Remagnetization processes in microwires and nanoscopic arrays. Przemysław Gawroński

Transcription

1 Remagnetization processes in microwires and nanoscopic arrays Przemysław Gawroński

2 Part I SUMMARY OF PROFESSIONAL ACCOMPLISHMENTS 2

3 Contents I SUMMARY OF PROFESSIONAL ACCOMPLISHMENTS 2 A PERSONAL DATA 5 B ACADEMIC DEGREES 5 C INFORMATION ABOUT PREVIOUS EMPLOYMENT IN SCIENTIFIC ESTABLISHMENTS 5 D INDICATION OF THE SCIENTIFIC ACHIEVEMENT CONSTITUT- ING THE AUTHOR S SIGNIFICANT CONTRIBUTION TO THE DE- VELOPMENT OF THE GIVEN SCIENTIFIC DISCIPLINE 5 D.1 Title of the scientific achievement D.2 List of publications that are the basis of the scientific achievement E DESCRIPTION OF THE SCIENTIFIC OBJECTIVES OF THE ACHIEVE- MENT WITH DISCUSSION OF THE OBTAINED RESULTS AND THEIR POSSIBLE DEVELOPMENTS AND APPLICATIONS 6 E.1 Introduction E.1.1 Fe-rich amorphous wires E.1.2 Co-rich amorphous microwires E.1.3 Arrays of nanodots E.1.4 Arrays of antidots E.2 Detailed discussion of all the publications that are the basis of the proposal. 11 E.3 Summary F DISCUSSION OF OTHER SCIENTIFIC AND RESEARCH ACHIEVE- MENTS 35 F.1 Before obtaining the PhD degree F.2 After obtaining the PhD degree II ACADEMIC ACTIVITY INCLUDING DIDACTIC ACHIEVE- MENTS, RESEARCH CO - OPERATION AND POPULARIZA- TION OF SCIENCE 43 G LIST OF PUBLICATIONS PROVIDING THE BASIS OF THE ACHIEVE- MENT 44 G.1 Publications in journals tracked by the Journal Citation Report (JCR) H LIST OF OTHER SCIENTIFIC PUBLICATIONS (NOT INCLUDED INTO THE ACHIEVEMENT, AS INDICATED IN D.1) AND BIBLIO- METRIC INDICATORS 46 H.1 Publications in journals tracked by the Journal Citation Report (JCR) H.2 Inventions and other objects of industrial property which are granted patent protection and were demonstrated at national or international exhibitions.. 52 H.3 Monographs, academic publications in international or national journals that are not tracked by the JCR database

4 H.4 Joint publications, collections catalogued, databases prepared, professional assessments H.5 Impact factor of all publications H.6 The total number of citations H.7 Hirsch index H.8 Leadership in international or national research projects or participation in such projects H.9 International and national awards for academic or artistic work H.10 Papers presented at thematic national or international conferences and scientific meetings I ACHIEVEMENTS IN TEACHING AND POPULARIZATION OF SCI- ENCE, RESEARCH CO-OPERATION 55 I.1 Participation in European programmes and other international or national programmes I.2 Active participation in international or national academic conferences I.3 Participation in organizing committees of international and national scientific conferences I.4 Prizes and awards other than those indicated in H I.5 Participation in consortiums and research networks I.6 Management of scientific research and development projects performed in collaboration with national and international partners from science and industry (other than those indicated in H.8) I.7 Participation in editorial boards and other editorial activities I.8 Membership in international or national organizations and scientific societies 58 I.9 Achievements in teaching and popularization of science I.10 Scientific assistance to students as scientific supervisor I.11 Scientific assistance and co-supervision to Ph.D. students I.12 Internships in foreign research or academic centres I.13 Evaluations performed or other scientific studies made to order I.14 Participation in expert and competition teams I.15 Reviews of international and national research projects I.16 Reviews of publications in international and national journals I.17 Activities other than those indicated in I.1 - I

5 A PERSONAL DATA Name and surname: Przemysław Gawroński B ACADEMIC DEGREES 1. Ph. D. in physics, 2003, Faculty of Physics and Nuclear Techniques, University of Science and Technology, dissertation title: Study of the dynamics of the nonlinear systems by coupled map lattices technique, supervisor: prof. dr hab. Krzysztof Kułakowski, reviewers: prof. dr hab. Andrzej Sukiennicki, prof. dr hab. Andrzej Maksymowicz. 2. M. Sc. in technical physics, 1998, Faculty of Physics and Nuclear Techniques, University of Science and Technology, thesis title: The dynamics of reversible many body systems, supervisor: dr hab. Krzysztof Kułakowski, reviewer: dr inż. Jacek Tarasiuk. C INFORMATION ABOUT PREVIOUS EMPLOYMENT IN SCIENTIFIC ESTABLISHMENTS IX II.2004, 20.VI VII postdoc in laboratory of magnetism in Faculty of Chemistry in University of Basque Country, San Sebastian, Spain II VIII teaching and research assistant, Faculty of Physics and Computer Science, University of Science and Technology, Cracow, Poland. 3. VIII XII personal investigator in laboratory of magnetism in Faculty of Chemistry in University of Basque Country, San Sebastian, Spain IX now, assistant professor, Faculty of Physics and Computer Science, University of Science and Technology, Cracow, Poland. D INDICATION OF THE SCIENTIFIC ACHIEVE- MENT CONSTITUTING THE AUTHOR S SIGNIF- ICANT CONTRIBUTION TO THE DEVELOPMENT OF THE GIVEN SCIENTIFIC DISCIPLINE D.1 Title of the scientific achievement Pursuant to Article 16(2) of the Act of March 14, 2003 on university degrees and university title in arts (Journal of Laws No 65, item 595, as amended), I would like to present the cycle of 9 scientific publications entitled: Remagnetization processes in microwires and nanoscopic arrays. D.2 List of publications that are the basis of the scientific achievement H1 P. Gawroński, A. P. Zhukov, V. Zhukova, J. M. Blanco, J. Gonzalez, K. Kułakowski, Distribution of fluctuations of switching field in Fe-rich wires under tensile stress, Appl. Phys. Lett. 88, (2006), IF(2006) = 3,977. 5

6 H2 P. Gawroński, A. Chizhik, J. M. Blanco, J. Gonzalez, Influence of the circular magnetic field and the external stress on the remagnetization process in Fe-rich amorphous wires, IEEE Trans. Magn. 46, (2010), 365. IF(2010) = 1,053. H3 P. Gawroński, V. Zhukova, A. Zhukov, and J. Gonzalez, Manipulation of domain propagation dynamics with the magnetostatic interaction in a pair of Fe-rich amorphous microwires, J. Appl. Phys. 114, (2013), IF(2013) = 2,185. H4 A. Chizhik, C. Garcia, A. Zhukov, J. Gonzalez, P. Gawroński, K. Kułakowski and J. M. Blanco, Relation between surface magnetization reversal and magnetoimpednace in Co-rich amorphouos microwires, J. Appl. Phys. 103, (2008), 07e742. IF(2008) = 2,201. H5 A. Chizhik, A. Zhukov, J. M. Blanco, J. Gonzalez, P. Gawroński, and K. Kułakowski, Experimental determination of relation between helical anisotropy and torsion stress in amorphous magnetic microwires, IEEE Trans. Magn., 44, (2008), IF(2008) = 1,129. H6 P. Gawroński and K. Kułakowski, Remanence and switching sensitivity in nanodot magnetic arrays, J. Nanosci. Nanotechnol., 8, (2008), IF(2008) = 1,929. H7 P. Gawroński, K. J. Merazzo, O. Chubykalo-Fesenko, A. Asenjo, R. P. del Real and M. Vaqzquez, Micromagnetism of dense permalloy antidot lattices from anodic alumina templates, EPL, 100 (2012) IF(2012) = 2,260. H8 C. Castan-Guerrero, J. Herrero-Albillos, J. Bartolome, F. Bartolome, L. A. Rodriguez, C. Magen, F. Kronast, P. Gawroński, O. Chubykalo-Fesenko, K. J. Merazzo, P. Vavassori, P. Strichovanec, J. Sese, and L. M. Garcia, Magnetic antidot to dot crossover in cobalt and permalloy patterned thin films, Phys. Rev. B 89, (2014) IF(2014) = 3,736. H9 P. Gawroński, K. J. Merazzo, O. Chubykalo-Fesenko, R. P. del Real and M. Vaqzquez, Micromagnetism of permalloy antidot arrays prepared from alumina templates, Nanotechnology 25, (2014), IF(2014) = 3,821. Total impact factor of [H1-H9]: 22,291. E DESCRIPTION OF THE SCIENTIFIC OBJECTIVES OF THE ACHIEVEMENT WITH DISCUSSION OF THE OBTAINED RESULTS AND THEIR POSSI- BLE DEVELOPMENTS AND APPLICATIONS E.1 Introduction Scientific achievement documented here consists of 9 publications [H1-H9]. The investigations of the remagnetization processes: of amorphous wires are presented in [H1-H5], nanodots arrays in [H6] and antidots arrays in [H7-H9]. The results of the experimental works are described in publications [H1-H3]. The publications [H3-H9] contain results of the numerical simulations. 6

