How To Create A Domain_6 Wall

I. INTRODUCTION

Section:

The domain structure of ferromagnets and ferrimagnets is conventionally explained as a result of minimizing the free energy, which in micromagnetic theory is expressed in the continuum approximation $\vec{M} (\vec{r})$ , a smoothly varying vector function of constant norm, supposing that its typical space scale essentially exceeds inter-atomic distance. Domain formation leads to the energy minimization in most cases. ^1,2 1. J. M. D. Coey, Magnetism and Magnetic Materials (Cambridge University Press, 2009). 2. A. Aharoni, Introduction to the Theory of Ferromagnetism (Clarendon Press, Oxford, 1996), p. 192. Going to the main aim of this work, we would note that many aspects of nucleation theory are presented in Ref. 2 2. A. Aharoni, Introduction to the Theory of Ferromagnetism (Clarendon Press, Oxford, 1996), p. 192. . The author, discussing the problem, starts with a general expression for energy and considers variational principle, with natural restriction for the unit normalized magnetization vector modulus. The result is given as a pair of equations under the name of Brown. In this monograph ² 2. A. Aharoni, Introduction to the Theory of Ferromagnetism (Clarendon Press, Oxford, 1996), p. 192. an expression for nucleation field H _n is derived on base of Brown equations, and it results in a simple relation of the field H _n in terms of the medium parameters. In the paper ³ 3. M. Ipatov, N. A. Usov, A. Zhukov, and J. Gonzalez, "Local nucleation fields of Fe-rich microwires and their dependence on applied stresses," Physica B 403, 379 (2008). https://doi.org/10.1016/j.physb.2007.08.054 the first term of the expression is estimated as prevailing and connected with formula for anisotropy constant K and applied for evaluation of H _n that is compared with results of measurements. This result is based on stationary version of the Brown theory, as it does not contain information about dynamics and domain wall (DW) form. Interest in this complex problem has grown in the last decade in connection with various applications, such as laser-induced switching ⁴ 4. A. Stupakiewicz, A. Chizhik, A. Zhukov, M. Ipatov, J. Gonzalez, and I. Razdolski, "Ultrafast magnetization dynamics in metallic amorphous ribbons with a giant magnetoimpedance response," Physical Review Applied 13, 4 (2020). https://doi.org/10.1103/physrevapplied.13.044058 and the principle features of the DW dynamics and forms a variety of investigation. ^5,6 5. A. Chizhik, A. Zhukov, J. Gonzalez, and A. Stupakiewicz, "Control of reversible magnetization switching by pulsed circular magnetic field in glass-coated amorphous microwires," Applied Physics Letters 112(7), 072407 (2018). https://doi.org/10.1063/1.5018472 6. A. Chizhik, J. Gonzalez, A. Zhukov, and P. Gawronski, "Study of length of domain walls in cylindrical magnetic microwires," Journal of Magnetism and Magnetic Materials 512, 167060 (2020). https://doi.org/10.1016/j.jmmm.2020.167060 A numerical realization of the problem solution (valid only in nanoscale) is given in Ref. 7 7. M. H. A. Badarneh, G. J. Kwiatkowski, P. F. Bessarab, "Mechanisms of energy efficient magnetization switching in a bistable nanowire," Nanosystems: Physics, Chemistry, Mathematics 11(3), 294–300 (2020). https://doi.org/10.17586/2220-8054-2020-11-3-294-300 .

To describe magnetization dynamics the Landau–Lifshitz–Gilbert (LLG) equation is used, which relaxation term guarantee align with the applied field. Its simplest version, without space derivatives included gives estimations of coherent reversal as an inherently fast process (0.01 ns). ¹ 1. J. M. D. Coey, Magnetism and Magnetic Materials (Cambridge University Press, 2009). Such form of the equation cannot give an answer to the question about creation of DW with sharp coordinate dependence. The reason to use the complete 3D LLG equation, however, is the very complicated nonlinear vector partial differential equation. ⁸ 8. M. Lakshmanan, "The fascinating world of the Landau–Lifshitz–Gilbert equation: An overview," Phil. Trans. R. Soc. A 369, 1280–1300 (2011). https://doi.org/10.1098/rsta.2010.0319 A statement of the problem with a DW creation and its general solution for such LLG is nontrivial and still have not been done either analytically or numerically.

