Keywords: high-pressure torsion, Mg-RE alloy, microstructure, recrystallization, texture

The evolution of microstructure, texture and mechanical properties of an Mg-1.43Nd (wt.%) alloy are investigated after processing by high-pressure torsion at room temperature through 5 turns and isochronal annealing for 1 h at 150, 250, 350 and 450 °C using Electron BackScatter Diffraction and Vickers microhardness. The alloy exhibits a good thermal stability up to an annealing at 250 °C, with mean grain size of ~0.65 μm. The microhardness shows an initial hardening after annealing at 150 °C and then a subsequent softening. The deformation texture, a basal texture shifted 60° away from the shear direction (SD), is retained during annealing up to 250 °C. By contrast, a basal texture with symmetrical splitting towards SD is developed after annealing at 350 °C. The precipitation sequence and their pinning effect are responsible for the age-hardening, stabilization of grain size and the texture modification. The kinetics of grain growth in the Mg-1.43Nd alloy follows two stages depending on the temperature annealing range, with an activation energy of ~26 kJ/mol in the low temperature range of 150-250 °C and ~147 kJ/mol in the high temperature range of 250-450 °C.

non-equilibrium grain boundaries in the deformed ultrafine-grained microstructure whereas at high temperatures the grain growth is governed by lattice self-diffusion.

6.

Introduction

Over the last decade magnesium-rare earth (Mg-RE) alloys have been widely investigated with the objective of replacing the conventional Mg-based alloy such as AZ31 (Mg-3Al-1Zn, wt.%) in numerous industrial applications. [1,2] In practice, the presence of RE elements in dilute binary Mg-RE alloys, or even as additional alloying elements in ternary Mg-based alloys, have proven their efficiency in improving the room temperature formability and enhancing the strength and creep resistance through the development of a weak texture that is different from the typical basal texture. [1][2][START_REF] Mackenzie | Magnesium Alloys and Their Applications[END_REF] For example, deformation by extrusion of Mg-RE alloys leads to the development of a new texture designated "rare earth texture" where the [21 ̅ 1 ̅ 1] of grains is parallel to the extrusion direction (ED). [START_REF] Stanford | [END_REF]5] By contrast, a texture named "TD-split texture", where the basal (0002) poles of the grains are tilted from the normal direction (ND) towards the transverse direction (TD), was reported in hot-rolled Mg-RE alloys. [6,7] It is recognized that recrystallization and grain growth processes are responsible for the weak and unusual textures in Mg-RE alloys. [1] Hence, many theories have been established in attempts to explain the mechanisms responsible for the special effect of RE elements on texture modification. First, it was attributed to a particle-stimulated nucleation (PSN) mechanism after processing. [8] Then the nucleation and growth of new oriented grains was highlighted at deformation features such as shear bands [9] or twins. [10] Also, it was proposed that the activation of non-basal slip systems and the changing of grain boundary mobilities by solute drag may also affect the final texture. [11,12] Another positive effect of RE elements on Mg-based alloys is the additional ductility provided by grain refinement [START_REF]Rokhlin in Magnesium alloys containing rare earth metals: Structure and Properties[END_REF] which ensures a high thermal stability of the refined microstructure during annealing due to a solute-grain boundary drag at lower annealing temperatures. For example, the low diffusivity (~10 -22 m 2 /s) of Nd in Mg at 177 °C [START_REF] Choudhuri | [END_REF] and the precipitation of different metastable and stable phases during annealing which retard the grain boundary mobility. [15][16][17] Nevertheless, it is necessary in practice to increase the performance of Mg-RE alloys in order to extend their use in the automotive and aerospace industries or for use in biomedical applications. Severe plastic deformation (SPD) by equal-channel angular pressing (ECAP) or high-pressure torsion (HPT) appears to be an excellent strategy for improving the mechanical and superplastic properties of Mg-based alloys through the production of excellent grain refinement, down to the sub-micrometer level, and by introducing a high density of defects. [18][19][20][21] Moreover, Mg-RE alloys have proven their efficiency in producing very significant grain refinement compared to conventional Mg-based alloy after SPD processing. [22][23][24][25] For example, the Mg-10Gd (wt.%) alloy exhibits an average grain size of ~100 nm against ~1 μm in the AZ31 alloy after similar HPT processing. [22,23] Several studies were conducted earlier on the binary Mg-RE processed by HPT in an attempt to extend the fundamental understanding of the effect of RE elements and SPD processing on the precipitation, recrystallization and gain growth phenomena. [START_REF] Bourezg | Proc. 26th International Conference on Metallurgy and Materials[END_REF][START_REF] Bourezg | [END_REF][28][29][30] First, the effect of HPT processing on the microstructural and textural evolution in Mg-1.43Nd, Mg-1.44Ce and Mg-1.44Dy (%.wt) alloys was investigated. [28,29] Second, the sequence and kinetics of pre-precipitation in a Mg-1.43Nd alloy was investigated after HPT processing using in situ synchrotron X-ray diffraction and differential scanning calorimetry (DSC). [START_REF] Bourezg | [END_REF] Third, the recrystallization process and thermal stability after HPT processing was characterized by DSC and EBSD in Mg-1.44Ce and Mg-1.44Dy alloys, respectively. [START_REF] Bourezg | Proc. 26th International Conference on Metallurgy and Materials[END_REF]30] However, the thermal stability of Mg-1.44Dy was investigated at only two annealing temperatures (200 and 400 °C for 1 h). [30] Thus, the main objective of the present study was to extend an investigation of the thermal stability of an HPT-processed Mg-RE alloys over a large range of annealing temperatures in order to attain a more comprehensive analysis of the recrystallization and grain growth phenomena of Mg-RE alloys. Accordingly, the evolution of microstructure, texture and mechanical properties of the Mg-1.43Nd alloy were investigated after HPT processing at room temperature through 5 turns and isochronal annealing for 1 h at temperatures of 150, 250, 350 and 450 °C using electron backscatter diffraction (EBSD) and Vickers microhardness. The grain growth kinetics of the alloy were also investigated during the isochronal annealing.

Materials and Methods

An as-cast Mg-1.43Nd (wt.%) alloy was supplied from the Institut für Metallkunde und Metallphysik (IMM-RWTH) in Aachen, Germany. The alloy was produced by induction melting and casting under a protective gas atmosphere of Ar/CO2 followed by a heat treatment for 20 h at 420 °C.

The HPT processing was conducted on discs with diameters of 10 mm and thicknesses of 0.9 mm at room temperature through 5 turns using an imposed applied pressure of 6.0 GPa and with a rotational speed of 1 rpm under quasi-constrained conditions. [31] Processing through 5 turns was selected based on an earlier investigation demonstrating that these conditions produce homogeneous ultra-fine grains in Mg-RE alloys. [29] After the processing by HPT, annealing was carried out at temperatures of 150, 250, 350 or 450 °C for 1 h in a radiation furnace followed by quenching in water.

The microstructures of the processed and annealed samples were investigated near the centres of the discs using electron backscatter diffraction (EBSD) in the RD-SD plane after mechanical and ionic polishing using a Gatan PECS II system at a high voltage of 5 kV for 15 min, where RD and SD denote the rotational and shear directions, respectively. The observations were carried out using a scanning electron microscope FEG-SEM SUPRA 55 VP operating at 20 kV. The scanned areas were 50 × 50 μm 2 with a 50 nm step size for samples annealed at 150, 250 and 350 °C and 300 × 300 μm 2 with a 0.5 μm step size for sample annealed at 450 °C. The EBSD data acquisition and analysis were undertaken using the TSL Orientation Imaging Microscopy, OIM TM software. The grain size data were obtained using a grain tolerance angle of 5° and the minimum grain size was chosen as 5 pixels. All datum points with a confidence index (CI) lower than 0.05 were excluded from the analysis, where the CI quantifies the reliability of the indexed pattern.

Quantitative texture analysis was carried out by calculating the Orientation Distribution Function (ODF) using the MTex software. [32] The ODF was calculated using the harmonic method (L = 22) with each orientation modeled with a Gaussian function having a half-width of 5°.

The Vickers microhardness was measured at the centre of the discs using a SHIMADZU type HMV-2 tester. An average microhardness value was obtained using at least three indentations under a load of 100 g (Hv0.1) with a dwell time of 10 s.

Results

.Microstructural evolution

The microstructures in terms of the orientation imaging micrographs in the inverse pole figure (IPF) maps of the HPT-processed and the annealed Mg-1.43Nd alloy after 1 h at 150, 250, 350 and 450 °C are illustrated in Figure 1. It is important to note that the as-cast microstructure exhibited very coarse grains with an average grain size of ~600 μm and HPT processing through 5 turns produced very strong grain refinement with an average grain size of ~0.6 μm. The mean grain size remains stable in annealing at 250 °C but then increases slowly to ~4.2 μm after annealing at 350 °C and thereafter rapidly increases to ~34 μm after annealing at 450 °C as revealed also in Figure 1(e). Unfortunately, precise identifications of the chemical compositions of these particles was not easy using energy-dispersive X-ray spectroscopy (EDS) analysis because of their very small size of <0.3 μm. Nevertheless, the volume fraction of these second phase particles was calculated by quantitative metallography as ~1.2 ± 0.3% in the deformed samples. Furthermore, their fractions were stable during annealing up to 250 °C and then increased to ~2.9 ± 0.4% after annealing at 450 °C. Table 1 shows the EDS results summarizing the relative proportions of the Mg and Nd elements (in wt.%) in different positions (points 1-3) on the deformed sample in Figure 3(b). Thus, the Mg matrix contains ~1.38% of the Nd element which is close to the anticipated concentration in the Mg-1.43Nd alloy. The particle in point 2 is probably Mg41Nd5 and for point 3 it appears to be the Mg12Nd phase. The presence of Mg12Nd particles is a direct consequence of the slow solidification rate during casting. [33] Table 1. EDS analysis in weight percentage in different positions in the microstructure of Mg-1.43Nd alloy processed by HPT through 5 turns (shown in Figure 3b).