7 E.1.1 Fe-rich amorphous wires The study of the magnetic properties of amorphous wires are motivated by their use in sensor technology [1]. Amorphous wires are manufactured by the modified Taylor-Ulitovsky technique [2, 3, 4]. During the production process, a specific distribution of internal stresses in the wire is formed, which determines the domain structure. The postulated domain structure of amorphous Fe-rich wires consists of a single-domain inner core and multidomain radially oriented outer shell [4, 5, 6]. The experimental results suggest that in the remagnetization process of the Fe-rich amorphous wire the main role is played by an axially oriented, monodomain inner core. Fe-rich amorphous wires are often referred as bistable, because the remagnetization process switches between two states of the inner core with opposite signs of magnetization. The domain wall is nucleated at the end of the wire and then propagates to the other end. The remagnetization occurs when an applied magnetic field is equal to the critical value called the switching field (H s ). In the case of currently manufactured glass coated amorphous microwires the ratio ρ of the diameter of the metallic core to the total diameter of microwire controls the distribution of internal stresses, direct measurement of which is extremely difficult. Ease of production of microwire for any given ratio ρ provokes the basic research questions: What parameters control the experimentally observed value of the switching field and with which external factors it can be modified? The magnetic properties of amorphous wires are investigated by the induction method that uses a modified version of the measuring system proposed by Sixtus and Tonks to observe movements of the domain walls. The standard measurement system consists of a solenoid or Helmholtz coils generating an alternating magnetic field where the pick-up coil with the microwire inside is situated. The propagating of the domain wall through the wire induces in the pick-up coil the signal which after integrating in the integrator is proportional to the change of magnetization. The investigation of the remagnetization process in Ref. [7, 8] revealed the existence of the switching field H s fluctuations in subsequent remagnetization cycles of the Fe-rich microwires and its strong dependence on temperature. Two mechanisms possibly responsible for the occurrence of switching field fluctuations have been identified. The subject of research in [H1-H3] are Fe-rich amorphous wires. The purpose of the work [H1] was to measure and analyze the distribution of the switching field H s fluctuations in Fe-rich amorphous wires as dependent on an external tensile stress. The standard induction measuring system was equipped with the ability to measure the hysteresis loops in the presence of an applied tensile stress. Since the values of the switching field do depend on the distribution of internal stresses [9, 10, 11], the application of the external tension should modify the distribution of internal stresses and affects the distribution of the switching field fluctuations. The result of [H1] extends our knowledge about mechanisms responsible for the fluctuations, as discussed in Ref. [8]. In [H2] the standard measuring system was enriched by the possibility of the hysteresis loop measurements of the amorphous wire through which an alternating current flows, generating additional circular magnetic field. The aim of the study was the experimental demonstration of the ability to control the remagnetization process of Fe-rich amorphous wire through additional circular magnetic field. The study also tested whether the circular magnetic field is able to initialize the remagnetization wire when the axial magnetic field is lower than the 7

8 switching field. The use of monodomain amorphous wire with the inner core of several centimeters of length allows the direct measurement of the velocity of the propagating domain wall. The previously described standard induction method setup is used but instead of one pick-up coil a system of three or four coils distributed along the entire length of the wire is employed. Generally it is assumed that the velocity of the domain wall is directly proportional to the applied magnetic field [12, 13, 14]. Due to the applications it is desired to use the wires of the maximal propagation velocity of the domain wall. It leads to questions about the dependence of the domain wall velocity of wire on the control parameters such as coefficient ρ or an external stress. Amorphous wires belong to the group of soft magnetic materials. It is therefore desirable to achieve the highest possible velocity of the domain wall, while maintaining at the same time the value of the switching field as small as possible. In [H3] the impact of the frequency of the magnetic field on the shape of the hysteresis loop for the system of two magnetostatically interacting microwires was measured and described within a phenomenological model, proposed there. The induction method setup was used to demonstrate an influence of the magnetostatic interaction between microwires on the velocity of the propagation of domain walls in the system. E.1.2 Co-rich amorphous microwires Amorphous Co-rich microwires are intensively studied because of the use of giant magnetoimpedance (GMI) effect in sensor technology [4, 15, 16]. Giant magnetoimpedance is a surface phenomenon, that is why in the case of Co-rich microwires we were investigating the remagnetization process in the surface layer. Domain structure of Co-rich amorphous microwires is composed of an inner core with the longitudinal easy axis, which is surrounded by an outer shell, where the helical anisotropy is present [4, 17]. The experimental studies are carried out by use of the transverse magneto-optical Kerr effect measuring system [17]. A polarized light from the He-Ne laser was reflected from the surface of the microwire of the curvature about 1 towards the detector. The intensity of the reflected light is proportional to the circular magnetization of the surface layer of the microwire which is perpendicular to the plane of polarization of light. The numerical calculations of the remagnetization process, presented in [H4-H5], were performed by minimizing the magnetic energy within the coherent rotation approach and taking into account the existence of the uniaxial helical magnetic anisotropy in the surface layers of Co-rich microwire. In [H4] it was assumed that the applied magnetic field is a superposition of two orthogonal components: the axial field (h axial ) and the circular field (h circ ), while in [H5] only the effect of the axial field (h axial ) was taken into account. The aim of the work [H4] was to find the correlation between the giant magnetoimpedance ratio, the jump of the circular magnetization M and the angle of helical anisotropy φ. In [H5] the surface hysteresis loops of the Co-rich amorphous microwires have been measured in the presence of the torsional stress. Numerical calculations of the remagnetization process of the surface layer of Co-rich amorphous microwire have been performed in order to determine the relationship between an applied torsion and the angle of the helical anisotropy, as well as to determine the possible range of the angles of the helical anisotropy induced by the applied torsion. 8

9 E.1.3 Arrays of nanodots Numerical simulations of the magnetic properties of the nanodots arrays were stimulated by the possibility of a potential use as magnetic recording media [18, 19, 20]. Each element of the experimental array from Ref. [21] is a one-domain cylindrical nickel dot of average diameter d = 57 nm, average height h = 115 nm, located on a square lattice with a period of p = 100 nm. Due to the high value of the shape anisotropy the direction of magnetization is perpendicular to the array. Remagnetization of a single nanodot occurs when the switching field of average value of H s = 710 Oe is reached. Nanodots are called bistable because the remagnetization takes place between two states of opposite directions of magnetization and the hysteresis loop of a single array element is approximately rectangular. Imperfections of the manufacturing process effect in differences in the shape of individual nanodots that lead to different values of the switching field. The nanodot array can be characterized by the standard deviation of the switching field, which for the experimental array from Ref. [21] is σ = 105 Oe. The application of the nanodot arrays for the information storage requires the stability of the magnetic state. The magnetic state of the array is stable when it is not damaged by the spontaneous remagnetization of a nanodot due to the magnetostatic interaction of the neighboring nanodots. The influence of the magnetostatic interaction cannot be limited by the increase of the distance between the nanodots, because this will decrease the information storage density, and the main research goal is the production of magnetic materials for high-density storage. The aim of the work [H6] was to investigate, using numerical simulations, the impact of the standard deviation σ of the switching field and the finite size of the arrays on the value of the remanence. The damage spreading technique has been used to study the stability of the magnetic state of the arrays of the magnetostatically interacting nanodots. The damage spreading technique [22] is to track the evolution of the two systems with a slight difference of initial conditions referred to as damage. The damage can spread up to the range of the system size, or vanishes immediately after the simulation start, or it remains localized. When the damage spreads over the whole array, the latter cannot be used for information storage. When the damage is localized, an application of a redundancy writing algorithm and information corrections provides the safety of the information storage in such arrays. The stability has been investigated in the alternating magnetic field, because we are interested in the impact of initial conditions of the array state on the subsequent cycles of information recording due to the changes of magnetization. E.1.4 Arrays of antidots Arrays of antidot can be used in magnetic recording [23, 24, 25], in sensor technology [26, 27], and to construct magnonic devices [28, 29, 30]. Antidots arrays allow to control magnetic properties of thin films by changing their geometrical parameters such as height of the arrays, the diameter of the antidots and the distance between antidots. The remagnetization process is investigated by micromagnetic simulation, i.e. by solving the Landau-Lifshitz-Gilbert equation in the following form[31, 32]: M = γ G M H eff + α ( M M ), t M s t 9

