Stripe domains reorientation in ferromagnetic films with perpendicular magnetic anisotropy

Ferromagnetic thin films with moderate perpendicular magnetic anisotropy (PMA) are known to support weak stripe domains provided film thickness exceeds a critical value. In this work, we performed both an experimental and theoretical investigation of a peculiar phenomenon shown by weak stripe domains: namely, the stripe domains reorientation when a dc magnetic field is applied in the film plane along the direction perpendicular to the stripes axis. We focus on bct α′-Fe8N1−x thin films obtained by N 2 + implantation of α-Fe films epitaxially grown on ZnSe/GaAs(001). By using different ion implantation and heat treatment conditions, we show that it is possible to tune the PMA values. Magnetic force microscopy and vibrating sample magnetometer measurements prove the existence of weak stripe domains at remanence, and of a threshold field for the reorientation of the stripes axis in a transversal field. Using a one-dimensional model of the magnetic stripe domains, where the essential parameter is the maximum canting angle of the stripe magnetization out of the film plane, the various contributions to the magnetic energy can be separately calculated. A linear increase of the reorientation threshold field on the PMA is obtained, in qualitative agreement with experimental data in our Fe–N films, as well as in other thin films with weak stripe domains. Finally, we find that also the rotatable anisotropy field linearly increases as a function of the PMA magnitude.


Introduction
The formation of stripe domain patterns has been observed in a variety of physical systems [1][2][3][4][5] as a consequence of competing interactions acting on different spatial scales. In thin ferromagnetic films with a perpendicular component in the magnetic anisotropy energy, magnetic domains appear as stripes with alternating (up and down) out-of-plane orientation of the magnetization. Magnetic stripe domains occur due to In systems with high PMA compared to the magnetostatic energy, i.e. with Q?1, stripe/band domains occur even for extremely low thickness. This kind of stripe domains is characterized by thin domain walls (that is, very sharp transitions between stripes with opposite magnetization orientations) and can be properly modeled by the domain approach of Kittel [7], such as in the work of Kooy and Enz [8]. In the beginning, this approach was employed to investigate bubble materials such as Ba-ferrite or yttrium-iron garnet plates, in which, when approaching magnetic saturation, stripe domains become energetically competitive with magnetic bubbles. The latter are cylindrical domains with high mobility that were explored for magnetic recording [8][9][10][11]. Similar stripe domain patterns were later found in other thin film materials with high PMA, such as Co ultrathin layers on Au, FePt thin films and Co/Pt(Pd) multilayers [12][13][14][15]. Several works have modeled the domains in these thin films, starting from the domain theory of [7,8] and extending it to magnetic multilayers [16] or exploring limiting cases, such as the ultrathin layer regime [17].
Stripe domains also occur in systems with moderate or low PMA, i.e. with Q<1, but above a critical thickness t cr which in general depends on Q [6]. (Note that, for Q=1, an approximate analytical estimate [18] is where A ex is the exchange energy per unit length). These 'weak' stripe domains appear as dense, straight stripes with wide domain walls, that is, with smooth variation of the magnetization profile between stripes with opposite magnetization orientations. Since the PMA contribution to the magnetic free energy is in this case comparable with the magnetostatic contribution, rigorously a micromagnetic approach is required to correctly model the behavior of the system [6]. In order to interpret experimental data in real systems, convenient approximations to a fully three-dimensional micromagnetic model can be adopted. Typically, within such approaches the magnetostatic interaction associated with the stripe pattern is properly evaluated by taking into account a realistic, although simplified, magnetization profile across the stripes, such as in [18][19][20][21][22][23][24] (1D models), and/or across the film thickness, such as in [25,26] (2D models). Alternatively, a numerical micromagnetic calculation can be performed, but the limitation of such an approach is related to the finite size of the film plane [27][28][29][30][31].
The subject is currently attracting wide interest, thanks to the central role played today by magnetic thin films with PMA in many applications, from memories to logic devices and sensors [52]. Current active research on PMA films and multilayers is focused onto the definition of magnetic phase diagrams into different systems where magnetic stripe and skyrmion or bubble phases compete [30,47,50,[53][54][55][56][57]. Nanoscale magnetic textures are today the focus of a wide area of research, thanks to the possibility of generating and controlling the propagation of domains, bubbles, skyrmions and other kinds of spin textures for realizing, e.g. logic circuits, racetrack memories, sensors and rf devices [58][59][60][61]. Different methods have been pursued for the generation and controlled propagation, including magnetic field, electric current, electric field or mechanical stress [56,59,[62][63][64][65][66]. In some cases, the geometry and local asymmetries of stripe domain patterns have been used to guide the propagation of skyrmions or bubbles [66,67]. Other technological concepts are based on the controlled propagation of spin waves in films and micro/nanostructures through various kinds of spin textures. Notable examples are magnonic waveguides and spin-wave filters exploiting domain walls or different types of nanoscale spin textures [68][69][70][71][72][73]. Moreover, recent studies showed that stripe domains can be used to control spin-wave propagation, by changing the relative orientation between the spin-wave wave vector and the domains axis [74,75], suggesting the possibility to exploit stripe domains to realize reconfigurable spin-wave devices.
