decreases.That indicates that the Kerr rotation and thus the magnetization gets smaller as temperature rises.The size of the magnetic domains and the distance between them does not seem to change with temperature evolution indicating that the lattice is robust in a great temperature range below the T c, in contrast to [18] in which similar domains present a significant shrinkage with increasing temperature.

Micromagnetic simulations 3.6.1. Stability
In order to monitor the stability of CDW domain we performed micromagnetic simulations where a 512 × 512 × 8 grid was used with the cell size set at 0.5 nm.We created an initial magnetic configuration of a CDW domain (figure 5) where the inner and outer diameters were set to 94 and 200 nm respectively so their ratio was 0.47.In all of our simulations, the effect of temperature was not taken into account.By combining a series of interfacial DMI values with an external applied magnetic field we explored the conditions under which this specific magnetic configuration is stable.The direction of the magnetic field is parallel to the direction of magnetization that corresponds to the 'dark' regions.As we can see in figure 5 the absence of interfacial DMI drives the CDW domains to a solid magnetic bubble and when an external field is applied, the system falls into a single domain phase.By increasing the interfacial DMI we see that for small magnetic fields a stripe like pattern is the energetically favorable configuration whereas the CDW domain emerges at magnetic field range before the system falls into a single domain phase, similarly to what we have observed experimentally.This range of stability for the CDWs is clearly affected by the interfacial DMI.The micromagnetic simulations suggest that by increasing the external field further, we do not go straight to a single domain state since the CDW domain collapse to a skyrmion like configuration.In our experiment we also noticed that there is a collapse of the CDW domain to a spot-like configuration but our MOKE spatial resolution does not let us delineate the exact magnetic configuration experimentally.The general trend is that small DMI values shift the range of stability to small magnetic fields, as the value of the DMI increases the stability range goes to very high magnetic field values.In our approximation we used an effective average DMI for the whole 4 nm thick simulation slab.It is well known that the interfacial DMI decreases rapidly as we move away from the interface.The most common approximation is that the DMI is reduced proportionally to d −1 where d is the depth inside of the magnetic layer starting from the interface [32].A series of recent experimental results [33,34] have challenged this approximation.Their results indicate that the strength of the DMI actually follows a non-monotonic dependence on the distance from the interface.It starts by increasing and the maximum value is reached after few atomic layers, after that it follows the typical trend to decrease.Using an inhomogeneous DMI with a depth profile similar to those found in [33,34] (see figure S8 in supplementary material), we calculated the stability map of the CDW for three depth profiles.By comparing figures S8 and 5, we get qualitatively similar results as if we had set the D mean as constant in the simulation slab.There are only variations in the magnetic field range at which each magnetic configuration is stabilized, but there is no difference regarding their type, shape and order of appearance.
In figure 6 we see 3D images of the calculated magnetic structure, which reveal the nature of the domain walls formed in each case.When the interfacial DMI is set to zero we see that the solid magnetic bubble has a Bloch type domain wall (figure 6(a)).On the other hand, when an interfacial DMI is present we observe a change to Néel type domain wall (figures 6(b) and (c)) for the CDW domain one (figure 6(b)) as well as for the stripe like phase (figure 6(c)).Furthermore, we observe that the spotlike configuration after the CDW stability region is Néel type skyrmion (figure 6(d)).
In order to evaluate the experimental size of the CDW domains we performed a sampling on a series of CDW domains from our MOKE images, at 77 K with 220 mT applied magnetic field.In figure 7 we see the corresponding inner, outer diameter and Inner/outer diameter ratio for each one of the 26 CDW domains that were used as test samples.As we can see there is a variation to the size of the CDW domains that can be associated with non-uniformity of the flake surface and/or the applied magnetic field.The average outer diameter (figure 7(a)) is 2.44 µm whereas the average inner one is 0.96 µm.The corresponding average inner to outer diameter ratio is 0.39.In table 2 we demonstrate the inner-outer diameter ratio for the calculated CDW domain.As we can see the presence of magnetic field changes significantly this ratio.An increase of an opposing magnetic field yields shrinkage of the CDW domains diameter.In our experiment we observed a qualitatively similar behavior.Based on the diameter and the magnetic field range where the CDW domain is stable and the experimental results, we can conclude that a representative value for the interfacial DMI is close to 0.75 × 10 −3 J m −2 .This value is in accordance to the other works in the literature [13,15,16].