10 where M - magnetization, M s - saturation magnetization, H eff - effective field, α material damping constant, γ G = γ L (1+α 2 ) - Gilbert gyromagnetic ratio, γ L - Landau gyromagnetic ratio. According to theory of micromagnetism proposed by Brown in [33], the effective field H eff is the functional derivative of the energy density with respect to the orientation of the magnetization. The total energy reads: E eff = E ex + E ani + E dem + E Zee. Exchange energy E ex is responsible for the spontaneous magnetization of ferromagnetic materials, by favoring the parallel alignment of neighboring magnetic moments. The exchange energy is taken into account in micromagnetic simulations by providing the exchange constant, which is A = J/m for the Permalloy antidots arrays in [H7, H9], and it is A = J/m for the cobalt antidots arrays in [H8]. Magnetocrystalline anisotropy energy E ani arises due to spin-orbit interaction and it forces the magnetic moments to align with an easy axis. The magnetocrystalline anisotropy is taken into account by providing anisotropy constant K1 specific for the simulated material and the anisotropy directions. In numerical modeling of [H7-H9] the contributions of the magnetocrystalline anisotropy was neglected. On the other hand the micromagnetic simulations of the remagnetization process of iron antidot arrays in [B31] require the implementation of the polycrystalline material by generation of two dimensional Voronoi diagram of 2000 random grains with the random directions of the cubic anisotropy. The energy of the demagnetization field E dem is related with the magnetostatic interaction and it governs the formation of the magnetic domains. Taking into account the demagnetization field usually induces a significant increase of the computational complexity of the micromagnetic simulations. Appropriate boundary conditions should be carefully selected and they should match the boundary conditions of the exchange energy calculations. Micromagnetic simulations presented in [H7-H9] have been done with the periodic boundary conditions. The Zeeman energy E Zee is the potential energy of a sample in an external magnetic field, and it favors an alignment of the magnetization parallel to the applied field. The simulations performed in order to obtained the results analyzed in [H7-H8] are called quasistatic, because for each value of the applied field the system evolves according to the Landau-Lifshitz-Gilbert equation until it reaches a stable magnetic state, and then the applied field is changed by given step. Within the quasistatic approach, the knowledge of the exact value of the damping constant α, otherwise hard to be found experimentally, is not necessary. Therefore the damping coefficient α is usually taken as 0.1 or 0.5. The micromagnetic simulations presented in [H7-H9] have been performed using a public domain micromagnetics package OOMMF developed at the National Institute of Standards and Technology [34], which performs the spatial discretization of Landau-Lifshitz-Gilbert equation using the finite difference method. The alternative tool is the MAGPAR package [35] due to the spatial discretization by means of the finite element method. OOMMF package was chosen to model the remagnetization of the antidots arrays due to the possibility of usage of the periodic boundary conditions, absent in the MAGPAR package. The spatial discretization means the division of the magnetic sample into cubes for the finite difference method or tetrahedrons in the case of the finite element method. The correct discretization requires that the length object on which we divide the sample was less than the magnetic exchange length [36], which for antidots arrays studied in [H7-H9] equals to l ex 5 nm. The main purpose of the research in [H7-H9] was the micromagnetic simulations of the remagnetization process depending on the geometrical parameters such as the antidot diameter, the distance between the antidots, or the height of antidot arrays in the case of [H9]. The result of each simulation was the upper part of the hysteresis loop which was used to evaluate the value of coercive field and assign it to geometrical control parameters 10

11 of the simulation. This allows for an interpretation of experimental results in the form of the dependence of the coercive field on the geometric parameters of the antidot arrays. Besides the remagnetization curves we obtained in micromagnetic simulations a more detailed information about the remagnetization process in the form of a spatial distribution of the magnetization for each value of an applied field. The spatial configurations of the magnetization enable a more thorough analysis of the magnetic structures that appear during the remagnetization of the antidots arrays. This allows to verify the assumed model of the remagnetization process of the antidot arrays by comparing the structures shown in the simulations with the images of the magnetic structures obtained by XPEEM technique. E.2 Detailed discussion of all the publications that are the basis of the proposal [H1] P. Gawroński, A. P. Zhukov, V. Zhukova, J. M. Blanco, J. Gonzalez, K. Kułakowski, Distribution of fluctuations of switching field in Fe-rich wires under tensile stress, Appl. Phys. Lett., 88, (2006), The switching field fluctuations have been observed experimentally in Fe-rich wires [8]. They manifest themselves as random deviations of the values of the switching field in subsequent remagnetization cycles. The remagnetization process of a bistable wire begins when the value of an external magnetic field reaches the value of the switching field H s. The remagnetization of the wire may also occur due to the thermal activation process even though the value of the external field is less than the value of the switching field. The switching field fluctuations have been investigated in [8, 37, 38] in the framework of the phenomenological model of thermally activated overcoming of a potential barrier by domain wall in an applied external magnetic field. The probability density w(h s ), that the domain wall will overcome the energy barrier at the measured magnetic field H s, is equal to the probability that the remagnetization was not observed in the lower field multiplied by the probability that the remagnetization occurs in H s. According to this model, there is a linear dependence (ln ( dw d h) ) of the logarithm of the probability density to observe the switching field fluctuations on ( h) 3/2, where h = (H s H)/H s - the reduced magnetic field. A more detailed studies in F e 77.5 Si 7.5 B 15 and Co 68 Mn 7 Si 15 B 10 microwires have shown that this dependence consists of two approximately linear parts [8]. This suggests that there are two different mechanisms responsible for the switching field distribution: the magnetoelastic coupling of the domain wall with the internal stresses, coming from the fabrication process, and the pinning of the domain wall at defects on the atomic scale. The measurements of the distribution of the switching field fluctuations as dependent on the external tensile stress were done in the magnetic laboratory of the University of the Basque Country in San Sebastian. In order to determine the value of the switching field H s, the hysteresis loops of the amorphous wire F e 77.5 Si 7.5 B 15 with a diameter of 125 µm and the length of 100 µm were measured by the inductive method [39]. During the measurements of the hysteresis loops one end of the wire was fixed to a sample holder, while the mechanical load, that varied from 0 to 1000 g, was attached to the other end of the wire. The obtained experimental dependence of the switching field fluctuations on the applied tensile stress was analyzed within the thermal activation model. The histogram of the measured values of the reduced switching field ( h) for the two extreme values of the applied stress are presented in the Figures 1 a and b. The application of the external stress causes an increase in the width of the histogram of the measured values of the reduced field. In the Figure 1c we have shown the renormalized distribution of the switching field fluctuations calculated in accordance with the model of thermal activation. The dependence of the probability density of the observed Barkhausen jump in the range between H and H + dh on the reduced magnetic field ( h) 11

12 a) b) c) Figure 1: Non-normalized distributions of the switching field fluctuations measured at a) σ = 0 MP a, b) σ = MP a c) Renormalized distributions of the switching field fluctuations for various values of the applied tensile stress. [H1] for each applied stress is divided into two linear regimes. Such nonlinear dependence can be attributed to a complex shape of the potential of the domain wall consisting of the long range contribution, mainly the effects of magnetoelastic interaction and the short range contribution arising from pinning of the domain wall at the structural defects on the atomic scale [8]. These results indicate that the distribution of the switching field fluctuations depends strongly on the applied stress. The part of the spectrum, presented in the Figure 1c, that contains large fluctuations is strongly affected by the magnetoelastic anisotropy. The slope of this part of the spectrum slightly changes and its position shifts towards higher values of h with the growth of the applied tensile stress. The domain wall energy is given by γ = 2(AK) 1/2, where A is the exchange constant and K is the magnetic anisotropy constant, that in case of the amorphous wires depends mainly on the magnetoelastic component expressed by K me = 3/2λ S (σ int + σ appl ), where σ int i σ appl are the internal and applied tensile stress, respectively. Therefore, the shift of the part of the spectrum, that contains large fluctuations, in the direction of greater values of h with the increase of the applied stress is probably due to the influence of the applied stress on the main energy potential related with the magnetoelastic contribution. It cannot be excluded that the applied stress can also affects the pair ordering mechanism related with the domain wall stabilization at the structural defects of the atomic scale, what may additionally affect the shape of the domain wall potential. The collected experimental results show that the measured maximum value of the switching field increases with the applied tensile stress. This increase is a consequence of the positive value of the magnetoelastic coupling constant. The applied stress affects the remagnetization process for small fields, which probably is due to the change of the shape of the potential through the magnetoelastic contribution. In the Figure 1c, the slope coefficient α m for the magnetoelastic mechanism is larger than the coefficient α p for the pinning agent. The coefficient α m decreases with the applied stress 12