The present work is devoted to the experimental and theoretical investigation of the peculiar phenomenon shown by weak stripe domains [19] in a thin ferromagnetic film when a dc magnetic field is applied in the film plane along the direction perpendicular to the stripes axis: namely, the stripe domains reorientation taking place abruptly, above a threshold value for the transversal field. We focus on bct α′-Fe 8 N 1−x thin films obtained by + N 2 implantation of α-Fe films epitaxially grown on ZnSe/GaAs(001). First we show that the PMA strength of the Fe-N samples can be tuned by using different ion implantation and heat treatment conditions. This PMA gives rise to weak stripe domains, which we have visualized at remanence by magnetic force microscopy (MFM). Using this technique and vibrating sample magnetometry, we have proved the existence of a threshold field for the stripe domains reorientation in a transversal magnetic field. Noticeably, the threshold field was found to increase on increasing the PMA of the Fe-N samples. Using a 1D model [19] of the magnetic stripe domains where the essential parameter is the maximum canting angle of the stripe magnetization out of the film plane, the various contributions to the magnetic energy have been separately calculated in the presence of an in-plane magnetic field. The advantage of the 1D model is that it provides a mechanism for the abrupt stripe reorientation, and a clear evidence for the linear dependence of the reorientation threshold field on the PMA.
Finally, the 1D model allows us to calculate another peculiar property of weak stripe domains, the so-called 'rotatable' anisotropy [20,[76][77][78][79]. The latter strongly affects the magneto-dynamic properties of the thin films [37,79], providing an energy barrier which prevents from the free in-plane rotation of the magnetization vector. We show that the rotatable anisotropy increases linearly with increasing the PMA, in qualitative agreement with experimental data in our Fe-N films, and in other thin films with weak stripe domains [42,80,81]. In addition, a linear correlation between the rotatable anisotropy field and the threshold field for stripe reorientation has been established, confirming that both phenomena are driven by the PMA, responsible for the onset of weak stripe domains.

Materials and methods
A body-centered cubic (bcc) α-Fe thin film with a thickness of 78 nm was epitaxially grown on a ZnSe-buffered GaAs (001) substrate by molecular beam epitaxy (MBE . Then, the film was in situ protected against oxidization with an 8 nm thick gold capping layer. Four different samples have been prepared performing ion implantation of four pieces of this iron film, following the procedure detailed in our previous study [39]. Three samples were implanted with + N 2 ions accelerated to 26 keV with fluences of 3.0×10 16 + N 2 cm −2 and 3.5×10 16 + N 2 cm −2 , and 40 keV with a fluence of 5.3×10 16 + N 2 cm −2 , respectively. The last sample was implanted with + N 2 ions accelerated to 40 keV with a fluence of 5.3×10 16 + N 2 cm −2 , and has undergone heat treatment at 150°C for 72 h, in an ultra-high vacuum chamber. X-ray diffraction (XRD) measurements were carried out using monochromatized Cu K α radiation in order to investigate the crystalline structure of the Fe-N films. The magnetization versus field of the samples was measured at room temperature by a vibrating sample magnetometer (VSM). A MFM was used to image the weak stripe domains of the magnetic thin films (phase detection).

Structural investigations
The out-of-plane XRD patterns of as-grown and implanted samples are reported in figure 1. For the unimplanted sample, the α-Fe(002) line is coherent with the epitaxial conditions given above (section 2.1).
After nitrogen implantation, out-of-plane XRD patterns disclose the presence of the body-centered tetragonal (bct) α′-Fe 8 N 1−x phase. No reflections of α′-Fe 8 N 1−x other than the one related to (002) crystal planes were observed. Hence, the c-axis of this bct compound is preferentially perpendicular to the film plane. Following the Bragg's law, the c-lattice constant of α′-Fe 8 N 1−x can be inferred considering the 2θ angle of maximum intensity of the (002) diffraction peaks. For the as-implanted samples, we found c≈3.12 Å. For the heat-treated sample, we found c≈3.14 Å, which equals the expected value for α′-Fe 8 N 1−x (x=0) [82]. The increase of the c-lattice constant of α′-Fe 8 N 1−x (previously formed by nitrogen implantation in our case) by a subsequent heat treatment is in agreement with previous studies [82]. Moreover, additional in-plane XRD measurements along the [100] and [110] directions of one of these samples (i.e. the one implanted with fluence 5.3×10 16 + N 2 cm −2 at 40 keV) showed that the a-lattice constant of α′-Fe 8 N 1−x is approximately equal to that of α-Fe, as expected [82]. Besides, the peak width of α′-Fe 8 N 1−x possibly reflects its low crystallinity and the variation of its c-lattice constant. The small shoulder around 2θ=64°in the diffraction pattern (b) of figure 1 can be attributed to a minor fraction of α-Fe in the film implanted with the lowest fluence and energy. This is due to a slight variation of the nitrogen concentration through the film thickness, according to TRIM simulations [83]. Furthermore, the nitrogen distribution is probably impacted by the heat treatment conditions, through diffusion. Figure 2 compares the magnetization curves for the as-implanted iron thin film with fluence of 3.0×10 16 + N 2 cm −2 at 26 keV in (a), and the as-grown iron thin film in the inset (b). Measurements were made along the inplane [100] and [110] directions, and the out-of-plane [001] one. For the unimplanted sample, figure 2(b), we found the typical behavior of a bcc iron film, with the easiest axis along the [100] in-plane direction, due to the magnetocrystalline anisotropy, and the hardest