Magnetic field dependence of the CDW diameter
In order to model the response of the CDW domain to an increasing magnetic field we kept the interfacial DMI fixed to the optimal value of 0.75 × 10 −3 J m −2 .In figures 8(a) and (b) we see that the outer diameter shrinks for the simulated CDW domains as the magnetic field increases while the inner one increases until 110 mT and then reaches a plateau.Overall, the inner to outer diameter ratio increases as we increase the magnetic field until the threshold of 280 mT is reached where the CDW domain collapse to a single Néel type skyrmion.It is interesting to notice that the diameter of the skyrmion is around ten times  smaller than that of the CDW domain (figure 8(a)).Based on this we can infer that if a CDW indeed collapses to a skyrmion, it may not be detectable since the average size of a CDW domain is of the order of 2-3 µm and the spatial resolution of our MOKE system is of the order of 500 nm at maximum.By comparing the inner/outer diameter ratio for the simulated CDW domains (figures 8(a) and (b)) and the experimental one 8(c) and (d)) we see that there are in agreement qualitatively, since both present (1) an inner/outer diameter plateau at a specific magnetic field threshold and (2) collapse  to a dot like feature (skyrmion) above this field threshold.

Interaction of two CDW domains
Our experimental MOKE results revealed that the CDW domains form a regular lattice where the distance between them is quite large.In order to understand the why the CDW domains prefer to stay in a specific distance, we performed micromagnetic simulation of two CDW domains for a range of distances (figure 9(a)) and we monitored the effective magnetic field that is generated (figure 9(b)).For 50 nm distance the two CDW merge.As we can see for 100, 140 and 180 nm distance between the center of the CDW domains, the effective magnetic field form close magnetic field lines with opposite orientation between them (red and blue lobes).Due to the Néel type character of the CDW domains (figure 6(b)) the in plane component of magnetization in the domain wall has a radial direction.This meaning that in each point of the CDW outer perimeter the magnetization of the domain wall points towards the CDW center.Thus, if two CDW approach each other the outer perimeter domain walls sense the effective repulsive magnetic field that is generated from its neighbor as if there were two magnets with opposing magnetic poles.So, the effective magnetic field resembles the case of two opposing magnetic poles, yielding a repulsive magnetic field.For 220 nm distance we see that each CDW domain start to be decoupled from the magnetostatic influence of its neighbor.The corresponding spacing between them is 158 nm (red arrow in figure 9(a)) which means that the ratio between the spacing and the CDW domain diameter is of the order of 2.7.In order to further illustrate the above statement, we performed finite element analysis simulation (see supplementary information) using a toy model of two donut shaped magnets, resembling the outer domain walls of two CDW.By testing different distance between the two magnets and using both attracting and repulsing arrangements we further illustrated that the emerged effective magnetic field and magnetic field lines has the same qualitative texture as in the case of two neighboring CDW.
The experimentally observed CDW domains have a variety of diameters as we mentioned above.An average spacing between the CDW domains in our experiment is 5-6 µm.So, the corresponding ratio between the CDW domain spacing and diameter is of the order of 2-3, which is in agreement with the distances that both micromagnetic simulations and finite element analysis predict to have non-interacting CDW domains.