13 what is associated with a reduction of the susceptibility, when magnetoelastic anisotropy increases a result of the increase of the stress. In summary, the distribution of the switching field fluctuations for the amorphous Fe-rich wires were measured and analyzed as dependent on the applied tensile stress. The shape of the distribution of the switching field fluctuation has been interpreted as the result of two mechanisms, a) the pinning of the domain wall associated with the magnetoelastic coupling of the domain wall with the internal stresses arising in the process of production of wire and b) the pinning the domain wall at the structural defects of the atomic scale. The applied stress dependence allows us to separate these two different mechanisms of the switching field fluctuations. H[2] P. Gawroński, A. Chizhik, J. M. Blanco, J. Gonzalez, Influence of the circular magnetic field and the external stress on the remagnetization process in Fe-rich amorphous wires, IEEE Trans. Magn. 46, (2010), 365. The main objective was to investigate experimentally the effect of the circular magnetic field produced by the AC current flowing through the wire and the applied tension on the remagnetization process of Fe-rich amorphous wires. The induction method was used to measure axial hysteresis loops of wire of nominal composition F e 77.5 B 15 Si 7.5, the length 10 cm and the diameter 125 µm. The measurements were carried out for the case when the amplitude of the applied axial magnetic field (H m = 15 A/m) was greater than the critical value needed to initialize the remagnetization process, called the switching field (H s ), that for the wire F e 77.5 B 15 Si 7.5 is H s = 6 A/m as well as for the case when the amplitude of the applied axial magnetic field (H m = 3 A/m) was smaller than H s. The frequency of the axial magnetic field in both cases was equal to 50Hz. The control parameters of the measurements were the amplitude of circular magnetic field (H circ ), that varied from 0 to 700 A/m and the frequency, that was set to 1, 5, 10, 25, 50, 75, 100 khz in the subsequent series of measurements. In the first case for H m = 15A/m, the experimental data presented in the Figure 2a indicate that the application of the additional circular magnetic field (H circ ) reduces the value of the switching field (H s ). For the low frequency (1 khz) the switching field H s decreases quite rapidly as the amplitude of circular magnetic field (H circ ) grows. The increase of the frequency of the circular magnetic field to 100 khz slows down the rate of the decline of value of H s with increasing amplitude of the circular magnetic field. The remanence of the wire presented in the Figure 2b remains almost constant for all the frequencies as long as the amplitude of the circular magnetic field (H circ ) is less than 350A/m. Above this value the decrease of the remanence with the increasing amplitude (H circ ) for the frequencies of 5kHz and 10kHz can be observed. The changes in the value of the switching field and the remanence with the increasing amplitude of the circular magnetic field lead to the loss of the bistability of the wire. The influence of the circular magnetic field on the remagnetization process in the presence of an external tensile stress was measured for the amplitude of the axial magnetic field H m = 15 A/m. This additional stress causes the reduction of both the switching field (H s ) and the remanence with the increase of the amplitude of the circular magnetic field (H circ ) is much smaller and has a similar nature for each tested frequency. The conditions were tested under which the circular magnetic field can initialize the remagnetization process despite the amplitude of the applied axial magnetic field being equal to 3 A/m was below the critical value of the switching field (H s = 6A/m) for wiref e 77.5 B 15 Si 7.5. The probability of the remagnetization of the wire was measured as a function of the amplitude and the frequency of the circular magnetic field (H circ ) for N = successive periods of the applied axial magnetic field. The value of the probability, presented in the Figure 3, was calculated as the number of the periods when the successful remagnetization 13

14 a) b) Figure 2: The influence of the applied circular magnetic field on a) the switching field H s and b) the remanence of the wire F e 77.5 B 15 Si 7.5 in the case, when the amplitude of the external axial magnetic field was H m = 15 A/m. [H2] occurred, divided by the number N. For a small amplitude of the circular magnetic field the probability of the remagnetization is zero because neither the axial magnetic field nor the thermal fluctuations of the switching field could initialize the remagnetization process. The probability of remagnetization increases monotonically with the increasing amplitude of the circular magnetic field, because the circular magnetic field reduces the energy barrier required to initialize the remagnetization process. As shown in the Figure 3, the slope of the dependence of the probability on H circ is decreasing as the frequency of the circular magnetic field grows. For each of the studied frequency we have found two threshold values of the amplitude of the circular magnetic field: (H circ1 ) - the value for which the probability of the remagnetization of the wire is greater than 0.01 and (H circ1 ) - the value for which the probability of remagnetization of the wire is equal to 1. These values are increasing with the frequency of the circular magnetic field. The reduction of the switching field (H s ) with the increasing amplitude of H circ observed for H m = 15 A/m has taken place for H m = 3 A/m, provided that for each frequency the data were collected when the amplitude was above H circ2. The experimental data presented here indicate that the circular magnetic field interacts with a radially oriented domains located in the outer shell, and it modifies the boundary between the single domain inner core and the multidomain outer shell. It lowers the energy barrier for depinning of the domain wall. The effect of the circular magnetic field on the value of the switching field is reduced by the applied external stress, which increases the anisotropy energy of the inner core, and by the decrease of the susceptibility of the wire with the increasing frequency of the circular magnetic field [40]. In summary, we have demonstrated experimentally the possibility of easy and reversible manipulation of the remagnetization process of the bistable amorphous wire F e 77.5 B 15 Si 7.5 by the circular magnetic field produced by the AC current flowing along the wire. [H3] P. Gawroński, V. Zhukova, A. Zhukov, and J. Gonzalez, Manipulation of domain propagation dynamics with the magnetostatic interaction in a pair of Fe-rich amorphous microwires, J. Appl. Phys. 114, (2013), The measurements of the remagnetization process and the dynamics of the domain wall of the amorphous, glass-coated microwires of the composition F e 75 B 15 Si 10 were made in the magnetic laboratory of the University of the Basque Country in San Sebastian. The 14

15 Figure 3: The dependence of the probability of remagnetization of the wire on the amplitude and the frequency of the circular magnetic field, in case when the amplitude of the axial magnetic field was equal to 3 A/m. [H2] samples characterized in Table 1 have the length of L = 10 cm and they differed in the ratio ρ = d/d of the diameter of the metallic core (d) to the total diameter (D) of the microwire including the glass coating. The measured values of the switching field (H s ), gathered in Table 1 are inversely proportional to the diameter of the metallic core (d) and they increase with the growth of the internal stress, which increases when the ratio ρ decreases. diameter total diameter of the metallic core (d) of the microwire (D) ρ = d/d H s (50Hz) próbka A 18.0 µm 24.8 µm A/m próbka B 6.8 µm 21.3 µm A/m próbka C 6.1 µm 25.2 µm A/m Table 1: The switching field (H s ), the diameter of the metallic core and the total diameter of the investigated microwires F e 75 B 15 Si 10. In the first measurement, the impact of the frequency of the applied magnetic field on the shape of the hysteresis loop and the value of the switching field of a single wire were measured. The shape of the hysteresis loop depends on the relation between the time of the flight of the domain wall through the microwire and the period of the change of the applied magnetic field. The typical hysteresis loop for a single Fe-rich amorphous microwire for the frequency of the magnetic field equal to 50Hz is presented in the Figure 4. As the frequency of the applied magnetic field increases the hysteresis loop loses its characteristic rectangular shape, while the increase of the switching field is observed. In another series of the measurements, the hysteresis loops of two parallel microwires were measured. The microwires were placed close to each other, so the distance between their metallic inner cores was a sum of their glass coating thickness. The typical hysteresis loops for two microwire systems and their comparison with the loops for single wires is shown in the Figure 4. A characteristic feature of the hysteresis loop for the two microwire system is an existence of two Barkhausen jumps, separated by a plateau, where the microwires are magnetized in opposite directions. In the system of two microwires both magnetized in the same direction e.g. down, the 15

16 Figure 4: The hysteresis loops for single microwires: A - single microwire of the ratio ρ = 0.72, C - single microwire of the ratio ρ = The hysteresis loops for two microwire systems: C C - two identical microwires having the same ratio, A C - two different microwires. The amplitude of the applied magnetic field was H m = 360 A/m, and the frequency 50 Hz. [H3] magnetostatic interaction between them manifests itself as an additional field δ, which helps the remagnetization of the first microwire during the first Barkhausen jump, reducing the value of the effective switching field (H CC1 = Hs C δ), where Hs C is a switching field for a single microwire. After the first Barkhausen jump the microwires are magnetized in the opposite directions. If the microwires have the same absolute value of the magnetization, then the magnetization of the plateau is zero. An example of such a loop in the Figure 4 is marked as C C. The magnetostatic interaction between the microwires tends to maintain the state of the magnetization in the opposite directions by increasing the value of the effective switching field during the second Barkhausen jump, when the second microwire changes the magnetization (H CC2 = Hs C + δ). The observed width of the plateau (H CC2 H CC1 = 2δ) for two identical microwires is a measure of the magnetostatic interaction between them. In the case when two microwires differ in diameter of the metallic core, and thus the value of the magnetization at the plateau separating two consecutive Barkhausen jumps can not be zero. An example of such a loop in the Figure 4 is marked as A C. In two microwire system A and C, the first to remagnetize is the microwire A in the field H AC1 = Hs A δ 1, where Hs A is the switching field of a single microwire A, and δ 1 is an additional field, which is the result by the magnetostatic interaction of the microwire C on the microwire A. The remagnetization of the microwire C takes place in the field of H AC2 = Hs C + δ 2, and this time the magnetostatic interaction is preventing the remagnetization and increasing the switching field of the microwire C by δ 2. Because in the system of two microwires A and C the magnetization saturation of the microwire A is greater than the one of the microwire C, the impact of the magnetostatic interaction of the microwire A on the microwire C is greater than C on A (δ 2 > δ 1 ). 16