axis along the [001] out-of-plane direction because of the thin film shape anisotropy. Whereas, for the as-implanted Fe-N sample, figure 2(a), the in-plane hysteresis loops are quasi isotropic (i.e. nearly independent of the applied field orientation) and the normalized in-plane remanence, M r /M s ≈0.5, is substantially smaller than for the unimplanted sample. Note that the bend observed near coercivity in the in-plane ([100] and [110]) hysteresis loops measured for the as-implanted Fe-N sample can be ascribed to the presence of α′-Fe 8 N 1−x phases with a slightly different nitrogen composition, induced by the depth variation of the nitrogen concentration. Moreover, the in-plane loops in figure 2(a) present a characteristic 'transcritical shape' [84]: namely, an intermediate-field region exists, where the magnetization linearly depends on the intensity of the in-plane applied magnetic field. This behavior is commonly regarded [19,36] as a fingerprint of weak stripe domain structures, and it has been observed for all the Fe-N thin films investigated in this work. The out-of-plane hysteresis loop has a nearly zero remanence owing to the presence of the stripe domains. Nevertheless, we note that the specific in-plane magnetic properties vary depending on the  preparation conditions of the samples, as shown in figure 3. More precisely we observed that, on increasing the ions energy and fluence, the in-plane saturation field increases, while the normalized in-plane remanence decreases, suggesting an increase of the PMA arising from the magnetocrystalline anisotropy of α′-Fe 8 N 1−x . In addition, the annealing procedure was found to induce a further increase of the PMA strength. The value of K PMA has been quantitatively estimated by measuring the area between the magnetization curves with the field along the in-plane and out-of-plane directions. We obtained K PMA in the range from 5 to 7×10 6 erg cm −3 (see figure 7(c) later on), consistent with the reported values for the magnetocrystalline anisotropy of α′-Fe 8 N 1−x [85,86], and in agreement with K PMA values measured by ferromagnetic resonance [87]. We note that, assuming M s =1700 emu cm −3 [31], the quality factor Q of the Fe-N samples investigated in this work is between 0.275 and 0.385, depending on the preparation conditions, while the critical thickness above which the stripe domains appear is about 40 nm.

Magnetic measurements
In figure 4, the presence of weak stripe domains at remanence in the as-implanted iron thin film with 3.0×10 16 + N 2 cm −2 at 26 keV is revealed by MFM. We observed very regular stripe domains, aligned along the in-plane direction of the last saturating magnetic field, and having a period of 110-140 nm. Similar stripe domain patterns have been found for the other samples. In particular, the stripes period was observed to be almost independent of the ions implantation parameters.  Furthermore, we carried out a detailed MFM study of the stripe domain reorientation when a dc magnetic field is applied in-plane along a direction perpendicular to the stripes axis. The MFM measurements (figure 5) were performed in the as-implanted sample with a fluence of 5.3×10 16 + N 2 cm −2 at 40 keV. The stripes were initially aligned along the [100] direction, then an in-plane magnetic field, H trans , was applied perpendicularly to the stripes axis and its maximum value H trans max was progressively increased. Each image was recorded at remanence. The dark/white dots in figure 5 highlight the presence of magnetic edge dislocations, namely topological defects spontaneously nucleated in the stripe domain structure [39]. We observed that the whole stripe structure remains unperturbed up to = H 800 Oe, the stripes were found to coherently rotate by a rather large angle (α 0 ≈60°) towards the direction of the applied field, apart from a small region where the straightness of the domains is not preserved (see figure 5(b)). For higher values of H trans max the stripe domains continued their reorientation, which appeared to be completed at the saturation field Oe, see figure 5(d)). Note that, in each panel of figure 5, the dark/white contrast of the MFM image is the same irrespective of H trans max , because each MFM image has been recorded at remanence. On the contrary, whenever MFM images are recorded in the presence of an in-plane dc magnetic field, a progressive loss of contrast is observed [88,89] with increasing the field intensity, because the in-plane component of the film magnetization progressively increases.
In order to better understand the reorientation process, VSM measurements have been performed applying a transversal magnetic field, H trans along the in-plane direction perpendicular to the stripes axis. All the Fe-N samples obtained from the same 78 nm thick iron film, differing (see figure 1) for technical specifications of ion implantation and heat treatment, were investigated. For each sample, a saturating field was first applied in plane to align the stripe domains along the [110] direction and was removed. Then, a transversal field, H trans , was applied in plane perpendicularly to the stripe axis, and reduced to zero while the magnetization component As shown in figure 6, even if the four investigated samples exhibit a similar behavior, they are characterized by a different value of the threshold field H thr . In figure 7, the threshold field for stripe reorientation is plotted as a function of the in-plane saturation field, H s , the normalized in-plane remanence, M r /M s , and the perpendicular magnetic anisotropy energy density, K PMA , for the four investigated Fe-N thin film samples. These findings clearly show that the threshold field for stripe reorientation linearly increases with the PMA, in accordance with the work by Coïsson et al [42], where such linear dependence has been observed for the stripe domains in Fe 78 Si 9 B 13 thin films.