Element

Microhardness evolution

Discussion

The present investigation deals with the evolution of microstructure, texture and mechanical properties in an HPT-processed Mg-1.43Nd alloy during isochronal annealing. As shown in Figs 1 and3, the alloy exhibits a good thermal stability when annealing at 250 °C.

However, the evolution of microhardness as a function of annealing temperature shows an initial hardening after annealing at 150 °C and then a subsequent softening. In practice, the precipitation sequence is responsible for the age-hardening as well as for the stabilization of grain size. The occurrence of softening is an indication of recrystallization and grain growth as well as the coarsening of precipitate phases.

Precipitation during isochronal annealing

The Mg-Nd alloy undergoes precipitation during the annealing temperature range of 150-450 °C as used in the present study. A precipitation sequence for a binary supersaturated solid solution (SSS) Mg-Nd alloy was proposed as follows: [34] SSS → GP zones → β " -Mg3Nd → β ' -Mg7Nd → β1-Mg3Nd → β-Mg12Nd → βe-Mg41Nd5

The equilibrium βe-Mg41Nd5 phase was observed only after a very high temperature annealing at 500 °C. [35] The Guinier-Preston (GP) zones, β " -Mg3Nd and β'-Mg7Nd phases were reported as formed during the early stage of ageing at low temperatures (150-200 °C). [START_REF] Choudhuri | [END_REF] The GP zones and metastable precipitates of the β " -Mg3Nd and β ' -Mg7Nd phases are responsible for the peak hardening in the Mg-Nd alloy. [36] Nevertheless, those phases were observed only by transmission electron microscopy (TEM), high-angle annular dark field scanning transmission electron microscopy (HAADF-STEM) and atom probe tomography (APT)

techniques [START_REF] Choudhuri | [END_REF]34,36,37] because of their fine-scale with a size ranging between ~5 and ~15 nm . [36] These small sizes explain the inability to observe these phases in the SEM micrographs from the present investigation as in Figure 3. Furthermore, it is reasonable to anticipate that the sizes of the precipitate phases will be finer in the HPT-deformed sample because of the higher density of nucleation sites. [38] It is worth noting that the peak hardening was shifted to a low temperature (150 °C) in the present alloy compared to earlier reports for the Mg-Nd alloy (170 °C) and a similar Mg-RE alloy (250 °C). [START_REF] Choudhuri | [END_REF]36,39] This is attributed to the enhanced diffusivity of Mg and Nd due to the high densities of dislocations and vacancies introduced by the severe plastic deformation.

Moreover, it should be noted that the metastable phases such as β ' -Mg7Nd and β1-Mg3Nd have a preference for dislocation nucleation. [START_REF] Choudhuri | [END_REF]37,40] Basically, the presence of dislocations and the excess of vacancies accelerates significantly the kinetics of the precipitation process in the Mg-Nd alloy by acting as heterogeneous nucleation sites and increasing the diffusion rate of solutes. [START_REF] Choudhuri | [END_REF][39][40][41] Simultaneously, the metastable phases were found to also nucleate around pre-existing particles. [39] Hence, it is expected that the present alloy undergoes a significant amount of precipitation since it was severely deformed by HPT processing and the initial deformed sample contains a reasonable fraction of second phase particles (1.2 ± 0.3%).

Thus, the good resistance against a coarsening of grain size when annealing at 150 and 350 °C is caused by the precipitation of metastable phases and a preferable nucleation at dislocations which prevents boundary movement and annihilation.

Ageing above 250 °C causes a rapid disappearance of GP-zones and the β ' -Mg7Nd phase together with the formation of coarse precipitates of the stable β1-Mg3Nd and β -Mg12Nd phases. [36,42] Furthermore, HPT processing increases the volume fraction of these precipitate phases compared to the non-deformed sample. [START_REF] Bourezg | [END_REF]41] The average diameter of the β-Mg12Nd phase was reported to increase from ~120 nm to ~300 nm when ageing at 250 and 350 °C, respectively. [39] It is interesting to note that the excess strain-induced vacancies were found to play a significant role in coarsening of the precipitate phases. [42] Based on these observations, the granular precipitates dispersed throughout the microstructure in samples annealed at 350 and 450 °C can be identified as the β-Mg12Nd phase, as denoted by the arrows in Figure 3(h). Close inspection of these microstructures in Figure 3(h) and (j) shows that the diameters of these particles increase with increasing annealing temperature.

In addition, Figure 3(j) shows that after annealing at 450 °C there are some agglomerates of particles in the Mg matrix, shown in the square at lower right, caused probably by the high annealing temperature. Therefore, the disappearance of metastable phases and the coarsening of the stable phases during annealing at 350 °C and 450 °C leads to an overall softening and the strong increase of grain size observed in the present alloy in Figure 2. The thermal stability in annealing at 250C is generally larger as reported in Mgbased alloys after conventional or severe plastic deformation [41,[43][44][45][46][47][48][49] . Nevertheless, processing by HPT allows the development of a stable UFG structure with a grain size of ~0.6 μm in the present study or between ~2-5 μm in an Mg-Al-Ca alloy. [43]

Texture evolution during isochronal annealing

The deformation texture developed in the present Mg-1.43Nd alloy, and as reported earlier in binary Mg-Ce and Mg-Dy alloys where the basal texture was shifted towards SD [28,29] , was different from that reported for pure Mg where there is a typical basal fiber texture in similar deformation conditions. [50] Such a difference is attributed to the effect of the RE elements in decreasing the stacking fault energy and hence facilitating the activation of <a> prismatic slip and <c+a> pyramidal slip during HPT processing at room temperature. [1,29] The rearrangement of the <a> prismatic and <c+a> pyramidal slip into a cell structure or subgrains by cross-slip and climb during the processing operation gives rise to a progressive rotation of the sub-grains from the parent orientation.

It is apparent from Figure 4 that the deformation texture was more or less similar until annealing at 350 °C for 1 h where a split basal texture was produced with peaks along the SD.

This texture is similar to the TD-split texture often reported in hot-rolled Mg-RE alloys. [6,7] Again, annealing at 450 °C gives a different texture with a significant spread along the SD compared to the texture after annealing at 350 °C as in Figure 4(e).

The textures reported in this investigation are different from those reported for the recrystallization textures in Mg-based alloys. Usually, the deformation texture is retained after a recrystallization treatment in Mg-based alloys due the occurrence of dynamic recrystallization during deformation processing. [51][52][53] As demonstrated in Figure 1(a), the high distorted zones in the microstructure of the HPT-deformed sample reveal the general lack of dynamic recrystallization. Nevertheless, it was reported previously that identical deformation processing and conditions with a Mg-0.44Dy alloy, with processing by HPT at room temperature through 5 turns, led to a homogeneous microstructure with 50% of dynamically recrystallized grains. [29] This difference is attributed to the presence of second phase particles of the Mg12Nd and Mg24Nd5 phases in the Mg-1.43Nd alloy used in the present experiments where these particles restrict the occurrence of dynamic recrystallization.

Another factor accounting for the difference between the two Mg-based alloys may be the electronic configuration and chemical differences between the Nd and Dy elements which will affect the deformation and recrystallization mechanisms of both alloys. [START_REF]Rokhlin in Magnesium alloys containing rare earth metals: Structure and Properties[END_REF] The results

shown in Figs 2 and3 indicate that changes in the crystallographic texture during annealing are associated also with the grain size stability.

The recrystallized grains may be identified using the Grain Orientation Spread (GOS) criterion where GOS is defined as the average deviation between the orientation of each point in the grain and the average orientation of the grain. [54] The recrystallized grains were separated from the deformed grains based on the GOS value where recrystallized grains have GOS values less than 1-2°. Figure 6 demonstrates the EBSD maps of deformed and recrystallized grains for the HPT-processed Mg-1.43Nd alloy and after annealing at 250 and 350 °C, respectively. The (0002) basal pole figure is also represented in Figure 6, where it is divided in terms of the single orientation scatter and recalculated pole figure of the recrystallized and deformed grains. Thus, the volume fraction of recrystallized grains increases rapidly from 15% to 80 % with increasing annealing temperature. The texture of the recrystallized grains after annealing at 250 °C is significantly scattered and obviously deviates from the deformed grains. However, it is also observed that the deformed grains remaining after annealing at 350 °C have different orientations from the recrystallized grains.

From these analyses, it is concluded that the preferred growth of recrystallized grains with deformed orientations after annealing at 250 °C leads probably to the nucleation of metastable phases that delay recrystallization and prevent any growth related to a texture modification. In addition, the pinning effect of precipitation is suppressed during annealing at 350 °C which permits the development of new recrystallized grains with different textures by oriented nucleation events which are most probably governed by nucleation at shear bands. [55] Annealing at the relatively high temperature of 450 °C causes a more angular spread of basal poles along RD and SD as a consequence of the extensive increase in grain size due the additional thermal activation and more rapid growth rate. The presence of extension twins in the microstructure of the annealed sample, as in Figure 1(e), also adds to the change in the texture. Finally, the occurrence of twins introduces high misorientation angles between the grains and the twinning and this also significantly changes the final texture. [56]

Hall-Petch relationship

The Hall-Petch relationship for the isochronally annealed Mg-1.43Nd alloy was used in order to examine the contribution of any grain boundary strengthening mechanism in the evolution of microhardness. The microhardness data from Figure 5 were re-plotted against d --1/2 in Figure 7 following the relationship: [57] 𝐻 𝑣 = 𝐻 𝑣0 + 𝐾 𝐻 𝑑 -1/2

where Hv is the measured hardness, Hv0 is the friction hardness which represents the resistance of the crystal lattice to the movement of dislocation and KH is a locking parameter which indicates the contribution of the grain boundaries to the hardening. [58] These parameters may be calculated directly from a linear fit to the curve in Figure 7. Thus, the fitted constants from It is interesting to note that this value of Hv0 is close to the microhardness value of the as-cast Mg-1.43Nd alloy (44 ± 3 Hv). Similar values were also documented (Hv0 = 38, KH = 42) in an AZ31 alloy processed by ECAP [59] and it was reported that reinforcing AZ31 alloy with Al2O3 nano-particles increased both the values of the friction hardness and the locking parameter (Hv0 = 38.8, KH = 60.9) compared to the conventional AZ31alloy (Hv0 =36.7, KH = 51.7). [58] Accordingly, the similar values found in the present binary alloy compared to the industrial ternary AZ31 alloy is attributed to the grain boundary strengthening from the pinning mechanism introduced by the precipitate particles and by their role in restricting dislocation movement during deformation induced by the penetration of the microhardness indenter.