17 An investigation of the impact of the frequency of the applied magnetic field on the shape of the hysteresis loop was carried out also for the systems of two microwires. As the frequency of the applied magnetic field grows, the width of the plateau gradually decreases. For each value of the amplitude of the applied magnetic field, a critical frequency ω cr was measured, where the plateau disappears. The value of the critical frequency decreases when the amplitude of the applied magnetic field increases, as seen in the Figure 5, because the velocity of the domain wall increases with the growth of the amplitude of the applied field. The dependence of a velocity of the domain wall on the amplitude of the applied field was measured using Sixtus-Tonks like experimental setup. This dependence presented in the Figure 6 for a single microwire is divided into three linear regimes of the different domain wall mobility. The mobility is given by [5, 13] S = 2µ 0 M s /β, where M s is the magnetization saturation and β is the domain wall damping. The damping increases when the applied stress increases as well as the internal stress controlled by ratio ρ. The mobility and, as a result, the domain wall velocity increase when the applied stress decreases. As the applied stress increases, the breaking points of the nonlinear dependence of the domain wall velocity on the applied field move to the higher fields, as well as the mobility domain wall decreases. The hysteresis loops for two microwire systems, presented in the Figure 4, suggest a possibility of measuring the velocity of the domain wall in the microwire, which is under the influence of the magnetostatic interaction of the second microwire. The remagnetization process of the first microwire is well separated from the remagnetization process of the second microwire, because they occur at the different values of the applied magnetic field. The dependencies of the velocity of the domain wall on the amplitude of the magnetic field Figure 5: The dependence of the critical frequency ω cr on the amplitude of the applied magnetic field: measured for the system of two different microwires B and C - red squares, measured for the system of two the same microwires B - green squares. The continuous lines are the simulated values: the red line for two microwires B and C, the green line for two microwires B. [H3] 17

18 Figure 6: The influence of the magnetostatic interaction between the microwires on the dependence of the velocity of the domain wall on the amplitude of the applied magnetic field. B and C - the velocity of the domain wall in a single microwire. C B - the velocity of the domain wall in the microwire B under the influence of the magnetostatic interaction with the microwire C. A B - the velocity of the domain wall in the microwire B under the influence of the magnetostatic interaction with the microwire A. C C - the velocity of the domain wall in the microwire C under the influence of the magnetostatic interaction with the microwire C. A C - the velocity of the domain wall in the microwire C under the influence of the magnetostatic interaction with the microwire A. [H3] marked in the Figure 6 as C C and C B were measured at the first Barkhausen jump. In this case, the domain wall propagates at the lower external field and with the greater velocity than for a single microwire. The dependencies marked as A C and A B were measured at the second Barkhausen jump, and therefore, only the velocity of the propagation of the domain wall at the high amplitudes of the applied magnetic field, could be observed. The obtained values of the velocities are smaller than for a single microwire due to the presence of the magnetostatic interaction between microwires. The application of the increasing tensile stress, during the measurement of the velocity of the domain wall, in two microwire system, additionally reduces the velocity of the propagating domain wall. The experimental dependence of the critical frequency (ω cr ) on the amplitude of the applied field, presented in the Figure 5, has been reproduced numerically within the proposed phenomenological model. The remagnetization process of two microwire system begins at t 0 given by the equation H m sin(ωt 0 ) = Hs A (ω) H AC1. The time t 1, when the remagnetization of the microwire A is completed, assuming the domain wall in microwire A propagates from one end to the other, is calculated by the integration of the equation dz dt = S A[H m sin(ωt) H A0 +H AC1 ], where H A0 - the experimentally estimated propagation field for the microwire A [41]. Similarly, the time to start the remagnetization process in the 18

19 Figure 7: The local magnetization profile of the microwire, when the domain wall is propagating along the microwire. [H3] second microwire t 2 is given by the equation H m sin(ωt 2 ) = Hs C (ω) + H AC2. The time t 3, when the remagnetization of the microwire C is completed, is calculated by integration of the equation dz dt = S C[H m sin(ωt) H C0 H AC2 ], where H C0 - the experimentally estimated propagation field for the microwire C. The numerical calculation of the dependence of the hysteresis loop on the frequency of the applied field requires the values of the magnetization for every position of the domain wall (z DW ) between the ends of the microwire. During the propagation of the domain wall the microwire is no longer uniformly magnetized, because the area magnetized parallel to the applied field is increasing at the expense of the area magnetized in the opposite direction. In the Figure 7 the postulated local magnetization profile is presented, when the domain wall, that propagates along the microwire, is at the point z DW. The magnetization is given by m(h(t)) = zdw 0 M up (z )dz + L z DW M down (z )dz L 0 M(z )dz Using the above mentioned relations and the equation for m(h(t)), the dependence of the shape of the hysteresis loops on the frequency of the magnetic field was numerically reproduced. The dependence of the critical frequency on the amplitude of the applied magnetic field can be determined assuming that the plateau will disappear if the domain wall in the second microwire starts to move at the same moment when the domain wall in 19

20 the first microwire finishes its motion along the microwire, that is t 1 = t 2. The equation [ ( S A ω cr = [H m cos arcsin H ) sa (ω cr ) H intca L H m ( cos arcsin H )] sc (ω cr ) + H intac H m ( (H A0 H intca ) arcsin H sc (ω cr ) + H intac H m arcsin H )] sa (ω cr ) H intca, (1) H m allows to determine the numerical relation between the critical frequency (ω cr ) and amplitude of the applied magnetic field. A good qualitative agreement has been obtained between the experimental and the numerical values, as presented in the Figure 5. In summary, the influence of the frequency on the shape of the hysteresis loop for a single microwire and two microwire system has been investigated experimentally. The results have been analyzed using the proposed phenomenological model, the condition of the disappearance of the plateau that separates the subsequent remagnetization processes in two microwire system has been derived and compared with the experimental data. It has been demonstrated experimentally that the remagnetization process can be controlled by varying the velocity of the domain wall by means of the magnetostatic interaction and applying the external tensile stress. The results obtained for two microwire system can be successfully extended to an array of multiple microwires. [H4] A. Chizhik, C. Garcia, A. Zhukov, J. Gonzalez, P. Gawroński, K. Kułakowski and J. M. Blanco, Relation between surface magnetization reversal and magnetoimpedance in Co-rich amorphouos microwires, J. Appl. Phys. 103, (2008), 07e742. The numerical calculations of the remagnetization process in the surface of the glass covered Co-rich amorphous microwires were inspired by the experimental studies of the correlation between the shape of the surface hysteresis loops and the giant magnetoimpedance (GMI) ratio. The transverse magnetooptical Kerr effect (MOKE) in the axial magnetic field were used to measure the remagnetization process of five microwires of the same chemical composition Co 69.5 F e 3.9 Ni 1 B 11.8 Si 10.8 Mo 2, but with different ratio of the diameter of the metallic core to the total diameter of the microwire ρ = 0.785, 0.88, 0.885, 0.90, 0.93, where the diameter of the metallic core of microwires were respectively 19.0 µm, 13.2 µm, 20.0 µm, 16.4 µm, 16.2 µm. The GMI ratio were also measured for these microwires. The numerical calculations of the remagnetization process were performed within the coherent rotation approach and taking into account the existence of the uniaxial helical magnetic anisotropy in the surface of Co-rich microwire [42]. The hysteresis loops were calculated by minimizing the magnetic energy U, defined by U = K U cos 2 (θ φ) h m = K U cos 2 (θ φ) h axial cos(θ) h circ sin(θ), (2) where K U - the uniaxial anisotropy constant, m - the saturation magnetization, ( m = 1), φ - the angle between the anisotropy axis and the microwire axis, θ - the angle between m and the microwire axis. The impact of the inner core of the microwire was omitted here, this issue was discussed in [B10]. For the sake of the calculation it was assumed that the applied magnetic field is a superposition of two orthogonal components: the axial (h axial ) and the circular (h circ ) field. The direction of the anisotropy was changed from axial to 20