One-dimensional model of stripe domains
In order to explain the linear dependence of the reorientation threshold field (H reor thr ) as a function of the in-plane saturation field (H s ) experimentally observed in our Fe-N films, as well as in other magnetic films [42] with weak stripe domains, we adopt a 1D model of the stripe domain structure originally proposed by Saito et al [19] for Permalloy films. In principle, a micromagnetic approach [6] would be required in order to correctly model the behavior of weak stripe domains. However, it is very difficult to investigate the peculiar phenomenon of stripe reorientation in a transversal magnetic field using numerical micromagnetic calculations, owing to the strong pinning effects induced by the finite size of the film plane exploited in the simulations. As it is explained in more detail in appendix A, where different 1D models for stripe domains are compared, the advantage of the Saito model [19] is that it allows us to obtain approximate analytical expressions for the magnetic energy contributions. In this way, it is easy to understand the role of the model parameters in the stripe domains reorientation, induced by a transversal magnetic field above a threshold value.
The 1D stripe domain structure is sketched in figure 8. The stripes are assumed to be infinite along the y direction of the film plane, xy. Consequently the magnetization, M, depends only on x, the in-plane direction perpendicular to the stripes axis. Hereafter y will be denoted as the longitudinal direction, and x as the transversal direction. The magnetization M=(M x , M y , M z ) forms an angle θ(x) with the film plane, while the in-plane projection, M IP =(M x , M y ), forms an angle j 0 , independent of x, with the stripes axis y. One has therefore The angle θ(x) is assumed [19] to be a periodic function of x with the 1D profile sketched in figure 8(c), namely where d is the stripes width and P=2d is the period; θ 0 denotes the maximum value assumed by the out-ofplane canting angle θ(x) along the transversal direction, x.
The total energy density, ò tot , is the sum of different magnetic contributions The contribution, ò A , of the exchange interaction, A ex , is The uniaxial contribution, ò K , comes from the perpendicular magnetic anisotropy, K PMA >0, which favors the out-of-plane direction z (see figure 8(a)) In order to calculate the contribution of the magnetostatic energy, ò M , we exploit an approach by Corciovei and Adam [90], which is valid even for a 2D periodic domain structure. Inside the film, the magnetization vector is expanded in Fourier series where, in general, the position vector is r IP =(x, y,0), and the wavevector is . P is the period of the domain pattern within the film plane, and n x , n y =0, ±1, ±2, L. Outside the film, one has M(r)=0. Next, the magnetostatic energy can be separated [90] into a surface and a volume contribution, ò M =ò S +ò V . Making the approximation, valid for not too thick films, that the magnetization does not depend on the perpendicular coordinate z, the two contributions take the form [90], respectively where t denotes the film thickness, k 0 is the unit vector along k=(k x , k y ,0), and i z the unit vector along z.
Specializing to the case of the 1D stripe domain structure in figure 8 and equation (2), one has k y =0 because the stripes are infinite along y. It then follows that = Note that the volume contribution ò V has been expressed, according to [20], in terms of an effective rotatable anisotropy, K rot . Keeping only the m=2 term in (12), the following estimate for K rot is obtained The effective rotatable anisotropy field [20] H rot , associated with K rot , takes the form : i.e. the magnetization lies entirely in the film plane. Finally, the Zeeman energy density, ò H , namely the contribution from a dc magnetic field applied in plane along a direction forming an angle ψ with the stripes axis y, is Once obtained the total energy density (3) as the sum of the various contributions (4), (5), (11), (12), and (17), the equilibrium values for the polar and azimuthal angles, q eq 0 and j eq 0 respectively, can now be determined by minimizing ò tot (θ 0 , j 0 ) with respect to its arguments. In the following, for simplicity we consider only two limiting cases, of a transversal magnetic field H x (ψ=90°) and a longitudinal magnetic field H y (ψ=0°) H long s y , 0 0 0 ( ) Our numerical calculations for the equilibrium configuration and the magnetic energy density are presented in section 4 and appendix B for transversal field, and in appendix C for longitudinal field, respectively. The calculations were performed by setting the material parameters equal to values obtained in previous investigations of the Fe-N films. The saturation magnetization is M s =1700 emu cm −3 [39]. The exchange interaction is A ex =1.8×10 −6 erg cm −1  [91]. The magnetocrystalline in-plane anisotropy was set to zero, while different values of the out-of-plane uniaxial anisotropy (with K PMA ranging from 5.7 to 6.3 ×10 6 erg cm −3 ) were exploited, in order to investigate the role of the PMA in the stripe reorientation process. The film thickness is the same for all samples, t=78 nm. Finally, the period of the stripes was kept constant, P=115 nm. Such an approximation is in good agreement with experimental observations showing that, in many film systems (NiFe [19], FeGa [37,38], FeN [39], and FeSiB [42]), the period of the stripe domains does not vary on changing the intensity of the in-plane applied field.