Grain growth kinetics

The grain growth rate, X, may be determined using the conventional Arrhenius equation: [60] 𝑋 = 𝐴𝑒𝑥𝑝 (-

𝑄 𝑅𝑇 ) (2)
where A is the pre-exponential constant, R is the gas constant, Q is the activation energy for grain growth and T is the absolute annealing temperature. The grain growth rate may be expressed as a function of the mean grain size so that: [60] 𝑋 =

𝑑 𝑛 -𝑑 0 𝑛 𝑡 ( 3
)
where d is the grain size after annealing, do is the grain size before annealing, t is the time for annealing where t = 1 h in these experiments and n is the grain growth exponent. Usually, the value of n is in the range from 2 to 4 for Mg alloys processed by SPD [61] and therefore the value was taken as equal to 2 in the present study. range is much lower than the anticipated value for boundary self-diffusion in pure Mg (~92 kJ mol -1 ) [START_REF] Frost | Deformation Mechanism Maps: The Plasticity and Creep of Metals and Ceramics[END_REF] and, moreover, it is expected that the activation energy of grain boundary diffusion will be higher in alloys than in pure metals. [58] In the high temperature range, the value of Q is very similar to the value for lattice self-diffusion in pure Mg (~135 kJ mol -1 ) [START_REF] Frost | Deformation Mechanism Maps: The Plasticity and Creep of Metals and Ceramics[END_REF] and the small difference may be caused by the presence of second phase particles. [START_REF] Lambri | [END_REF] Thus, it is concluded that lattice self-diffusion is the dominant mechanism for grain growth in the severely deformed Mg-1.43Nd alloy in the high temperature range. This is in good agreement with the assumption that lattice self-diffusion is the dominant process of grain growth when the alloy consists of large grains [64] which is correct for the present alloy after annealing at 350 and 450 °C. In fact, a strong basal texture was reported as also responsible for an increase in the activation energy for grain growth [65] and, as can be seen from Figure 4, the texture of the present alloy starts to change after annealing at 250 °C where a basal texture is well developed.

It appears that the activation energy in the low temperature range corresponds to the energy needed for the recovery and recrystallization of ultrafine grains, as demonstrated by the stabilization of the mean grain size between the deformed state and annealing at 250 °C as shown in Figure 2. Furthermore, the energy in the high temperature range is required for grain growth from a fine recrystallized microstructure to a coarse structure, as is evident from the increase in the mean grain size from ~0.65 μm after annealing at 250 °C to ~34 μm after annealing at 450 °C.

There are several investigations reporting similar low activation energies for grain growth (~19-50 kJ/mol) in Mg-based alloys. [58,59,[66][67][68] For example, an activation energy of 24.5 kJ/mol was obtained in the temperature range of 250-400 °C in AZ31 processed by ECAP at 200 °C though 4 passes using route Bc and then annealing for 30 min. [59] An exceptionally low activation energy of 19.2 kJ/mol was reported in the temperature range of 257-427 °C for AZ31 reinforced with Al2O3 nano-particles after ECAP processing at 227 °C for 4 passes using route Bc and then annealing for 30 min. [58] Additionally, the activation energy of a hot extruded Mg-Gd-Y-Zr alloy was reported as ~50 kJ/mol in the relatively high temperature range of 400-450 °C. [66] Detailed examination shows that the temperature range of the low activation energy in the present study is much lower (150-250 °C) than in these earlier reports (~200-400°C). [58,59,66] This difference may be associated with the different types of processing (HPT vs. ECAP) and the different deformation temperatures (room temperature vs. hot deformation) which will significantly affect the mean grain size and thus the grain growth process. It is well known that HPT processing has the capability of causing greater microstructural instability because of the inherent high dislocation density and/or the high concentration of vacancies compared to other SPD processing procedures such as ECAP. [69] It was demonstrated earlier that processing by HPT can lower the recrystallization temperature and lead to a decrease in the stored energy with increasing HPT turns in Mg-RE alloys. [START_REF] Bourezg | Proc. 26th International Conference on Metallurgy and Materials[END_REF] Two main hypotheses were proposed to provide an explanation of this very low activation energy. The first hypothesis was that grain growth under unrecrystallized conditions, caused by the presence of a high extrinsic dislocation density in the nonequilibrium grain boundaries, may lead to a higher atomic mobility compared to conventional equilibrium grain boundaries. [58,70] Vacancies are needed for the climb of dislocations and therefore these extra vacancies will assist the climb process and this will tend to enhance the recrystallization. This is in good agreement with the assumption that SPD processing may decrease the activation energy of grain boundary diffusion in UFG alloys. [71,72] The second hypothesis was that grain growth under recrystallized conditions may lead to a low activation energy due to a progressive decrease in the dislocation density through enhanced recovery with an increase in the annealing temperature. [59,67] A consideration of these alternatives suggests that the first hypothesis is more appropriate to explain the low activation energy in the present study since no recrystallized grains were observed within the deformed microstructure as in Figure 1(a). Another factor is the presence of precipitate particles during annealing. These particles act as obstacle to the movement of dislocations and delay their annihilation which will cause a reduction in the internal energy [73] and, in addition, a part of this stored energy will be consumed by the growth of the precipitates during annealing.

Conclusions

1. The thermal stability of an Mg-1.43Nd alloy was investigated using EBSD and Vickers microhardness measurements after processing by HPT trough 5 turns and subjecting to isochronal annealing at temperatures ranging from 150-450 °C.

2. The HPT-processed alloy exhibits good thermal stability at lower temperatures.

Processing through 5 turns produces significant grain refinement with an average grain size of ~0.6 μm. This mean grain size remains stable on annealing up to 250 °C but there is major grain growth to ~34 μm after annealing at 450 °C.

3. HPT processing produces a basal texture inclined by 60° from RD towards SD. This texture is retained after annealing at 150 and 250 °C but a TD-split texture develops starting from annealing at 350 °C.

Figure 1 :

 1 Figure 1: Inverse pole figure maps of Mg-1.43Nd alloy after (a) 5 HPT turns and subsequent

Figure 2

 2 Figure2shows the evolution of the mean grain size of the Mg-1.43Nd alloy processed

Figure 2 :

 2 Figure 2: Evolution of mean grain size of Mg-1.43Nd alloy processed by HPT through 5 turns as function of annealing temperature.

Figure 3

 3 Figure 3 presents the scanning electron microscopy (SEM) micrographs in

Figure 3 :

 3 Figure 3: Micrographs by scanning electron microscopy in backscattering mode of the Mg-

Figure 4

 4 Figure 4 shows the recalculated pole figure (0002) for the Mg-1.43Nd alloy after 5

Figure 4 :

 4 Figure 4: Recalculated basal (0002) pole figures of the Mg-1.43Nd alloy after (a) 5 HPT

Figure 5

 5 Figure5shows the evolution of the microhardness of the Mg-1.43Nd alloy processed

Figure 5 :

 5 Figure 5: Evolution of the microhardness of Mg-1.43Nd alloy processed by HPT through 5 turns as function of annealing temperature.

Figure 6 :

 6 Figure 6: Orientation imaging micrographs and texture data of deformed and recrystallized

Figure 7

 7 Figure 7 are: Hv0 = 38.6 and KH = 42.9 μm 1/2 .

Figure 7 :

 7 Figure 7: Microhardness plotted against mean grain size representing the Hall-Petch relationship for Mg-1.43Nd alloy processed by HPT through 5 turns and subsequent annealing.

Figure 8 :

 8 Figure 8: Evolution of ln(X) as a function of 1000/T (K -1 ) for Mg-1.43Nd alloy processed by HPT through 5 turns and subsequent annealing.

Figure 8

 8 Figure 8 presents the evolution of ln(X) as a function of 1000/T( K -1 ) for Mg-1.43Nd

4 . 5 .

 45 The microhardness shows an initial hardening after annealing at 150 °C due precipitation and then a subsequent softening caused by recrystallization and grain growth phenomena. HPT processing decreases the optimum ageing condition (150 °C) compared to the conventional treatment in Mg-based alloys. The kinetics of grain growth in this alloy follows two stages depending on the temperature annealing range. Activation energies of ~26 and ~147 kJ/mol were measured in the low temperature range (150-250 °C) and the high temperature range (250-450 °C), respectively. At low temperatures the grain growth is controlled by the higher mobility of the

°C are very similar to the deformed texture. However, the low texture intensity observed in the sample annealed at 150 °C is related to the small number of grains taken from the EBSD map in Figure1(b) compared to the other EBSD maps. By contrast, a typical basal texture develops starting from annealing at 350 °C with symmetrical splitting towards the SD.The texture intensity remains stable after annealing at 450 °C (~6.2 mrd corresponding to multiples of a random distribution) but the distribution of the (0002) poles are less symmetrical and more spread away from the SD.

HA gratefully acknowledges Dr. Talal Al-Samman, Institut für Metallkunde und Metallphysik (IMM-for supplying the Mg-RE alloys. YH and TGL were supported by the European Research Council under ERC Grant Agreement No.

267464-SPDMETALS.

Received: ((will be filled in by the editorial staff))

Revised: ((will be filled in by the editorial staff)) Published online: ((will be filled in by the editorial staff))


Andre Thiaville
email: andre.thiaville@universite-paris-saclay.fr

Jacques Miltat

Stanislas

André Thiaville

Stanislas Rohart

Magnetism and Topology

published or not. The documents may come

I. INTRODUCTION

The application of topology to describe real-space structures in condensed matter has developed over the years, in fact as topology itself was being rigorously constructed. The first structures treated by this approach were the defects, like dislocation lines in crystal lattices, or the lines observed in liquid crystals, at the beginning of the 20th century. The methodology was then extended to structures that contain no defect. We shall keep this historical order for the first half of this chapter, dealing with the static description of magnetic structures (also called magnetic textures). The second half will deal with the dynamics of the magnetic structures.