We act in the spirit of Brown micromagnetics, but use a rather asymptotic concept of initial - final states using the notion of hysteresis curve to mimic state equation, but omit intermediate states details. Therefore we restrict our consideration to such a problem, stated for the DW creation in a cylindrical microwire that demonstrates important features of a 3D picture. Meanwhile an inclined elliptic DW breaks the overall cylindrical symmetry of a wire. The corresponding statement of problem for DWs as the asymptotic states gives a possibility to study individual process of a DW creation. The resulting state of the wire 3D field of magnetization being the specific solution of the LLG equation is characterized by the stationary motion with the velocity that is determined by equilibrium between external field action and Gilbert relaxation, as, e.g., in Ref. 9 9. M. Vereshchagin, "Structure of domain wall in cylindrical amorphous microwire," Physica B: Condensed Matter 549, 91 (2018). https://doi.org/10.1016/j.physb.2017.10.065 , that we plan to explore. There are set of publications that outlines the problem and provide information about the DW propagation and, however not much, about its form. ^10,11 10. A. Janutka and P. Gawroski, "Structure of magnetic domain wall in cylindrical microwire," IEEE Transactions on Magnetics 51, 1–6 (2015). https://doi.org/10.1109/tmag.2014.2374555 11. A. Jiménez, R. P. del Real, and M. Vázquez, "Controlling depinning and propagation of single domain walls in magnetic microwires," Eur. Phys. J. B 86, 113 (2013). https://doi.org/10.1140/epjb/e2013-30922-9

We choose a starting point of the theory, based on energy conservation law, adding a thermodynamic expression for the minimal work, necessary for DW creation, that we equalize the DW energy itself. The DW form, defined as the magnetic moment distribution in space, we take as an exact stationary solution of LLG equation, ^9,10,12 9. M. Vereshchagin, "Structure of domain wall in cylindrical amorphous microwire," Physica B: Condensed Matter 549, 91 (2018). https://doi.org/10.1016/j.physb.2017.10.065 10. A. Janutka and P. Gawroski, "Structure of magnetic domain wall in cylindrical microwire," IEEE Transactions on Magnetics 51, 1–6 (2015). https://doi.org/10.1109/tmag.2014.2374555 12. M. Vereshchagin, I. Baraban, S. Leble, and V. Rodionova, "Structure of head-to-head domain wall in cylindrical amorphous ferromagnetic microwire and a method of anisotropy coefficient estimation," Journal of Magnetism and Magnetic Materials 504, 166646 (2020). https://doi.org/10.1016/j.jmmm.2020.166646 extracting the necessary anisotropy parameter from explicit expression for mobility as in Ref. 12 12. M. Vereshchagin, I. Baraban, S. Leble, and V. Rodionova, "Structure of head-to-head domain wall in cylindrical amorphous ferromagnetic microwire and a method of anisotropy coefficient estimation," Journal of Magnetism and Magnetic Materials 504, 166646 (2020). https://doi.org/10.1016/j.jmmm.2020.166646 . Note, that it is, perhaps, the only possibility for amorphous microwire now. We also attribute the energy balance relation to the starting point (z = 0) of the double DW nucleation.

The scheme of the modeling looks as follows. We switch on the external magnetic field with a critical value H ₀, named as the nucleation field, considering the switching process as a quick. We suppose that the DW pair excitation is doing by a short coil, ¹³ 13. V. Rodioniova et al. , "The defects influence on domain wall propagation in bistable glass-coated microwires," Physica B 407, 1446–1449 (2012). https://doi.org/10.1016/j.physb.2011.09.125 such that the induced magnetic moment at the wire is localized to create the opposite phase domain. After the short transition time interval we suppose the field to be constant till the decay begins. The growing magnetic moment upto a time of DWs start reaches a value that defines the work that converts into two-walls energy, propagating in opposite sides.