Mechanism of stripe domains reorientation in Fe-N thin films
The equilibrium configurations of the film magnetization when an external magnetic field is applied along the in-plane x direction, perpendicular to the stripes axis, y, have been calculated as a function of the field intensity, H x . For increasing values of H x , we hypothesize three configurations, schematically illustrated in figures 8-10.
Low field configurations (I, II) without stripe rotation. For sufficiently low values of H x , we look for two possible equilibrium configurations, I and II, both characterized by the direction of the stripes axis being y, i.e. the same as at remanence. In the configuration I (figure 8), the in-plane uniform canting angle, formed by the inplane magnetization component M IP with the stripes axis y, is small: j 0 =90°. In contrast, in the configuration II (figure 9), the in-plane angle is exactly j 0 =90°: i.e. M IP is completely rotated towards the direction of the applied field, but the axis of the up/down magnetic stripe domains is not rotated. Also the out-of-plane maximum canting angle θ 0 is expected to be different, in general, for the two configurations I and II. The total energy density is given by where for both configurations I and II the various contributions on the rhs of (19) are explicitly given by (4), (5), (11), (12), and (18a). By minimizing ò tot with respect to its arguments, the equilibrium angles θ 0 eq and j eq 0 are obtained.
In figure 11, the contour plots of ò tot (θ 0 , j 0 ) are reported for a fixed value (K PMA =6.3×10 6 erg cm −3 ) of the out-of-plane uniaxial anisotropy and selected values of the transversal field H x , ranging from 0 to 400 Oe. At H x =0 ( figure 11(a)) and 100 Oe ( figure 11(b)), the total energy density has only one minimum, with small inplane uniform canting angle, j  0 eq 0 , and large out-of-plane maximum canting angle, q   90 eq 0 , corresponding to the configuration I ( figure 8). Consequently, both at H x =0 and 100 Oe the normalized average in-plane magnetization turns out to be sensibly smaller than 1 (m≈0.64). At H x =200 Oe . The second minimum thus corresponds to the configuration of type II (figure 9), and it turns out to be higher in energy than the first minimum. Note that at H x =200 Oe the normalized average in-plane magnetization is m≈0.92 for configuration II, thus sensibly larger than m≈0.65, the value for configuration I at the same field intensity. On further increasing H x , the situation is reversed. In fact, at H x =300 Oe (figure 11(d)) and 350 Oe (figure 11(e)), the deeper minimum corresponds to the configuration II, and the shallower one to the configuration I. Eventually, at H x =400 Oe ( figure 11(f)), the minimum corresponding to the configuration I disappears.  High field configuration (III) with stripe rotation. For sufficiently high values of H x , we look for the equilibrium configuration III (figure 10), where the direction of the stripes axis is rotated by a large angle α 0 with respect to y, the direction at remanence. The total energy density is now expressed as The first four terms on the rhs of equation (20) are respectively given by (4) By minimizing the total energy density ò tot (θ 0 , α 0 ) with respect to its arguments, one thus obtains that, in the absence of in-plane anisotropies, the equilibrium value for the stripe rotation angle is a =  90 eq 0 for any value of H x . Whereas, the equilibrium out-of-plane maximum canting angle, q eq 0 , depends on the in-plane field intensity, H x .
In figure 12, the contour plots of ò tot (θ 0 , α 0 ) are reported for selected values of H x , ranging from 100 Oe to 1750 Oe. Note that the latter value turns out to be the in-plane saturation field H s for K PMA =6.3× 10 6 erg cm −3 . As it can be seen, the total energy density ò tot (θ 0 , α 0 ) has only one minimum, corresponding to the configuration III in figure 10, with a =  90 eq 0 . On increasing H x , α 0 eq remains constant, while q eq 0 decreases. At the saturation field H s =1750 Oe, the out-of-plane maximum canting angle vanishes, q  0 eq 0 , and the ( )(see figure 13(d) later on). The existence of a threshold field for the reorientation of the stripes axis can be explained looking at figure 13, where the equilibrium angles, the normalized average in-plane magnetization, and the magnetic energy density are reported versus the transversal field intensity, H x , and compared for the three different configurations. However, it is important to note that, for a given field intensity where different metastable states are possible, the configuration assumed by the system cannot be predicted only on the basis of energetic considerations, but the history of the sample must also be taken into account. We observe that, in the low field range (200-380 Oe) where the three configurations are locally (I and II) or globally (III) stable, the configuration I (red symbols) and the configuration III (blue symbols) have similar values of the out-of-plane maximum canting angle q eq 0 ( figure 13(b)), and thus similar values of the normalized average in-plane magnetization q q = m sin eq 0 0 ( ) ( figure 13(c)). In contrast, the configuration II (gray symbols) has a sensibly lower value of q eq 0 ( figure 13(a)) and consequently an almost saturated in-plane magnetization (m0.9 in figure 13(c)). Starting from remanence with the stripes aligned along the y axis, the system is allowed to remain in the configuration I even when the intensity of the transversal field H x is increased, as far as this configuration remains locally stable. Oe. The reorientation of the stripes axis towards the field direction is associated with a large rotation angle (a =  90 eq 0 in figure 13(b)). In contrast, the out-of-plane maximum canting angle q eq 0 has a small (though finite) variation, see figure 13(b). Note that, passing from I to III at H trans thr , a noticeable lowering of the total energy density ò tot is realized, as it can be seen in figure 13(d).