In this chapter, mathematical formalism will be kept to a minimum. The reader completely unaware of this field of mathematics should here and there refer to textbooks [START_REF] Kléman | Points, Lines and Walls[END_REF][START_REF] Hatcher | Algebraic Topology[END_REF] or review papers [START_REF] Braun | [END_REF] and, if a deeper mathematical view is desired, to the last chapter of this book.

Defects can appear only when there is an underlying order. In condensed matter physics there are many cases where phases with different types or degree of order appear in sequence, for example when temperature or pressure is changed. In the Landau theory of phase transitions, order is characterized by the existence of a certain physical quantity called order parameter, which can be a scalar, a complex number, a vector, a tensor, etc. This order parameter is also endowed with a certain number of degrees of freedom, defining a space V called the manifold of internal states. To be specific, let us enumerate the cases for magnetism. The order parameter is the magnetic moment. Depending on the type of magnetism (localized or itinerant), magnetic order (ferromagnetic, ferrimagnetic [Note that we do not consider in this chapter the antiferromagnets, for which the order parameter is different]), and also atomic structure (single crystal, polycrystal, amorphous), the distribution of the magnetic moment on the atomic scale will be very different. However, on the mesoscopic lengthscale where micromagnetics [START_REF] Brown | Micromagnetics[END_REF][START_REF] Hubert | Magnetic Domains[END_REF] reigns, these atomic subtleties are polished out, leaving only a magnetic moment that is a continuous function of space and time. This moment has a zero thermodynamical average above the Curie temperature, and a non-zero one below this temperature, leading to a spontaneous magnetization M s (T ) a sole function of temperature T . The degrees of freedom correspond to the orientation of this moment, and they are described by a unit vector m( r, t), a function of position r and time t. Our world being three-dimensional, three cases are considered.

• Heisenberg model: Magnetization has n = 3 components as m can take any orientation in 3D, the manifold of internal states is V = S 2 , the unit sphere in three dimensions which is described by two variables (for example the polar angle θ and the azimuthal angle φ), hence the exponent 2. Physically this should be the only case, in general. It is however sometimes interesting to consider the two other limiting cases.

• XY mode: This case corresponds to an easy plane, magnetization has n = 2 components, the out of plane anisotropy energy being (mathematically) infinite. Physically it means that other energy terms are much smaller. This can be a description of a soft magnetic thin film, or of a weak ferromagnet (the orthoferrites, for example) where the Dzyaloshinskii-Moriya interaction confines the ferromagnetic moment to the plane normal to its vector. The manifold of internal states is V = S 1 , the unit cercle in the (easy) plane, described by just one variable (the angle).

• Ising model: The last case is complementary to the preceding one, with an easy axis of (mathematically) infinite anisotropy, so that magnetization has just n = 1 component. The manifold of internal states is V = S 0 , this notation meaning just the two points +1 and -1.

The internal space V plays a capital role in the topological formalism.

II. TOPOLOGICAL DEFECTS IN MAGNETISM

In a medium with order, a defect is a point or a collection of points where the order parameter is not defined; one can have a point-like defect, or a defective line, or a defective surface (etc. for space dimensions d larger than 3). The defect is said to be topological when, whatever the continuous modifications exerted on the order parameter distribution over space, it cannot be removed. In short, this defect cannot be cured.

The defect dimensionality is called d = 0 for a point, d = 1 for a line and d = 2 for a surface. The dimension of the physical space being denoted d, one has d < d. Note that the defect, in this idealization of the order parameter as a continuous function, has no width, it is just a discontinuity of the order parameter. But how can the presence or not of a defect with no size be detected ? This can be done with topology. The method is inspired by the Burgers circuit used to detect the presence of dislocation lines in atomic arrangements [START_REF] Friedel | Dislocations[END_REF]. The generalization of this method is to consider a closed contour of dimension r in the physical space (i.e. the sphere S r , or any continuous deformation of it). To look for a defect of dimension d in a space of dimension d, the contour dimension r has to satisfy [START_REF] Toulouse | [END_REF]8]

d + r + 1 = d. (1)
The case of a dislocation line (d = 1) in the usual space with d = 3 corresponds to a closed loop r = 1. Similarly, to look for a point defect d = 0 in the 3D space (d = 3) requires a sphere-like surface hence r = 2 (Fig. 1). Every magnetization texture, by the orientation of the magnetization vector, gives rise to an image of the r-dimensional closed contour on the parameter space V. In mathematical terminology this is called a mapping. A continuous deformation of the magnetization texture leads to a continuous deformation of the image on the parameter space V. The branch of topology called homotopy precisely deals with that. Two images are said to be equivalent (belonging to the same homotopy class) if they can be continuously transformed one into the other. For every space V there exists an object called the r-th homotopy group, denoted π r (V), and each element of the group is a homotopy class. The group has an obvious 'zero' (it is called neutral) element, the class of the contours whose image is a point. The group is said to be trivial if it consists only of the neutral element. This means that all contours can be continuously contracted to a point. In that case, there can be no topological defect of dimension d = dr -1. Indeed, there will be no obstacle in shrinking in the physical space the r-dimensional contour, magnetization will be always defined on the contour, and will finally converge to the magnetization vector at the center of the contour when its size reaches zero. The topologically interesting case is the non-trivial one, when the homotopy group has more than one element. In that case, if a contour has an image on V which belongs to a non-neutral homotopy class, when shrinking this contour in physical space, at some moment the magnetization orientation has to be undefined: the defect will be met.

For the case of magnetism where V = S n-1 , with n the number of magnetization components (see Sec. I), the non-trivial homotopy groups with r < 3 are • π 0 (S 0 ) = Z 2 ≡ {0, 1}: for Ising spins, there exist topological defective surfaces in 3D, topological line defects in 2D, and topological point defects in 1D;

• π 1 (S 1 ) = Z ≡ {0, +1, -1, +2, -2, ...}: for XY spins, there exist topological linear defects in 3D space, and topological point defects in 2D;

• π 2 (S 2 ) = Z: for Heisenberg spins, there are topological defects in the form of points in 3D space. The well-known examples of the second case are the Kosterlitz-Thouless vortices and antivortices (Fig. 3), whereas in the third case it is the Bloch point (Fig. 2). Therefore, given that the XY and Ising models are only approximations, the only true topological defect in ferromagnetism is the Bloch point.

A. The Bloch point

The structure and the name were invented by Ernst Feldtkeller in Ref. [9], a pioneering paper for the application of topology to magnetism. It is the first topological magnetic structure studied; it has the defining property that around it every magnetization orientation appears once exactly. Some schematic structures of the magnetization around it are drawn in Fig. 2 A surprising property of the Bloch point is that it has a finite energy. In the frame of micromagnetics, even if the exchange energy density diverges like A/r 2 around the Bloch point [A is the micromagnetic exchange constant, proportional to the exchange energy J and inversely proportional to the lattice constant. [START_REF] Hubert | Magnetic Domains[END_REF]], the integral of this density is finite, amounting to 8πAR, R being the radius up to which the Bloch point profile holds. On the other hand, going back to the atomic scale, having a Bloch point amounts to forcing to zero the magnetization in a volume of the size of an atom. Thus, if J is the exchange energy per nearest-neighbor bond, and Z is the number of nearest neighbors, the cost of a Bloch point is about Z J. For a simple cubic lattice, the two formulae are close for R = a/2. This is the 'core' cost of a BP, consisting of exchange only.

The magnetostatic cost of a Bloch point was evaluated by W. Döring [10]. It was found that the hedgehog Bloch point (Fig. 2(a)), being a monopole, has the highest energy. Conversely, the 'combed' Bloch point (Fig. 2(b)) carries much less energy. There is also an anisotropy cost of a Bloch point. Altogether, the insertion energy of a Bloch point in a material with anisotropy (K) dominating magnetostatics is given by a micromagnetic radius R ≈ √ A/K [11]. In the framework of atomic micromagnetics, the exchange energy cost as a function of the position of the mathematical center of the Bloch point (in between the atoms) was studied by Reinhardt, who found the best interstitial positions for the Bloch point as well as its saddle point when moving from an atomic cell to another [12]. The friction linked to these saddle points was shown to be detectable in the motion of so-called Bloch point walls in nanowires [13].

B. The singular vortex

As is apparent from Fig. 3, a vortex topological point defect for the XY model (therefore in 2D) can be seen as a cut through the center of a Bloch point, the point topological defect for the Heisenberg model, so in 3D.

Regarding the vortex energy, the exchange energy density in the continuous micromagnetic formulation diverges again as A/r 2 , but now the radial integral also diverges logarithmically [14]. The divergence at small size is removed, in the theory papers, by introducing a finite lower bound for the integration over the radius. This indeed corresponds to the atomic micromagnetic calculation, which gives a finite result.

III. TOPOLOGICALLY STABLE STRUCTURES (TOPOLOGICAL SOLITONS) IN MAGNETISM

In the same pioneering paper by E. Feldtkeller [9], another topological effect relative to the magnetic structures was introduced. Consider a regular magnetic structure, i.e. without any defect, as well as another structure, also without defect. The question is whether or not it is possible to continuously transform one structure into the other (a property called accessibility). This is again a topological question. It is not at all academic, as E. Feldtkeller was considering the possible obstacles to the magnetization reversal in ferromagnetic rings, used as bits in the ferrite core memory.

When it is not possible to transform continuously one structure into another, the structures are said to be topologically stable or, equivalently, they are called topological solitons [15]. The idea behind the latter name is that, similarly to a soliton that keeps its shape while moving, a topological soliton keeps its topology while deforming (moving being one way of deforming).

In order to apply homotopy arguments to this situation, we need to realize mappings from r-dimensional spheres S r to the order parameter space V. A perfect case is therefore when the sample itself is a sphere, like a film grown on a sphere realizing S 2 , or a nanoring realizing S 1 . The first case is rare presently, it could be realized by core-shell magnetic particles with large diameter. The second case has been studied in detail [16], but the magnetic structures (domain walls) are the same as when the nanoring is not closed, discussed in Sec. III B 3.

