The profile of the domain walls in amorphous glass-covered microwires

Accepted Manuscript The profile of the domain walls in amorphous glass-covered microwires F. Beck, J.N. Rigue, M. Carara PII: DOI: Reference:

S0304-8853(16)32775-5 http://dx.doi.org/10.1016/j.jmmm.2017.03.003 MAGMA 62524

To appear in:

Journal of Magnetism and Magnetic Materials

Received Date: Revised Date: Accepted Date:

25 October 2016 8 February 2017 3 March 2017

Please cite this article as: F. Beck, J.N. Rigue, M. Carara, The profile of the domain walls in amorphous glasscovered microwires, Journal of Magnetism and Magnetic Materials (2017), doi: http://dx.doi.org/10.1016/j.jmmm. 2017.03.003

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

The profile of the domain walls in amorphous glass-covered microwires F. Beck¹, J.N. Rigue¹ and M. Carara² ¹Universidade Federal de Santa Maria, Campus Cachoeira do Sul, RS, Brazil ²Departamento de Física, Universidade Federal de Santa Maria, Santa Maria, RS, Brazil We have studied the domain wall dynamics in Joule-annealed amorphous glass-covered microwires with positive magnetostriction in the presence of an electric current, in order to evaluate the profile and shape of the moving domain wall. Such microwires are known to present magnetic bi-stability when axially magnetized. The single domain wall dynamics was evaluated under different conditions, under an axially applied stress and an electric current. We have observed the well known increasing of the domain wall damping with the applied stress due to the increase in the magnetoelastic anisotropy and, when the current is applied, depending on the current intensity and direction, a modification on the axial domain wall damping. When the orthogonal motion of the domain wall is considered, we have observed that the associated velocity present a smaller dependence on the applied current intensity. It was observed a modification on both the domain wall shape and length. In a general way, the domain wall evolves from a bell shape to a parabolic shape as the current intensity is increased. The results were explained in terms of the change in the magnetic energy promoted by the additional Oersted field. Index Terms — glass-coated microwires, domain structure, domain wall dynamics, domain-wall shape, domain wall energy

Introduction The potential application of domain walls (DW) in logical devices has attracted a great deal of attention to this issue in the past few years [1-3]. Many of those works are focused on the domain wall dynamics. The ability of microwires with positive magnetostriction (λS) to reverse its magnetization upon the displacement of just one domain wall, running from one end to the other, provides an additional way to the study of domain wall dynamics which, finally, can be applied to other systems [4]. However, there are still some doubts about the specific shape or profile of the domain wall and such understanding would help the design of new devices based on magnetization reversion in microwires. It is well known that the domain wall structure of a magnetic material is due to the minimization of the system’s free magnetic energy, and also, the shape of an individual magnetic domain wall must obey such minimization. While we have some insight on the domain wall structure, meaning a core-shell one [5, 6], or a single domain axially magnetized [7], the local description of the magnetic free energy is a hard task for this kind of wire, by the same reason that gave to the microwires its outstanding properties: the complex distribution of frozen stress, induced by the fabrication process and the presence of the glass cover [8]. There are some works treating the shape and the dynamics of the domain walls in microwires and some of them introduce domain walls with different shapes. Some of these works are based on the study of the induced emf in the pick-up coil during the passage of the DW. Studying the conventional amorphous wires (without glass cover), D.-X. Chen and coworkers [9] have tested the emf when the pick-up coils are crossed by different wall types such as tubular shrinking and planar. S. A. Gudoshnikov et al [10] have also analyzed the DW structure from the induced emf in the pickup coils, having found that the DW is much larger than the microwire radius and it has not any abrupt frontier. In reference [11] M. Tibu et al. have designed a combined method, Kerr effect and Sixtus-Tonks, to analyze the shape of the propagating domain wall. For FeSiB microwires they have concluded that the DW presents a parabolic profile. In ref. [12] J. Gonzalez et al have made direct observation of the domain structure in glass-covered microwires by using the magneto-optical Kerr-effect (MOKE) having related the magnetization reversal with the peculiarities of the surface domain structure. From different methods transverse [13], flexible transverse [14,15], conical [16] and vortex [17] are some examples of the proposed shapes for the head-to-head DW occurring in the core of the microwires.