The following study starts with the energy balance equation formulation, adjusted to the conventional ferromagnetic micro-wire geometry and DW initiation experiment. ^3,14 3. M. Ipatov, N. A. Usov, A. Zhukov, and J. Gonzalez, "Local nucleation fields of Fe-rich microwires and their dependence on applied stresses," Physica B 403, 379 (2008). https://doi.org/10.1016/j.physb.2007.08.054 14. A. Zhukov, M. Vázquez, J. Velázquez, A. Hernando, and V. Larin, "Magnetic properties of Fe-based glass-coated microwires," Magn. Magn. Mater. 170, 323 (1997). https://doi.org/10.1016/s0304-8853(97)00041-3 The resulting expression for the critical magnetic field H ₀ (H _n at Ref. 3 3. M. Ipatov, N. A. Usov, A. Zhukov, and J. Gonzalez, "Local nucleation fields of Fe-rich microwires and their dependence on applied stresses," Physica B 403, 379 (2008). https://doi.org/10.1016/j.physb.2007.08.054 ) contains material parameters that for amorphous ferromagnetic micro wire cannot be measured by conventional methods. In this relation enters also the DW mobility, that is also measurable as in Ref. 13 13. V. Rodioniova et al. , "The defects influence on domain wall propagation in bistable glass-coated microwires," Physica B 407, 1446–1449 (2012). https://doi.org/10.1016/j.physb.2011.09.125 . If in some case all necessary parameters are known, a comparison of the evaluated and measured H ₀ confirms the developed theory.

II. ENERGY BALANCE EQUATION FOR THE CYLINDRICAL AMORPHOUS MICROWIRE

Section:

A. Principle relation energy-work. Energy balance of a double DW creation

If we neglect the electromagnetic waves emission contribution in a relatively slow transition of the double DW switching, the thermodynamics gives for elementary work for unit volume

We denote a DW energy as

If, after short transition stage, the magnetic field $\vec{H}$ is constant, the work is made due to the magnetization vector growth. Let us remind that the few-turn coil, placed far from ends, launches two DW in opposite directions. So, finally the energy balance equation reads

that, for a minimal work, necessary for nucleation, we shall consider as a basic equation to determine H = H ₀, as a minimal field, that could create the double DW. ¹³ 13. V. Rodioniova et al. , "The defects influence on domain wall propagation in bistable glass-coated microwires," Physica B 407, 1446–1449 (2012). https://doi.org/10.1016/j.physb.2011.09.125

To close the description we need equation of state $\vec{M} (\vec{H})$ , that, for a ferromagnetic, is presented by hysteresis loop. Such function is not unique, hence strongly depends on initial state of the matter. In this work we take the ascending branch of the hysteresis loop curve of the loop, because the initial magnetisation has opposite direction with respect to the initiation field.

The energy of a domain wall with cylindrical symmetry in cylindrical coordinates and SGS units, which we will hold (besides few cases for convenience for a comparison with a source), is evaluated by integration over the domain wall, taking surface energy density

where, A is the exchange stiffness, K is the anisotropy constant, see e.g., Ref. 15 15. L. V. Panina, M. Ipatov, V. Zhukova, and A. Zhukov, "Domain wall propagation in Fe-rich amorphous microwires," Physica B: Condensed Matter 407, 1442–1445 (2012). https://doi.org/10.1016/j.physb.2011.06.047 .

We, in this paper, would concentrate on the case of plane DW. ^9,12 9. M. Vereshchagin, "Structure of domain wall in cylindrical amorphous microwire," Physica B: Condensed Matter 549, 91 (2018). https://doi.org/10.1016/j.physb.2017.10.065 12. M. Vereshchagin, I. Baraban, S. Leble, and V. Rodionova, "Structure of head-to-head domain wall in cylindrical amorphous ferromagnetic microwire and a method of anisotropy coefficient estimation," Journal of Magnetism and Magnetic Materials 504, 166646 (2020). https://doi.org/10.1016/j.jmmm.2020.166646 Its energy is obviously minimal, compared with others, so it is excited first. For a plane wall of elliptic form with the semi-minor and semi-major axes a, b the energy is approximately equal to

where

By definition, a coincides with the radius of the amorphous ferromagnetic core of a wire, then, introducing the DW length Δ along z, we derive

Then, plugging in double (2), yields

Let H ₀ be the nucleation field ¹³ 13. V. Rodioniova et al. , "The defects influence on domain wall propagation in bistable glass-coated microwires," Physica B 407, 1446–1449 (2012). https://doi.org/10.1016/j.physb.2011.09.125 at wire axis in the generating coil plane, having hysteresis loops of glass-covered microwire with the saturation parameter M _s value. For the energy balance we write

H_{0} M_{s} V_{s} = 2 E_{D W} = 4 π a \sqrt{K A} \sqrt{4 a^{2} + Δ^{2}},

(4)

A comment on the relation between value H ₀ and one directly measured by the solenoid current is placed at the end of the supplementary material I. A modification of the formula (1) is presented it the supplementary material Eq. (2).