A. Uniform boundary conditions

If the physical space is the infinite euclidian space R d , we cannot directly apply homotopy arguments as R d is not equivalent to S d , even if they have the same number of dimensions. But if, in addition, it is known that magnetization is uniform at infinity, then by a transformation similar to the stereographic projection of the sphere onto the plane, one can transform R d to S d .

In such situation we see that all magnetic structures realize mappings of S d to the order parameter space V. So they can be classified by the homotopy group π d (V). Let us be more explicit by specifying d, for V = S n-1 (except the case corresponding to π 0 (S 0 ) = Z 2 that is not interesting).

1. Two-dimensional physical space (d = 2)
This case is non trivial as π 2 (S 2 ) = Z. This means that the physical space is bi-dimensional (one can think of a film where nothing changes along the thickness, for example an ultrathin film with a thickness not much above the magnetic exchange length Λ = 2A/µ 0 M 2 s ). So topology tells us that for 3D moments (n -1 = 2) on a plane, which are uniform at infinity (i.e. the structures considered are of finite extent), there exists an infinity of topologically different structures, labelled by the signed integers (called the topological index). Two structures with different index are mutually inaccessible by continuous deformation. Among these structures, only those with index zero can be continuously transformed to the uniform state. The simplest structures are drawn in Fig. 4. The structure with index +1 is called a skyrmion, that with index -1 an antiskyrmion, and the others are higher order skyrmions.

The topological index is easy to calculate in this situation, as it is simply the number of times that the sphere S 2 is covered by the mapping. If the plane R 2 is described by (x, y) coordinates, we see from geometry that the differential surface covered on the sphere reads

dΩ = ∂ m ∂x × ∂ m ∂y • m dxdy. (2)
Indeed, m is the local normal to the sphere, both partial derivatives of m belong to the local tangent plane to the sphere, the modulus of their vector product is the sine of their angle, which is giving the surface of the associated parallelogram. Note that the covered surface can be positive or negative. Altogether, the topological index is the integral of dΩ, divided by 4π which is the surface of the sphere. It is notable that, here, the topological index can be obtained by the space integral of a 'topological density'.

In the case of cylindrical symmetry, namely m = (sin θ cos φ, sin θ sin φ, cos θ) with the spherical angles θ and φ relative to the magnetization at infinity being sole functions of the radius r and the in-plane angle ϕ, respectively, one obtains the useful relation

Ω = sin θ dθ dr dr dφ dϕ dϕ = 4πS p. (3)
The two quantities involved are p = [cos θ(0)cos θ(∞)]/2, the polarity of the core of the structure, and S = 2π 0 dφ dϕ dϕ/(2π), the winding number of the planar magnetization component. Note that, if the structure considered does not possess cylindrical symmetry, it can be continuously deformed so as to show it. This formula is therefore general.

One-dimensional physical space (d = 1)

This is another non trivial case, due to the fact that π 1 (S 1 ) = Z. The physical space is now one-dimensional (a nanowire), the moments are XY (n -1 = 1), and they are identical at both ends, so that the wire can be thought as a closed loop. A physical realization of this situation could for example be a nanowire where the moments are constrained to lie in the transverse plane, so that a magnetic structure is locally a helix. In this case, the topological index is simply the algebraic number of turns of the helix. Another realization of this situation could be an ultrathin film (normal to z) where the moments vary only along one direction x (e.g. a series of parallel domain walls). Then again we could have helices (moments in the (y, z) plane), or also cycloids (moments in the (x, z) plane), these planes being fixed by a strong Dzyaloshinskii-Moriya energy term. The topology is the same for both realizations. (b) non-topological structure with an in-plane winding number S = 0. This structure can be continuously destroyed by a magnetization rotation around the vertical in-plane axis, the angle of rotation being π at the center, π/2 at the wall and 0 at infinity. (c) a Néel skyrmion with positive chirality. Note that, by π rotation of the moments around the perpendicular axis, this structure can be continuously transformed into a Néel skyrmion with negative chirality, as well as into Bloch skyrmions with positive or negative chirality by ±π/2 rotation. (d) a higher order skyrmion with an in-plane winding number S = +2.

Three-dimensional physical space (d = 3)

There exists also π 3 (S 2 ) = Z. Whereas the previous homotopy groups were rather intuitive, this one is not. It was only reported in 1931 by the mathematician Heinz Hopf [17], and a mapping of S 3 to S 2 is called a Hopf fibration, the name fiber coming from the fact that each point on S 2 is the image of a line of points in S 3 , a fiber. By the same argument as before, S 3 is R 3 , just with identical moments at infinity, i.e. we discuss finite three-dimensional magnetic structures within a uniform background.

The structures with index 1 are now called hopfions; they have been observed in hybrid magnetic liquid crystals [18] and predicted in nanostructures of non-centrosymmetric magnets [19]. The topological index was described by Hopf, it is the linking number between any two fibers. It can also be calculated by integration of a density, albeit non-local [20], related to the presently much studied Berry curvature [21]. The grey torus figures the locus of the in-plane oriented moments (the 'wall' of the hopfion). The orientation of magnetization on this surface is depicted by arrows, the loops with the same color drawing the fibers of the structure, the loci of the points with the same magnetization. These loops are all linked once, this being the topological index of the structure. (b) cut view through a vertical plane, with the cut of the torus superposed, allowing to see the magnetization inside the torus, which is akin to a reverse domain.

B. Non-uniform boundary conditions

Non-uniform boundary conditions are frequent. The obvious case is the domain wall, where on both sides magnetization goes to different limits at infinity. Finite samples (like patterned microstructures) are another case, the boundary conditions being imposed by the energetically preferred magnetization orientations at the edges. Two emblematic cases of this are the disk, and the nanostrip.

When additional constrains on the order parameter exist at the edge, another mathematical object exists, called the relative homotopy group. However, as the emphasis of this chapter is physics rather than mathematics, we shall directly see the consequences of the boundary conditions, case by case.

The domain walls

If it is enforced that magnetization goes to m 1 at x → -∞ and to a different m 2 at x → +∞, then a domain wall has to exist. In topological terms, it means that all paths in physical space connecting x = -∞ to x = +∞ have to go through m 1 and m 2 , so that they cannot be contracted to a point: a domain wall has indeed to exist. But we learn nothing by this tautology.

The more interesting question is whether all walls satisfying this condition can be continuously transformed one into the other. This depends on the nature of the first homotopy group of the order parameter space π 1 (V): if it is trivial then all these walls are topologically equivalent, but if it is not then topologically different walls exist. The latter case occurs for the XY model only (Fig. 6).

The vortex in a soft magnetic film

In a soft magnetic film, the demagnetizing energy dominates the anisotropy term, so that magnetization is confined to the plane of the film. A perfect confinement would mean the XY model, for which we have seen that topological defects (the vortices, antivortices, etc.) exist for d = 2, the present case. As confinement is not perfect in usual samples (finite magnetization, non-zero exchange), these topological defects are regularized by having the core magnetization 'escape' in the third dimension, over a size proportional to the micromagnetic exchange length Λ = 2A/(µ 0 M 2 s ) (see Ref. [START_REF] Hubert | Magnetic Domains[END_REF]22] for an analytic calculation of the vortex core profile in the limit of vanishing sample thickness). The question then is: are these structures topologically stable ?

The answer is Yes, if the magnetization is assumed to stay perfectly in the plane at infinity. In such a case, similarly to the plane-to-sphere transformation of Sec. III, the situation is the same as a finite disk with magnetization assumed to belong to the plane at the disk edge.

Let us look at the edge structure first. It realizes a mapping of S 1 (the edge) to S 1 (the equator on the unit sphere, the locus of the in-plane magnetic moments). So this structure is classified by π 1 (S 1 ) = Z: it is topologically stable regarding the relevant topological index, which is the winding number of the in-plane magnetization. Thus, from the winding number of the (in-plane) edge magnetic structure, we know if we have a vortex (S = +1), an antivortex (S = -1), nothing (S = 0), or more complex vortices or antivortices (|S | > 1). Physically, as the edge constrain arises from magnetostatics, the avoidance of surface magnetic charges dictates that m be tangent to the edge. So, for a disk-shape sample, the stabilized winding number is 1.

As we have 3D spins inside the disk, the core of the vortex is not singular, but regular with a purely out-of-plane moment m z ≡ p = ±1, p being called the polarity of the vortex. From the topology of the edge magnetization, we already know that the vortex cannot be continuously erased into a uniform structure. But could we continuously go from p = +1 to p = -1 ? The answer is No. Indeed, if this were the case, one would at some moment have the top surface of the film with p = 1, and the bottom one with p = -1. Then mapping the full surface of the disk (the top surface, glued to the bottom surface at the disk edge), which is like S 2 , to the parameter space V = S 2 would give a full coverage, meaning that the disk contains a Bloch point in its interior: continuity would be broken.

We thus see that additional constrains at the sample edge let other topologically stable structures appear. As they cover half of the sphere, the Heisenberg vortex or antivortex are also called merons. Note that this half-integer coverage index of S 2 is not quantized: if a perpendicular field is applied to a disk with a vortex, the image of the edge on S 2 leaves the equator, increasing or decreasing the coverage depending on the field direction with respect to the core polarity (see Ref. [23] for an experimental demonstration). The green points at the edge correspond to topological defects of π 0 (S 0 ) (the edge when tangential magnetization is assumed), which become merons of π 1 (S 1 ) (the edge when planar magnetization is assumed, with values fixed at infinity). The orange points correspond to topological defects of π 1 (S 1 ) (in-plane moments assumed in the sample), which become merons of π 2 (S 2 ) (Heisenberg spins, with in-plane spins far from the core), namely a vortex (a) and an anti-vortex (c).

The soft magnetic nanostrips

This case illustrates nicely how the topological objects transform when one progressively lifts the constrains on the magnetization orientation. The topological description provides a general scheme to understand the magnetic structures in the soft nanostrips [24,25]. It has led to insightful 'mechanical' models of the domain walls in these samples [26].

The nanostrip geometry is similar to the disk case, but now the edge is not closed. At an edge, combining the magnetostatic conditions of planar magnetization for the film and tangent magnetization at the edge, one obtains that the lowest energy magnetizations are in S 0 . From the fact that π 0 (S 0 ) = Z 2 , there exists one associated type of topological point defect, which is just a wall in the edge magnetization.