In order to contribute to this discussion, we have measured the DW dynamics by applying, with the magnetizing field, a current along the wires axis and an axial stress. Associated to this current there is a circumferential field (Hφ) which actuates in a different way according to the DW general shape. In a conical DW, the Hφ will be in the wall’s plane, otherwise for a transverse domain, Hφ would be transversal which, by its time, changes the DW mobility. We have demonstrated a dependence of the axial and orthogonal DW velocity with the applied current. The results have shown that the DW dynamics changes are due to modifications in the magnetic energy which, by its time, modifies the DW length and shape. Based on the work of Panina et al. [18] which considers the magnetic energies to form the DW and the eddy current damping on the orthogonal DW motion, we have proposed, in a very simplified way, a general shape for the DW profile for both cases with and without an applied current.

Experimental The amorphous glass-covered microwires studied in this work were produced by the Taylor-Ulitovsky technique, with nominal composition Fe77.5Si7.5B15. Reference [19] presents a review on the preparation method and main characteristics of this kind of wire. The samples diameters are 47 µm for the whole wire and 25 µm for metallic core (r0). Pieces with 30 cm in length were studied. The axial DW velocity (v) was measured using a Sixtus-Tonks based experiment [20, 21]. The experimental setup consists of four coaxial coils: one to excite the DW, two sensing coils to detect the DW transit and a pinning coil to guarantee that the DW will move just when the exciting field is stabilized. One of the extremes of the sample was kept out of the exciting coil to ensure that just one DW is put in motion. A current (I) was also applied to the microwire, with different intensities and directions, during the DW velocity measurements. Figure 1 presents the signal detected from the sensing coils, superposed to the exciting magnetic field. Note that the DW is depinned only when the exciting magnetic field is stabilized. The DW velocity is calculated as v=∆x/∆t. ∆x (= 3 cm) is the distance between the sensing coils and ∆t is the time interval between the maximums of the signal detected by the sensing coils when the DW passes them. For each applied field ∆t was obtained by averaging ten measurements. The DW length Lw is directly proportional to the pulse duration ∆τ and the axial velocity

Lw = v ∆τ – 0.2 cm.

(1)

The 0.2 cm value refers to the pickup coils length. Likewise, for each applied field the pulse duration was obtained by averaging ten measurements. The relaxation time of the wall’s detecting system (sensing coils and amplification) is less than 20 µs, meaning at least one order of magnitude lower than the pulse length (∆τ). Such relaxation time was not considered in expression (1) to calculate the DW length. The experimental setup it is better described in ref. [22, 23]. ∆t 10

dφ /dt (a.u.)

Field (Oe)

5 Field dφ1/dt

Delay

dφ2/dt

0

-5

-10

∆τ 0

2

4

6

8

Time (ms) Figure 1 – Induced emf in the pick-up coils, indicated by the closed (blue) and open (red) symbols, superposed to the exciting magnetic field, black open circles. Dotted lines indicate the time interval (∆t) between the peaks detected when the DW crosses the pick-up. Solid black line shows the delay provided by the pinning coil in order to ensure that the DW is depinned only when the magnetic field is stabilized. ∆τ is the pulse length used to calculate the DW length.

In order to apply the axial stress on the samples, one end was fixed to a sample holder and a known weight was attached to the other end. The applied stress was calculated considering the cross section of the sample, meaning metal core plus glass cover. The maximum applied stress was 400 × 107 dyn/cm2. To apply an electric current in the amorphous microwires it is the easiest way to promote the relief of the frozen stress, thus modifying its magnetic properties via magnetoelastic anisotropy. All samples were pre-annealed during 1 hour at a current 60% higher than the maximum applied current (25 mA) used during the measurements of the DW velocity. This was done in order to avoid additional structural modifications

on the samples during the DW velocity measurements. The current direction is that of the charge carriers. In our system this means that for I > 0, the charge carriers follow against the DW propagation and vice-versa. The magnetization curves were obtained by electronic integration of the voltage generated in a pair of compensated sensing coils, wounded in opposite phases. One of them involves the central portion of the sample and the other is empty. The whole system is placed in a long solenoid in order to apply the magnetizing field. The magnetization curves were measured with the simultaneous application of mechanical stress and current.

