- Email: [email protected]
Geometry dependent crossover of Barkhausen statistics in iron thin films
Physica E: Low-dimensional Systems and Nanostructures 106 (2019) 19–24
Contents lists available at ScienceDirect
Physica E: Low-dimensional Systems and Nanostructures journal homepage: www.elsevier.com/locate/physe
Geometry dependent crossover of Barkhausen statistics in iron thin films ∗
T
Arnab Roy , P.S. Anil Kumar Department of Physics, Indian Institute of Science, Bangalore 560012, India
A B S T R A C T
We report the crossover of Barkhausen noise statistics upon a change in the measurement geometry. The behaviour is akin to the angle dependent critical to subcritical crossover of avalanche statistics as present in avalanche models with random anisotropy directions. The samples studied were 15–25 nm thin films of iron, grown on (001) GaAs by pulsed laser ablation. Use of planar Hall effect as a probe for the magnetic state facilitated this work, which would be impractical to do in a MOKE setup. We compare our data with simulations of a random anisotropy Ising model, and demonstrate striking similarity between experimental and simulated avalanche size distributions. Further, through micromagnetic simulations, we conclude that crystallite misorientations were the origin of domain wall pinning in our system leading to Barkhausen noise.
1. Introduction A close connection exists between magnetic hysteresis and disorder. Disorder in a magnetic material gives rise to inhomogeneity in the energy landscape encountered by a magnetic domain wall during magnetization reversal. The consequence is that, in the course of its motion, the magnetic domain wall may get trapped at local energy minima, which can be overcome only after a change in the driving magnetic field. This trapping and untrapping manifests itself as Barkhausen noise, when magnetization of a sample changes in an intermittent and discontinuous manner upon driving with a smoothly changing magnetic field. The sizes and durations of these avalanches are distributed according to power-laws of the form
P (s ) = As−τ
(1)
Here P(s) is the probability of occurrence of a magnetization ‘jump’ of size ‘s’, and the form persists up to a cutoff size above which the probability falls off exponentially. The magnetization steps during a field sweep are stochastic and non-reproducible even under identical initial conditions. τ is referred to as a critical exponent, whose value depends on the universality class to which the system belongs. The study of such phenomena which fall under the general category of crackling noise, has received a lot of attention in the past few decades because of its connection to disorder-induced critical phenomena. A number of models have been put forward aimed at reproducing the statistical properties of this noise starting from a minimum of ingredients, the earliest among which is the ABBM model [1,2]. It was the first attempt aimed at specifically predicting the statistical properties of domain wall motion over a rough energy landscape. It predicted τ = 3/2 – c/2D, where c is the driving rate and D is a disorder parameter.
∗
Our present understanding of Barkhausen phenomena is based on two classes of models: Front propagation models like the CZDS [3] model, deal with the motion of a 2D domain in a lattice unbounded in one direction under an external magnetic field changing smoothly in time. Along with this, there exist site-dependent random magnetic fields and exchange energy modeled as a ‘surface tension’ of a domain wall. Its prediction is similar to that of the ABBM model, with τ = 3/2 – c/2. Another class of models, namely the nucleation models [4–14], have a finite lattice with site dependent disorder in the form of a random magnetic field or random magnetocrystalline anisotropy directions. The first studies of avalanche dynamics in such models emerged around 1995 due to Perković, Dahmen and Sethna [5]. One of the predictions of both the classes of models is the presence of a critical point. In the ABBM model, the critical point is dependent on the ratio of the magnitude of the disorder to the magnetic field sweeping rate [1,2]. In nucleation models the critical point is arrived at by tuning the magnitude of the random disorder. The critical point appears as a transition from a single-step magnetization reversal to a collection of discrete steps corresponding to a large number of pinning and depinning events. The order parameter in this transition is the size of the largest single avalanche that results in the magnetization reversal, and its magnitude decreases to zero at disorders higher than the critical disorder. Though easily accessible through the models, experimental evidence of such phenomena are not common [15–18] and the conclusions are often fairly indirect. The present authors have recently observed the crossover from sub-critical (non-universal) to critical (universal) behavior by increasing the disorder level in permalloy thin films [19]. In this article, a similar crossover is reported in Fe thin films, arising out of purely geometrical reasons. Our study consists of the experimental investigation of the effect of measurement geometry on
Corresponding author. E-mail address: [email protected] (A. Roy).
https://doi.org/10.1016/j.physe.2018.10.023 Received 16 July 2018; Received in revised form 26 September 2018; Accepted 15 October 2018 Available online 16 October 2018 1386-9477/ © 2018 Published by Elsevier B.V.
Physica E: Low-dimensional Systems and Nanostructures 106 (2019) 19–24
A. Roy, P.S.A. Kumar
the four-fold anisotropy. The anisotropy field was approximately 1Oe, which was a fairly small value for iron at room temperature [26]. Preliminary electrical characterization comprised resistivity and magnetoresistance measurements at low temperatures. The residual resistivity was fairly low (∼6 μΩ cm), which indicated a largely defectfree film growth. High-field (B > 2 T) and high temperature (T > 100 K) magnetoresistance was dominated by the negative linear magnon magnetoresistance, while at low temperatures and high fields, signatures of Lorentz magnetoresistance could be seen. This was easily distinguishable from the MR due to localization effects because of its non-saturating behavior even at high magnetic fields. The presence of Lorentz MR was yet another indicator of good film quality, as it requires a substantial electronic mean-free-path to become noticeable [22]. The importance of having good quality samples for this study will become apparent in the discussion on Barkhausen noise. For the Barkhausen noise study, planar Hall voltage was measured at 300 K in the magnetic field of an air-core coil. Magnetic field was swept in steps of ∼0.01Oe and the signal was acquired synchronously after every step. DC constant current measurements yielded a typical S/ N ratio of 104 and a typical voltage swing of ∼1 mV. Each sweep was repeated 1000 to 2000 times to gather statistics. Fig. 2 shows a typical PHE curve during a field sweep. The sharp minima correspond to the magnetization reversal fields of the Fe thin films, which is the field at which there is a maximum abundance of domains. A finer field step-size reveals a multitude of avalanches (inset), the positions and sizes of which varied randomly between different field sweeps. Barkhausen avalanche statistics were extracted in 2 stages: An initial processing accumulated all signal jumps whose magnitude was above a preset threshold. Following this, Barkhausen noise was separated from instrumental noise by choosing a cut-off size below which the probability of ‘upward’ and ‘downward’ jumps became comparable. Avalanche statistics were measured for different rotations of the sample relative to the applied magnetic field, the field being applied both in-plane and out-of-plane to the sample. In Fig. 3, ‘zoomed-in’ views of switching curves for 6 different orientations of a 100 μm wide and 25 nm thick Hall bar are shown. Four in-plane configurations were chosen: 0O, 45O, 90O and 135O, (Fig. 3:A,D,E,F) between the applied field and current direction. These angles correspond to the positions of maxima, minima and ‘zero's for the planar Hall voltage at saturation ie, when α is the same as the measured angle (Eqn. (1)). Notwithstanding the differences in the ‘direction’ of the voltage changes, the switching process for all the four in-plane configurations were qualitatively very similar: a single large switching event dominated the magnetization reversal accompanied by a number of smaller events in each case. The consequence was that the size distributions with the applied field in-plane consisted of two regions: a power-law probability distribution for the small sizes and characteristic ‘hump's towards the tail of the distributions representing the large jumps (Fig. 4: A,D,E,F). The two other orientations that are presented here are two angles of