Energy analysis of CDW formation
In order to understand the origin of the CDW domains stabilization we monitored the variation of the balance between the various energy terms as a function of interfacial DMI and applied magnetic field.For the energy relaxation we used the steepest descent algorithm to minimize the total energy which is expressed as: where m(r,t) is the function describing the magnetization and the various energy terms are: where û is the unitary vector along surface normal For D i = 0 mJ m −2 (figure 10(a)) we see that the most significant (by one order of magnitude) energy term that determine the total energy is the anisotropy term followed by the Zeeman term which presents the usual linear dependence with the magnetic field.Also the demagnetization term plays the role of an energy penalty term since we have an out of plane magnetized flake with its lateral dimension much bigger than its width.The exchange term, is of course zero at the single domain phase, but for B = 0 mT, where a Bloch type bubble domain is stabilized, it has a small positive value.
For all the D i ̸ = 0 mJ m −2 case (figures 10(b)-(d)) we see that upon the transition to the CDW domain phase (marked with red dashed lines in figure 10) there is an abrupt change to all energy terms.The anisotropy term presents an increase in its absolute value after the transition, thus playing a more stabilizing role since it is negative (equation ( 5)).The fact that the anisotropy contribution becomes more stabilizing can be associated with the mutual annihilation of all in plane components of the magnetization (figures S10(a) and (b)) due to the circular shape of the CDW domain.Furthermore, the stabilization of the CDW restricts the extension of domain walls compared to the stripe like phase.By comparing the extend of the in plane components of the magnetization vector field (figures S11(a)-(h)) we see that upon transition from the stripe like pattern to the CDW lattice there is a rapid decrease in the number of regions of the simulated system that have non zero in plane components, thus we can conclude that by stabilizing the CDW lattice the system diminish the portion of the material that has in plane components of magnetization by drastically reducing the domains walls.
The demagnetization term presents an abrupt increase upon transition to CDW phase thus playing the role of an energy penalty term.This can be explained by the increase of the out of plane magnetization component upon transition to the CDW phase (figure S10(c)).
Since the exchange term depends only of the spatial variation of the magnetization it is safe to assume that the main contribution comes from the domain walls, where we have a spatial variation of magnetization.Thus, we can say that this energy term gives us information about the energetics of domain wall creation-annihilation.As we can see upon transition from the stripe like domains to the CDW domains the exchange term presents an abrupt decrease in its absolute values and falls to zero (figures 10(b)-(d)).Note also that the sign reversal of the exchange term is attributed to the presence of the DMI term.
When we have a non-zero DMI its contribution is included in the exchange term so the new total exchange term combining equations ( 2) and ( 6) is: Based on the 3D images of the spin configuration depicted in figure 6 we can assume that the variation of magnetization vector field along the z component is less significant than the in plane components and thus we assume that ∂mz ∂z → 0 .Let us now consider the spin variation as we across one side of the CDW perimeter (figure 6(b)) depicted in figure 11.
Due to circular symmetry of the CDW the spin configuration depicted above is the same for all radial cuts of the CDW domain.Based on figure 11 we can conclude that the deviation of the magnetization vector field is negative across the CDW domain (∇ • m < 0 ⇒ ∂mx ∂x + ∂my ∂y < 0) since the magnetization vector field presents a 'sink' in the core of the CDW domain (all arrows pointing inwards to the core which is the red tile in figure 11).Based on equation (7) and figure 11 we can conclude that the divergence of the magnetization across each side of the CDW is always negative due to its specific spin configuration.The symmetric part of the exchange is always positive (equation ( 2)) and plays the role of a penalty energy when we move away from a uniform magnetize phase.This means that the symmetric exchange favors the minimization of domain walls.On the other hand, the antisymmetric (interfacial DMI) part (equation ( 6)) is a bit more complex.Its first term, when is integrated will be always positive since, based on our assumptions about the direction of the magnetization, the integral of z component of the magnetization is negative.The second term gives the in plane spatial variation of the z component of the magnetization.When we consider a cut across a CDW domain (figure 6(b) red line) we can say, in a crude way, that the variation of m z has the following spatial variation 1 ↓ 2 ↑ 3 ↓ 4 ↑ 5 ↓.For simplicity reasons we will present the differentiation of m z along the x-direction but the case is the same for any given combination of differentiation along, x-and y-direction due to the circular symmetry of the CDW and the Néel type nature of the domain wall (the magnetization vector is always radial with respect to the CDW).So, we will have: This means that the second term in the antisymmetric exchange (equation ( 6)) falls to zero upon transition to the CDW phase.This is the reason why the exchange term presents a sudden increase when the CDW is stabilized because now only the first term of equation ( 6) stands.So the antisymmetric exchange term tends to favor the formation of many domain walls with magnetization configurations that do not make the (m • ∇)m z go to zero, since it is a negative (stabilizing) term.