Results and discussion Figure 2 presents a schematic diagram the glass-covered microwire, also showing the estimated domain structure, based on the frozen internal stress and magnetostatic energy [5, 6]. The domain structure of these λS > 0 microwires consists of a large axially oriented domain surrounded by radially magnetized domains. Also, in order to reduce the local magnetostatic field, there exists a closure domain in the extremes of the sample, whose dynamics is studied in this work. It is displaced by the applied magnetic field reversing the wires’ magnetization.

Figure 2 - Schematic diagram of the glass-covered microwire with positive magnetostriction. The Domain structure is composed of a core with axial magnetization surrounded by a shell radially magnetized. The sketch is not in scale. The domain wall at the wires’ core was put in motion by the external field (H) in the presence of a circumferential field (Hφ) associated to the applied current (I).

The accepted value for the magnetostriction of microwires with the composition used in this work is 30×10 -6, measured by small-angle magnetization rotation (SAMR) [24]. This method is based on the changes of the anisotropy field with the applied stress. Here the magnetostriction and frozen internal stress were estimated from the evolution of the anisotropy field as function of the axial applied stress in the magnetization loop, considering the saturation field as the anisotropy one. As it can be seen on the Figure 3, the magnetization loop is typical of a material with uniaxial anisotropy endorsing the above consideration. The as cast sample has presented a λS = 22 × 10 -6 and a frozen stress (σf) of 220×107 dyn/cm2. On the other hand, the Jouleheated sample has such values reduced; the frozen stress is 28 ×107 dyn/cm2 and λS = 14 × 10-6. Considering that the applied current during the measurement of the DW dynamics increases the samples’ temperature due to the Joule effect, the magnetization loops were measured with the application of different current intensities. When MS value is compared to the one obtained without current, almost no changes were detected even for the higher applied current. Also, the remanence (Mr) has presented no changes and the whole curve is a fingerprint of an axially magnetized wire. On the other hand, as can be seen in Fig. 3, the applied current acts as a bias field, shifting the entire magnetization loop to negative fields with positive applied current and vice-versa. These shifts and changes in the inversion fields are related to a modification in the DW energy due to the presence of the Oersted field associated to the applied current [25].

1.0

M/Ms

0.5

Applied current -25mA 0mA 25mA

0.0

-0.5

-1.0 -1

0

1

Field (Oe) Figure 3 - Magnetization loops of the studied microwire under different applied currents. Observe the field shift of the loop according to the current direction.

The study of the DW dynamics is made by considering the DW as a part of a damped and forced harmonic oscillator [26]. In a viscous medium, the DW reaches its terminal velocity in a very short time, compared to the time for the DW cross the sensing coils [27]. When the DW is forced to dislocate by low applied fields (H) just higher HC, the DW dynamic can be well described by a linear dependence as [4, 14, 18] v = S axial ( H − H 0 ) =

2M S (H − H0 ) , β axial

(2)

where Saxial, βaxial and H0 are, respectively, the mobility and damping associated to the axial motion of the DW and the critical field necessary to release the DW from its pinning position. Figure 4 presents measurements of DW axial velocity as a function of the applied field for different applied currents. The applied stress was 200×10 7 dyn/cm2.

Velocity (cm/s)

150k

I (mA) -25 -20 0 20

100k

50k 1.0

2.0

3.0

Field (Oe) Figure 4 - Field evolution of the domain wall axial velocity measured under different applied currents as indicated. The axial applied stress was 200×107 dyn/cm2.