Barkhausen noise of iron thin films measured by planar Hall effect. It also attempts to arrive at an explanation primarily based on simulations of a random anisotropy Ising model (RAIM). Certain finer points are clarified through micromagnetic simulations. Planar Hall effect was used to avoid the inherent difficulties associated with an optical (MOKE) measurement setup upon changes to the measurement configuration. 1.1. Planar Hall effect Planar Hall effect is a phenomenon related to anisotropic magnetoresistance which occurs due to spin-orbit interaction in ferromagnetic materials [20]. In thin films, it is essentially an in-plane voltage developed perpendicular to the direction of the current density, when the magnetic field is in the plane of the sample along with the current. Empirically, the Hall voltage is expressed [21] as:
V = J . d. (
ρ|| − ρ⊥ 2
)sin 2 α
(2)
where α is the angle between the magnetization and the current density J. The relation holds for a single magnetic domain. In an actual sample, there are multiple ferromagnetic domains pointing in different directions, so that the effective α is the weighted average of the angle over different domains, modulo 180O (as the signal is periodic in 2α) [22,23]. This 'effective medium' approach may be summarized as:
V = J . d. (
ρ|| − ρ⊥ 2
3
)
∑ fi sin 2αi i=1
(3)
where the summation is over the 3 spatial DOF's and fi is the volume fraction of domains parallel to the ith direction in the region of interest. Thus, an abrupt change in the domain structure represented by [fi] shows up as a jump in the Hall voltage. 2. Experiment Samples used for this study were thin films of Fe grown on GaAs (001) by pulsed laser deposition (PLD). GaAs was chosen since it is known that Fe grows epitaxially on GaAs(001) [24,25]. The lattice mismatch between BCC Fe (2.87 Å) and GaAs (5.65 Å) is about 1%, with two unit cells of Fe growing on one unit cell of GaAs. PLD was carried out at room temperature in a UHV chamber with a base pressure of 2 × 10−10 Torr, and 5 × 10−8 Torr during deposition. Thicknesses of 15 nm and 25 nm were deposited, each with a capping layer of about 5 nm of Au to prevent oxidation. Patterning to 100 μm Hall bars was done using laser direct writing and lift-off. The primary characterization was done with MOKE magnetometry. L-MOKE with in-plane sample rotation showed a predominantly fourfold anisotropy with 〈110〉 and 〈11¯0〉 being easy directions. This information was obtained from the angular dependence of the coercivity (Fig. 1a) and remnant magnetization ratios (Fig. 1b). A closer look revealed that there was also a weak uniaxial anisotropy superimposed on
Fig. 2. Typical planar Hall hysteresis curve for100 μm wide Fe Hall bar. Inset: ‘zoomed in’ view of the region which shows Barkhausen noise (right), Schematic diagram of the hall bar sample (left).
Fig. 1. Coercivity (a) and remanence (b) as a function of angle for Fe (15 nm) on GaAs obtained from MOKE hysteresis loops. Four-fold magnetic anisotropy is indicated. 20
Physica E: Low-dimensional Systems and Nanostructures 106 (2019) 19–24
A. Roy, P.S.A. Kumar
3. Simulations To examine the striking similarity of the experimental data with a disorder-induced critical phase transition, simulation of a popular nucleation model, the 2D Gaussian random field Ising model (2D-GRFIM) was carried out. 3.1. Random field ising model (RFIM) The Gaussian Random Field Ising model deals with an ensemble of classical spins in a 2D lattice. The Hamiltonian of the system is [ [8]]
Η = −J ∑ Si Sj − Hext i, j
Fig. 3. PHE switching curves of Fe Hall bar for 4 in-plane (A,D,E,F) and 2 outof-plane (C,D) orientations of the driving magnetic field. The large switching event is observable in all the in-plane configurations and is absent for the outof-plane configurations.
∑ Si − ∑ Hi Si i
i
(4)
Here, J (> 0) is the ferromagnetic coupling constant between the neighboring spins, Hext is the external field and Hi is a site dependent random field, with a Gaussian distribution centered at zero and an adjustable standard deviation R. Spins are allowed to take the values +1 and −1. For our simulations, we chose a square lattice with only ‘nearest’ neighbor interactions, so that the number of spins interacting with any given spin is 4. There were no long-range interactions. Periodic boundary conditions were used to satisfy the above criterion for a spin at the edge. We used 3 different lattice sizes: 100 × 100, 200 × 200 and 1000 × 1000. The system was initialized with all the Si = 1 and the external field was swept from a sufficiently large negative value to a sufficiently large positive value to ensure reversal of all the ‘spin's. Since the model parameters are scalable by an arbitrary scale factor, we fixed J = 1. Simulations of the 2D-GRFIM presented in literature had been performed in the adiabatic limit, so that only a single avalanche was triggered at each field step. However, to satisfy the real experimental conditions more closely, the adiabatic limit was discarded and the simulation was carried out at a constant field step. Following a well-established protocol, after each field step, the system was allowed to evolve to a metastable configuration over several iterations, with changed nearest neighbor configurations due to spins flipped in the previous iteration. The accumulated value of the total number of spins flipped due to a field step was taken as the size of a jump. To collect statistics, the sweeps were repeated 1000 to 5000 times with different realizations of the random field. Fig. 5 shows four experimentally obtained size distributions (top row) matched with those obtained from simulations of 2D GRFIM (bottom row). The experimental parameters and the simulation parameters that generate the matching size distributions are shown in Table 1. Even though the matching was close, there could be no reasonable physical argument that could connect the scalar RFIM to angle dependent experimental observation. It was intuitively obvious that the minimum requirement was a model that incorporated direction dependent ‘randomness’, at the same time incorporating all the characteristic features of the RFIM, particularly that of the disorder-induced critical phase transition. For this, we invoke a relatively less studied nucleation model: the random anisotropy Ising model (RAIM).
Fig. 4. Size distribution of Barkhausen avalanches in Fe Hall bar for 4 in-plane (A,D,E,F) and 2 out-of-plane (C,D) orientations of the magnetic field. Sub critical to critical crossover clearly observable when going to an out-of-plane configuration.
rotation out-of plane starting from the 0O in-plane configuration. We refer to this angle as ω and have the rotation direction illustrated in Fig. 2. In principle, a rotation out of plane of sample should not have much significance when it comes to AMR and PHE, as the magnetization is strongly confined in-plane due to shape anisotropy originating from demagnetization effects. The only observable change expected would that of the coercivity value. However, as can be seen in Fig. 3(B,C), the switching behavior was altered in a much more profound manner, so that a switching curve now consisted of only small jumps. Correspondingly, the size distributions were also changed and the ‘familiar’ power law behavior with a cut-off was observed (Fig. 4B and C). The power-law exponent (τ′) values for the 25 nm thick sample were 1.5 and 2.2 for ω = 45O and 67O respectively. The behavior was similar for the sample of 15 nm thickness, except that the disappearance of the single large switching event did not occur at 45O tilt, and the transition was only complete at 67O. Comparing with theoretical studies on hysteresis loop criticality, it is easy to map the single large switching event to a spanning avalanche, which refers to one that traverses the whole sample from boundary to boundary. The state corresponding to the presence of a spanning avalanche is a sub-critical state with respect to disorder. On the other hand, a condition where avalanche sizes are distributed according to a powerlaw is close to a disorder-induced critical point. It is not clear a priori as to what is bringing about this transition, since the phenomenon of the ‘disorder’ level changing with measurement condition is somewhat contra-intuitive.