21 Figure 8: The calculated dependence of the jump of the circular magnetization M on the angle of helical anisotropy φ, and the examples of the calculated hysteresis loops of the circular magnetization in the axially applied magnetic field for various values of φ. [H4] circular direction. For each angle of the helical anisotropy (φ) the energy was minimized in the following way: every time the value of the axial field was changed the new value of the angle θ that minimizes the energy term (2) was found. The Figure 8 shows the calculated dependence of the jump of the circular magnetization M on the angle of helical anisotropy φ. This jump M is maximal for φ = 62. The comparative analysis of the experimental and the numerical values of M has led to a mutual assignment of ρ and φ. The experimental loops ρ = 0.785, 0.88, 0.90, 0.93, have corresponded to the numerical loops φ = 88, 85, 62, 54, 53. Assuming that the giant magnetoimpedance effect is closely associated with the circumferential permeability, which is determined by the jump of the circular magnetization M [43, 44], a direct relation have been established between the angle of helical anisotropy φ and the ratio of the giant magnetoimpedance, that reachable for the microwire of given ratio ρ. The maximum of GMI ratio equals to 80% was measured for the microwire of ρ = 0.9. This corresponds to the calculated value of the angle of helical anisotropy equals to φ = 62. In summary, the numerically obtained hysteresis loops for the Co-rich amorphous microwires allowed us to find a correlation between the giant magnetoimpedance ratio, the jump of the circular magnetization M, and the angle of helical the anisotropy φ. Comparative analysis of the numerically calculated and the experimentally measured hysteresis loop has shown a strong correlation between the shape of the hysteresis loop and the measured value of GMI ratio. [H5] A. Chizhik, A. Zhukov, J. M. Blanco, J. Gonzalez, P. Gawroński, and K. Kułakowski, Experimental determination of relation between helical anisotropy and torsion stress in amorphous magnetic microwires, IEEE Trans. Magn., 44, (2008), The calculations were inspired by the experimental studies of the surface magnetiaztion reversal of Co-rich amorphous glass covered microwires in the presence of a torsional stress. The remagnetization process of the microwire Co 69.5 F e 3.9 Ni 1 B 11.8 Si 10.8 Mo 2 of the diameter of the metallic core d = 19.0 µm and the thickness of the glass coating of 2.6 µm in the presence of the torsion stress were measured by the magnetooptic Kerr effect (MOKE) 21

22 in an axial magnetic field. The numerical calculations of the remagnetization process were performed within the coherent rotation approach and taking into account the existence of the uniaxial helical magnetic anisotropy in the surface of Co-rich microwire. The hysteresis loops were calculated by minimizing the magnetic energy U, defined by U = K U cos 2 (θ φ) h m = K U cos 2 (θ φ) h axial m cos(θ), (3) where K U - the uniaxial anisotropy constant, m - the saturation magnetization, h axial - the applied axial magnetic field, φ - the angle between the anisotropy axis and the microwire axis, θ - the angle between m and the microwire axis. In our calculations we assumed that the microwire surface was a two-dimensional system, because in the MOKE measurement the curvature of the area of microwire surface, from which the light was reflected was approximately about 1. The surface remagnetization process of Co-rich microwire consists of two steps: the magnetization rotation from axial to circular direction in the outer shell of the microwire and the jump of the surface circular magnetization ( M) between two states with opposite directions [45]. The calculated dependence of the normalized value of the jump of the circular magnetization M circ /M MAX as a function of the angle of the helical anisotropy φ are presented in the Figure 9a. These values are negative for 45 < φ < 90, and positive for 90 < φ < 135. For φ = 90, the jump of the circular magnetization M circ equals zero, a) b) c) Figure 9: a) The calculated dependence of the normalized value of the jump of the circular magnetization M circ /M MAX as a function of the angle of the helical anisotropy φ. b) The experimental dependence of the normalized the jump of the circular magnetization M circ /M MAX as a function of the applied torsion stress. c) The dependence of the angle of the helical anisotropy φ on the applied torsion stress. [H5] 22

23 that corresponds to the case of the transverse anisotropy. Two extremes of M circ have been found: the first one for φ = 62, and the second one for φ = 118. The simulated dependence of the jump of the circular magnetization M circ on the angle of the helical anisotropy φ corresponds to the experimentally observed dependence of the jump of the circular magnetization M circ on the torsion stress, presented in the Figure 9b. The analysis of the numerically obtained hysteresis loops for angles φ close to 90 suggest that the remagnetization process is determined mainly by a fluent rotation of the magnetization. For the angles φ close to 60 and 120 the jump of the magnetization M circ occurs at the moment when the direction of the magnetization is close to the direction of the circular magnetization and it results in the quite large values of the jump M circ. The results of the comparative analysis of the numerical and the experimental hysteresis loops have been used to construct the dependence of the helical anisotropy on the angle of the applied torsion stress, shown in the Figure 9c. In the absence of the applied stress the anisotropy was directed almost to the transverse direction. This minimal deviation from 90 was eliminated by the application of small torsional stress of 2.2 π rad m 1. The increase of the absolute value of the stress up to 40πradm 1 causes the growth of the absolute value of the angle of helical anisotropy φ. The dependence shown in the Figure 9c, suggests that the inclination of the helical anisotropy, induced by the application of a torsion stress, does not exceed 45, and this is in agreement with previously considered models [46, 47], but it has never been verified experimentally for Co-rich amorphous microwire. In summary, the numerical results of the simulations of the remagnetization process in Corich amorphous microwires combined with the experimental hysteresis loops have been used to establish a direct relation between the applied torsion stress and the angle of the helical anisotropy. The range of possible the angles of the helical anisotropy (45 < φ < 135 ) induced by the applied torsion stress have been determined. [H6] P. Gawroński and K. Kułakowski, Remanence and switching sensitivity in nanodot magnetic arrays, J. Nanosci. Nanotechnol., 8, (2008), The study of the impact of the finite size of the magnetic nanodot arrays on the remanence was performed. The damage spreading technique was used to investigate the stability of the magnetic state of the magnetic nanodot arrays in AC magnetic field. Each element of the experimental array from Ref. [21] is a one-domain cylindrical nickel dot of the average diameter d = 57 nm, the average height h = 115 nm, located on a square lattice with a period of p = 100 nm. The saturation magnetization was M s = 370emu/cm 3, the average switching field was 710 Oe, and the standard deviation of the switching field was δ = 105 Oe. The computer modeling of the dynamics of the nanodot array was done using the Pardavi- Horvath algorithm [48], the magnetostatic interaction between the nanodots was taken into account using the Rectangular Prism Approximation [49]. During the calculations the periodic boundary conditions were not applied. The results of the numerical calculations presented in the Figure 10a indicate, that the remanence of the nanodot array decreases with the increase of the size of the system, defined by the number nanodots. The calculations were performed for two values of the saturation magnetization M s = 370 emu/cm 3 and M s = 290 emu/cm 3. The effect of the array size on the remanence is smaller when the saturation magnetization is smaller M s, because the interaction energy is proportional to Ms 2. The results of the simulations of the impact of the standard deviation σ of the switching field on the remanence are shown in the Figure 10b. For both investigated arrays the remanence decreases when σ is increasing, because when the distribution of the switching field is broadened, the values of the 23

24 switching field of an increasing number of nanodots become smaller than the magnetostatic interaction, and then more and more nanodots are being remagnetized. That is why, less and less states of the array are stable at zero field. These unstable states of the nanodot arrays are useless for the potential application in the information storage [50]. The stability of the magnetic state of the nanodot array, the ability of the system to store information was analyzed using the damage spreading technique according to the following algorithm. The simulation starts with a random configuration of magnetic moments in the nanodot array. The nanodot array evolves until the stable state is reached, then a magnetic moment of one of the nanodots is modified. The external oscillating magnetic field is applied. The field is parallel to the magnetic moments of the nanodots and perpendicular to the plane of the array. The remagnetization of the nanodot occurs when the effective magnetic field, consisting of the contributions from the external field and the magnetostatic interaction, exceeds the value of the switching field of the nanodot. The simulations are carried out in parallel and independently for the original array and the modified one. The damage is defined as the Hamming distance between these two arrays, i.e. the number of the magnetic moments having different orientation in these two arrays. a) b) c) Figure 10: a) The remanence as a function of the nanodots (N) for: (+) the square array M s = 290 emu/cm 3, (x) the square array M s = 370 emu/cm 3, ( ) the rectangular array of aspect ratio 2 : 3 M s = 370 emu/cm 3. b) The remanence as a function of the standard deviation σ of the switching field for the array the upper curve, the lower curve. c) The average damage < D > as a function of the standard deviation σ of the switching field for the array [H6] The simulation of the damage spreading were done for the array and the control parameter was the standard deviation of the switching field. The average damage < D > presented in the Figure 10c decreases when σ increases. Thus, the array with the more diverse switching fields (larger σ) shows the greater stability of the magnetic state, because the average damage is smaller. It is worth noting that in some cases the damage does not spread at all but disappears immediately after the start of the simulation. The probability of the disappearance of the damage for σ = 15 Oe equals 33 % and for σ = 105 Oe increases to 63 %. 24