In Fig. 4 it can be seen a linear dependence of the axial velocity of the DW with the applied field, independent of the intensity or direction of the current. The Saxial value is about 43×103 cm/sOe, according to expression (1) taken from the curve with I = 0. It can also be seen a dependence of S axial with the current intensity and direction. While Saxial is not very sensitive to positive current, it appears to increase for negative current. It would be expected a higher dependence of Saxial with the positive current when the DW dynamic is measured with H < 0.

By considering the time needed to the DW cross just one of the sensing coils (∆t ~ 600 µs) at the above measured velocity, we can infer that Lw >> r0. This feature has already been verified, as for example in [18]. In view of this, it is possible to relate the DW dynamics to the motion of the DW in the orthogonal direction and the associated velocity (vn) can be calculated as [18, 23]

orthogonal velocity (cm/s)

vn = v

r0 . Lw

(3)

300

250

200

Applied Current +10mA 0 mA -10mA

150

1.0

2.0

3.0

Field (Oe) Figure 5 - Domain wall orthogonal velocity calculated from the axial velocity using expression (4), for the indicated currents and 200×107 dyn/cm2 of axial stress.

Figure 5 presents the field dependence of vn, calculated from the measured axial velocity using expression (3), for different values of applied current and 200×107 dyn/cm2 of applied stress, the same applied stress in the measurements presented in the Fig. 4. By comparing the velocity ranges seen in the figures 4 and 5, it can be seen that vn << v, as reported in [18]. Also, it is clear from the figure 5, that vn presents a linear dependence with the exciting field so its dynamic should be described by expression (2), [4, 14]. Furthermore, vn has a smaller dependence with applied current intensity and direction. Such dependency on the DW dynamic with the applied current is related to changes in the DW length [23] and shape, as will be discussed below. As stated by expression (2), from the vn × H data it can be obtained the total DW damping in the orthogonal direction (β). On the other hand, β is given by the sum of the acting damping mechanisms: structural relaxation (βx) [32], spin relaxation (βspin) [33]

and eddy currents (βeddy) [9, 28]. In our discussion the damping related to the structural relaxation is neglected as it is relevant only at low temperatures [32]. The βspin parameter it is related to how fast the magnetic moments can modify its orientation into the DW during the wall motion. The Landau-Lifshitz damping parameter (χ) and the DW thickness (tDW) are the main factors contributing to this term.

β spin ∝

χ t DW

∝ χM S

3 A 2

λS (σ f + σ app )

(4)

In the above expression tDW it is related to the exchange interaction (A) and anisotropy written in terms of λS, σf and σapp. This dependence of βspin with the applied stress has been already verified, for example, in refs. [22, 31-33]. The βeddy parameter it is related to the energy loss by eddy-currents during the DW motion and it is governed by the wire’s conductivity (κ), radius (r0) and DW shape (F). A mathematical expression for this damping term was derived by Panina et al. [18], in fact in terms of the associated mobility, or µ eddy. In [18] the authors consider that the eddy-current loss is balanced by the energy supplied by the work of the magnetic field and the orthogonal mobility due to eddy-currents is expressed as

µ eddy =

2M s

β axial

=

c2 , 16πκMr0 F

(5)

with 2

d  1 F = ∫ dy x 2  ln   .  dy   x 

(6)

Where x = r / r0 and y = l / Lw are normalized lengths measured along to the radial and

axial directions, respectively. The tentative shape for the DW profile is introduced in the above expression (6) by an x(y) function and that the whole DW present a cylindrical symmetry. The above discussed damping parameters can be simply added, or in terms of the inverse of the mobility 1 1 3 = +χ A λS (σ f + σ app ) . S S eddy 8

(7)

The procedure used to estimate the DW shape profile was to fit the expression (2) to measured data of vn × H under different applied stress (Fig. 5) to thus obtaining the S(σapp, I). After that, from a fitting of S(σapp, I) × σapp (shown in the Fig. 6) to the

expression (7) and by knowing σf, and λS values we can, finally, separate the eddycurrent contribution from the total mobility and to estimate the DW shape from expressions (5) and (6). The obtained results of damping parameter and F values for different applied currents are presented in the Table I.