In appendix B, the various contributions to the total energy density ò tot were calculated versus H x , and separately reported in figure B1 (panel (b), and its enlargement, (c)). From figure B1, it appears that the decrease of the equilibrium out-of-plane maximum canting angle q eq 0 on increasing H x (panel (a)) determines, in the magnetic energy density, the increase of the PMA contribution, ò K , and the decrease of all the other contributions (in particular, the transversal field contribution, ò H, trans , and the surface magnetostatic contribution, ò S ). In order to minimize the cost in PMA energy, it is then energetically preferred for the system to undergo, at H trans thr , a small variation of the out-of-plane maximum canting angle q eq 0 accompanied by a large inplane rotation (a =  90 eq 0 ) of the stripes axis (see figure 13(b)), rather than a large variation of q eq 0 without stripe rotation (see figure 13(a)). Namely, the configuration II is expected not to be realized, in spite of its being locally stable and lower in energy than I in the low field range (H x =200-380 Oe, see figure 13(d)).
In order to better understand the dependence of H trans thr on the intensity of the perpendicular magnetic anisotropy, we have determined the equilibrium configurations of the system in a transversal magnetic field, H x , using different values of K PMA in the range from 5.5 to 6.3×10 6 erg cm −3 . (Note that within the 1D model of stripe domains [19], an upper limit to K PMA is imposed by the condition q   90 eq 0 .) In figures 14(a)-(c), the calculated out-of-plane maximum canting angle q eq 0 is shown versus H x (below and above H trans thr , red and blue symbols, respectively) for three increasing values of the out-of-plane uniaxial anisotropy K PMA . As it can be seen, the higher K PMA , the further the configuration I remains stable, and as a consequence the higher is the reorientation threshold field H trans thr , marked by a vertical dashed line in figure 14. Similarly, the higher K PMA , the higher is the in-plane saturation field H s , marked by a vertical short-dashed-dotted line in figure 14.
In figure 15, the calculated threshold field for stripe reorientation, H trans thr , and in-plane saturation field, H s , are reported versus the intensity of the perpendicular magnetic anisotropy. For both quantities, a linear dependence is obtained in the investigated range of K PMA , although with a different slope. (A third quantity reported in figure 15, also showing a linear dependence on K PMA , is the rotatable anisotropy field, H rot,0 , discussed later on in section 5.) These results are in qualitative agreement with experimental data obtained for the out-of-plane maximum canting angle, q eq 0 , the in-plane uniform canting angle, j eq 0 , and the stripe rotation angle, a eq 0 , calculated at equilibrium for different configurations, of type I (red symbols), type II (gray symbols) and type III (blue symbols), respectively. (c) and (d): the normalized average in-plane magnetization, m, and the total energy density, ò tot , calculated for the configurations I, II, and III. In (d), the vertical dashed line marks a possible, abrupt transition, at the reorientation threshold field H thr trans ≈380 Oe, from the metastable state I to the lowest energy state III; the vertical short-dashed-dotted line marks the position of the saturation field (H s ≈1750 Oe).
both Fe-N ( figure 7(a)) and Fe 78 Si 9 B 13 [42] films. Clearly, in order to obtain a quantitative estimate of the reorientation threshold field intensity H trans thr for a given film system, one should definitely resort to a more realistic 2D model of the stripe domains. A 2D model with closure stripe domains should be required (as it was proved in the very similar case of a Fe-Ga film [38]) to account also for other subtle features, such as the linear dependence of M ON /M s on H reor thr experimentally observed (figure 6(a)) in our Fe-N samples. Nevertheless, the 1D model [19] is valuable since it provides a straightforward explanation of the linear dependence of H trans thr and H s on the PMA in terms of the equilibrium out-of-plane maximum canting angle q eq 0 , which would hardly be obtained by numerical methods and/or more refined theoretical models.

The rotatable anisotropy in Fe-N films with weak stripe domains
A further advantage of the 1D model [19] is that it allows us to theoretically investigate another peculiar property of weak stripe domains, the so-called 'rotatable' anisotropy, which consists in the possibility of selecting the easy magnetization direction in the film plane simply by applying a sufficiently large magnetic field along this direction [20,[76][77][78]. The rotatable anisotropy strongly affects the magneto-dynamic properties of the thin films [37,79] since it provides an energy barrier which prevents the free in-plane rotation of the magnetization vector. As a matter of fact, the rotatable anisotropy has been termed as a pseudo-uniaxial in-plane anisotropy [20] because its magnitude is not constant, but depends on the intensity of the applied magnetic field [37,79].