Discussion
Our MOKE results revealed that by imposing a field sweep protocol a CDW lattice emerges.Assisted by our micromagnetic simulations we can infer that the CDW are formed due to the present of an Interfacial DMI from an oxidized FGT epilayer and have a Néel type domain wall.The magnetic configuration of the CDW, as suggested by the simulations, resembles the magnetic configuration of a skyrmionium.Furthermore, there is a magnetic field threshold after which the CDW collapses to a Néel type skyrmion.Although our MOKE images are limited in spatial resolution, they confirm that the CDW collapse to a dot like feature after a specific field, that can be a precursor of a Néel type skyrmion.The formation of topological non-trivial magnetic structures by an oscillating magnetic field has been reported also to other systems [16,35].Since our MOKE experiments suggested that the crucial parameter for the stabilization of the CDW lattice is the external magnetic field we monitored the dependence of the various energy terms by it (figure 10).Based on that we can say that the stabilization of the CDW lattice is a result of an antagonism between two energy term groups.The first is the high out of plane anisotropy, the symmetry exchange and the Zeeman term (when a non-zero magnetic field is applied) which favor the creation of large uniform magnetized areas whereas the second group consists of the antisymmetric exchange (DMI) and the demagnetizing energy term, which favor the formation of a dense domain pattern.In the absence of DMI only the symmetric exchange term stands, which is positive so it plays the role of energy penalty when domain walls are present.In this case the system prefers to make large uniform magnetized regions, with limited domain walls and fall to uniform magnetized phase even for small external fields.When the interfacial DMI term is introduced (antisymmetric exchange), the formation of regions with canted magnetization is favored.Also, the demagnetizing term tends to favor the formation of a dense domain pattern and counteract the formation of large areas with uniform out of plane magnetization.For low external magnetic fields this antagonism between the various energy terms settles with the formation of a stripe-like pattern (figure 5).By carefully observing the total energy for all D i values (figure 10) we notice that there is actually a small energy barrier between the stripe pattern phase and the CDW phase.This energy barrier is crossed with the assistance of the Zeeman term.In figures 10(b)-(d) we see that upon transition to the CDW phase the Zeeman term starts to dominate over the exchange term.This is the reason why we need the external magnetic field to generate the CDW phase and we do not see the spontaneous formation of CDW.By stabilizing the CDW domains, the system manages to 'tame' the high out of plane anisotropy, because in this Néel type domain configuration the in plane component of magnetization are limited (figures S10(a) and (b)), thus lifting the penalty of having a series of domain walls with in plane components as was the case for the stripe domain phase.The beneficial role of interfacial DMI in this process is that it enables the transition from a Bloch type domain wall to a Néel type one and thus enabling the formation of this CDW domain.Also, other experimental works in the literature have verified that an Interfacial DMI can tune the domain walls from Bloch to Néel type [12][13][14][15][16][17][18][19]36].By combining the simulation data presented in figure 5 we can sketch the phase diagram of the external magnetic field versus interfacial DMI for FGT thick flakes.In figure 12 we see that four distinct regions are present namely the bubble-line domain, CDW, skyrmion and single domain phase.By carefully engineering the interlayer DMI and using an external magnetic field one could navigate from one phase to another.The formation of a CDW domain lattice, and their subsequent collapse to Néel type skyrmions, offers opportunities in the development of various spintronic applications in the areas of data storage with for example racetrack memory devices and logic devices for in memory computing applications where an array of magnetic tunneling junctions with FGT as a free layer would be used.For such applications the main advantage of the CDW domain lattice is that it has a well-defined and robust pattern of domain walls, which allows for precise manipulation and control.This arrangement provides the means to encode and store information in a compact manner.The CDW lattice formation results in an organized and predictable motion of the CDW, contributing to the reliable operation of potential memory and logic devices.