1/S (sOe/cm)

0.02

Current (mA) 0 - 20 +20

0.01

0

250M

500M

750M 2

Applied stress (dyn/cm ) Figure 6 - Total damping of the DW in the orthogonal direction as function of the applied stress and different applied currents. Solid lines are fitting to the expression (7).

Table I - Eddy current mobility (Seddy) obtained for different applied currents. The F values are associated to a proper domain wall profile as discussed in the text.

Appl. current Seddy

F

F

(mA)

(cm/sOe) (measured) (tentative)

20

2340

0.64

F1

10

2310

0.65

F1

0

3550

0.42

F4

-10

2170

0.69

F1

-20

2060

0.73

F1

In order to estimate the shape of the DW, different profiles for the DW (x(y) functions) were tested and the respective F values were calculated with the expression (6). The results are shown in the Table II. Fig. 7 presents a sketch of the DW profile for each tentative F function.

Table II - DW shape (x(y) profile) associated to the F values. Sketches of the proper F’s are in the Fig. 6.

DW Shape

F (Calculated)

(Tentative functions)

F1

(1 − y 3 / 2 )

0.67

F2

(1 − y )

0.50

(1 − y)

F3

0.44

(1 − y1/ 2 )

F4

0.41

By comparing the experimental F values to the calculated ones, it can be seen that with no applied current, the DW present a profile similar to that of F4, a bell shape as suggested in the sketch of the Fig. 2 and the blue solid line in the Fig. 7. A DW with a F4 profile is orthogonal to the wire axis at the surface (r/r0 = 1), resulting in a reduced magnetostatic energy when compared to another approaching angles [34]. Besides that, the F4 DW presents a smooth curvature at the central portion of the wire, reducing the exchange energy in this region when compared to the conical shape F3 as suggested in [11] or even the profile proposed in [18].

r/r0 (wire's radius)

1.0

0.5

F1 F2 F3 F4

0.0

-0.5

-1.0 0.0

0.5

1.0

l/L (domain wall lenght)

Figure 7 - Suggested domain wall profiles and the respective F’s. Radius and Length scales are normalized in the figure. The domain wall length is much larger than the wire’s radius.

When the current is applied to the wire, the associated Hϕ acts in a way to reduce the DW superficial energy [23, 18]. However this reduction is not uniform as Hϕ increases linearly with the radial coordinate. So it is expected that the DW be dislocated to a region nearer the surface of the wire where the superficial density energy is lower. By comparing the obtained F values of the wire with applied current to the estimated shapes, it can be seen that the DW shape gets closer to the shapes described by F1. The DW surface area is higher for the profiles F1 and F2 when compared to F3 and F4. On the other hand this increase in the DW superficial area is compensated by the reduced energy needed to form the DW near the wire’s radius. Combined to these increase in the surface area, the DW also increases its length as already discussed in [23].

Conclusion

It was verified that the DW orthogonal velocity is weakly influenced by the applied current and that the main feature on the DW dynamics is to change the DW shape and length. Such changes were justified by a reduction on the DW energy density promoted by the Oersted field. From our data, and considering the eddy current contribution to the DW dynamics combined to the free magnetic energy to form the DW [18], it was proposed a Bell shape for the DW profile when no current is applied. This evolves to a parabolic shape as the current is applied and increased.

ACKNOWLEDGMENT The authors would like to thank to prof. H. Chiriac, National Institute of R&D for Technical Physics, Iasi, Romania, for the production of the samples. This work was supported by the Brazilian agencies CNPq, CAPES and FAPERGS.