Fig. 5. Experimental avalanche size distributions for 15 nm thick Fe Hall bars (A–D). Simulated avalanche size distributions using 2D-GRFIM (E–H). 21
Physica E: Low-dimensional Systems and Nanostructures 106 (2019) 19–24
A. Roy, P.S.A. Kumar
Table 1 Experimental and simulation parameters corresponding to plots in Fig. 5. Figure
Experiment details
5A&E 5B & F 5C & G 5D&H
15 nm, 15 nm, 25 nm, 25 nm,
0O 67O 0O 45O
L
R/J
sweep range
step
Power-law exponent
1000 1000 100 100
0.01 50 0.001 0.33
± 1000 ± 1000 ± 200 ± 200
2 0.2 0.2 0.2
3.65 × 2.24 1.26
3.2. Random anisotropy ising model (RAIM) Random anisotropy models were first used in the context of amorphous ferromagnets by Harris, Plischke and Zuckermann [27]. Since then, this class of models has found wide applications in various fields of study, the most important applications being in the theory of spinglasses [28,29] and disordered ferromagnets [30–34] and in the theory of nematics and liquid crystals [35,36]. A study of the avalanche properties of this model under driving with an external field was presented by Vives and Planes [37,38], in the limit of strong anisotropy. In this limit, the model is reduced to the random anisotropy Ising model, which, according to Vives et al., is expected to belong to the same universality class as the RFIM. They demonstrated that the critical exponent τ had a value slightly above 2, and varied slightly depending on the details of the representing of disorder. A general Hamiltonian of the system with random anisotropy directions and without long-range interactions can be
Fig. 6. Simulated magnetization reversal curves of RAIM as a function of out-ofplane angle of the applied field. The dispersion in the anisotropy axes is 10O. Disappearance of the spanning avalanche between 80O and 82.5O indicates the sharp crossover to critical behavior.
→→ → → → H = −J ∑ Si · Sj − Hext · ∑ Si − D ∑ (nˆ i · Si )2 i≠j
i
i
(5)
where nˆ i is the local anisotropy direction and D is the anisotropy con→ stant. In the limit of large D, the spins Si are constrained to lie either parallel or antiparallel to nˆ i , and the model is reduced to an Ising model in which the spins can only have values of ± 1. Under this approximation, the reduced Hamiltonian of the system becomes
Η = − ∑ Jij Si Sj − Hext i, j
∑ gi Si i
(6)
where Jij = Jnˆ i . nˆ j and gi = qˆ. nˆ i . This was the form of the model used for our simulations. The effective disorder in this case is gi, and plays a role analogous to Hi in the RFIM. However, in this case, it is angle dependent. In fact, a random anisotropy model is a more realistic representation of an actual sample than a random field model, as the former can be linked readily to the mosaicity of the crystalline material comprising the sample. The dynamics were metastable and the evolution at a given field was stopped after all the spins had locally relaxed. The disorder was a uniform distribution of the anisotropy axes within a circular region around the x axis. This was equivalent to having an ‘average’ anisotropy → direction nˆ 0 parallel to the x axis and a random anisotropy vector Δn i → was unity. The angles made by nˆ such that the norm of nˆ i = nˆ 0 + Δn i i with nˆ 0 were uniformly distributed within an angular range of ± σ. σ was synonymous with the disorder magnitude. Without loss of generality, the external magnetic field was confined to the z-x plane. The program was used to attempt to reproduce the experimental size distributions as a function of angle of the applied field relative to the xaxis. As in the RFIM case, sub-critical to critical (power law) behavior was observed when the external field direction was rotated away from the x-axis. At sub-critical angles, practically the entire sample was reversed in a single avalanche, while above the critical angle the reversal was through multiple discrete steps (Fig. 6). Extraction of the size distribution of Barkhausen jumps showed the existence of a power-law region even when the system was sub-critical, the exponent τ′ decreasing to a minimum of about 1.7 at the critical effective disorder and increasing on both sides of it. The value of this
Fig. 7. Power-law exponent of avalanche size distributions (τ′) in the RAIM as a function of angle for two system sizes and two disorder magnitudes.
minimum was practically independent of the system size and disorder magnitude, while its position shifted to lower angles as the disorder magnitude was increased (Fig. 7). The somewhat surprising observation was that the power-law exponent increased after the critical angle was exceeded, a hint of which was also observed in our experiment with the 25 nm film, where it increased from 1.5 to 2.2 as ω was increased from 45O to 67O. As expected, the shapes of the simulated size distributions matched the experimental distributions closely, both in the critical and sub-critical regimes (Fig. 8). At the set disorder level (10O), the transition from subcritical to critical was complete within an angle change of 2O. The RAIM simulations did not generate the very large τ′ values ( > 4) obtained in experiment at low angles for the 15 nm sample, though that does not hamper the physical understanding of the system through this model. To the best of our knowledge, this is the first study that compares experimental Barkhausen noise statistics with a random 22
Physica E: Low-dimensional Systems and Nanostructures 106 (2019) 19–24
A. Roy, P.S.A. Kumar
Fig. 9. Simulated M − H half-loops for a 100 μm × 100 μm × 15 nm thin film without demagnetizing effects with two different types of disorder (b & c). The field is swept from left to right. M − H curves of the same system without disorder are also shown for comparison (a).
obtained from this software, which was needed as unusually large cell sizes were being used. The magnetization reversal of ‘perfect’ systems with no disorder was also mediated by domain nucleations, which were produced purely as a result of demagnetizing fields, and had no connection with sample homogeneity. The hysteresis curves generated by micromagnetic simulations had virtually no dependence on the angle of the applied field when the demagnetization contribution was ‘enabled’. Only if the demagnetization field contribution was ‘turned off’ that we were able to see hysteresis loops having a strong-enough dependence on the out-of-plane angle of the applied field. It may be noted that two types of demag field calculations are provided in the OOMMF package: for open and periodic boundary conditions, both of which led to qualitatively similar observations. Thus, it was clear that in our systems, demagnetizing fields did not play a deciding role in the case of Barkhausen noise, ie, the pinning sites were not active through the demagnetizing fields around them. The remaining question, regarding whether the disorder was ‘directional’ or not, could be verified by performing simulations without demagnetizing effects, and comparing between the two types of disorder used. The results are shown in Fig. 9. As expected from the RAIM simulations, hysteresis loops with random anisotropy showed a transition from single-step hysteresis to multiple-step hysteresis as the applied field direction was turned more and more out-of-plane. The system with disordered magnetization values, on the other hand, showed exactly the opposite trend, with Barkhausen noise disappearing in favor of a single switching event as the magnetic field was rotated out of plane. Though reason qualitatively different behaviors of the simulations with the two types of disorder is not clearly understood, the results are sufficient to conclude that domain wall pinning in our samples originated from the mosaicity of the film, which led to a distribution of the magnetocrystalline anisotropy axes.
