Shape instability in out of equilibrium magnetic domains observed in ultrathin magnetic films with perpendicular anisotropy

Shape instability in out of equilibrium magnetic domains observed in ultrathin magnetic films with perpendicular anisotropy

Journal of Magnetism and Magnetic Materials 192 (1999) 409— 418 Shape instability in out of equilibrium magnetic domains observed in ultrathin magnet...

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Journal of Magnetism and Magnetic Materials 192 (1999) 409— 418

Shape instability in out of equilibrium magnetic domains observed in ultrathin magnetic films with perpendicular anisotropy R. Hoffmann , Y. Samson *, A. Marty , V. Gehanno , B. Gilles, J.E. Mazille CEA Grenoble, De& partement de Recherche Fondamentale sur la Matie% re Condense& e, Service de Physique des Mate& riaux et Microstructures, 17 Avenue des Martyrs, 38054 Grenoble, France CNRS, LTPCM / ENSEEG, BP 75, 38402 Saint Martin d’He% res, France CEA Grenoble, CEREN/DEM/SPCM, 38054 Grenoble, France Received 15 July 1998; received in revised form 1 December 1998

Abstract The magnetic configurations induced by the growth process in a thin film with perpendicular magnetisation have been observed by magnetic force microscopy (MFM). The FePd thin film has been grown by molecular beam epitaxy. A high uniaxial chemical ordering of the alloy into the tetragonal L1 structure induces the development of a large perpendicular  anisotropy. As the growth process is performed below the Curie temperature of the FePd alloy, domain nucleation occurs during the growth process. The magnetic configuration has been imaged in the as grown state. As the equilibrium size of the magnetic domains decreases when the thickness of the layer increases, the domains obtained from spontaneous nucleation at the beginning of the growth of the thin film are submitted to very large strains as the layer thickness increases. At low thicknesses (low strains), the domain wall instability gives rise to an undulation of the domain walls. Thereafter, it leads to the formation of well-defined magnetic fingers, thus giving birth to the coexistence of two length scale in the domain structure. A quantitative estimation of the strain leading to the fingering instability is obtained. Last, the implications of these observations on the kinetic of domain wall distortion in ultrathin layers are discussed.  1999 Elsevier Science B.V. All rights reserved. PACS: 75.70 Keywords: Magnetic force microscopy; Magnetic domains; Perpendicular anisotropy; Shape instability; Thin films; Ordered alloy;

1. Introduction

* Corresponding author. Tel.: 33-4-76-88-35-62; fax: 33-4-7688-50-97; e-mail: [email protected]

The magnetic configuration induced by perpendicular magnetic anisotropy has recently been observed in thin and ultrathin magnetic films [1—3]. This has been achieved through the use of magnetic

0304-8853/99/$ — see front matter  1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 8 ) 0 1 0 7 5 - 0

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force microscopy (MFM). Indeed, this new tool has made possible the investigation of the magnetic configuration in thin films with unprecedented resolution and sensitivity: magnetic domains down to 50 nm have been observed with a high contrast [2,3] and down to very low thicknesses of the magnetic layer (2 nm in Cu/Ni/Cu [1]). These experiments have attracted attention as well for their fundamental interest as for the potential applications of such layers as data storage media. We have observed the magnetic domains resulting from spontaneous nucleation at the beginning of the growth process of an ultrathin magnetic layer. As the thickness of the layer is increased, the competition between the domain wall energy and the magnetostatic energy implies an exponential diminution of the equilibrium domain size [4]. As a result, the pre-existing magnetic pattern resulting rom the nucleation processes has to adapt to very high strains. The analysis of the MFM images obtained in both the as-grown and the demagnetised state of the sample allowed us to understand the process leading to the magnetic patterns observed at each thickness. This provides a new insight into the means that an out of equilibrium magnetic configuration adopts to shift toward equilibrium in a ultrathin magnetic layer. This includes the observation of domain wall buckling at low thicknesses and the development of magnetic fingers from highly out of equilibrium magnetic domains at larger thicknesses.