B. Model verification

To test the proposed theory on example of the H ₀ measurements results we need values of all parameters in the right-hand-side (rhs) of the resulting formula (4). The paper ¹¹ 11. A. Jiménez, R. P. del Real, and M. Vázquez, "Controlling depinning and propagation of single domain walls in magnetic microwires," Eur. Phys. J. B 86, 113 (2013). https://doi.org/10.1140/epjb/e2013-30922-9 contains rich information about the DW parameters, including the length, estimated via Faraday pulse in a pickup coil. The Fe-rich composition (Fe ₇₉ Si ₁₀ B ₈ C ₃) of the microwire ensures the magnetic bistability, and its metal radius $a_{met} = \frac{d_{met}}{2} = 10.25 μ m$ quite fit the DW creation theory conditions. The problem is how to extract the mobility from the figure that is drawn in the paper: ¹¹ 11. A. Jiménez, R. P. del Real, and M. Vázquez, "Controlling depinning and propagation of single domain walls in magnetic microwires," Eur. Phys. J. B 86, 113 (2013). https://doi.org/10.1140/epjb/e2013-30922-9 the small angle of the curve v-H inclination for the linear part. One more question is to define the volume in which DW creation take place, because the DW under consideration in Ref. 11 11. A. Jiménez, R. P. del Real, and M. Vázquez, "Controlling depinning and propagation of single domain walls in magnetic microwires," Eur. Phys. J. B 86, 113 (2013). https://doi.org/10.1140/epjb/e2013-30922-9 appears in a vicinity of the wire end. It is produced by long solenoid, hence the scale of phenomenon is determined, perhaps, by the DW dimension. Third, as it is shown in the Suppl. II of our letter, we need detailed information about hysteresis loop.

In Ref. 13 13. V. Rodioniova et al. , "The defects influence on domain wall propagation in bistable glass-coated microwires," Physica B 407, 1446–1449 (2012). https://doi.org/10.1016/j.physb.2011.09.125 all the parameters besides the DW length Δ may be extracted from the experimental data. The value of DW mobility is evaluated from the plot Fig. 2a (red rounds) and estimated as $S = \frac{1200 - 411}{285 - 90} = 4.05$ (SI units). It gives for K = 7.58 × 10⁷, SGS.

The DW length is evaluated by authors of Refs. 16 16. A. Zhukov, "Design of the magnetic properties of Fe-rich, glass-coated microwires for technical applications," Adv. Funct. Mater. 16, 675 (2006). https://doi.org/10.1002/adfm.200500248 and 17 17. S. A. Gudoshnikov, Y. B. Grebenshchikov, B. Y. Ljubimov, P. S. Palvanov, N. A. Usov, M. Ipatov, A. Zhukov, and J. Gonzalez, "Ground state magnetization distribution and characteristic width of head to head domain wall in Fe-rich amorphous microwire," Physica Status Solidi A 206(4), 613–617 (2009). https://doi.org/10.1002/pssa.200881254 as for different microwires, the results of measurements vary from 70a to 150a; we, for the wire of diameter 2a = 19.4 μm ¹³ 13. V. Rodioniova et al. , "The defects influence on domain wall propagation in bistable glass-coated microwires," Physica B 407, 1446–1449 (2012). https://doi.org/10.1016/j.physb.2011.09.125 and the mean (along the wire) magnetic field value (about 380 A/m=4.6 Oe), estimate as Δ = 50a, by the corresponding plot extrapolation. The rhs of (16) is calculated for M _s = 1000 emu/cm ³, see Ref. 3 3. M. Ipatov, N. A. Usov, A. Zhukov, and J. Gonzalez, "Local nucleation fields of Fe-rich microwires and their dependence on applied stresses," Physica B 403, 379 (2008). https://doi.org/10.1016/j.physb.2007.08.054 . The volume of the double DW generation may be estimated as cross-section of the ferromagnetic core πa ², multiplied by the product of number of turns of the exiting coil and its wire diameter as 10 × 0.05 cm.

The expression (4) for the nucleation field gives

H_{0} = \frac{4 π a}{M_{s} V_{s}} \sqrt{K A} \sqrt{4 a^{2} + Δ^{2}} = \frac{8}{M_{s} a^{2}} \sqrt{10 K A} .

(5)

Numerically it gives in SI: $H_{0} = \frac{4}{π} \sqrt{10 K A} = 158,8$ A/m. The generating coil of radius R = 0.25 mm was used in experiments of Ref. 13 13. V. Rodioniova et al. , "The defects influence on domain wall propagation in bistable glass-coated microwires," Physica B 407, 1446–1449 (2012). https://doi.org/10.1016/j.physb.2011.09.125 . So, finally, we obtain the theoretical value for H ₀, while the mean value of measured H ₀ along the wire gives 381 A/m, with the good coincidence with the estimation by (5).