These topological defects are regularized by lifting the tangency constrain, i.e. by considering, at the edge, V = S 1 instead of S 0 . Assuming that the magnetization in the two domains separated by the wall is fixed (see Sec. III B 1), the topological defects of S 0 become topological solitons, in fact merons of π 1 (S 1 ).

Going one step further, by comparing the topological characters of the two edges (which is easy to do e.g. when, on the left and right of the nanostrip area considered, it is known that magnetization is uniform), one can know if there exists a topological defect of π 1 (S 1 ) inside. The typical cases are the vortex wall (with a vortex inside), the transverse wall (no vortex inside), the transverse wall in the process of reverting (with an antivortex inside), see Fig. 7.

The vertical Bloch line

The vertical Bloch line (VBL) is the second magnetic topological object studied, and it is a topological soliton. Lines in the domain walls of garnet films were intensively studied in the 70's, in the wake of research about the bubble memories. These samples are single crystals of (at that time) few-micron thickness, epitaxially grown on a single crystal non-magnetic garnet substrate. As the lattice parameters can be precisely matched, the crystalline defect density is extremely low (1 cm between dislocations, typically). The samples are, however, not perfectly uniform magnetically as the large unit cell of garnets (a cube of 1.3 nm edge) accommodates many substituants, whose concentration can vary from place to place. The magnetic garnets are also ferrimagnetic (in fact, ferrimagnetism was discovered in the garnets), with a tunable low magnetization density. They have a growth-induced perpendicular anisotropy originating from the atomic ordering of the substituants during the growth. The magnetic structure consists of parallel domain walls with a globally disordered structure, with as limiting cases the straight parallel domains array and the bubble lattice.

As the sample thickness (few micrometers) is much larger than the domain wall width (a few tenths of a micrometer), in the The tilt of the domain wall magnetization at the surfaces, from a Bloch to a chiral Néel orientation, due to the stray field of the domains and that occurs in films thick enough compared to the exchange length [START_REF] Malozemoff | Magnetic Domain Walls in Bubble Materials[END_REF], is not drawn. The lines can be magnetized in both directions. Only the vertical Bloch line is stable in statics.

(111) epitaxy case with small in-plane anisotropy one gets Bloch walls. In the vicinity of the surfaces they turn to the Néel orientation due to the stray field of the domains (twisted Néel-Bloch-Néel structure, [START_REF] Malozemoff | Magnetic Domain Walls in Bubble Materials[END_REF]).

In turn, these walls can contain the two orientations of the Bloch wall, being separated by a line (a 'wall in a wall'). This line has its magnetic moment oriented along a Néel direction, hence the proposition by E. Feldtkeller to call them Néel lines. But lines in domain walls had been observed earlier in soft thin films supporting Néel walls, and called Bloch lines, so that this generic name was kept for all types of lines. As shown in Fig. 8, in a garnet film horizontal Bloch lines (HBL) and vertical Bloch lines (VBL) exist. The HBL is an intermediate state during the dynamics, whereas the VBL is a stable structure. Moreover, in the 80's a Bloch line memory was developed, taking advantage of the properties, including the topological ones, of the VBLs [START_REF] Konishi | [END_REF]. So we concentrate on VBLs here.

The topological stability of a VBL within a domain wall is analyzed similarly to the case of the vortex. Magnetization is fixed in the domains far from the wall, as well as in the wall far from the VBL, so that in fact far away (in 2D) from the VBL magnetization is fixed. In the absence of fields transverse to the easy axis, these fixed magnetizations belong to a great circle on S 2 , so that the VBL is also a meron of π 2 (S 2 ).

C. Role of topological defects for topological solitons

The preceding sections have shown relations between topological defects and topological solitons. These can be seen by comparing the situations described by the same homotopy group. Take π 2 (S 2 ) as a start. It describes topological point defects in a 3D medium, as well as topological solitons for a 2D medium, both cases being for Heisenberg spins. The relation is simple: in the transformation from a topological soliton to a different one, the breaking of continuity is realized by the presence of a topological defect corresponding to the difference of their topological indices. In this comparison, one goes from a 2D medium to a 3D one either by time-stacking the 2D media, or by considering the different slices of its thickness. For example, annihilating a skyrmion requires a Bloch point. Two examples of this are the merging of two skyrmions [29], and the so-called 'bobber' structure where a skyrmion exists in the vicinity of one surface of the film only, a Bloch point terminating the structure [30].

Similarly, π 1 (S 1 ) describes topological point defects (vortex, antivortex, etc.) in a 2D medium and for XY spins, as well as topologically stable knots of these spins along a line. Going from a line with one knot to a line without knot requires, similarly, a vortex or an antivortex (see Fig. 9).

IV. EFFECT OF TOPOLOGY ON MAGNETIZATION DYNAMICS

Up to now we have discussed the statics of magnetic structures. As we are going to show in this section, the dynamics of magnetic structures is also affected by topology, in some cases.

In general, magnetization dynamics in continuous media is governed by the Landau-Lifshitz-Gilbert (LLG) equation [START_REF] Hubert | Magnetic Domains[END_REF]31,32], supplemented by additional torque terms in presence of spin-polarized currents [33,34]. In this equation, the energy density E gives rise to an effective magnetic field H eff defined by a variational equation [31]. [Following the use of the magnetic CGS system in the older literature, one continues to use the name of an effective field, hence the µ 0 factors appearing at (too) many places. It would be simpler to employ the effective induction B eff = µ 0 H eff , but for homogeneity with the other chapters of this (a) (b) FIG. 9. Soft magnetic strip where on the lower edge magnetization is uniform, whereas on the upper edge it makes a 2π turn, one way (a) or the other (b). For XY moments, the 4 sides of the structures drawn constitute a closed path, on which the moments turn by -2π (a) or +2π (b). Hence there must be a topological defect of π 1 (S 1 ) inside, the antivortex (a) or the vortex(b). For Heisenberg moments, if we assume that moments far away are in the plane, the structures are topological solitons (merons) of π 2 (S 2 ). book we will use the effective field.]

dE ≡ dE = -µ 0 M s H eff • d m. ( 4
)
The LLG equation then reads

∂ m ∂t = γ 0 H eff × m + α m × ∂ m ∂t + γ 0 µ 0 M s T . (5)
In this equation γ 0 = µ 0 γ is the gyromagnetic factor as adapted to fields, with the negative sign of the electron charge absorbed in the reversal of the vector product, α is the dimensionless damping coefficient as introduced by Gilbert [32,[START_REF] Gilbert | Formulation, foundations and applications of the phenomenological theory of ferromagnetism[END_REF], and T represents the current-induced torques (spin transfer torques [36], spin-orbit torques [37]), as generally they do not derive from an energy density. As the effective field has contributions from the energy densities of exchange, magnetic anisotropy, applied field, demagnetizing field as well as Dzyaloshinskii-Moriya anti-symmetric exchange etc., this equation is quite complex and in most cases not solvable analytically. Presently many codes, open-source or proprietary, exist for the efficient numerical solution of this integro-differential equation. But it is always valuable to have approximate analytical solutions in order to understand the physics. For magnetic textures, the Thiele equation [38] is such a tool.

A. The Thiele equation

Suppose that (it is possible to assume that) a structure moves as a solid object, without deforming. It means that there exists a function m 0 ( r) such that

m r, t = m 0 r -R(t) , (6)
where R(t) is the position vector of the structure. This allows expressing the time derivative of magnetization

∂ m ∂t = -( V • ∇) m 0 , (7)
where V = d R/dt is the structure's velocity vector. We then 'solve' the LLG equation in terms of the effective field, and get

H eff = 1 γ 0 m × ∂ m ∂t + α γ 0 ∂ m ∂t - 1 µ 0 M s m × T + λ m, (8)
where the last term appears because the component of the effective field along the magnetization cannot be determined by this way.

This effective field is used to compute the force acting on the structure, due to the energy density considered. It is given by mechanics expression F = -dE/d R. We can evaluate this derivative using the definition of the effective field

F i = - ∂E ∂R i = µ 0 M s H eff • ∂ m ∂R i = -µ 0 M s H eff • ∂ m 0 ∂r i . ( 9
)
The next step is to put the expression Eq. 8 of the effective field into this expression of the force [Note that the unknown function λ disappears at this moment], replacing the time derivatives by space derivatives according to the rigid motion assumption Eq. 7. This gives, using the convention of summation on repeated indices and abbreviating ∂ m/∂r i to ∂ i m,

F i = µ 0 M s γ 0 m 0 × V j ∂ j m 0 • ∂ i m 0 + α µ 0 M s γ 0 V j ∂ j m 0 • ∂ i m 0 + m 0 × T • ∂ i m 0 . ( 10
)
This can be rewritten, term by term, into the Thiele balance of forces equation [38]

F + F gyro + F dissip + F torques = 0. ( 11
)
The gyrotropic force F gyro and the dissipation force F dissip are proportional to the velocity. The dissipation force can be written

F dissip,i = -αD i j V j ,
where the elements D i j of the dissipation matrix D appear. The gyrotropic force can be rewritten F gyro = G × V where the components of the gyrovector G are

G k = - µ 0 M s γ 0 i jk 2 ∂ i m 0 × ∂ j m 0 • m 0 d 3 r, ( 12
)
where i jk is the Levi-Civita totally antisymmetric tensor. Comparison with the expression of the surface covered on the 2dimensional sphere of Sec. III A 1 allows to see that, for a texture only varying in two dimensions (in a film of thickness h), the gyrovector has only a perpendicular component, that is directly proportional to the topological index of π 2 (S 2 ) of the structure:

G z = - µ 0 M s γ 0 4πhN π 2 (S 2 ) . (13)
This relation, together with the Thiele balance of forces equation, constitute the basis of the link between the topology of magnetic structures and their dynamics. Note that it singles out the π 2 (S 2 ) topology. For example, for a pure XY model where m belongs to a plane, the gyrovector is automatically zero: Heisenberg spins are required.