Figure 2 .
Figure 2. (a) The configuration of domains structure after ZFC.(b), (c) The corresponding line scans showing characteristic dimensions for the domains with the magnetization points parallel or antiparallel.Red line corresponds to right-up linescan whereas yellow line to right down linescan.

Images 1 →
2) and then increasing the magnetic field again (242 mT → 254 mT) (figure3(a) Images 2 → 3) the line pattern starts to break forming small linear segments.A second round of magnetic field sweeping (254 mT → 238 mT → 274 mT → 256 mT) (figure 3(a) Images 3 → 4 → 5 → 6) results in the formation of even smaller line segments that start to branch and curl.As we move close to saturation by increasing the magnetic field (256 mT → 266 mT → 267 mT → 274 mT) (figure 3(a) Images 6 → 7 → 8 → 9) the small curly line segments morph into C-shaped segments (and not the straight segments shown before) that become almost a point like feature at 274 mT, just before the saturation.By reducing the field further to 260 mT (figure 3(a) Images 9 → 10 → 11) a CDW pattern emerge.The CDW domains are arranged in a lattice and once they are formed, they are stable in specific range of magnetic fields

Figure 3 .
Figure 3. (a) Polar-MOKE images for the generation and evolution of the CDW domains and the lattice configuration they form number 1-11 in images are only for enumeration, their colors correspond to the respective magnetic field color code.(b) Effect of the magnetic field on the stabilized CDW domain lattice.

Figure 4 .
Figure 4. Temperature dependence of CDW domain pattern.

Figure 5 .
Figure 5.The stability of the CDW for different combinations of interfacial DMI-external magnetic fields up to 1150 mT.For each simulation the starting configuration is the same.Dark regions correspond to down spin, bright to up.The color code for the different magnetic configuration is bubble or stripe like domains-yellow, CDW domain-violet, skyrmion-pink and single domain-green.

Figure 7 .
Figure 7. Histograms of the (a) outer diameter, (b) inner diameter and (c) outer to inner diameter ratio for a series of samples with CDW domains.The x-axis represents the number of CDWs that was measured and found to have a specific inner (outer) radius.

Figure 8 .
Figure 8. Magnetic field dependence of the (a) inner and outer diameter and (b) the ratio of the inner/outer diameter of the simulated CDW domain, (c) inner and outer diameter and (e) the ratio of the inner/outer diameter of the experimental CDW domain.

Figure 9 .
Figure 9. (a) Simulation of two CDW domains in a series of distances between their centers.(b) The effective magnetic field for each case, pink and blue arrows indicate the direction of the effective field (right and left respectively).

Figure 10 .
Figure 10.The energy terms for D i = 0.00 mJ m −2 (a) D i = 0.00 mJ m −2 , (b) D i = 0.65 mJ m −2 , (c) D i = 0.75 mJ m −2 and (d) for D i = 0.85 mJ m −2 is depicted.Lines are just guide to the eye.The notation L-B, SD, CDW and Sk stand for line-bubble, single domain, circular wall domain and skyrmion phase respectively.

Figure 11 .
Figure 11.Spatial spin variation across one side of the CDW.

Figure 12 .
Figure 12.Phase diagram of the magnetic structure of FGT for different values of interfacial DMI and applied magnetic field.

Table 2 .
Calculated outer and inner diameter of the CDW domain.