III. CALCULATIONS AND COMPARISON WITH EXPERIMENT

Section:

A. Matter parameters estimation

To apply the formula (4), apart from geometry parameters a, R, Δ, we need the matter parameters K, A.

Let us reproduce the approximate relation for the anisotropy constant K

that is derived on base of LLG equation solution, from the formula for DW mobility S, which includes the gyro-magnetic relation γ and the Gilbert damping coefficient α. ¹² 12. M. Vereshchagin, I. Baraban, S. Leble, and V. Rodionova, "Structure of head-to-head domain wall in cylindrical amorphous ferromagnetic microwire and a method of anisotropy coefficient estimation," Journal of Magnetism and Magnetic Materials 504, 166646 (2020). https://doi.org/10.1016/j.jmmm.2020.166646

We take the exchange stiffness numerical value in SGS as A(Fe) = 2 × 10⁻⁶, that is given by many sources, see e.g., Refs. 15 15. L. V. Panina, M. Ipatov, V. Zhukova, and A. Zhukov, "Domain wall propagation in Fe-rich amorphous microwires," Physica B: Condensed Matter 407, 1442–1445 (2012). https://doi.org/10.1016/j.physb.2011.06.047 and 17 17. S. A. Gudoshnikov, Y. B. Grebenshchikov, B. Y. Ljubimov, P. S. Palvanov, N. A. Usov, M. Ipatov, A. Zhukov, and J. Gonzalez, "Ground state magnetization distribution and characteristic width of head to head domain wall in Fe-rich amorphous microwire," Physica Status Solidi A 206(4), 613–617 (2009). https://doi.org/10.1002/pssa.200881254 . We need also γ and dimensionless α = 0.016, see, for example, ¹⁸ 18. S. Atalay and P. T. Squire, "Magnetomechanical damping in FeSiB amorphous wires," Journal of Applied Physics 73(2), 871–875 (1993). https://doi.org/10.1063/1.353299 and for the product in SGS we have γα = 2.94 × 10⁵ sec ⁻¹ Oe ⁻¹.

Estimation for the mobility in SGS units, taking, for Example S = 4, as for glass-coated microwire in Ref. 12 12. M. Vereshchagin, I. Baraban, S. Leble, and V. Rodionova, "Structure of head-to-head domain wall in cylindrical amorphous ferromagnetic microwire and a method of anisotropy coefficient estimation," Journal of Magnetism and Magnetic Materials 504, 166646 (2020). https://doi.org/10.1016/j.jmmm.2020.166646 , gives K = 7.78 × 10⁷. See also supplementary material 1.

B. Comparison with experiments

1. On parameters extraction. Anisotropy constant

We take the results of DW mobility measurements from Ref. 13 13. V. Rodioniova et al. , "The defects influence on domain wall propagation in bistable glass-coated microwires," Physica B 407, 1446–1449 (2012). https://doi.org/10.1016/j.physb.2011.09.125 , where two cases are considered. In this paper the authors investigated ten pieces of the Fe ₇₄ B ₁₃ Si ₁₁ C ₂ microwires from the same batch with the metallic nucleus diameters d = 19.4 μm and total microwires diameters D = 26.6 μm. The length of each sample was 10 cm. All studied samples exhibited rectangular hysteresis loops, as previously observed in Fe-rich microwires. ¹⁶ 16. A. Zhukov, "Design of the magnetic properties of Fe-rich, glass-coated microwires for technical applications," Adv. Funct. Mater. 16, 675 (2006). https://doi.org/10.1002/adfm.200500248 In this section we take data from Ref. 13 13. V. Rodioniova et al. , "The defects influence on domain wall propagation in bistable glass-coated microwires," Physica B 407, 1446–1449 (2012). https://doi.org/10.1016/j.physb.2011.09.125 . Minimal values of magnetic field are about 80 A/m (sample a) and 100 A/m (sample b), see again Ref. 13 13. V. Rodioniova et al. , "The defects influence on domain wall propagation in bistable glass-coated microwires," Physica B 407, 1446–1449 (2012). https://doi.org/10.1016/j.physb.2011.09.125 , that we use for DW mobility evaluation.

The results of measurements of distribution of the local nucleation fields are represented by Figs. 1 and 2 here for a reader convenience.