Fig. 8. RAIM avalanche size distributions for four angles of the applied field separated by 1O.
anisotropy model. It was a question, though as to which anisotropy of the real-world system was to be mapped to the ‘average’ anisotropy direction in the RAIM. In the case of the thin film Fe samples, two major sources of magnetic anisotropy were (1) shape anisotropy, that arises due to magnetostatic effects and attempts to confine the magnetization in-plane, and (2) magnetocrystalline anisotropy, that is fixed to the crystallographic directions, and for a well oriented film of Fe, will also point in-plane with four-fold rotational symmetry. Another question was regarding the nature of the disorder. Earlier we had categorically ruled out any type of scalar disorder, as they would be incapable of producing angle dependent effects. However, in the presence of a demagnetizing field originating from the magnetization of the sample, the pinning potential around even ‘scalar’ defects can have angle dependence. 3.3. Micromagnetic simulations Micromagnetic simulations were carried out to address the above somewhat subtle aspects. We used the popular micromagnetic package OOMMF [39] for our simulations. Micromagnetic simulation packages would not have been suitable for the calculation of Barkhausen noise statistics due to the prohibitive computational requirements. However, it was possible to calculate single hysteresis curves for qualitative study. Our key question was regarding the role of the demagnetizing field which was not incorporated inside our RAIM. We also wanted to check whether the angular dependence of the nature of the hysteresis loops was retained for realistic values of sample parameters. Since the microscopic nature of the disorder in our samples was unknown to us, we used two different types of disorder. One consisted of random anisotropy directions, identical to the one used for the RAIM, the second was a spatially randomized value of MS. The latter was a scalar quantity, which would also help to address our questions about demagnetizing fields around ‘non-directional’ defects. For a 100 μm × 100 μm system size, we used a cell size of 500 nm, which was adequate as the Barkhausen noise phenomena occurred over much larger length scales. We used literature values for the magnetic parameters of Fe: MS = 1.7 × 106, A = 2.1 × 10−11, K1 = K4 = 4.8 × 104 S.I. units. Both uniaxial and cubic anisotropies were investigated. The degree of disorder was kept between 1% and 33%. Preliminary OOMMF runs showed that, regardless of the type of disorder, magnetization reversal occurred through domain nucleation and growth. Uniaxial anisotropy resulted in stripe domains and four-fold anisotropy produced predominantly triangular closure domains. Both of these outcomes were expected on physical grounds, and helped to validate the results
4. Conclusion In this paper we studied Barkhausen noise of iron thin films through planar Hall effect. The films had weak in-plane four fold magnetic anisotropy. We observed a crossover in Barkhausen noise statistics the films by changing the direction of the driving magnetic field. The crossover is similar to a disorder-induced crossover in avalanche statistics from sub-critical to critical observed in random field models of Barkhausen noise. The Barkhausen noise of the high-quality films was sub-critical for all in-plane orientations of the driving magnetic field relative to the sample. Application of an out-of-plane field brought about a phase transition in the statistics which now showed critical behavior. An initial comparison with a random field Ising model simulation showed striking similarity between the observed and measure size distributions. Different orientations of the sample corresponded to different values of disorder. However, to establish the physical origins of the crossover in statistics, a random anisotropy Ising model was studied. The angle dependent crossover in statistics shown by the random anisotropy model was found to closely match the trend shown by the 23
Physica E: Low-dimensional Systems and Nanostructures 106 (2019) 19–24
A. Roy, P.S.A. Kumar
sample. The origin of the Barkhausen noise in the actual sample could be attributed to the presence of misoriented crystallites which gave rise to domain wall pinning. A few other possibilities were eliminated by micromagnetic simulations.
[16] A. Berger, M.J. Pechan, R. Compton, J.S. Jiang, J.E. Pearson, S.D. Bader, Phys. B Condens. Matter 306 (2001) 235. [17] J. Marcos, E. Vives, L. Mañosa, A. Planes, E. Duman, M. Acet, J. Magn. Magn Mater. 272–276 (2004) E515. [18] J. Marcos, E. Vives, L. Mañosa, M. Acet, E. Duman, M. Morin, V. Novák, A. Planes, Phys. Rev. B 67 (2003) 224406. [19] A. Roy and P.S.Anil Kumar P S, To Be Published. [20] J. Smit, Physica 17 (1951) 612. [21] A. Roy, P.S. Anil Kumar, J. Phys. D Appl. Phys. 43 (2010) 365001. [22] U. Ruediger, J. Yu, S. Zhang, A.D. Kent, S.S.P. Parkin, Phys. Rev. Lett. 80 (1998) 5639. [23] W. Gil, D. Görlitz, M. Horisberger, J. Kötzler, Phys. Rev. B 72 (2005) 134401. [24] G.A. Prinz, Appl. Phys. Lett. 39 (1981) 397. [25] J.J. Krebs, B.T. Jonker, G.A. Prinz, J. Appl. Phys. 61 (1987) 2596. [26] S.D. Hanham, A.S. Arrott, B. Heinrich, J. Appl. Phys. 52 (1981) 1941. [27] R. Harris, M. Plischke, M.J. Zuckermann, Phys. Rev. Lett. 31 (1973) 160. [28] K.H. Fischer, J.A. Hertz, Spin Glasses vol. 1, Cambridge university press, 1993. [29] K. Binder, A.P. Young, Rev. Mod. Phys. 58 (1986) 801. [30] R.J. Elliott, J.A. Krumhansl, P.L. Leath, Rev. Mod. Phys. 46 (1974) 465. [31] R. Alben, J.J. Becker, M.C. Chi, J. Appl. Phys. 49 (1978) 1653. [32] M.E. McHenry, M.A. Willard, D.E. Laughlin, Prog. Mater. Sci. 44 (1999) 291. [33] M. Gruyters, Europhys. Lett. 77 (2007) 57006. [34] J.R. Iglesias, J.I. Espeso, N. Marcano, J.C. Gómez Sal, Phys. Rev. B 79 (2009) 195128. [35] S. Kralj, V. Popa-Nita, Eur. Phys. J. E. Soft Matter 14 (2004) 115. [36] A. Ranjkesh, M. Ambrožič, S. Kralj, T.J. Sluckin, Phys. Rev. E 89 (2014) 022504. [37] E. Vives, A. Planes, J. Magn. Magn Mater. 221 (2000) 164. [38] E. Vives, A. Planes, Phys. Rev. B 63 (2001) 134431. [39] M.J. Donahue, D.G. Porter, Oommf User's Guide, Version 1.0. Interagency Report NISTIR 6376, National Institute of Standards and Technology, Gaithersburg, MD, 1999.
Acknowledgements One of the authors (A.R.) acknowledges C.S.I.R, India for financial support. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
B. Alessandro, C. Beatrice, G. Bertotti, A. Montorsi, J. Appl. Phys. 68 (1990) 2901. B. Alessandro, C. Beatrice, G. Bertotti, A. Montorsi, J. Appl. Phys. 68 (1990) 2908. S. Zapperi, P. Cizeau, G. Durin, H. Stanley, Phys. Rev. B 58 (1998) 6353. J. Sethna, K. Dahmen, S. Kartha, J. Krumhansl, B. Roberts, J. Shore, Phys. Rev. Lett. 70 (1993) 3347. O. Perković, K. Dahmen, J. Sethna, Phys. Rev. Lett. 75 (1995) 4528. M. Kuntz, J. Sethna, Phys. Rev. B 62 (2000) 11699. E. Vives, J. Goicoechea, J. Ortín, A. Planes, Phys. Rev. E 52 (1995) R5. F. Pérez-Reche, E. Vives, Phys. Rev. B 67 (2003) 134421. F. Pérez-Reche, E. Vives, Phys. Rev. B 70 (2004) 214422. E. Vives, F.-J. Pérez-Reche, Phys. B Condens. Matter 343 (2004) 281. X. Illa, E. Vives, Phys. Rev. B 74 (2006) 104409. D. Spasojević, S. Janićević, M. Knežević, Phys. Rev. E 84 (2011) 051119. O. Perković, K. Dahmen, J. Sethna, Phys. Rev. B 59 (1999) 6106. D. Spasojević, S. Janićević, M. Knežević, Phys. Rev. E 89 (2014) 012118. A. Berger, A. Inomata, J. Jiang, J. Pearson, S. Bader, Phys. Rev. Lett. 85 (2000) 4176.