REFERENCES

1 - D.A. Allwood G. Xiong, C.C. Faulkner, D. Atkinson, D. Petit, R.P. Cowburn, “Magnetic Domain-Wall Logic,” Science, vol. 309, p. 1688-1692, 2005. 2 - A. Zhukov and V. Zhukova, Magnetic Sensors and Applications Based on Thin Magnetically Soft Wires with Tunable Magnetic Properties (International Frequency Sensor Association Publishing, Barcelona, 2014). 3 - S.S.P. Parkin, M. Hayashi and L. Thomas, “Magnetic Domain-Wall Racetrack Memory,” Science, vol. 320, pp. 190-194, 2008. 4 - P. Klein, R. Varga and M. Vázquez, “Stable and fast domain wall dynamics in nanocrystalline magnetic microwire,” Journal of Alloys and Compounds, vol.550, pp. 31-34, 2013. 5- M. Vázques. A Zhukov, “Magnetic properties of glass-coated amorphous and nanocrystalline microwires”, J. Magn. Magn. Mat., v. 160, p. 223-228, 1996. 6- H. Chiriac, T.-A. Óvari, “Magnetic properties of amorphous glass-covered wires”, J. Magn. Magn. Mater., vol. 249, pp. 46-54, 2002. 7 - Jingfan Y., R.P. del Real, G. Infante and M. Vazquez, “Local magnetization profile and geometry magnetization effects in microwires as determined by magneto-optical Kerr effect,” J. Appl. Phys., vol. 113, pp. 043904 1-5, 2013. 8 - H. Chiriac, J. Yamasaki, T.O. Óvari and M. Takajo, “Magnetic Domain Structure in Amorphous Glass-Covered Wires with Positive Magnetostriction,” IEEE Trans. on Magnetics, vol. 35, pp. 3901-3903, 1999. 9 - D.-X Chen, N.M. Dempsey, M. Vázquez, A. Hernando, “Propagating domain wall shape and Dynamics in iron-rich amorphous wires,” IEEE Trans. Magnetics, vol. 31, pp. 781-790, 1995. 10 - S. A. Gudoshnikov, Yu. B. Grebenshchikov, B. Ya. 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” Phys. Status Solidi A, vol. 206, pp.613-617, 2009. 11- M. Tibu, M. Lostun, T.A. Óvari, H. Chiriac, “Simultaneous, magnetooptical Kerr effect and Sixtus-Tonks method for analyzing the shape of propagation domain wall in ultrathin magnetic wires”, Rev. Sci. Instrum., v. 83, p. 064708, 2012. [12] J. Gonzalez, A. Chizhik, A. Zhukov and M. Blanco, “Surface magnetization reversal and magnetic domain structure in amorphous microwire”, Phys Status Solidi A v. 208, p. 502, 2011. 13 - P. A. Ekstrom and A. Zhukov, “Spatial structure of the head-to-head propagating domain wall in glass-covered FeSiB microwire,” J. Phys. D: Appl. Phys., vol. 43, pp. 205001 1-6, 2010.