substrate is also below the Curie temperature of the equiatomic alloy, thus allowing domain nucleation during the growth process. The L1 structure is based on a tetragonal cubic face centered lattice (Fig. 1). The L1 phase can also be seen as consisting in pure Pd and pure Fe alternated atomic layers. This phase exhibits a large unixial magnetic anisotropy along the c-axis. Here, the chemical ordering is purely uniaxial with the pure atomic planes perpendicular to the growth direction. As a result, the chemical ordering of the alloy leads to the development of a strong perpendicular anisotropy (K/2pM &1.6—2) [4]. In order to perform extensive ex situ measurements as a function of the layer thickness, a 15 mm long FePd wedge whose thickness ranges from 0 to 120 nm has been prepared. It has been obtained by progressively masking the sample with a motorised shutter during the growth process. The distance between the shutter plate and the sample is around 1 mm whereas the distance between the sources and the sample exceeds 200 mm. As a result, imperfect shadowing effects near the shutter edge are negligible. A 2 nm Pd capping was finally deposited on the layer in order to protect it from air oxidation. All MFM measurements have been performed in air using a Nanoscope III A from Digital Instruments and commercial or locally prepared MFM tips (with CoCr coatings). These tips were magnetised

2. Experimental The sample has been prepared using molecular beam epitaxy on MgO(0 0 1) substrates. The growth process and the structural properties of the sample have been described precisely in another paper [4]. A 50 nm Pd buffer layer is first grown on the MgO substrate. Thereafter, Fe and Pd are simultaneously evaporated from two independent e-beam evaporators using identical rates of 0.16 ML/s for each element. The temperature of the substrate (600 K) is below the order—disorder transition temperature of the equiatomic alloy, but high enough to induce a high L1 chemical order during the epitaxial growth. The temperature of the

Fig. 1. Scheme of the L1 lattice. Full circle: A atoms, open  circles: B atoms. A"Fe (or Pd), B"Pd (or Fe). The c-axis is aligned with the direction of easy magnetisation. On a larger scale, the L1 phase can be seen as pure Fe and pure Pd atomic  plane alternating with a bi-atomic period.

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perpendicularly to the surface of the observed sample. This results in a coupling with the perpendicular component of the field emerging from the sample. The uncertainty on the position on the sample for each measurement results in an error of less than 0.4 nm on the exact thickness of the imaged parts of the magnetic layer. Last, we observed that the sample exhibits an unchanged magnetic configuration over weeks, thus demonstrating that the magnetic configurations presented in this paper are stable within observable time scales. STM investigations of the FePd surface (without capping) have been performed in UHV. For low thicknesses of the layer (below 15 nm), these investigations reveal a very low surface roughness, due to atomic (or bi-atomic) steps. The observed corrugations are then negligible with respect to the layer thickness, except for the thinnest part of the wedge observed by MFM (h"1.6 nm). However, the width of the atomic terraces falls around a few tens of nanometers, well below the width of the magnetic domains (a few hundreds of nanometers) observed in the same thickness range. As a result, the influence of the surface steps on the magnetic configuration should be negligible and the sample can be considered as a nearly ideal flat one, with negligible roughness, into the following study.

3. The observed magnetic patterns The as-grown magnetic domains have been imaged on the sample for thicknesses of the magnetic layer between 1.6 and 10 nm (see Fig. 2 and Fig. 3). The dark and bright areas correspond to up and down magnetic domains with respect to the layer plane. The magnetic pattern may be still present below 1.6 nm, but then we have been unable to image it without perturbing the magnetic configuration by the field emerging from the magnetic tip. Conversely, it has been carefully verified that the magnetic configuration can be imaged repeatedly without any perturbation for all the presented images (obtained at thicknesses larger than 1.6 nm). Between 1.6 and 2.4 nm, the magnetic pattern can be described as a mixed structure involving bubbles and stripes, both of rather well-defined

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width (Fig. 2). This kind of pattern is commonly observed in a variety of physical systems such as magnetic or organic (Langmuir) thin films and type I superconducting thin films2 It results from a competition between line tension and a longrange interaction [5], here, respectively, the domain wall energy and the demagnetising field. Above 3.1 nm, a new pattern appears, involving two different length scales (Fig. 3). Indeed, magnetic fingers now emerge from the large domains into the neighbouring ones with opposite magnetisation. The number of these fingers increases with increasing thicknesses and they progressively invade the whole sample. Above 10.5 nm, the images reveal only a classical serpentine structure, with a welldefined length scale.

