Domain wall mobility engineering by a perpendicular magnetic field in microwires with a gradient of perpendicular anisotropy

Fast domain wall (DW) motion is a vital factor of modern magnetic sensors and spintronic devices.^1–5 Among other things, current applications require small size, low cost, and multifunctional materials. These properties are typical for amorphous magnetically bistable microwires, which appear to be a desirable material for efficient use in practice. The study of the magnetization process of the DW motion still offers unexpected effects. Several studies confirmed a single domain structure of amorphous microwires with positive magnetostriction. The axial magnetic domain is covered by a thin shell of radial domains and closure domains at the wire ends.^6–9 The magnetization reversal runs through single DW propagation along the entire length of a wire. Various parameters can effectively control the DW motion.^10–13 One of them is a perpendicular magnetic field.^14–16 

Despite the microwires’ cylindrical shape, the DW velocity increases in a perpendicular field applied in one direction and decreases in the opposite direction.¹⁴ Additionally, a perpendicular field enhances the DW velocity at a particular rotation of microwire.¹⁵ A gradient of a perpendicular magnetic anisotropy can explain such an effect. Such a gradient is induced to microwire during the fabrication process because the wire is cooled asymmetrically from one side by a flowing water stream.¹⁷ 

Our work’s main goal is to extend current knowledge of the DW dynamics in wires with a perpendicular anisotropy gradient. The effect of the perpendicular field plays an essential role in DW mobility.

The effect of the perpendicular field is studied on samples with composition Fe₇₆Si₉B₁₀P₅. Three samples have a different ratio of metallic core d, and a glass coat D. They were prepared by the Taylor-Ulitovski method.¹⁷ The length of samples is 10 cm and the dimensions are given as follows: Sample no.1 d = 24 µm, D = 81 µm (d/D = 0.296), sample no.2 d = 16.5 µm, D = 51 µm (d/D = 0.324), and sample no.3 d = 23 µm, D = 59 µm (d/D = 0.389).

A common approach for measurements of a DW dynamics is a simple Sixtus-Tonks method.¹⁸ This method is based on an excitation coil providing the axial magnetic field and two pick-up coils located at a well-defined distance. Perpendicular fields are generated by Helmholtz coils (Fig. 1(a)). The measurements are done at the highest perpendicular field that allows single-domain wall propagation. At higher fields, an offset of the perpendicular field generates the axial field component that leads to the multiple DW propagation. The experiment is designed to rotate the microwire with a coil system (primary coil and sensing coils) around its axis in Helmholtz coils. The angle of rotation θ is between the normal to the axis of the microwire (the y-axis direction, see Fig. 1(b)) and the vector of the perpendicular magnetic field H_T (the z-axis direction).

In the viscous regime of a DW propagation, the DW velocity is described by a linear equation of motion:¹⁹ 

where S is the DW mobility, and H₀ is the critical magnetic field. The relation is also valid in the presence of the perpendicular field. However, as shown in our work,¹⁵ the DW velocity and mobility in circular wires vary with the angle θ. Moreover, variation of the DW velocity and mobility is different for wires with different d/D ratio (Fig. 2). The thinner the glass-coating, the DW mobility varies more (Fig. 2(a)), in comparison to the sample with the thicker glass-coating (Fig. 2(b)).

Fig. 3 shows angular dependencies of the DW mobilities extracted from Fig. 2. DW mobilities of samples with larger or medium d/D show periodic oscillations. On the other hand, the sample with the smallest value of the d/D ratio exhibits almost no angular dependence on mobility. For larger d/D, a small perpendicular magnetic field has the highest effect on the DW mobility (for H_T = 95 A/m data taken from,¹⁵ see Fig. 3). On the other hand, the sample with the smallest d/D shows a small angular dependence of mobility even in a high perpendicular magnetic field (3100 A/m).

One possible explanation of the angular dependence of mobility in the perpendicular magnetic field could be attributed to the perpendicular magnetoelastic anisotropy.¹⁵ Direct observation of such a gradient is not possible in amorphous samples. However, after annealing above its crystallization temperature, the crystalline structure is clearly visible in a wire’s cross-section. The direction of crystal growth follows internal stresses resulting from the fabrication process (Fig. 4). There are also other possible sources of such crystallization like defects,²⁰ frozen crystallization centres,²¹ surface irregularities,²² etc. Anyway, even in their case, they will provide a gradient of homogeneity in the transversal direction of the wire that could finally influence the dependence of the domain wall dynamics on the transversal field’s radial rotation.

Such gradient looks weak compared to the axial stresses induced by drawing and cannot define the microwire’s domain structure. However, it looks high enough to define the direction of magnetization in the centre of the domain wall (where magnetic moments are in delicate equilibrium being rotated out of the easy axis).

Much work has been done to understand the distribution of internal stresses from the fabrication process.^23–28 Knowledge of stress distributions is certainly essential for the magnetic properties of wires. The internal stresses arise from the cooling process due to the glass cover and metallic core’s thermal coefficients. Therefore, the strength of induced stresses depends on the radius of the metallic core d and total diameter D of the microwire.²⁹ When the glass coat is thin, the metallic nucleus cools faster. In turn, faster cooling results in a larger cross-sectional gradient of the perpendicular anisotropy (see Fig. 4(a) – fine longitudinal crystals after annealing oriented in a shell-like structure). Thus, even a low perpendicular field has a significant effect. A decrease in the d/D ratio is associated with a lower cooling rate of the wire. Consequently, a smaller gradient is induced in the microwire’s cross-section, which is visible by larger crystals after the annealing process (Fig. 4(b)). In the case of a very thick glass coat (i.e., smaller d/D ratio), the cooling process is so slow that the induced gradient is negligible so that a minor effect of the perpendicular magnetic field is observed (see Fig. 4(c) – larger crystals after annealing with vanishing shell-like structure).

It is worth mentioning that depinning fields of domain walls in the closure domain structure naturally limit the perpendicular magnetic field’s application. For a sufficiently large perpendicular field, a multiple domain wall propagation occurs. In turn, single domain wall propagation is persisting up to certain fields (marked as the maximum perpendicular fields). Fig. 5 shows that the maximum perpendicular field varies with the d/D ratio as well. Thus, perpendicular fields can control DW motion with high efficiency in microwires with thicker glass coating.

In conclusion, we studied the domain wall motion in amorphous glass-coated microwires with gradients of the perpendicular magnetic anisotropy. Rotations of wires in the perpendicular magnetic field lead to a periodic dependence of DW mobilities. Wires with larger d/D ratio are more sensitive to perpendicular fields than wires with lower (or intermediate) d/D ratio. The larger the d/D ratio, the larger the gradient of the perpendicular magnetoelastic anisotropy, and consequently, the lower perpendicular magnetic field is necessary to obtain the effect. The measured data suggest a maximum perpendicular field at which there is still a single-domain structure. Such a maximum perpendicular field decreases with increasing d/D ratio.

Correlation between d/D ratio and the amplitude of DW mobility in the perpendicular magnetic field can be used in applications. Microwires with a high d/D ratio offer a possibility of well-controlled domain wall dynamics by the perpendicular magnetic field. On the other hand, microwires with small d/D are ideal for applications where the perpendicular magnetic field’s influence is not desirable.

This work was supported by Slovak Grant Agency VEGA 1/0053/19; Slovak Grant Agency grant number APVV-16-0079, APVV-17-0184, APVV SK-FR-2017-024 and projects supported by Pavol Jozef Šafárik university VVGS-2016-320.

All authors contributed equally to this work.

The data that support the findings of this study are available from the corresponding author upon reasonable request.