Using the equilibrium values for the angles θ 0 and j 0 , we calculated the rotatable anisotropy field H rot , as given by equation (14), for three different values of K PMA (the same as in figure 14). In figure 16, H rot is reported versus the intensity of an applied magnetic field, either transversal (H x , red circles) or longitudinal (H y , black line) to the stripes axis. One can see that the rotatable anisotropy field assumes the maximum value H rot,0 at remanence (i.e. for H=0). On increasing the transversal field intensity H x , H rot starts to decrease because the angle q eq 0 decreases (namely, the in-plane magnetization increases), up to vanish when the stripe pattern disappears for θ 0 =0 (namely, the in-plane magnetization becomes saturated). In addition, H rot versus H x exhibits a discontinuity at the threshold field H trans thr , which becomes more marked on increasing K PMA . Above the reorientation threshold field, there is no difference between the value of the rotatable anisotropy field calculated for transversal (H x ) or longitudinal (H y ) applied field. In fact, in the configuration III the stripes have rigidly rotated by an angle α 0 =90°, and therefore the stripes axis has become parallel to the direction of the applied field. In the longitudinal field configuration, it is worth noticing that our calculations within the 1D model [19] predict (see black line in figure 16) a smooth, monotonic decrease of H rot versus the applied field intensity, H y . A similar behavior has recently been found in thin films with weak stripe domains, such as permalloy films investigated by ferromagnetic resonance (FMR, see figure 2(b) in [79] and figure 4 in [80]), and Fe-Ga films investigated by Brillouin light scattering (BLS, see figure 4 in [37]). Moreover, from figure 16 it appears that H rot,0 increases on increasing the perpendicular magnetic anisotropy. In particular, using equation (14) we have calculated H rot,0 and found a linear dependence on K PMA , as shown in figure 15 (red circles). Such a feature turns out to be in qualitative agreement with a recent experimental study [81] in ferrite doped FeNi thin films, where an increase of the rotatable anisotropy field H rot on increasing the perpendicular magnetic anisotropy was observed by FMR measurements.
Finally, we have found that a linear correlation holds, see figure 17, between the rotatable anisotropy field at remanence, H rot,0 , and the reorientation threshold field, H trans thr . This finding indicates that, although the rotatable anisotropy field (namely, the field proportional to the energy barrier preventing from the free in-plane rotation of the magnetization vector) and the threshold field (namely, the transversal field to be overcome in order to induce the reorientation of stripe domains) are in principle distinct, they are both driven by the PMA, which is responsible for the onset of the weak stripe domains.

Conclusions
In this work, the reorientation of weak stripe domains, driven by a transversal magnetic field greater than a threshold value, has been investigated in Fe-N thin films, both experimentally and theoretically. The samples have been prepared by ion implantation of nitrogen molecular ions + N 2 on an α-Fe film, epitaxially grown on ZnSe/ GaAs(001). We found that the PMA strength of the Fe-N films can be tuned on changing both ion implantation and heat treatment conditions. MFM images disclosed very regular stripe domains which at remanence are aligned along Figure 15. Black squares: the reorientation threshold field, H trans thr , versus the intensity of the perpendicular magnetic anisotropy, K PMA . Open squares: The saturation field, H s , versus K PMA . Red circles: the rotatable anisotropy field calculated at remanence, H rot,0 , versus K PMA . The lines are guides to the eye. the last saturation direction. By applying in plane a magnetic field H trans , perpendicular to the stripe axis and having an increasing intensity, the stripe domains reorientation has been observed when H trans is larger than a threshold field H trans thr . Interestingly, a linear increase of the H trans thr value on increasing the PMA strength has been observed by VSM measurements. This linear behavior of H trans thr has been interpreted by using a one-dimensional model of the weak stripe domains, which allowed us to calculate the different terms in the total energy density. Various possible equilibrium magnetization congurations have been considered, depending on the intensity of the magnetic field applied perpendicularly to the stripe domain axis. The maximum canting angle of the stripe domain magnetization out of the film plane has been shown to be the essential parameter to explain the linear dependence of both H trans thr and H rot,0 as a function of the PMA strength. In addition, the effective rotatable anisotropy field at remanence, H rot,0 , was also found to linearly increase with the PMA intensity. These findings indicate that both H trans thr and H rot,0 are driven by the PMA, although these two quantities affect different magnetic properties of the stripes domains: i.e. the static and dynamic magnetic behavior, respectively. We expect that the present work will provide the basis for a better understanding of the reorientation mechanism of stripe magnetic domains, paving the way to the possible Figure 16. The rotatable anisotropy field, H rot , versus the intensity of an in-plane magnetic field, applied either in the transversal direction (H x , red points) or in the longitudinal one (H y , black line) with respect to the stripes axis. The calculation was performed for three different values of K PMA , the same as in figure 14. exploitation of stripe patterns to guide and control the propagation of spin waves, or spin textures such as bubbles and skyrmions, in thin magnetic films.

Acknowledgments
We thank F Fortuna for ion implantation performed at CSNSM Université Paris Sud-CNRS, P Atkinson for the MBE growth of buffer layers of GaAs at INSP and the staff of the MPBT (physical properties-low temperature) platform of Sorbonne University for their support. M Coïsson at INRIM is gratefully acknowledged for fruitful discussions on the threshold field for the reorientation of the stripe domain pattern in FeSiB films, as well as for sending us a preprint of the paper [42].