4. Image analysis In order to allow a quantitative discussion of the observed phenomena, the MFM images have been submitted to numerical image analysis. For the sake of brevity, the ‘distance between neighbouring domain walls’ will be referred to as ‘domain width’ in the following. The image analysis has been done in order to extract the evolution of the width of the magnetic stripes (the diameter of the bubbles), as a function of the layer thickness. The process is briefly described here: first, the grey-level image is improved by a smoothing Gaussian filter, followed by a non-linear delineation. After binarisation, the image is cleaned by removal in both phases of the smallest objects, which may be considered as artefacts as their size is by large below the domain size. Then, the distance mapping is applied to the phase of interest (on the images the dark one). The value of this function is the distance to the nearest boundary inside the phase, zero elsewhere. This provides for each point of the image the distance to the nearest domain wall. The phase of interest is skeletonised. The skeleton is the set of the points of maximal breadth. The breadth distribution (domain width) is twice the value of the distance function on the skeleton. When both large domains and thin fingers are observed, two peaks appear in the breadth distribution. This method is reliable, as long as phases are well separated. When they are too

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Fig. 2. 16 lm MFM images of the as-grown magnetic structure of a thin magnetic film of FePd alloy. The MFM tip is magnetised perpendicularly to the sample surface. Dark and bright areas corresponds to perpendicular domains of opposite magnetisation (up and down). The thickness of the magnetic layer is: (a) 1.6 nm, (b) 1.9 nm, (c) 2.4 nm, (d) 3.1 nm. The white lines provide an estimation of the length scale of the instability wavelength.

much intricated, the concept of breadth vanishes. In such cases (configurations observed at the largest thicknesses (Fig. 3c and Fig. 3d)), the width of the large domains may be reconstructed by applying a series of smoothing—segregating filters to eliminate the fingers. In addition, the width of the small stripes and fingers can be easily measured on higher resolution (smaller) images taken in the same area of the sample. A few instances of the width distributions are presented in Fig. 4.

A slight asymmetry between the two phases (clear and dark domains) justifies the choice of the dark phase for image analysis: the clear domains are connected together with a very few exceptions while the dark ones are closed. However, the two phases occupy the same portion of the space. The origin of this topological asymmetry remains unclear and it has only one noticeable consequence on the phenomena described here: domain walls as seen from dark domains exhibits an higher number

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Fig. 3. 16 lm MFM images of the as-grown magnetic structure of a thin magnetic film of FePd alloy. The MFM tip is magnetised perpendicularly to the sample surface. Dark and bright areas corresponds to perpendicular domains of opposite magnetisation (up and down). The thickness of the magnetic layer is: (a) 3.8 nm, (b) 4.2 nm, (c) 8.5 nm, (d) 10.5 nm.

of positive curvature areas. As the magnetic fingers are emerging more easily from these areas, most of the magnetic fingers emerge from dark domains into the neighbouring clear ones (as expected, the reverse observation is true on the few closed clear domains such as the one observed in Fig. 3b). Hence we have measured the domain width into the dark phase in order to access easily to the width of both the large domains and the magnetic fingers. Last, it must be pointed out that the number of dark magnetic domains (per unit area) remains

unchanged whatever the layer thickness (about 1$0.15 10 cm\) except at the largest investigated thicknesses (from h"8.5 nm) where nucleation begins to occur. This demonstrates that the observed evolutions are induced only by a distortion of the domain walls (and not by nucleation processes). If one admits that the nucleation energy of bubbles is decreased on defects or impurities in the layer, this can be seen as an additional clue of the high quality of the observed sample. Such an energy barrier preventing the nucleation of new

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Fig. 4. Distribution of the distances between domains walls (for dark domains). These data have been extracted from 16 lm images (512;512 pixels) using image analysis procedures. The corresponding sample thicknesses h are indicated in the figures. The line indicates the position of the maximum in the distribution observed at the lowest investigated thicknesses (1.6 and 1.9 nm).

domains was not observed in previous studies of ultrathin films with perpendicular anisotropy (Ni or FePd thin films [1,4] where bubble nucleation occurring at the middle of the large pre-existing stripes is very clear on the presented MFM images).