24
Contents lists available at ScienceDirect
Physica E: Low-dimensional Systems and Nanostructures journal homepage: www.elsevier.com/locate/physe
Geometry dependent crossover of Barkhausen statistics in iron thin films ∗
T
Arnab Roy , P.S. Anil Kumar Department of Physics, Indian Institute of Science, Bangalore 560012, India
A B S T R A C T
We report the crossover of Barkhausen noise statistics upon a change in the measurement geometry. The behaviour is akin to the angle dependent critical to subcritical crossover of avalanche statistics as present in avalanche models with random anisotropy directions. The samples studied were 15–25 nm thin films of iron, grown on (001) GaAs by pulsed laser ablation. Use of planar Hall effect as a probe for the magnetic state facilitated this work, which would be impractical to do in a MOKE setup. We compare our data with simulations of a random anisotropy Ising model, and demonstrate striking similarity between experimental and simulated avalanche size distributions. Further, through micromagnetic simulations, we conclude that crystallite misorientations were the origin of domain wall pinning in our system leading to Barkhausen noise.
1. Introduction A close connection exists between magnetic hysteresis and disorder. Disorder in a magnetic material gives rise to inhomogeneity in the energy landscape encountered by a magnetic domain wall during magnetization reversal. The consequence is that, in the course of its motion, the magnetic domain wall may get trapped at local energy minima, which can be overcome only after a change in the driving magnetic field. This trapping and untrapping manifests itself as Barkhausen noise, when magnetization of a sample changes in an intermittent and discontinuous manner upon driving with a smoothly changing magnetic field. The sizes and durations of these avalanches are distributed according to power-laws of the form
P (s ) = As−τ
(1)
Here P(s) is the probability of occurrence of a magnetization ‘jump’ of size ‘s’, and the form persists up to a cutoff size above which the probability falls off exponentially. The magnetization steps during a field sweep are stochastic and non-reproducible even under identical initial conditions. τ is referred to as a critical exponent, whose value depends on the universality class to which the system belongs. The study of such phenomena which fall under the general category of crackling noise, has received a lot of attention in the past few decades because of its connection to disorder-induced critical phenomena. A number of models have been put forward aimed at reproducing the statistical properties of this noise starting from a minimum of ingredients, the earliest among which is the ABBM model [1,2]. It was the first attempt aimed at specifically predicting the statistical properties of domain wall motion over a rough energy landscape. It predicted τ = 3/2 – c/2D, where c is the driving rate and D is a disorder parameter.
∗
Our present understanding of Barkhausen phenomena is based on two classes of models: Front propagation models like the CZDS [3] model, deal with the motion of a 2D domain in a lattice unbounded in one direction under an external magnetic field changing smoothly in time. Along with this, there exist site-dependent random magnetic fields and exchange energy modeled as a ‘surface tension’ of a domain wall. Its prediction is similar to that of the ABBM model, with τ = 3/2 – c/2. Another class of models, namely the nucleation models [4–14], have a finite lattice with site dependent disorder in the form of a random magnetic field or random magnetocrystalline anisotropy directions. The first studies of avalanche dynamics in such models emerged around 1995 due to Perković, Dahmen and Sethna [5]. One of the predictions of both the classes of models is the presence of a critical point. In the ABBM model, the critical point is dependent on the ratio of the magnitude of the disorder to the magnetic field sweeping rate [1,2]. In nucleation models the critical point is arrived at by tuning the magnitude of the random disorder. The critical point appears as a transition from a single-step magnetization reversal to a collection of discrete steps corresponding to a large number of pinning and depinning events. The order parameter in this transition is the size of the largest single avalanche that results in the magnetization reversal, and its magnitude decreases to zero at disorders higher than the critical disorder. Though easily accessible through the models, experimental evidence of such phenomena are not common [15–18] and the conclusions are often fairly indirect. The present authors have recently observed the crossover from sub-critical (non-universal) to critical (universal) behavior by increasing the disorder level in permalloy thin films [19]. In this article, a similar crossover is reported in Fe thin films, arising out of purely geometrical reasons. Our study consists of the experimental investigation of the effect of measurement geometry on
Corresponding author. E-mail address: [email protected] (A. Roy).
https://doi.org/10.1016/j.physe.2018.10.023 Received 16 July 2018; Received in revised form 26 September 2018; Accepted 15 October 2018 Available online 16 October 2018 1386-9477/ © 2018 Published by Elsevier B.V.
Physica E: Low-dimensional Systems and Nanostructures 106 (2019) 19–24
A. Roy, P.S.A. Kumar
the four-fold anisotropy. The anisotropy field was approximately 1Oe, which was a fairly small value for iron at room temperature [26]. Preliminary electrical characterization comprised resistivity and magnetoresistance measurements at low temperatures. The residual resistivity was fairly low (∼6 μΩ cm), which indicated a largely defectfree film growth. High-field (B > 2 T) and high temperature (T > 100 K) magnetoresistance was dominated by the negative linear magnon magnetoresistance, while at low temperatures and high fields, signatures of Lorentz magnetoresistance could be seen. This was easily distinguishable from the MR due to localization effects because of its non-saturating behavior even at high magnetic fields. The presence of Lorentz MR was yet another indicator of good film quality, as it requires a substantial electronic mean-free-path to become noticeable [22]. The importance of having good quality samples for this study will become apparent in the discussion on Barkhausen noise. For the Barkhausen noise study, planar Hall voltage was measured at 300 K in the magnetic field of an air-core coil. Magnetic field was swept in steps of ∼0.01Oe and the signal was acquired synchronously after every step. DC constant current measurements yielded a typical S/ N ratio of 104 and a typical voltage swing of ∼1 mV. Each sweep was repeated 1000 to 2000 times to gather statistics. Fig. 2 shows a typical PHE curve during a field sweep. The sharp minima correspond to the magnetization reversal fields of the Fe thin films, which is the field at which there is a maximum abundance of domains. A finer field step-size reveals a multitude of avalanches (inset), the positions and sizes of which varied randomly between different field sweeps. Barkhausen avalanche statistics were extracted in 2 stages: An initial processing accumulated all signal jumps whose magnitude was above a preset threshold. Following this, Barkhausen noise was separated from instrumental noise by choosing a cut-off size below which the probability of ‘upward’ and ‘downward’ jumps became comparable. Avalanche statistics were measured for different rotations of the sample relative to the applied magnetic field, the field being applied both in-plane and out-of-plane to the sample. In Fig. 3, ‘zoomed-in’ views of switching curves for 6 different orientations of a 100 μm wide and 25 nm thick Hall bar are shown. Four in-plane configurations were chosen: 0O, 45O, 90O and 135O, (Fig. 3:A,D,E,F) between the applied field and current direction. These angles correspond to the positions of maxima, minima and ‘zero's for the planar Hall voltage at saturation ie, when α is the same as the measured angle (Eqn. (1)). Notwithstanding the differences in the ‘direction’ of the voltage changes, the switching process for all the four in-plane configurations were qualitatively very similar: a single large switching event dominated the magnetization reversal accompanied by a number of smaller events in each case. The consequence was that the size distributions with the applied field in-plane consisted of two regions: a power-law probability distribution for the small sizes and characteristic ‘hump's towards the tail of the distributions representing the large jumps (Fig. 4: A,D,E,F). The two other orientations that are presented here are two angles of