B. Applications of the Thiele equation

The Thiele equation applies to a great variety of situations, much beyond its original scope of describing the steady-state motion of magnetic bubbles in bubble garnet films, as a function of the structure of their walls.

Steady-state motion of domain walls

Consider a one-dimensional domain wall, i.e. magnetization is a sole function of the x coordinate. The gyrovector is zero (this can be seen by analysis, from Eq. 13, or from the fact that a curve covers no surface on the unit sphere). Driving the domain wall by an easy axis field H, the steady state dynamics is simply given by F + F dissip = 0. If the wall is a 180 • wall, the force per unit domain wall surface under an easy axis field is along the wall normal (x), and reads F x = 2µ 0 M s H. The dissipation matrix has only an xx element. The balance of forces thus reads 2µ 0 M s H = αD xx V. We therefore recover the well-known steady-state dynamics equation V = (γ 0 ∆ T /α)H, in which instead of the Bloch wall width parameter ∆ appears the so-called Thiele domain wall width ∆ T given by [39] 2

∆ T = d m dx 2 dx. ( 14
)
This relation is valid whatever the profile of the domain wall. It explains why, remarkably, vortex walls in nanowires move more slowly than transverse walls under an easy axis field [40,41], even if their lateral extent is larger. Indeed, the large magnetization gradient around the vortex core leads to a smaller ∆ T .

Dynamical deflection of topological structures

When discussing regular structures, we have seen the π 2 (S 2 ) topological solitons of integer topological index for uniform boundary conditions, and the solitons with non-integer index in the case of non-uniform boundary conditions (in simple terms, the skyrmion on the one hand, the vortex and the VBL on the other hand). Consequently, these structures have a non-zero gyrovector. Hence, when moving at a velocity V, the balance of forces includes the gyrotropic force F gyro = G × V, by construction transverse to the velocity. As in the two-dimensional case the gyrovector is along the film normal, the velocity and the gyrotropic force are in the film plane, and orthogonal.

This phenomenon has been known since the invention of the Thiele equation, first in the field of bubble garnet films [38,39], and then in the field of vortex dynamics [42], under the name of gyrotropic force (to which a gyrotropic deflection is associated). In the context of skyrmions, the effect has been called 'skyrmion Hall effect'. The derivation of the Thiele equation shows that the effect is intrinsic to the magnetization dynamics, independent of the stimulus used to drive the texture: it exists in insulators as well as in metals, or when the structure is driven by field or by electrical current. Thus, the description of this effect as due to Newton's action-reaction law in the case of current-driven motion, because of the electron deflection by the topological Hall effect [43], is misleading.

In the case of bubbles, their skew propagation with respect to an applied field gradient has been used to identify the state of the bubble, i.e. the number and type of VBLs present in the bubble's domain wall [START_REF] Malozemoff | Magnetic Domain Walls in Bubble Materials[END_REF]. The reversal of the gyrotropic force on the VBL as its core magnetization was flipped has been observed by real time magneto-optical imaging [44]. Similarly, for vortices the sign of the gyrotropic force has been imaged in real time, as well as its reversal upon flipping the vortex core polarity [45].

Topological Brownian motion

The Thiele equation can be used to describe the Brownian motion, within the Langevin model where random forces, with an autocorrelation fixed by the fluctuation-dissipation theorem, are used to represent thermal agitation [46]. In the case of magnetism, white-noise random magnetic fields are introduced, which by the integral expression Eq. 10 lead to random forces on the structure. Solving the Thiele equation for the velocity vector, and performing the statistical average results in a diffusion constant D (leading to average quadratic displacements < X 2 (t) >=< Y 2 (t) >= 2Dt after a time t) given by

D = k B T αD G 2 + (αD) 2 , (15)
where D = D xx = D yy for a revolution-symmetric structure is the dissipation matrix diagonal element, and G the gyrovector z component [47]. The relation agrees with the Einstein relation D = k B T µ with µ = V/F the viscous mobility. This expression shows that, at low damping, the diffusion of topological structures (G 0) is completely different from the diffusion of non-topological structures (G = 0). Recent experiments on skyrmions [48] indicate however that, with the present samples quality, this intrinsic diffusion regime is masked by much slower skyrmions trapping and escape processes.

V. TOPOLOGY VERSUS ENERGETIC STABILITY

By definition, a topological soliton cannot be erased (a special case of transformation to another topologically different structure) by a continuous deformation of the magnetization. This is sometimes called 'topological protection'. But does it mean that this structure cannot be erased at all ? The answer is No as we have seen: erasing this soliton requires breaking the continuity by creation of a topological defect (either by having a topological defect enter from one edge of the sample, or by creating a pair of topological defects with opposite topological indices within the sample). We have moreover seen that the Bloch point, the only topological defect of Heisenberg spins, living in 3D space, has a finite energy even in (almost everywhere, mathematically speaking) continuous micromagnetics.

So the physical answer is that, in order to erase a topological soliton, a finite energy barrier has to be overcome. And the only physical question left is how large this barrier is: by the Arrhenius law this tells us if the structure is stable on the relevant timescale (the experiment interaction time, the lifetime of a storage device, our life, the lifetime of Earth, etc.).

We describe below some cases where this erasure process has been observed, and the associated barrier evaluated.

A. The collapse of magnetic bubbles

In the magnetic bubble memory [START_REF] Malozemoff | Magnetic Domain Walls in Bubble Materials[END_REF][START_REF] Bobeck | Proceedings of the International Conference on Ferrites 1970[END_REF][START_REF] Eschenfelder | Magnetic bubble technology[END_REF], the information 1/0 is coded by the presence/absence of a cylindrical magnetic domain, called 'bubble'. To write and rewrite information, bubbles have therefore to be created and destroyed, at higher than MHz rates. To make things worse, the bubbles used are of the topological type (S = 1 winding number), because these propagate faster as no VBLs slow down their motion (think of the Thiele domain wall width, see Eq. 14).

Experimentally, one characteristic of a bubble material is the so-called collapse field, above which no bubbles exist (except those pinned at structural defects, which are very rare and optically visible by strain-induced birefringence). Therefore, the 'topological protection' has never been mentioned in this context. Moreover, the collapse field can be computed from the energy versus radius profiles, as a function of the bias field, and the value of this field is used to extract the parameters of the sample. Indeed, depending on the bias field, the bubble is stable or metastable with respect to the uniform state, and at the collapse field the local minimum in the profile disappears, at a non-zero radius called the collapse radius. The shape of the energy profile is given by the energies of applied field, demagnetizing field, and domain wall (for the materials used for memories, the bubble radius is much larger than the domain wall width, so that the domain wall can be treated like a mathematical surface, with a surfacic energy). The excess energy of a bubble at collapse being typically 0.1µ 0 M 2 s h 3 [51], one sees that as soon as h > 3Λ this energy is larger than the insertion cost of a Bloch point. The bubble collapse is therefore explained by these evaluations.

Experiments revealed additionally that bubbles with significantly larger collapse fields, called hard bubbles [START_REF] Malozemoff | Magnetic Domain Walls in Bubble Materials[END_REF]52], exist. This was linked to the presence of many VBLs in the wall of the bubble, all with the same rotation sense so that such bubbles are skyrmions with a large topological index (values up to 100 were observed [START_REF] Malozemoff | Magnetic Domain Walls in Bubble Materials[END_REF]53]). The larger collapse field results from the progressive annihilation of pairs of VBLs under compression as the bubble shrinks under bias field. A winding VBL pair having a topological index of 1, it also involves a Bloch point during its annihilation.

Note that this observed stability of the VBL pairs has led to the concept of a Bloch line memory [54], in which the writing process of a VBL pair, necessarily discontinuous, was appropriately called Bloch point writing [55].

B. The vortex core reversal

To reverse the core of a vortex in a soft thin film, a Bloch point is required as we have seen. The vortex core reversal in such samples has been the subject of several studies.

Statically, the core polarity in NiFe disks has been observed by magnetic force microscopy, various fields being applied ex situ so as to determine a vortex core switching field [56]. The micromagnetic modelling of this experiment [11] has shown that the Bloch point injection has to be made easier by defects; this was checked by using, instead of NiFe, an amorphous CoFeB layer in which structural defects are anticipated to be present.

The study of the dynamical switching of the vortex core polarity has revealed a quite interesting behavior. The first experiment [45] used an a.c. in-plane magnetic field to excite the resonant vortex motion in a patterned microstructure, with time-resolved transmission X-ray microscopy in stroboscopic mode for observation. As directly expected from the gyrotropic force, a rotation of the vortex core is obtained, even if the driving field is uniaxial, the rotation sense being given by the sign of the gyrovector which is, as the winding number is S = +1, the same as the polarity of the core. It was also observed that, upon submitting the sample to a sufficiently intense a.c. field pulse, the vortex core polarity could switch. The surprise was that, compared to the static fields of a few 100 mT required to switch the vortex core, much smaller a.c. fields (about 1 mT) were required. The second experiment [57] used an a.c. spin polarized current driving the vortex by spin-transfer torque, to obtain the same polarity switching. The micromagnetic analysis showed that a moving vortex has a distorted profile, with a dip of magnetization perpendicular component appearing next to the core [58,59]. This dip separates into two parts when it reaches m z = -p, one part annihilating with the m z = +p vortex core in a discontinuous process, leaving the other part as a m z = -p vortex core.

C. The skyrmion collapse

The annihilation of a skyrmion towards a uniform magnetic structure is equivalent to collapsing an S = 1 bubble. The energetics of a skyrmion is however different: whereas the bubble is stabilized by the demagnetizing field, this term is suppressed as the sample thickness goes down to one atomic monolayer. Thus, at intermediate thicknesses, both small radius skyrmions and larger radius bubbles can be stabilized [60,61], with a continuous evolution from one structure to the other as the micromagnetic parameters are varied. This is in full agreement with topology, as both structures are equivalent in this respect. Needless to say, experiments have shown that skyrmions can be created, most directly by localized current [62,63], and also may annihilate spontaneously [64].