FIG. 1. The values of H ₀ in A/m as function of coordinate z in mm, sample a.

High-resolution

FIG. 2. The values of H ₀ in A/m as function of coordinate z in mm, sample b.

High-resolution

Now we take the mobility directly from data prepared for the plots of velocity a and b in Ref. 13 13. V. Rodioniova et al. , "The defects influence on domain wall propagation in bistable glass-coated microwires," Physica B 407, 1446–1449 (2012). https://doi.org/10.1016/j.physb.2011.09.125 . The metal wire radius $a = \frac{19.4}{2} = 9 . 7 μ m$ . The number of turns of the exciting coil at Ref. 13 13. V. Rodioniova et al. , "The defects influence on domain wall propagation in bistable glass-coated microwires," Physica B 407, 1446–1449 (2012). https://doi.org/10.1016/j.physb.2011.09.125 is ten, the coil's wire diameter is d = 0.05 cm so, the volume V _s may be estimated as V _s = 1.5 × 10⁻⁶ cm ³.

The data from Ref. 13 13. V. Rodioniova et al. , "The defects influence on domain wall propagation in bistable glass-coated microwires," Physica B 407, 1446–1449 (2012). https://doi.org/10.1016/j.physb.2011.09.125 are taken for coils of measurements by the coils shifted on 2 cm to the left and right from the generating one. The results for pairs of coils, at the figures from Ref. 13 13. V. Rodioniova et al. , "The defects influence on domain wall propagation in bistable glass-coated microwires," Physica B 407, 1446–1449 (2012). https://doi.org/10.1016/j.physb.2011.09.125 are marked by as "red" (for the right) and as "black" (for the left) points. Let us present the corresponding numbers at the Table I. The details of the parameters evaluation is given at the supplementary material.

Table icon

TABLE I. Mobility (S) evaluated by the dependence of DW velocity on magnetic field H: v(H); and anisotropy coefficient (K) via the expression (6) in SGS. ¹³ 13. V. Rodioniova et al. , "The defects influence on domain wall propagation in bistable glass-coated microwires," Physica B 407, 1446–1449 (2012). https://doi.org/10.1016/j.physb.2011.09.125 The initial point of the dependence v(H) marked as "min (v.H)" and the final measured point, as "max (v,H)." The cases a and b differ by samples and by results of estimation of mobility by means of a plots inclination for the left (black) and right (red) coils, as marked in Ref. 13 13. V. Rodioniova et al. , "The defects influence on domain wall propagation in bistable glass-coated microwires," Physica B 407, 1446–1449 (2012). https://doi.org/10.1016/j.physb.2011.09.125 .

Sample	2a (black)	2a (red)	2b (red)	2b (black)
min (v, H)	90, 411	90, 423	99, 411	99, 423
max (v, H)	230, 1023	230, 1004	450, 1372	460, 1172
S	4.37	4.15	2.74,	2.07
K	6.51 × 10⁷	7.22 × 10⁷	1.66 × 10⁸	2.89 × 10⁸

2. Comparison with experiment. Final table

The theoretical estimations and averaged measurements (by Figs 1 and 2 ) of the nucleation field H ₀ are presented in the Table II.

Table icon

TABLE II. Comparison with experiment. ¹³ 13. V. Rodioniova et al. , "The defects influence on domain wall propagation in bistable glass-coated microwires," Physica B 407, 1446–1449 (2012). https://doi.org/10.1016/j.physb.2011.09.125 Theoretical values for the nucleation field are calculated by the expression (5), the cases a and b differ by samples and results of estimation of mobility via v(H) plots for the left (black) and right (red) coils, and, next - via the anisotropy coefficient K (see the Table I).

Field	H _0ab	H _0ar	H _0br	H _0bb
Theory	363.5	359.1	580.4	765.9
Experiment	381	381	633	633

As it is seen from the results, the nucleation field estimation on base of formula (5) looks better for the sample "a," for which the v(H) dependence is linear, compared to the sample "b" that deviate from linear dependence, i.e. exhibit (small) acceleration, see the "Discussion."