Barkhausen noise of iron thin films measured by planar Hall effect. It also attempts to arrive at an explanation primarily based on simulations of a random anisotropy Ising model (RAIM). Certain finer points are clarified through micromagnetic simulations. Planar Hall effect was used to avoid the inherent difficulties associated with an optical (MOKE) measurement setup upon changes to the measurement configuration. 1.1. Planar Hall effect Planar Hall effect is a phenomenon related to anisotropic magnetoresistance which occurs due to spin-orbit interaction in ferromagnetic materials [20]. In thin films, it is essentially an in-plane voltage developed perpendicular to the direction of the current density, when the magnetic field is in the plane of the sample along with the current. Empirically, the Hall voltage is expressed [21] as:
V = J . d. (
ρ|| − ρ⊥ 2
)sin 2 α
(2)
where α is the angle between the magnetization and the current density J. The relation holds for a single magnetic domain. In an actual sample, there are multiple ferromagnetic domains pointing in different directions, so that the effective α is the weighted average of the angle over different domains, modulo 180O (as the signal is periodic in 2α) [22,23]. This 'effective medium' approach may be summarized as:
V = J . d. (
ρ|| − ρ⊥ 2
3
)
∑ fi sin 2αi i=1
(3)
where the summation is over the 3 spatial DOF's and fi is the volume fraction of domains parallel to the ith direction in the region of interest. Thus, an abrupt change in the domain structure represented by [fi] shows up as a jump in the Hall voltage. 2. Experiment Samples used for this study were thin films of Fe grown on GaAs (001) by pulsed laser deposition (PLD). GaAs was chosen since it is known that Fe grows epitaxially on GaAs(001) [24,25]. The lattice mismatch between BCC Fe (2.87 Å) and GaAs (5.65 Å) is about 1%, with two unit cells of Fe growing on one unit cell of GaAs. PLD was carried out at room temperature in a UHV chamber with a base pressure of 2 × 10−10 Torr, and 5 × 10−8 Torr during deposition. Thicknesses of 15 nm and 25 nm were deposited, each with a capping layer of about 5 nm of Au to prevent oxidation. Patterning to 100 μm Hall bars was done using laser direct writing and lift-off. The primary characterization was done with MOKE magnetometry. L-MOKE with in-plane sample rotation showed a predominantly fourfold anisotropy with 〈110〉 and 〈11¯0〉 being easy directions. This information was obtained from the angular dependence of the coercivity (Fig. 1a) and remnant magnetization ratios (Fig. 1b). A closer look revealed that there was also a weak uniaxial anisotropy superimposed on
Fig. 2. Typical planar Hall hysteresis curve for100 μm wide Fe Hall bar. Inset: ‘zoomed in’ view of the region which shows Barkhausen noise (right), Schematic diagram of the hall bar sample (left).
Fig. 1. Coercivity (a) and remanence (b) as a function of angle for Fe (15 nm) on GaAs obtained from MOKE hysteresis loops. Four-fold magnetic anisotropy is indicated. 20
Physica E: Low-dimensional Systems and Nanostructures 106 (2019) 19–24
A. Roy, P.S.A. Kumar
3. Simulations To examine the striking similarity of the experimental data with a disorder-induced critical phase transition, simulation of a popular nucleation model, the 2D Gaussian random field Ising model (2D-GRFIM) was carried out. 3.1. Random field ising model (RFIM) The Gaussian Random Field Ising model deals with an ensemble of classical spins in a 2D lattice. The Hamiltonian of the system is [ [8]]
Η = −J ∑ Si Sj − Hext i, j
Fig. 3. PHE switching curves of Fe Hall bar for 4 in-plane (A,D,E,F) and 2 outof-plane (C,D) orientations of the driving magnetic field. The large switching event is observable in all the in-plane configurations and is absent for the outof-plane configurations.
∑ Si − ∑ Hi Si i
i
(4)
Here, J (> 0) is the ferromagnetic coupling constant between the neighboring spins, Hext is the external field and Hi is a site dependent random field, with a Gaussian distribution centered at zero and an adjustable standard deviation R. Spins are allowed to take the values +1 and −1. For our simulations, we chose a square lattice with only ‘nearest’ neighbor interactions, so that the number of spins interacting with any given spin is 4. There were no long-range interactions. Periodic boundary conditions were used to satisfy the above criterion for a spin at the edge. We used 3 different lattice sizes: 100 × 100, 200 × 200 and 1000 × 1000. The system was initialized with all the Si = 1 and the external field was swept from a sufficiently large negative value to a sufficiently large positive value to ensure reversal of all the ‘spin's. Since the model parameters are scalable by an arbitrary scale factor, we fixed J = 1. Simulations of the 2D-GRFIM presented in literature had been performed in the adiabatic limit, so that only a single avalanche was triggered at each field step. However, to satisfy the real experimental conditions more closely, the adiabatic limit was discarded and the simulation was carried out at a constant field step. Following a well-established protocol, after each field step, the system was allowed to evolve to a metastable configuration over several iterations, with changed nearest neighbor configurations due to spins flipped in the previous iteration. The accumulated value of the total number of spins flipped due to a field step was taken as the size of a jump. To collect statistics, the sweeps were repeated 1000 to 5000 times with different realizations of the random field. Fig. 5 shows four experimentally obtained size distributions (top row) matched with those obtained from simulations of 2D GRFIM (bottom row). The experimental parameters and the simulation parameters that generate the matching size distributions are shown in Table 1. Even though the matching was close, there could be no reasonable physical argument that could connect the scalar RFIM to angle dependent experimental observation. It was intuitively obvious that the minimum requirement was a model that incorporated direction dependent ‘randomness’, at the same time incorporating all the characteristic features of the RFIM, particularly that of the disorder-induced critical phase transition. For this, we invoke a relatively less studied nucleation model: the random anisotropy Ising model (RAIM).
Fig. 4. Size distribution of Barkhausen avalanches in Fe Hall bar for 4 in-plane (A,D,E,F) and 2 out-of-plane (C,D) orientations of the magnetic field. Sub critical to critical crossover clearly observable when going to an out-of-plane configuration.
rotation out-of plane starting from the 0O in-plane configuration. We refer to this angle as ω and have the rotation direction illustrated in Fig. 2. In principle, a rotation out of plane of sample should not have much significance when it comes to AMR and PHE, as the magnetization is strongly confined in-plane due to shape anisotropy originating from demagnetization effects. The only observable change expected would that of the coercivity value. However, as can be seen in Fig. 3(B,C), the switching behavior was altered in a much more profound manner, so that a switching curve now consisted of only small jumps. Correspondingly, the size distributions were also changed and the ‘familiar’ power law behavior with a cut-off was observed (Fig. 4B and C). The power-law exponent (τ′) values for the 25 nm thick sample were 1.5 and 2.2 for ω = 45O and 67O respectively. The behavior was similar for the sample of 15 nm thickness, except that the disappearance of the single large switching event did not occur at 45O tilt, and the transition was only complete at 67O. Comparing with theoretical studies on hysteresis loop criticality, it is easy to map the single large switching event to a spanning avalanche, which refers to one that traverses the whole sample from boundary to boundary. The state corresponding to the presence of a spanning avalanche is a sub-critical state with respect to disorder. On the other hand, a condition where avalanche sizes are distributed according to a powerlaw is close to a disorder-induced critical point. It is not clear a priori as to what is bringing about this transition, since the phenomenon of the ‘disorder’ level changing with measurement condition is somewhat contra-intuitive.