Appendix A. Wide-wall and narrow-wall 1D models for weak and strong stripe domains In this appendix, we compare a few, paradigmatic 1D models which have been proposed in the literature to describe stripe domain patterns in magnetic films. At the same time, we suggest a harmonization of the nomenclature according to the shape of the periodic wave θ(x), where θ is the out-of-plane canting angle of the magnetization and x is the in-plane direction perpendicular to the stripes axis (see figure 8).
A first distinction has to be made between wide-wall and narrow-wall models, depending on the ratio between the wall width, δ, and the stripe width, d. In a ferromagnetic film with exchange A ex per unit length and a perpendicular magnetic anisotropy energy density K PMA , an estimate for the wall width is d p Hence, it follows that films with a weak PMA (low Q) will have wide domain walls, while films with a strong PMA (high Q) will have narrow domain walls. For a given 1D angle profile, θ(x), with period P=2d, see figure A1, we adopt the following definition of the wall width [92] where θ 0 is the maximum of θ(x) at the middle of the magnetic domain (i.e. for x=d/2 in figure A1), and θ 0 ′ is the slope of θ(x) at the middle of the wall (i.e. for x=0 in figure A1).
The 1D profile we have chosen in this work for the out-of-plane canting angle θ(x) (see equation (2) and figure 8(c)) is the 1D triangular-wave angle profile originally proposed by Saito et al [19] for Permalloy films with a weak PMA. We note that the term 'triangular-wave' appears to be more appropriate than the 'sawtooth-wave' one frequently used in the literature [20,22,80]. Indeed, the triangular wave is the absolute value of the sawtooth wave. The triangular-wave profile is reported in panel (a1) (black line), and represents an extreme case of a widewall model because the wall width is just equal to the domain width: δ=d. The corresponding out-of-plane magnetization profile, . Figure 17. The rotatable anisotropy field at remanence H rot,0 versus the reorientation threshold field H trans thr , calculated for different values of K PMA , ranging from 5.5 to 6.3×10 6 erg cm −3 (see figure 15).
In contrast, a 1D square-wave profile for θ(x) was originally proposed by Kittel [93] for films with a strong PMA. The square-wave angle profile is reported in panel (a2) (blue line), and it generates a square-wave profile also for the magnetization M(x), see panel (b2). The square-wave profile represents an extreme case of a narrowwall model because the wall width is exactly zero: δ=0.
Later on, other 1D models of stripe magnetic domains have been proposed, which allow one to consider the intermediate case 0<δ<d In contrast, Virot et al [22] and subsequently Wei et al [80] suggested a 1D trapezoidal-wave profile for θ(x), reported in panel (a2). The only difference between the two authors is the fact that Virot et al imposed the out-ofplane maximum canting angle to be θ 0 =π/2, while Wei et al allowed for θ 0 <π/2. Virot et al showed that the trapezoidal-wave profile is especially well suited for films with strong stripe domains, i.e. for 0.5<Q<1.0, and it can be applied even to the case Q>1. Thus, the trapezoidal-wave profile can be considered as a narrow-wall model. As a matter of fact, a recent analysis [92] of the in-plane magnetization curves in 70 nm thick Fe-Ga films with weak stripe domains (Q=0.3) confirmed that a 1D trapezoidal-wave angle profile provides a less consistent description of the experimental data than a sine-wave profile.
In the present article, our main aim is to investigate the reorientation of weak stripe magnetic domains when a dc magnetic field is applied in plane perpendicularly to the stripes axis, with intensity H trans greater than a threshold  , d), respectively for a wide-wall (δ=d) triangular-wave profile [19] and a narrow-wall (δ=d) trapezoidal-wave profile [22,80]. The sine-wave profile [20] in (a1) has d = p d 2 (not shown), while the square-wave profile [93] in (a2) has δ=0.
value. Our Fe-N thin films are characterized by a quality factor Q comprised in the range between 0.28 and 0.38, depending on the preparation conditions. Therefore, our films are in the weak-stripe regime, and a wide-wall model is expected to provide better results than a narrow-wall model. Our adoption of the simplest wide-wall model, i.e. the 1D triangular-wave angle profile [19] in equation (2), is motivated by the fact that it allows us to obtain approximate expressions for the magnetic energy density contributions in the transparent, analytic form of equations (4), (5) (11), (12), and (18a). In this way, understanding the role of the model parameters in the stripe domains reorientation induced by a transversal magnetic field turns out to be easier than using any other periodic wide-wall waveform. Furthermore, we do not expect that using more sophisticated 1D models would lead to substantial improvements in the theoretical interpretation of the stripe reorientation mechanism.
Appendix B. Energy density contributions in a transversal magnetic field In this appendix, we show the calculated equilibrium configurations and magnetic energy density of the magnetic film versus the the intensity (H x ) of a transversal field: i.e. a magnetic field applied in plane perpendicular to y, the direction of the stripes axis at remanence. The total magnetic energy density, ò tot , and its various contributions (see text). Please note that (c) is a zoom of (b) around ò=0.