3. Discussion

We choose a ferromagnetic microwire as the first theory realization for few reasons. First, there is rich experimental information about values of the parameter H ₀, under the name "critical propagation field" of a DW pair excitation by few coils ¹³ 13. V. Rodioniova et al. , "The defects influence on domain wall propagation in bistable glass-coated microwires," Physica B 407, 1446–1449 (2012). https://doi.org/10.1016/j.physb.2011.09.125 or as "nucleation field"; ³ 3. M. Ipatov, N. A. Usov, A. Zhukov, and J. Gonzalez, "Local nucleation fields of Fe-rich microwires and their dependence on applied stresses," Physica B 403, 379 (2008). https://doi.org/10.1016/j.physb.2007.08.054 the second reason is the cylindrical symmetry and small cross-dimension that simplifies theoretical relations. Third, there are publications that propose a good zoo of DW forms, their parameters and dynamics. ^6,9,19 6. A. Chizhik, J. Gonzalez, A. Zhukov, and P. Gawronski, "Study of length of domain walls in cylindrical magnetic microwires," Journal of Magnetism and Magnetic Materials 512, 167060 (2020). https://doi.org/10.1016/j.jmmm.2020.167060 9. M. Vereshchagin, "Structure of domain wall in cylindrical amorphous microwire," Physica B: Condensed Matter 549, 91 (2018). https://doi.org/10.1016/j.physb.2017.10.065 19. A. Chizhik, A. Zhukov, J. Gonzalez, and A. Stupakiewicz, "Basic study of magnetic microwires for sensor applications: Variety of magnetic structures," Journal of Magnetism and Magnetic Materials 422, 299–303 (2017). https://doi.org/10.1016/j.jmmm.2016.09.011

We fix our main efforts of this and Sec. IV on the case of few coils excitation of DW far from a wire ends. A DW observation is performed by pickup coils posed close to generating one. ^13,17 13. V. Rodioniova et al. , "The defects influence on domain wall propagation in bistable glass-coated microwires," Physica B 407, 1446–1449 (2012). https://doi.org/10.1016/j.physb.2011.09.125 17. S. A. Gudoshnikov, Y. B. Grebenshchikov, B. Y. Ljubimov, P. S. Palvanov, N. A. Usov, M. Ipatov, A. Zhukov, and J. Gonzalez, "Ground state magnetization distribution and characteristic width of head to head domain wall in Fe-rich amorphous microwire," Physica Status Solidi A 206(4), 613–617 (2009). https://doi.org/10.1002/pssa.200881254 As it was already said, a possibility to evaluate a parameter depends on access to all others in a measurement and, in frame of the theory, fix absence of acceleration. The presence of defects change the DW dynamics, the basic LLG equation should also be modified, see the paper. ²⁰ 20. S. B. Leble and V. V. Rodionova, "Dynamics of domain walls in a cylindrical amorphous ferromagnetic microwire with magnetic inhomogeneities," Theor. Math. Phys. 202(2), 252 (2020). https://doi.org/10.1134/s0040577920020087 In example of the DW length extraction necessity, we should measure the mobility, having material parameters of the given wire, such as A and M _s. More accessible geometry parameters as wire radius and pickup coil dimension are implied as known. The quality of the result is determined by errors of measurements and approximations made in formulas, those, in this paper, were following.

1.	The DW form is taken as simplest (plane). Another forms, e.g., the flexural plane ¹² 12. M. Vereshchagin, I. Baraban, S. Leble, and V. Rodionova, "Structure of head-to-head domain wall in cylindrical amorphous ferromagnetic microwire and a method of anisotropy coefficient estimation," Journal of Magnetism and Magnetic Materials 504, 166646 (2020). https://doi.org/10.1016/j.jmmm.2020.166646 should be investigated and compared with the plane one.
2.	In the approximate expression (6) for K we could take the next term from Ref. 12 12. M. Vereshchagin, I. Baraban, S. Leble, and V. Rodionova, "Structure of head-to-head domain wall in cylindrical amorphous ferromagnetic microwire and a method of anisotropy coefficient estimation," Journal of Magnetism and Magnetic Materials 504, 166646 (2020). https://doi.org/10.1016/j.jmmm.2020.166646 .