5. Discussion: low thicknesses, low gap from equilibrium We will first consider the low thicknesses part of the sample (up to 3.1 nm). Here, the distribution of the domain sizes is unimodal as can be observed in Fig. 4a and Fig. 4b. Fig. 5 displays the lateral dimension (width) of the domains which decreases as the thickness of the layer increases. As can be seen in Fig. 2, this is allowed by an increasing distortion of the pre-existing domain walls. This phenomenon, commonly called ‘buckling’, occurs in frustrated systems when it is difficult to increase the number of domains in order to accommodate a strain. Here,

Fig. 5. Lateral dimension of the observed structures (bubbles stripes and fingers) as a function of the thickness of the magnetic layer. (D): stripes and bubbles observed below 3 nm, large substructures observed above, (F): fingers, (S) stripes after demagnetisation. The line D provides the mean value of (D) above

3.8 nm. D also defines the initial state from which all the

observed magnetic configurations have evolved (see text and Fig. 3).

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the strain can be understood as the gap between the actual and the equilibrium domain size. When the observed domains are too large with respect to their equilibrium width, the stripe buckling brings the domain walls close together, thus diminishing the cost in demagnetising energy of the magnetic configuration. Such a behaviour has also been observed in analogous systems such as smectic-A liquid crystals [6] and garnet films [7,8]. In addition, it may be noticed that the formation of a zigzag pattern has recently been obtained in the same classical way in a thin (30 nm) cobalt magnetic film [9]. This phenomena is quite similar to the one observed here on the first images (at the lowest investigated thicknesses (Fig. 2)) and may then be reminded. The authors [7,9] have decreased the number of domains in a parallel stripe pattern by applying a perpendicular field to the sample (see Fig. 6). Indeed, when a perpendicular (up) magnetic field H is applied to the layer, the increase of the width of the favoured (up) domains is larger than the diminution of the width of the opposite (down) domains [11]. This implies that the period of the magnetic pattern P(H) increases. In this way, they

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eliminate the dislocations in the magnetic structure and, when the applied field is removed, the number of domains has decreased. Then, the preferred width of the remaining domains is preserved through undulation of the domain walls. This undulation allows that too less numerous stripes fully occupy the space without changing their width and without the nucleation of new domains. However, when the values of the applied field are increased, this methods quickly leads to a completely disordered magnetic pattern. This prevents the observation of magnetic configuration under large strains as the ones we will describe below. However, between 1.6 and 2.4 nm, these phenomena are quite similar to the ones described here: buckling is used as a means for large domains to decrease their width without paying the cost for the creation of new magnetic domains.

6. The driving force for the instability The driving force for this evolution can easily be understood. Indeed, in a recent paper, Kaplan and Gehring [10] have expressed the energy density of a periodic stripe pattern as the sum of the domain wall energy and of the dipolar energy: º"px#2pM h(1#bx#(2/p)x ln(x)) where x"h/S, p is the domain wall energy per unit area, h the thickness of the magnetic layer, and S the stripe width. By minimisation with respect to x, an analytical relation between the thickness of the magnetic layer and the preferred size S of the magnetic domains may be obtained: SJh exp(nD/2h)

Fig. 6. (a) The parallel stripe structure in zero field, (b) with an (up) field applied perpendicular to the layer plane: the increase of the size of the favoured (up) domains is larger than the corresponding decrease of the size of the opposite (down) domains. This leads to an increase of the global period and then to the elimination of some magnetic domains. (c) Once the applied field is removed, an undulation of the domain wall allows that now less numerous domains fully occupy the space, while recovering the equilibrium distance between domain wall.

where D"p/2nM is the dipolar length. This relation is valid within the limit of small h/S. An experimental confirmation has recently been obtained qualitatively [12] and quantitatively [4]. Then, as the thickness of the magnetic layer increases, the competition between the line tension and the dipolar energy results in an exponential decrease in the equilibrium domain size. This is the origin of the observed buckling. Indeed, the magnetic domains which have nucleated at a low

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thickness during the growth of the layer have thereafter to adapt to a smaller equilibrium size. Obviously, this implies that the actual domain size is larger than the equilibrium one. In order to confirm this assumption, we have submitted the sample to an in plane oscillating field of decreasing amplitude, starting above the saturation field. In this way, the magnetic configuration of the sample is expected to shift toward the equilibrium one. As a result, we observed a classical magnetic configuration consisting in a disordered array of bubbles and stripes, but of rather well defined size. This allowed us to conclude that the equilibrium domain sizes (which have been reported in Fig. 5) are indeed well below the ones measured on the as-grown sample.