In the limiting case with no long-range demagnetizing field and no applied field [65], the skyrmion is always metastable, the energy profile being impacted by a curvature term which was negligible for bubbles. Similarly to bubbles, the stability of a skyrmion requires the calculation of the energy barrier to overcome. Moreover, the prefactor τ 0 of the Arrhenius law for the skyrmion lifetime τ (τ = τ 0 exp ∆E/(k B T )) needs also to be calculated. Indeed, it has been shown that it is not a constant, but varies rapidly with applied field [66].

Regarding the energy barrier, the situation is more complex than for bubbles as the maximum energy point is more difficult to find. A first approach used atomic-scale micromagnetics for a cobalt monolayer [67], together with the chain of configurations minimization method known as nudged elastic band technique. [Note that the initial Ref. [67] contains a methodological error [69], which was corrected afterwards [68].] As shown in Fig. 11, the top of the barrier is attained when the skyrmion reaches an atomic size, hence the necessity of an atomic micromagnetic model. The precise value of the energy barrier is found to depend on all energy terms, even if it duly increases when the Dzyaloshinskii-Moriya exchange term increases, as expected. In fact, based on the Belavin-Polyakov theorem [70] stating that the exchange energy of a structure has a minimum value fixed by the π 2 (S 2 ) topological index, a fair estimation of the energy maximum is this exchange energy limit [60]. The difficulty of the energy barrier evaluation is then transferred to the calculation of the skyrmion energy relative to that of a uniform background. This is exemplified in Fig. 14 In the numerical calculation, it is notable that the configuration with the maximum energy is not realized when the skyrmion number is just 1/2, but close to that value. Regarding this 1/2 value, one ought to realize that the definition of the topological index for atomic micromagnetics, which is fundamentally discrete, is problematic. The approach chosen [71] is to keep the geometrical signification of this index, the surface covered on the sphere being evaluated as the sum of the area of all spherical triangles involved (see Fig. 11). This area of a spherical triangle is always defined except when the three summits belong to a plane going through the sphere center. At this mathematical point, the topological index is both 0 and 1 depending on how the area of this triangle is counted, a situation denoted by a 1 2 topological index. Going back to topological arguments, a Bloch point cannot exists in such a sample as it is one atomic layer thick, hence really two-dimensional. Nevertheless, the change from one configuration with topological index 1 to the next with index 0 can be seen as the passage of a Bloch point through the atomic monolayer.

The energy barriers obtained by this chain minimization technique were compared to dynamic atomic scale simulations including temperature-induced fluctuations, in a narrow field window where the skyrmion annihilation is visible. In such conditions, the two determinations of the energy barrier were in agreement.

Note finally that, as the skyrmion energy at collapse depends sensitively on exchange, more complex calculations incorporating many-neighbors exchange interactions have predicted non negligible changes of the this barrier [72]. This might be a path to designing materials with atomic and electronic structures more favorable to skyrmion stability.

VI. CONCLUSION AND PERSPECTIVES

This chapter has tried to expose all the directions along which topology contributes to our understanding of the magnetic structures, be it in statics or in dynamics. As topology rests on the assumption of continuity and matter is fundamentally discontinuous, it might be tempting to dismiss all topology arguments. However, as the main energy term in ferromagnets is the exchange energy, which promotes continuity, applying topology to magnetic structures is justified. In this usage, the guide is to consider mathematics as a tool, and keep a physical approach.

Along this chapter, we have seen that the machinery of topology can be applied with profit in the description of the statics of magnetic textures, and of their transformations, with all the possible homotopy groups for spins with various degrees of freedom being involved. For the intrinsic effect of topology on magnetization dynamics, however, only the full-fledged spin with three components is relevant. This singles out the π 2 (S 2 ) homotopy group, and consequently the magnetic skyrmion, as well as the associated merons, the magnetic vortex and the vertical Bloch line.

As new types of samples are invented, the arguments exposed here can be extended as needed. This applies also to more complex magnetic orders.

FIG. 1 .

 1 FIG. 1. The contour dimensionality r is adapted to the defect dimensionality d according to Eq. 1, as illustrated here for (a) d = 2, (b) d = 1 and (c) d = 0. The defects are drawn in red, and the associated contours in gray.

FIG. 2 .

 2 FIG.2. The orientations of the magnetization on a sphere surrounding the Bloch point are schematically drawn. (a) so-called hedgehog Bloch point, with radial magnetization everywhere, covering the unit sphere +1 time. The variant with opposite magnetization everywhere covers the unit sphere -1 time, so is topologically different. (b) so-called combed Bloch point, with a lower magnetostatic energy[10], which is found in most structures (see Sec. V). This structure can be derived from (a) by a π/2 rotation of the magnetizations around the vertical axis, hence is topologically equivalent. (c) structure derived from (a) by a π rotation around the vertical axis, so again topologically equivalent to it. (d) structure which has a S = -1 winding number on the equator, with a p = -1 axial magnetization on the poles, hence also covering +1 time the sphere. It is also the same as (c), rotating the whole sample by π/2 around a horizontal axis.

FIG. 3 .

 3 FIG.3. The orientations of the magnetization around the vortex core, located at the origin, are figured by arrows. (a) Vortex, here in the radial variant. This structure can be continuously converted into the circulating variants (like the velocity field of a vortex in fluid dynamics), with both circulation directions. All these variants are described by the constant C in the relation φ = ϕ + C, with ϕ the angle of the position vector, and φ the angle of the magnetization. (b) Antivortex, described by φ = -ϕ (any additional constant can be absorbed by a rotation of the coordinate axes).

FIG. 4 .

 4 FIG.[START_REF] Brown | Micromagnetics[END_REF]. The in-plane components of the magnetization are depicted by black arrows, whereas the out of plane component is coded in color (blue=down, red=up, white= in-plane). (a) anti-skyrmion with an in-plane winding number S = -1. (b) non-topological structure with an in-plane winding number S = 0. This structure can be continuously destroyed by a magnetization rotation around the vertical in-plane axis, the angle of rotation being π at the center, π/2 at the wall and 0 at infinity. (c) a Néel skyrmion with positive chirality. Note that, by π rotation of the moments around the perpendicular axis, this structure can be continuously transformed into a Néel skyrmion with negative chirality, as well as into Bloch skyrmions with positive or negative chirality by ±π/2 rotation. (d) a higher order skyrmion with an in-plane winding number S = +2.

  FIG.5. (a) perspective view. The grey torus figures the locus of the in-plane oriented moments (the 'wall' of the hopfion). The orientation of magnetization on this surface is depicted by arrows, the loops with the same color drawing the fibers of the structure, the loci of the points with the same magnetization. These loops are all linked once, this being the topological index of the structure. (b) cut view through a vertical plane, with the cut of the torus superposed, allowing to see the magnetization inside the torus, which is akin to a reverse domain.

FIG. 6 .

 6 FIG.6. (a) Two Néel walls with opposite rotation sense, between the same domains, cannot be converted continuously one into the other, if the magnetization in the domains is assumed to be fixed, as the moments are XY. (b) Corresponding paths on the unit circle S 1 . For Heisenberg moments, path a2 can be transformed to path a1 by going over a pole, above or below the plane of the drawing.

FIG. 7 .

 7 FIG. 7. (a) the vortex wall; (b) the (symmetric) transverse wall; (c) the symmetric transverse wall during its dynamical reversal of the transverse moment.The green points at the edge correspond to topological defects of π 0 (S 0 ) (the edge when tangential magnetization is assumed), which become merons of π 1 (S 1 ) (the edge when planar magnetization is assumed, with values fixed at infinity). The orange points correspond to topological defects of π 1 (S 1 ) (in-plane moments assumed in the sample), which become merons of π 2 (S 2 ) (Heisenberg spins, with in-plane spins far from the core), namely a vortex (a) and an anti-vortex (c).

FIG. 8 .

 8 FIG.8. The schematic drawings (perspective view) show (a) a vertical Bloch line, and (b) a horizontal Bloch line. The tilt of the domain wall magnetization at the surfaces, from a Bloch to a chiral Néel orientation, due to the stray field of the domains and that occurs in films thick enough compared to the exchange length[START_REF] Malozemoff | Magnetic Domain Walls in Bubble Materials[END_REF], is not drawn. The lines can be magnetized in both directions. Only the vertical Bloch line is stable in statics.

FIG. 10 .

 10 FIG.10. The diffusion constant of a skyrmion, divided by temperature and by atomic numerical factors, is plotted in perspective view as a function of skyrmion radius R (normalized to the domain wall width parameter ∆), and Gilbert damping constant α. The crest obeys Rα = ∆. Above this crest line, the normal behavior where diffusivity decreases as damping increases is observed. Below the crest line, the 'topological diffusion' dominates. Adapted from Ref.[47].

FIG. 11 .

 11 FIG.11. Calculations were performed using the nudged elastic band technique, with 41 linked configurations, in an atomic micromagnetic code[67,68]. The DMI energy per Co-Co bond is 1.8 meV. (top) Variation of total energy, and topological number with number of the configuration along the path (the path is shown by its length in configuration space, and the full squares correspond to the images shown in the rest of the figure) (bottom) Dual representation of chosen configurations along the path, here with number 26 and 29 (continued on the next pages with numbers 30-33, and 34-37). For each configuration, the left column shows a real space view, with the surface height and color displaying the perpendicular m z component (red: positive; blue: negative). The triangular mesh corresponds to the atomic lattice and the nearest-neighbour bonds. The right column depicts the image of the configuration on the unit sphere S 2 , as seen from the North direction. The color of the nearest neighbor bonds expresses the latitude on the sphere (white: North, black: South, blue: Equator).

  FIG.14. For a cobalt monoatomic layer, and as a function of the strength of the bond DMI, the analytical model and the atomic micromagnetic model results are compared. (top) skyrmion absolute energy, (middle) saddle point absolute energy and, (bottom) resulting energy barrier. One sees that, even if the absolute energies are fairly described, their difference is much poorly obtained by the analytical model.

VII. ACKNOWLEDGEMENTS

This chapter is based on a tutorial talk delivered at the 61st Annual Conference on Magnetism and Magnetic Materials (New Orleans (USA), 31/10-04/11/2016), for which the presenter (A.T.) is thankful to the conference chairpersons. A.T. thanks F. Piéchon for communicating Refs. [20,21].