14 - R. Varga, K. Richter, A. Zhukov and V. Larin, “Domain Wall Propagation in Thin Magnetic Wires,” IEEE Trans. on Magnetics, vol. 44, pp. 3925-3930, 2008. 15 - J.P. Sethna, K.A. Dahmen and O. Perkovic, Random field ising models of hysteresis, in: G. Bertotti, I. Mayergoyz (Eds.), The Science of Hysteresis, Academic Press, New York, 2006, pp. 107. 16 - G. Durin and S. Zapperi, The Barkhausen effect, in: G. Bertotti, I. Mayergoys (Eds.), Science of Hysteresis, Academic Press, New York, 2006, pp. 181. 17 - R. Varga, A. Zhukov, V. Zhukova, J.M. Blanco and J. Gonzalez, “Supersonic domain wall in magnetic microwires;” Phys. Rev. B, vol.76, pp. 132406 1-3, 2007. 18 - L.V. Panina, M. Ipatov, V. Zhukova and A. Zhukov, “Domain wall propagation in Fe-rich amorphous microwires;” Physica B; vol. 407; pp. 1442– 1445, 2012. 19 - H. Chiriac and T. A. Óvari, “Amorphous glass-covered magnetic wires: Preparation, properties and applications”, Prog. In Mat. Sci., vol 40, pp 333 – 407, 1996. 20 - J. Sixtus, L. Tonks, “Propagation of large Barkhausen discontinuities,” Phys. Rev., vol. 37, pp. 930- 959, 1931. 21 - R. Varga, K.L. Garcia, A. Zhukov, M. Vazquez, M. Ipatov, J. Gonzalez, V. Zhukova, P. Vojtanik, “Magnetization processes in thin magnetic wires,” J. Magn. Magn. Mater., vol. 300, pp. e305-e310, 2006. 22 - F. Beck, R.C. Gomes, K.D. Sossmeier, F. Bohn, M. Carara, “Stress dependence of the domain wall dynamics in the adiabatic regime”, J. Magn. Magn. Mater., vol. 323, pp. 268-271, 2011. 23 - F. Beck, J.N. Rigue, M. Carara “Effect of electric current on domain wall dynamics”, IEEE Trans. on Magnetics, vol. 49, pp.4699, 2013. 24 - M. Churyukanova, V. Semenkova, S. Kaloshkin, E. Shuvaeva, S.Gudoshnikov, V. Zhukova, I. Shchetinin and A. Zhukov, “Magnetostriction investigation of soft magnetic microwires”, Phys. Status, Solidi A, v. 213, p. 363-367, 2016. 25- L. V. Panina, H. Katoh, K. Mohri and K. Kawashima, “Magnetization processes in amorphous wires in orthogonal fields”, IEEE Trans. on Magnetics, vol. 29, pp.2524, 1993. 26- C. Chikazumi, Physics of Magnetism, 2nd ed., Robert E. Krieger Publishing Company, Florida, 1964. 27 - R. Varga A. Zhukov, J. M. Blanco, M. Ipatov, V. Zhukova J. Gonzalez, P. Vojtaník, “Fast magnetic domain wall in magnetic microwires,” Phys. Rev. B, vol. 74, pp. 212405 1-4, 2006.

28- L. V. Panina, M. Mizutani, K. Mohri, F.B. Humprey and L. Ogasawara, “Dynamics and relaxation of large Barkhausen discontinuity in amorphous wires,” IEEE Trans. on Magnetics, vol. 27, pp. 5331-5333, 1991. 29- R. Varga, K. L. Garcia, M. Vàzquez, P. Vojtanik, “Single-domain wall propagation and damping mechanism during magnetic switching of bistable amorphous microwires”, Phys. Rev. Lett. v. 94, p. 017201, 2005. 30 - C. Kittel, J. K. Galt, Solid State Physics. Academic Press, New York, 1956. 31 - J. M. Blanco, V. Zhukova2, M. Ipatov and A. Zhukov, “Effect of applied stresses on domain wall propagation in glass-coated amorphous microwires” Phys. Status Solidi A, vol. 208, 545–548, 2011. 32 - A. Zhukov, J. M. Blanco, M. Ipatov, A. Chizhik1 and V. Zhukova, “Manipulation of domain wall dynamics in amorphous microwires through the magnetoelastic anisotropy”, Nanoscale Research Letters, vol. 7, p. 223, 2012. 33 - K. Richter, R. Varga and A. Zhukov, “Influence of the magnetoelastic anisotropy on the domain wall dynamics in bistable amorphous wires”. J. Phys.: Condens. Matter vol. 24, p. 296003, 2012. 34 - J. E. L. Bishop, “The shape, energy, eddy current loss and relaxation damping of magnetic domain walls in glassy iron wire”, IEEE Trans. on Magnetics, v. 13, p. 1638-1645, 1977.

Ref. Manuscript MAGMA_2016_1847 “The profile of the domain walls in amorphous glass-covered microwires” Highlights • We have evaluated the profile and shape of the moving domain wall in glasscovered microwires with positive magnetostriction. • The single domain wall dynamics was evaluated under different conditions, under an axially applied stress and an electric current • The domain wall evolves from a bell shape to a parabolic shape as the current intensity is increased.