We understand, that the measurement of the critical field H ₀ implies account for the number of turns of generating coil and its dimension because a difference between solenoid formula and real value of the field in the coil axis. A best possibility to avoid of the discrepancy is to have information about current in the coil. ²¹ 21. J. D. Jackson, Classical Electrodynamics (Wiley, 1962). There is a model that describes this difference on base of hypothesis that the field is calculated via solenoid formula. It means that we should estimate the current by H ₀ and, afterwards, recalculate the field at the microwire axis. In the paper ¹³ 13. V. Rodioniova et al. , "The defects influence on domain wall propagation in bistable glass-coated microwires," Physica B 407, 1446–1449 (2012). https://doi.org/10.1016/j.physb.2011.09.125 that we mainly address, the generating coil radius is R = 0.25mm, the turn number is 10. Estimations of the field inside a coil by the textbook formula and discussion of the last section show the direction of the theory upgrade to improve its coincidence with experiment that we perform for data from. ^13,17 13. V. Rodioniova et al. , "The defects influence on domain wall propagation in bistable glass-coated microwires," Physica B 407, 1446–1449 (2012). https://doi.org/10.1016/j.physb.2011.09.125 17. S. A. Gudoshnikov, Y. B. Grebenshchikov, B. Y. Ljubimov, P. S. Palvanov, N. A. Usov, M. Ipatov, A. Zhukov, and J. Gonzalez, "Ground state magnetization distribution and characteristic width of head to head domain wall in Fe-rich amorphous microwire," Physica Status Solidi A 206(4), 613–617 (2009). https://doi.org/10.1002/pssa.200881254

IV. CONCLUSION

Section:

Investigation of wires with a significant amount of defects exhibits in-homogeneity of magnetic properties. These are nucleation field variation along the wire as shown in the figures of this letter. Next we return to the paper ¹³ 13. V. Rodioniova et al. , "The defects influence on domain wall propagation in bistable glass-coated microwires," Physica B 407, 1446–1449 (2012). https://doi.org/10.1016/j.physb.2011.09.125 where the local nucleation field is studied in the context of the DW dynamics. The details of the magnetic properties definitely change the kinematics of DW movement. ²⁰ 20. S. B. Leble and V. V. Rodionova, "Dynamics of domain walls in a cylindrical amorphous ferromagnetic microwire with magnetic inhomogeneities," Theor. Math. Phys. 202(2), 252 (2020). https://doi.org/10.1134/s0040577920020087 Therefore a link of the nucleation field value and the DW parameters is of interest. We suggest the use of the derived relations [(5), Eq. (2) -supplementary material] for the directly observed values for nucleation field and such parameters as DW length and thickness. Keeping in mind the DW kinematics in a given wire we can evaluate the set of necessary parameters for manipulating the wire in a device. To the motivation we already mentioned, we would add the following: 1. Considering the simplest theory, we take the plane DW example, motivated by minimal energy value compared against rich variety of magnetic structures. ¹⁸ 18. S. Atalay and P. T. Squire, "Magnetomechanical damping in FeSiB amorphous wires," Journal of Applied Physics 73(2), 871–875 (1993). https://doi.org/10.1063/1.353299 For example for a conic DW its lateral area is equal πLa > πab: The value of H0 grows but dependence on mobility rests the same. 2. Let us add that for a non-plane DW the second term in (1) should be taken into account, which would change the DW energy value.

The suggested rather simple model may be significantly improved from a more exact theoretical elements account, pointed in the text, as from the experimental side that is also quite visible. Estimations of the field inside a coil by the textbook formula as in Ref. 21 21. J. D. Jackson, Classical Electrodynamics (Wiley, 1962). and discussion of the last section show the direction of the theory upgrade to improve its coincidence with an experiment that we perform for data from Refs. 13 13. V. Rodioniova et al. , "The defects influence on domain wall propagation in bistable glass-coated microwires," Physica B 407, 1446–1449 (2012). https://doi.org/10.1016/j.physb.2011.09.125 and 17 17. S. A. Gudoshnikov, Y. B. Grebenshchikov, B. Y. Ljubimov, P. S. Palvanov, N. A. Usov, M. Ipatov, A. Zhukov, and J. Gonzalez, "Ground state magnetization distribution and characteristic width of head to head domain wall in Fe-rich amorphous microwire," Physica Status Solidi A 206(4), 613–617 (2009). https://doi.org/10.1002/pssa.200881254 .

We would add, that the suggested theory may be directly generalized to account electric field action in equation of state as in Ref. 22 22. S. Leble, "Heisenberg chain equations in terms of Fockian covariance with electric field account and multiferroics in nanoscale," Nanosystems: Physics, Chemistry, Mathematics 10(1), 18–30 (2019). https://doi.org/10.17586/2220-8054-2019-10-1-18-30 on a way to multiferroics.

SUPPLEMENTARY MATERIAL

Section:

DATA AVAILABILITY

Section:

The data that support the findings of this study are available within the article and its supplementary material.

REFERENCES

Section:

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