25 The nanodot array can be used as a magnetic recording medium only when the recorded information will not disappear after the external field is switched off. In particular, in the remanent state the spontaneous remagnetization of the nanodots under the magnetostatic interaction cannot be present. In order to reduce the damage spreading the remanence should be as close as possible to saturation. The numerical calculations indicate, that the range of possible values of the standard deviation σ of the switching field is divided into two regimes. For small σ the remanence is close to saturation magnetization, which means that the saturation state of the nanodot array is stable or approximately stable, because for most of the nanodots the switching field is greater than the magnetostatic interaction. All the magnetic states of the system can be used to store information. On the other hand, for the array of small value of σ, the damage in the form of changed magnetic configuration spreads very quickly throughout the system in the presence of the oscillating magnetic field. However, in the case of the large σ the damage is localized and does not spread between nanodots, that is desirable because of the stability of the stored information. Unfortunately, a large value of σ causes a decrease in the remanence of the nanodot array, and the spontaneous remagnetization of the nanodots due to the magnetostatic interaction occur. In summary, the numerical simulations of the impact of the standard deviation σ of the switching field on the remanence and the application of the damage spreading technique have revealed that the optimization of the stability of the magnetic state of the nanodot array, requires efforts to achieve the remanence as close as possible to the saturation magnetization while maintaining a relatively high value of the standard deviation σ of the switching field. [H7] P. Gawroński, K. J. Merazzo, O. Chubykalo-Fesenko, A. Asenjo, R. P. del Real and M. Vaqzquez, Micromagnetism of dense permalloy antidot lattices from anodic alumina templates, EPL, 100 (2012) The motivation for the micromagnetic calculation was an increase of the coercive field (H c ) of the permalloy (Py, Ni 80 F e 20 ) antidot array with the antidots diameter (d), experimentally observed, in the laboratory of Instituto de Ciencia de Materiales in Madrid. The effect is presented in the Figure 11. The investigated antidot arrays were prepared by sputtering Py onto aluminum membrane templates. The diameters of the antidots were d = 11, 55, 62 and 71 nm and the distance between the antidots was D = 105 nm. The sputtering method allows to achieve a denser packing of antidots than previously achievable using lithography, for which the typical distances between antidots range from 200 nm to 2 µm. In the case of the arrays obtained by the lithography, inhomogeneous magnetic structures are often observed in the remanence state in the area between the antidots. These structures like e.g. zig-zag structures are responsible for the remagnetization process and they are perfectly visible using the magnetic forces microscopy [51, 52]. In the case of arrays obtained by the sputtering technique, there is often not enough space between the antidots for the formation of the inhomogeneous magnetic structures, and therefore one can expect a different type of micromagnetic processes than those observed in the case of arrays obtained by lithography. The micromagnetic simulations were carried out for the antidot arrays of a total size of nm 2, the discretization was 2 nm and the periodic boundary conditions were applied. The diameters of the antidots were d = 16, 25, 35, 45, 55, 65, 75, 85 nm, and the distances between the centers of the antidots were D = 100 and 200 nm. We used typical material constants for Py, namely the exchange constant A = J/m, the saturation magnetization M s = A/m and we neglect the magnetocrystalline anisotropy. Numerical calculations were performed for two types of arrays. In the first case we implemented the antidot arrays with the perfect hexagonal ordering in the entire array; such a case we called a perfect single crystal. In the second case, we broke the symmetry 25

26 of the perfect hexagonal lattice, by dividing it into two parts along the diagonal, then we rotated one of the halves of the lattice by 30 and we filled the empty spaces with the antidots. An example of such simulated antidot array is presented in the Figure 12b. The polycrystalline antidot arrays implemented in this way are a better representation of the real arrays investigated experimentally and presented in the Figure 12a, because the real arrays exhibit both the hexagonal ordering on the area of several square micrometers and the dislocations, which mark the boundaries between the ordered areas. The simulated hysteresis loops for the polycrystalline arrays show that the remagnetization process for the arrays D = 100nm and d < 45nm occur at one value of the external magnetic field; also the ratio of the remanence to the saturation magnetization is equal to one. When the diameter d of the antidots increase above 45 nm, the remanence decreases, and for d > 65 nm the remagnetization process does not finish at the value of the external magnetic field at which it begins, but requires a further increase of the applied magnetic field. The numerically calculated values of the coercive fields (H c ), as shown in the Figure 11, grow as a function of the diameter of antidots (d) in a good agreement with the experimental results. The simulated hysteresis loops for monocrystalline arrays show that the remagnetization process for D = 100 nm and d < 65 nm occurs at one value of the external magnetic field but for the larger diameters an increase of the applied magnetic field is required. All numerically simulated hysteresis loops for the monocrystalline antidot arrays have a ratio of remanence to saturation magnetization equal to one. The values of the numerically calculated coercive fields for the single crystal monocrystalline antidot array are greater than these obtained for the polycrystalline arrays for d 75 nm and they decrease with the increase of d, on the contrary to the relation observed in the experiment. The simulated spatial magnetization configuration indicate that, in the case of both types of the arrays for d < 65 nm the coercive field is defined by the nucleation process, where the nucleation area usually contains many antidots, as shown in the Figure 13a. In the case of the polycrystalline arrays the nucleation process begin in the area rotated by 30, where the external field is applied parallel to the hard axis and thus the coercive field is much Figure 11: The measured coercivity H c as a function of the antidot diameter d, and the simulated data for the polycrystal arrays, when the distance between the centers of the antidots was D = 100 nm. [H7] 26

27 a) b) Figure 12: a) SEM image showing the hexagonal array with d = 55 nm and the existence of domains. b) The spatial configuration of the magnetization for the simulated antidot array (D = 100 nm, d = 85 nm). The hexagonal lattice is divided diagonally into two halves, the upper half is rotated about 30 relative to the bottom half. The field was applied along the x-direction. The red color indicates magnetization with mx > 0, the white one with mx = 0 and the blue one with mx < 0. [H7] b) a) Figure 13: The spatial configuration of the magnetization for the simulated antidot array : a) d = 35 nm, D = 100 nm the nucleation process, b) d = 35 nm, D = 100 nm - the domain wall pinning. The magnetic field was applied along the x-direction. The red color indicates magnetization with mx > 0, the white one with mx = 0 and the blue one with mx < 0. [H7] smaller than in the case of monocrystalline antidot arrays. The numerical results show that for both types of arrays the remagnetization takes place by the propagation of domain walls, and during that process the pinning of the domain walls at antidots occurs, as shown in the Figure 13b. In most of cases the depinning of the domain wall occurs dynamically during the relaxation process at the same field value equal to the coercive field, while in some cases an increment of the applied field is necessary. For the polycrystalline arrays for the antidot diameter above d > 65nm the nucleation becomes more difficult which effects in an increase of the coercive field as a function of the diameter of antidots d. In the case when the diameter of the antidots d becomes comparable to the distance between the centers of antidots D, d > 75 nm, the numerically calculated hysteresis loops consist of several Barkhausen jumps. The nucleation area becomes comparable to the size of the separation of antidots D d. The simulated spatial configurations of the magnetization indicate that nucleation takes place in many different areas of the array and each subsequent one requires an increase of 27

28 the applied magnetic field; at the same time strong pinning is observed. The effect is similar for the polycrystalline and monocrystalline arrays, however for the polycrystalline arrays the disorder at the grains boundary promotes the nucleation and therefore the values of the coercive field for the polycrystalline arrays are smaller than for the monocrystalline arrays. During the study, the micromagnetic simulations of the arrays of the distances between antidots D = 200 nm were performed and the zig-zag structures were obtained, that initialized the remagnetization, in the remanence. The simulated magnetic properties of such arrays were found to be consistent with the properties of the arrays obtained in the lithography process described in the literature [53, 54]. In summary, the dependence of the coercive field (H c ) on the antidots diameter (d) obtained during the micromagnetic simulation of the polycrystalline antidot arrays is in good agreement with the experimental dependence (Fig. 11). The analysis of the numerically obtained spatial configuration of magnetization allows to describe the mechanism of remagnetization process in the polycrystalline Permalloy antidot arrays with dense hexagonal ordering. [H8] C. Castan-Guerrero, J. Herrero-Albillos, J. Bartolome, F. Bartolome, L. A. Rodriguez, C. Magen, F. Kronast, P. Gawronski, O. Chubykalo-Fesenko, K. J. Merazzo, P. Vavassori, P. Strichovanec, J. Sese, and L.M. Garcia, Magnetic antidot to dot crossover in cobalt and permalloy patterned thin films, Phys. Rev. B 89, (2014) The aim of the work was to study the continuous transition between the behavior characteristic for the antidot arrays and the behavior characteristic of isolated dots caused by varying the geometry of the system. The experimental part of the work was done in the laboratory of Instituto de Ciencia de Materiales de Aragón in Saragossa, where the series of antidot arrays with the fixed diameter d and the variable distance between the antidots centers p from p >> d to p < d, were produced by the focused ion beam etching. An additional ring of reduced magnetization surrounding the antidot appeared as an effect of the gallium ions etching. This damaged area caused a broadening of the effective diameter of the antidots d eff and allowed to create an antidot arrays, for which the distance between the centers of the antidots was smaller than their effective diameter (p < d eff ). The antidots intersect with each other forming an array that has the magnetic properties that are characteristic for an array of isolated dots. The motivation for the micromagnetic simulations was the experimentally measured dependence of the coercive field (H c ) on the distance between the antidots, presented in the Figure 14. The dependence was accompanied by a significant change of the domain structure observed by the x-ray photoemission electron microscopy (XPEEM). The micromagnetic simulations were performed for the arrays consisting of 25 antidots of fixed diameter d = 80, 90, 100, 110, 120 nm, arranged on a square lattice of the distances between the antidots centers between 70 nm < p < 250 nm and the thickness of 10 nm. In order to optimize the computation time the discretization of the small arrays was set to 2 nm, but for the largest arrays it was set to 4nm. The periodic boundary conditions were applied during the micromagnetic simulation. The following micromagnetic parameters for cobalt were used: the exchange constant A = J/m, and the saturation magnetization M s = A/m. The magnetocrystalline anisotropy is negligibly small for the polycrystalline structure of the Cobalt antidot arrays, therefore we set this parameter to zero in the simulations. The simulated dependencies of the coercive field (H c ) on the distance between the edges of the antidots λ = p d are presented in the Figure 14. For all examined antidot diameters (d) a maximum of the coercive field (H c ) has been found for λ 0, e.g. when p is slightly bigger than the d. Both for the experimental and the numerical curves in the Figure 14 28