Fig. 5. Experimental avalanche size distributions for 15 nm thick Fe Hall bars (A–D). Simulated avalanche size distributions using 2D-GRFIM (E–H). 21
Physica E: Low-dimensional Systems and Nanostructures 106 (2019) 19–24
A. Roy, P.S.A. Kumar
Table 1 Experimental and simulation parameters corresponding to plots in Fig. 5. Figure
Experiment details
5A&E 5B & F 5C & G 5D&H
15 nm, 15 nm, 25 nm, 25 nm,
0O 67O 0O 45O
L
R/J
sweep range
step
Power-law exponent
1000 1000 100 100
0.01 50 0.001 0.33
± 1000 ± 1000 ± 200 ± 200
2 0.2 0.2 0.2
3.65 × 2.24 1.26
3.2. Random anisotropy ising model (RAIM) Random anisotropy models were first used in the context of amorphous ferromagnets by Harris, Plischke and Zuckermann [27]. Since then, this class of models has found wide applications in various fields of study, the most important applications being in the theory of spinglasses [28,29] and disordered ferromagnets [30–34] and in the theory of nematics and liquid crystals [35,36]. A study of the avalanche properties of this model under driving with an external field was presented by Vives and Planes [37,38], in the limit of strong anisotropy. In this limit, the model is reduced to the random anisotropy Ising model, which, according to Vives et al., is expected to belong to the same universality class as the RFIM. They demonstrated that the critical exponent τ had a value slightly above 2, and varied slightly depending on the details of the representing of disorder. A general Hamiltonian of the system with random anisotropy directions and without long-range interactions can be
Fig. 6. Simulated magnetization reversal curves of RAIM as a function of out-ofplane angle of the applied field. The dispersion in the anisotropy axes is 10O. Disappearance of the spanning avalanche between 80O and 82.5O indicates the sharp crossover to critical behavior.
→→ → → → H = −J ∑ Si · Sj − Hext · ∑ Si − D ∑ (nˆ i · Si )2 i≠j
i
i
(5)
where nˆ i is the local anisotropy direction and D is the anisotropy con→ stant. In the limit of large D, the spins Si are constrained to lie either parallel or antiparallel to nˆ i , and the model is reduced to an Ising model in which the spins can only have values of ± 1. Under this approximation, the reduced Hamiltonian of the system becomes
Η = − ∑ Jij Si Sj − Hext i, j
∑ gi Si i
(6)
where Jij = Jnˆ i . nˆ j and gi = qˆ. nˆ i . This was the form of the model used for our simulations. The effective disorder in this case is gi, and plays a role analogous to Hi in the RFIM. However, in this case, it is angle dependent. In fact, a random anisotropy model is a more realistic representation of an actual sample than a random field model, as the former can be linked readily to the mosaicity of the crystalline material comprising the sample. The dynamics were metastable and the evolution at a given field was stopped after all the spins had locally relaxed. The disorder was a uniform distribution of the anisotropy axes within a circular region around the x axis. This was equivalent to having an ‘average’ anisotropy → direction nˆ 0 parallel to the x axis and a random anisotropy vector Δn i → was unity. The angles made by nˆ such that the norm of nˆ i = nˆ 0 + Δn i i with nˆ 0 were uniformly distributed within an angular range of ± σ. σ was synonymous with the disorder magnitude. Without loss of generality, the external magnetic field was confined to the z-x plane. The program was used to attempt to reproduce the experimental size distributions as a function of angle of the applied field relative to the xaxis. As in the RFIM case, sub-critical to critical (power law) behavior was observed when the external field direction was rotated away from the x-axis. At sub-critical angles, practically the entire sample was reversed in a single avalanche, while above the critical angle the reversal was through multiple discrete steps (Fig. 6). Extraction of the size distribution of Barkhausen jumps showed the existence of a power-law region even when the system was sub-critical, the exponent τ′ decreasing to a minimum of about 1.7 at the critical effective disorder and increasing on both sides of it. The value of this
Fig. 7. Power-law exponent of avalanche size distributions (τ′) in the RAIM as a function of angle for two system sizes and two disorder magnitudes.
minimum was practically independent of the system size and disorder magnitude, while its position shifted to lower angles as the disorder magnitude was increased (Fig. 7). The somewhat surprising observation was that the power-law exponent increased after the critical angle was exceeded, a hint of which was also observed in our experiment with the 25 nm film, where it increased from 1.5 to 2.2 as ω was increased from 45O to 67O. As expected, the shapes of the simulated size distributions matched the experimental distributions closely, both in the critical and sub-critical regimes (Fig. 8). At the set disorder level (10O), the transition from subcritical to critical was complete within an angle change of 2O. The RAIM simulations did not generate the very large τ′ values ( > 4) obtained in experiment at low angles for the 15 nm sample, though that does not hamper the physical understanding of the system through this model. To the best of our knowledge, this is the first study that compares experimental Barkhausen noise statistics with a random 22
Physica E: Low-dimensional Systems and Nanostructures 106 (2019) 19–24
A. Roy, P.S.A. Kumar
Fig. 9. Simulated M − H half-loops for a 100 μm × 100 μm × 15 nm thin film without demagnetizing effects with two different types of disorder (b & c). The field is swept from left to right. M − H curves of the same system without disorder are also shown for comparison (a).
obtained from this software, which was needed as unusually large cell sizes were being used. The magnetization reversal of ‘perfect’ systems with no disorder was also mediated by domain nucleations, which were produced purely as a result of demagnetizing fields, and had no connection with sample homogeneity. The hysteresis curves generated by micromagnetic simulations had virtually no dependence on the angle of the applied field when the demagnetization contribution was ‘enabled’. Only if the demagnetization field contribution was ‘turned off’ that we were able to see hysteresis loops having a strong-enough dependence on the out-of-plane angle of the applied field. It may be noted that two types of demag field calculations are provided in the OOMMF package: for open and periodic boundary conditions, both of which led to qualitatively similar observations. Thus, it was clear that in our systems, demagnetizing fields did not play a deciding role in the case of Barkhausen noise, ie, the pinning sites were not active through the demagnetizing fields around them. The remaining question, regarding whether the disorder was ‘directional’ or not, could be verified by performing simulations without demagnetizing effects, and comparing between the two types of disorder used. The results are shown in Fig. 9. As expected from the RAIM simulations, hysteresis loops with random anisotropy showed a transition from single-step hysteresis to multiple-step hysteresis as the applied field direction was turned more and more out-of-plane. The system with disordered magnetization values, on the other hand, showed exactly the opposite trend, with Barkhausen noise disappearing in favor of a single switching event as the magnetic field was rotated out of plane. Though reason qualitatively different behaviors of the simulations with the two types of disorder is not clearly understood, the results are sufficient to conclude that domain wall pinning in our samples originated from the mosaicity of the film, which led to a distribution of the magnetocrystalline anisotropy axes.