7. Discussion: large thicknesses, large gap from equilibrium Fig. 3 demonstrates that the magnetic layer exhibits a completely different magnetic configuration above 3.1 nm. Indeed, the pre-existing bubbles and stripes are progressively invaded by an increasing number of fingers emerging from the neighbouring domains. Most of these fingers are actually connected to a domain wall from which they seem to have emerged. Occasionally, some stripes are isolated in the middle of large domains. These ones may have occurred through bubble nucleation and a stripe-out instability [13]. This is observed only at the largest thicknesses (from Fig. 3c). Then, the demagnetising field in the middle of the remaining large domains may be large enough to overcome the energy barrier preventing bubble nucleation [14]. However, up to h"8 nm, the domain configuration evolves only through a distortion of the domain walls (wall undulation, formation of magnetic fingers). Due to the occurrence of these fingers, the distribution of the domain widths exhibits two maxima, one corresponding to the large stripes and bubbles and the other one to the magnetic fingers (Fig. 4c and Fig. 4d). It is noticeable that the width of the large domains (D) is now independent of the layer thickness (see Fig. 3 and Fig. 5) while the width (F) of the fingers decreases rapidly with the increase in the thickness. Finally, for thicknesses larger than

the one displayed in Fig. 3d (i.e. 10.5 nm), the large stripes and bubbles are no longer distinguishable. The magnetic pattern consists of the highly connected stripe pattern which has already been observed in thin magnetic layers with perpendicular magnetisation [2,4]. The width of the magnetic fingers is larger than the one observed for the domains in the demagnetised sample. This was to be expected as isolated domains within a large area of opposite magnetisation are submitted to a large demagnetising field (with respect to the one experienced by domains of the same width in a periodic pattern). As a result, the balance between wall energy and magnetostatic energy leads to larger domain widths. However, the actual difference could be smaller than observed. Indeed, during the MFM measurement, the cantilever flies above the sample at a constant height (around 25 nm here) chosen by the operator. This height has to be large enough so that a distortion of the image due to topography related forces (contact and van der Waals) of shorter range than the magnetic one can be avoided. Above an isolated domain, the emerging lines of field widen when the distance from the sample increases. This adds to the effect of the (non zero) radius of curvature of the tip and leads to an overestimation of the width of the fingers. Here the flight height is between 20 and 30 nm and the apparent increase in the width of the fingers cannot exceed a few tens of nanometers. This is not negligible but does not change the physical analysis. Though the involved topology is quite different, both buckling and fingering have the same origin. Indeed, both occur due to a wall instability which leads to the progressive distortion of the domains walls below 2.4 nm (buckling) and thereafter to the magnetic fingers as already pointed out, the wall instability is induced by the increase of the thickness of the layer. A significant parameter is the spatial wavelength of this instability. This wavelength may be defined as the period of the undulation, taken along a domain wall (see the white lines in Fig. 2d). This wavelength appears clearly from place to place in Fig. 2c and Fig.2d and in Fig. 3a. Qualitatively, it may be observed that it diminishes with increasing thickness. However, due to the complicated magnetic pattern, which results

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in various local configurations, its exact value varies inside a given image. This prevents any quantitative estimation but does not hinder the major tendency: the wavelength of the undulation decreases with increasing sample thickness. As far as the length scale of the instability is larger than the domains width, the observed phenomenon is reminiscent of buckling. Fig. 2d corresponds to the intermediate situation of a wall instability of the same length scale than the width of the domains. Thereafter, the wall instability results in the growth of the magnetic fingers, which is obviously a non-linear process. However, the physics of the instability may qualitatively be understood within the framework provided by Sornette [15]. He suggested that the undulation instability of domain walls can be treated using a smectic analogy. As described above, one can remove some domains in a parallel stripe pattern of period P by applying an external field H. Then, the remaining stripes experience a strain defined as X" (P(H)!P(H"0))/P(H). If X is the critical strain leading to the appearance of the wall undulation, their wavelength is proportional to 1/(X!X). Here, it would be interesting to estimate the strain. However, this implies to determine from which magnetic pattern the sample has evolved.