29 H c [Oe] d = 80 nm d = 90 nm d = 100 nm d = 110 nm d = 120 nm experimental 200 D INT AD λ = p - d [nm] Figure 14: The dependence of the coercive field (H c ) on the distance between the edges of the antidots λ = p d obtained in the micromagnetic simulations and experimentally measured for the cobalt antidot arrays of the effective diameter of 114 ± 4 nm. The dependence H c (λ) is divided into 3 regimes: AD - the antidots regime, INT - the intermediate regime and D - the dot regime. [H8] Figure 15: The fragments of the numerically simulated hysteresis loops for three regimes: AD - the antidots regime, INT - the intermediate regime and D - the dot regime. [H8 - unpublished] 29

30 three regimes of the magnetic properties have been distinguished: a) AD - the antidots regime, λ > 28 nm, b) INT - the intermediate regime, 0 < λ < 28 nm, c) D - the dot regime, λ < 0 nm. The spatial configurations of the magnetization for the antidot regime have revealed that the remagnetization process occurred through the nucleation and the propagation of domain walls. Also the pinning of the domain wall on the antidots has been observed. As a result, the numerical hysteresis loops presented in the Figure 15 consist of several jumps of the magnetization. The calculated values of the coercive field (Hc) in the antidot regime grow as the domain wall mobility decreases when the distance between the edges of the antidots λ decreases. This is a typical behavior which is often described by an empirical relation H c λ 1 [55, 56]. An existence of large elongated domains which include several antidots, is characteristic for the experimental as well as the numerical spatial configurations of the magnetization. In analogy to the experimental data, for the simulated results as well the value λ = 28 nm can be treated as a border between the antidot regime and the intermediate regime. The simulated configurations of the magnetization indicate that when the distance between the edges of the antidots λ is gradually decreasing, but remaining still positive, the remagnetization takes place mainly by the nucleation, because the propagation of domain walls is rather difficult due to the strong pinning on the antidots. Thus, the dependence of the coercive field H c on λ changes. For all the simulated diameters of the antidots d the coercive field H c reaches a maximum at λ 12 nm, and then decreases due to the reduction of the space between the antidots where the nucleation take place. When λ < 0, the simulated arrays are no longer continuous, as the area between the neighboring antidots is divided into isolated astroid shaped dots, like the one presented in the Figure 16. The simulations show that the remagnetization occurs in a noncoherent magnetization rotation, that starts at one of the cusps and then propagates towards the astroid center. In the dot regime the coercive field increases when the absolute value of λ decreases, because the astroid cusp angle becomes less-acute, and the dots are more similar in shape to squares, then it is easier to initialize the remagnetization process. The Figure 16 shows the numerically calculated nonhomogeneous magnetization distribution of the dot in the remanence, which we have named the astroid state. The magnetization in the central part is directed diagonally, and close to the cusps it is set vertically or horizontally. Figure 16: The astroid state - the dot in the remanent state - the spatial configuration of the magnetization obtained in the micormagnetic simulations for p = d = 110 nm. [H8] The simulated values of the coercive field presented in the Figure 14 are greater than the experimental values, due to the presence of random defects in the experimental arrays, which facilitate the nucleation and reduce the coercive field. The simulated curves for d = 110 nm and d = 120 nm are in a good agreement with the experimentally measured dependence of the coercive field on the parameter λ for the effective antidot diameter d eff of the cobalt antidot arrays equal to 114 ± 4 nm. In summary, the simulated dependence of the coercive field of the cobalt antidot arrays on the 30

31 distance between the antidots is in a good agreement with the experimentally measured one. The numerical calculations confirm the existence of the experimentally observed transition with change of p between the antidot regime, where the coercive field is proportional to λ 1, and the dot regime, where the coercive field increases when the absolute value of λ decreases, as well as the presence of the intermediate regime between them. [H9] P. Gawroński, K. J. Merazzo, O. Chubykalo-Fesenko, R.P. del Real and M. Vaqzquez, Micromagnetism of permalloy antidot arrays prepared from alumina templates, Nanotechnology 25, (2014), This work is a continuation of research initiated at work [H7]. The micromagnetic simulations of the remagnetization process of the Permalloy (P y) antidot arrays were performed as previously just the list of the control parameters that typically consists of the antidot diameter (d) and the distance between the antidots (D = 100 nm) was supplemented by the thickness of the antidot arrays (h). The numerical calculations for the antidot arrays of a total area of nm 2 were made using the OOMMF package, taking into account the periodic boundary conditions. In the micromagnetic simulations the typical material constants for Permalloy were used, namely the exchange constant A ex = J/m 3 and the saturation magnetization M s = Am 1. The magnetocrystalline anisotropy was neglected. a) b) c) Figure 17: The simulated dependence of the coercive field on the antidot diameter for various thicknesses of the antidot array, in the case of a) an ideal hexagonal lattice, b) a lattice with the hexagonal symmetry broken. c) The dependence of the coercive field on the thickness of the antidot array for the fixed antidot diameter of 45 nm calculated numerically for: an ideal hexagonal lattice - red dots, a lattice with the hexagonal symmetry broken - green dots. The experimental data - blue dots. [H9] The simulated dependence of the coercive field on the antidot diameter d and the thickness of the array h for an ideal hexagonal lattice is presented in the Figure 17a. A characteristic feature of the obtained dependencies is an existence of the maximum of the coercive field 31

32 for the diameter d between 35 nm a 55 nm, when the thickness is bigger than 10 nm. The spatial distributions of the magnetization, obtained during the simulation, allow to understand the dependence of the remagnetization process on the geometric parameters of the antidot arrays. For the diameters less than 45 nm the remagnetization process occurs by means of the anticlockwise rotation of the magnetization, starting with the nucleation at the antidots borders, and it leads to the formation of a periodic stripe-like structure presented in the Figure 18a. a) b) c) d) e) f) Figure 18: The spatial configuration of the magnetization in XY plane for antidot arrays M of an ideal hexagonal ordering for the thicknesses h = 34 nm: a) d = 35 nm, M s = 0.042, H app = 121 mt, b) d = 55 nm, M M s = 0.707, H app = 240 mt, c) d = 55 nm, M M s = 0.145, H app = 240 mt, d) d = 75 nm, M M s = 0.157, H app = 71 mt, e) d = 85 nm, M M s = 0.180, H app = 100 mt. f) The spatial configuration of the magnetization in XY plane for antidot arrays of a broken hexagonal ordering h = 60 nm, d = 35 nm, M M s = 0.394, H app = 22 mt. The magnetic field was applied along the x-direction. The red color indicates magnetization with m x > 0, the white one with m x = 0 and the blue one with m x < 0. [H9] For the diameters 45 and 55 nm the remagnetization occurs through the rotation of the magnetization in both clockwise and anticlockwise directions. This leads to a zig-zag-like structure, presented in the Figure 18b and then the formation of domain walls in different regions of the antidot arrays, as presented in the Figure 18c. Another characteristic feature of the remagnetization process of the arrays of the antidot diameters 45 and 55 nm is a strong pinning of the magnetization to the antidot, which is responsible for the maximum value of the coercive field. An increasing the diameter of the antidots to values of 65 nm - 75 nm brings the magnetic poles closer, which leads to a uniform magnetization perpendicular to the applied field in the region between antidots and reaching a periodic stripe-like structure presented in the Figure 18d. In the case of the antidot arrays of a diameter of 85 nm, antidots 32