Fig. 8. RAIM avalanche size distributions for four angles of the applied field separated by 1O.
anisotropy model. It was a question, though as to which anisotropy of the real-world system was to be mapped to the ‘average’ anisotropy direction in the RAIM. In the case of the thin film Fe samples, two major sources of magnetic anisotropy were (1) shape anisotropy, that arises due to magnetostatic effects and attempts to confine the magnetization in-plane, and (2) magnetocrystalline anisotropy, that is fixed to the crystallographic directions, and for a well oriented film of Fe, will also point in-plane with four-fold rotational symmetry. Another question was regarding the nature of the disorder. Earlier we had categorically ruled out any type of scalar disorder, as they would be incapable of producing angle dependent effects. However, in the presence of a demagnetizing field originating from the magnetization of the sample, the pinning potential around even ‘scalar’ defects can have angle dependence. 3.3. Micromagnetic simulations Micromagnetic simulations were carried out to address the above somewhat subtle aspects. We used the popular micromagnetic package OOMMF [39] for our simulations. Micromagnetic simulation packages would not have been suitable for the calculation of Barkhausen noise statistics due to the prohibitive computational requirements. However, it was possible to calculate single hysteresis curves for qualitative study. Our key question was regarding the role of the demagnetizing field which was not incorporated inside our RAIM. We also wanted to check whether the angular dependence of the nature of the hysteresis loops was retained for realistic values of sample parameters. Since the microscopic nature of the disorder in our samples was unknown to us, we used two different types of disorder. One consisted of random anisotropy directions, identical to the one used for the RAIM, the second was a spatially randomized value of MS. The latter was a scalar quantity, which would also help to address our questions about demagnetizing fields around ‘non-directional’ defects. For a 100 μm × 100 μm system size, we used a cell size of 500 nm, which was adequate as the Barkhausen noise phenomena occurred over much larger length scales. We used literature values for the magnetic parameters of Fe: MS = 1.7 × 106, A = 2.1 × 10−11, K1 = K4 = 4.8 × 104 S.I. units. Both uniaxial and cubic anisotropies were investigated. The degree of disorder was kept between 1% and 33%. Preliminary OOMMF runs showed that, regardless of the type of disorder, magnetization reversal occurred through domain nucleation and growth. Uniaxial anisotropy resulted in stripe domains and four-fold anisotropy produced predominantly triangular closure domains. Both of these outcomes were expected on physical grounds, and helped to validate the results
4. Conclusion In this paper we studied Barkhausen noise of iron thin films through planar Hall effect. The films had weak in-plane four fold magnetic anisotropy. We observed a crossover in Barkhausen noise statistics the films by changing the direction of the driving magnetic field. The crossover is similar to a disorder-induced crossover in avalanche statistics from sub-critical to critical observed in random field models of Barkhausen noise. The Barkhausen noise of the high-quality films was sub-critical for all in-plane orientations of the driving magnetic field relative to the sample. Application of an out-of-plane field brought about a phase transition in the statistics which now showed critical behavior. An initial comparison with a random field Ising model simulation showed striking similarity between the observed and measure size distributions. Different orientations of the sample corresponded to different values of disorder. However, to establish the physical origins of the crossover in statistics, a random anisotropy Ising model was studied. The angle dependent crossover in statistics shown by the random anisotropy model was found to closely match the trend shown by the 23
Physica E: Low-dimensional Systems and Nanostructures 106 (2019) 19–24
A. Roy, P.S.A. Kumar
sample. The origin of the Barkhausen noise in the actual sample could be attributed to the presence of misoriented crystallites which gave rise to domain wall pinning. A few other possibilities were eliminated by micromagnetic simulations.
[16] A. Berger, M.J. Pechan, R. Compton, J.S. Jiang, J.E. Pearson, S.D. Bader, Phys. B Condens. Matter 306 (2001) 235. [17] J. Marcos, E. Vives, L. Mañosa, A. Planes, E. Duman, M. Acet, J. Magn. Magn Mater. 272–276 (2004) E515. [18] J. Marcos, E. Vives, L. Mañosa, M. Acet, E. Duman, M. Morin, V. Novák, A. Planes, Phys. Rev. B 67 (2003) 224406. [19] A. Roy and P.S.Anil Kumar P S, To Be Published. [20] J. Smit, Physica 17 (1951) 612. [21] A. Roy, P.S. Anil Kumar, J. Phys. D Appl. Phys. 43 (2010) 365001. [22] U. Ruediger, J. Yu, S. Zhang, A.D. Kent, S.S.P. Parkin, Phys. Rev. Lett. 80 (1998) 5639. [23] W. Gil, D. Görlitz, M. Horisberger, J. Kötzler, Phys. Rev. B 72 (2005) 134401. [24] G.A. Prinz, Appl. Phys. Lett. 39 (1981) 397. [25] J.J. Krebs, B.T. Jonker, G.A. Prinz, J. Appl. Phys. 61 (1987) 2596. [26] S.D. Hanham, A.S. Arrott, B. Heinrich, J. Appl. Phys. 52 (1981) 1941. [27] R. Harris, M. Plischke, M.J. Zuckermann, Phys. Rev. Lett. 31 (1973) 160. [28] K.H. Fischer, J.A. Hertz, Spin Glasses vol. 1, Cambridge university press, 1993. [29] K. Binder, A.P. Young, Rev. Mod. Phys. 58 (1986) 801. [30] R.J. Elliott, J.A. Krumhansl, P.L. Leath, Rev. Mod. Phys. 46 (1974) 465. [31] R. Alben, J.J. Becker, M.C. Chi, J. Appl. Phys. 49 (1978) 1653. [32] M.E. McHenry, M.A. Willard, D.E. Laughlin, Prog. Mater. Sci. 44 (1999) 291. [33] M. Gruyters, Europhys. Lett. 77 (2007) 57006. [34] J.R. Iglesias, J.I. Espeso, N. Marcano, J.C. Gómez Sal, Phys. Rev. B 79 (2009) 195128. [35] S. Kralj, V. Popa-Nita, Eur. Phys. J. E. Soft Matter 14 (2004) 115. [36] A. Ranjkesh, M. Ambrožič, S. Kralj, T.J. Sluckin, Phys. Rev. E 89 (2014) 022504. [37] E. Vives, A. Planes, J. Magn. Magn Mater. 221 (2000) 164. [38] E. Vives, A. Planes, Phys. Rev. B 63 (2001) 134431. [39] M.J. Donahue, D.G. Porter, Oommf User's Guide, Version 1.0. Interagency Report NISTIR 6376, National Institute of Standards and Technology, Gaithersburg, MD, 1999.
Acknowledgements One of the authors (A.R.) acknowledges C.S.I.R, India for financial support. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
B. Alessandro, C. Beatrice, G. Bertotti, A. Montorsi, J. Appl. Phys. 68 (1990) 2901. B. Alessandro, C. Beatrice, G. Bertotti, A. Montorsi, J. Appl. Phys. 68 (1990) 2908. S. Zapperi, P. Cizeau, G. Durin, H. Stanley, Phys. Rev. B 58 (1998) 6353. J. Sethna, K. Dahmen, S. Kartha, J. Krumhansl, B. Roberts, J. Shore, Phys. Rev. Lett. 70 (1993) 3347. O. Perković, K. Dahmen, J. Sethna, Phys. Rev. Lett. 75 (1995) 4528. M. Kuntz, J. Sethna, Phys. Rev. B 62 (2000) 11699. E. Vives, J. Goicoechea, J. Ortín, A. Planes, Phys. Rev. E 52 (1995) R5. F. Pérez-Reche, E. Vives, Phys. Rev. B 67 (2003) 134421. F. Pérez-Reche, E. Vives, Phys. Rev. B 70 (2004) 214422. E. Vives, F.-J. Pérez-Reche, Phys. B Condens. Matter 343 (2004) 281. X. Illa, E. Vives, Phys. Rev. B 74 (2006) 104409. D. Spasojević, S. Janićević, M. Knežević, Phys. Rev. E 84 (2011) 051119. O. Perković, K. Dahmen, J. Sethna, Phys. Rev. B 59 (1999) 6106. D. Spasojević, S. Janićević, M. Knežević, Phys. Rev. E 89 (2014) 012118. A. Berger, A. Inomata, J. Jiang, J. Pearson, S. Bader, Phys. Rev. Lett. 85 (2000) 4176.
24