8. The initial state An unexpected observation may help our analysis: below 3 nm, the width of the large stripes and bubbles decreases regularly with the thickness, but it does not exhibit the same behaviour once the fingers have appeared. This is striking if one compares Fig. 2 images to Fig. 3c or Fig. 3d. As a result, the one-maximum distribution of the domain size observed below 2.4 nm exhibits thereafter two maxima (Fig. 5). The first aligment F corresponds to the fingers and exhibits a continuous evolution of the domain size. The second one D, which represents the large remaining bubbles and stripes is indicative of a constant domain width (denoted D in Fig. 5). This has a strong implication: the pattern observed above 3.1 nm has not evolved from this latter one, but directly from one pattern similar to the one observed at the lowest thickness

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(Fig. 2a). This can easily be understood within the following scheme: the magnetic pattern constituted of large bubbles and stripes has nucleated during the growth of the sample at a small thickness (below 1.6 nm), and the magnetic configuration has evolved thereafter on a larger time scale, larger than the time needed for the growth process. More precisely, the nucleation of the initial magnetic pattern occurs very soon when the layer is very thin (less than 1.6 nm) and this process is very fast with respect to the growth process, as expected. However, the distortion of the domain wall implied by the evolution towards equilibrium may be slower in very thin layers, as the one observed here. This scheme is strongly supported by the fact that, at each thickness, the magnetic configuration exhibits only two kind of structures: the pre-existing large domains with a constant size D , and the fingers with a width depending on the actual thickness of the layer. For instance, it is clear in Fig. 3 that the magnetic pattern does not exhibit the domains of intermediate widths observed for instance in Fig. 2d, this is also striking in Fig. 5: the remaining parts of large domains exhibit a size similar to the one observed at the lowest investigated thickness (about D ). Domains with continuously decreasing width should have emerged if the wall instability had developed continuously during the growth process. At first sight, it is surprising that the occurrence of the first fingers prevents the complete disappearance of large pre-existing domains. This may be due to the fact that the wall instability is no longer of limited amplitude. Due to their length, a few fingers occupy a significant portion of the space. This leads to a decrease of the demagnetising field in their vicinity, thus changing the balance between the line tension and the demagnetising energy. This may be the reason why, once a few fingers have appeared, the remaining parts of the pre-existing stripes and bubbles are preserved with their initial width. It might be outlined that, in previous publications on garnet films, the authors have called fingers the magnetic structures emerging from place to place perpendicularly to the direction of the parallel stripes. In these previous studies, both fingers and the domains from which they emerge exhibits the same width. Here, it is quite remarkable that two

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well defined widths coexist in the same image. This is clearly not an equilibrium situation. As pointed out above, this may be related to the slow kinetic of the domain wall distortion in the samples described here. Now, the strain to which the domains are submitted can be evaluated using the analysis from Sornette [15]. If one takes the domain size observed after demagnetisation (S in Fig. 4) as the preferred one, here X"(D !S)/D varies from about 0.5 at the lowest investigated thickness (1.6 nm) to about 0.93 at 10 nm. As X approaches unity, a fingering instability has been predicted as observed here. According to Fig. 5, the fingering instability appears when the layer thickness exceeds h"3 nm. Then, the ratio of the width of the initial domain to the equilibrium width reaches 5—6.

This may be the first observation of a magnetic structure strained as a result of the growth process of the layer and not after the application of an external magnetic field. This has been made possible by the recent mastering of perpendicular anisotropy upon a wide thickness range in ultrathin magnetic layers and by the use of magnetic force microscopy. Indeed, the very high strains observed on spontaneously nucleated domains are induced by the exponential variation of the equilibrium size of the domains which is specific to very low thicknesses of the layer. This has permitted a clear observation of the fingering instability, together with a quantitative estimation of the gap from equilibrium. Now, further work should be devoted to the understanding of the kinetic of the evolution of the magnetic configuration in ultrathin films.

9. Conclusion

References

The magnetic force microscopy imaging of the as-grown magnetic configuration of a FePd thin films allowed the observation of strongly out of equilibrium magnetic domains. These domains exhibit a shape instability as their equilibrium size shifts away from the pre-existing size when the thickness of the layer increases. Indeed, very high strains are generated at the highest investigated thicknesses (h"10.5 nm): the domain size is about 12 times the equilibrium one. The effect of the wall instability on the magnetic configuration of the sample has been investigated. Above a critical thickness (about h"3 nm corresponding to a ratio of initial to equilibrium domain widths around 3.5), the amplitude of the domain wall undulation diverges from place to place, leading to the formation of well defined magnetic fingers. Then, the gain in the demagnetising energy allowed by the development of a few fingers from the domain wall instability is high enough to stabilise the remaining parts of the pre-existing large magnetic domains.

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