POLITECNICO DI MILANO Dipartimento di Fisica
TESI DI DOTTORATO XVI CICLO
TEMPERATURE DEPENDENCE OF THE STATISTICAL PROPERTIES IN THE MAGNETIZATION PROCESS
Coordinatore: Prof. Lucio BRAICOVICH Tutor: Prof. Franco CICCACCI
Candidato: Maurizio ZANI Matr. Nr. D01039
Marzo 2004
Ai miei "insegnanti" Ezio e Franco
2 Summary
Acknowledgements ...... 5 Introduction...... 6 1. Magnetism...... 8 1.1 Magnetic domains...... 8 Weiss domains...... 8 Bloch walls...... 10 Neel walls...... 11 1.2 Magnetization process ...... 14 Stoner-Wohlfarth model...... 14 Barkhausen noise ...... 21 Criticality ...... 24 1.3 Magneto-optical Kerr effect ...... 28 Magneto-optics...... 28 Geometrical configuration...... 30 Kerr effect: general case...... 34 1.4 Bibliography ...... 36 2. Instrumentation...... 38 2.1 Inductive technique...... 38 History...... 38 Apparatus ...... 39 Measurements ...... 40 2.2 Magneto-optical technique ...... 42 Laser source...... 43 Magnetic source...... 44 Polarizers ...... 45 Photo-elastic modulator...... 46 Cryostat...... 49 2.3 Acquisition programs ...... 51 Inductive measurements...... 51 Magneto-optical measurements ...... 52 Ising simulations ...... 53 2.4 Bibliography ...... 54
3 3. Experimental results and simulations ...... 55 3.1 Metastable states...... 55 Experiment ...... 55 Model ...... 60 Reliability problem...... 62 Conclusions...... 64 3.2 Magnetization jumps vs temperature...... 66 Experiment ...... 66 Coercive field ...... 66 Metastable states...... 68 Magnetization jumps ...... 70 Conclusions...... 72 3.3 Simulations ...... 73 Ising model...... 73 Random coercive field Ising model...... 78 3.4 Bibliography ...... 85 A. Appendix: other measurements and collaborations ...... 86 A.1 Negative jumps ...... 86 Inductive measurements...... 87 Magneto-optical measurements ...... 89 A.2 Trilayer Fe/NiO/Fe ...... 93
A.3 Magnetite Fe3O4 ...... 96 A.4 Electrodeposited CoPt "in-situ"...... 97 A.5 Bibliography ...... 99 B. Appendix: publications ...... 100
4 Acknowledgements
Mi fa un immenso piacere cominciare a scrivere questa tesi partendo dai ringraziamenti, che credo un doveroso tributo verso coloro che ho conosciuto, che mi hanno guidato in questa cammino di tre anni e che con me hanno condiviso la bellezza della ricerca: grazie!
GRAZIE ad Ezio (prof. Puppin) per quello che mi ha insegnato in laboratorio, non solo la parte teorica e pratica, ma soprattutto l'intuizione e la passione da mettere in quello che la ricerca chiede: farsi domande e darsi risposte.
GRAZIE a Franco (prof. Ciccacci), al prof. Lucio Braicovich ed al prof. Bruno De Michelis, severi e leali coordinatori di questo mio periodo di Dottorato, che sempre mi hanno stimolato a perseguire e conoscere sempre più l'affascinante mondo della fisica.
GRAZIE a tutto il Gruppo di Ricerca (da noi chiamato Suspenx), che in più occasioni si è dimostrato una vera squadra. GRAZIE ai compagni di viaggio, a chi ha già terminato, a chi termina con me questa tappa ed a chi la tappa la sta raggiungendo. GRAZIE alle varie persone del Dipartimento di Fisica a vari ruoli, che ho avuto l'onore di conoscere e con i quali lavorare a fianco. GRAZIE agli studenti, pedine importanti nel mondo della didattica (loro e mia). GRAZIE ai ricercatori di altri dipartimenti od istituti, con i quali ho collaborato a diverso titolo nelle misure magneto-ottiche.
GRAZIE a Maggy per tutto lo stress sopportato, soprattutto negli ultimi mesi di scrittura della tesi.
Ed infine GRAZIE a me stesso, di aver preso coscienza sempre più che questa è la strada che mi piace percorrere: della ricerca e della didattica...
5 Introduction
This Ph.D. thesis work started with the idea of investigating the properties of magnetic materials at low temperature. There is a simple reason for doing research on this subject: our understanding of magnetism at low temperature is terribly poor. This statement is probably surprising also for scientists working in the field of magnetism but, nevertheless, it is substantially true. It is certainly true that temperature has a crucial role in determining the magnetic properties of any material, and the existence of a critical value of T which separates the paramagnetic from the ferromagnetic phase is the obvious proof of the previous sentence. Modern days statistical physics achieved its more celebrated victory with the Onsager solution of the Ising model, i.e., the theoretical extrapolation of the critical temperature value for a 2D system of spin on a matrix. But this is the high temperature part of the story, where the average energy available is sufficient for promoting the excited states of the system with a dynamical rate which is compatible with the scientists lifetime. Things dramatically change when the available energy is much smaller with respect to the height of the energy barriers involved, and the system evolution takes place in a completely different way. Under these circumstances the available knowledge is scarce both on the experimental and on the theoretical side. In other word, low temperature magnetism is a frontier field of investigation and all the activity described in the following pages is a preliminary attempt to obtain some experimental information on the statistical properties of a magnetic system at low temperature.
It is well understood that, at a sufficiently small length scale, magnetism manifests its statistical nature with fluctuations, a discovery which dates back to 1919 when Barkhausen first observed the noise produced during the magnetization process in a piece of iron. This Barkhausen noise is a direct manifestation of the existence of magnetic domains which, as we learned during the last 80 years, are a direct consequence of the complex nature of the disorder quenched in any real material. This disorder, due to randomly distributed imperfection in an otherwise perfect crystal lattice, act as a friction force which opposes to the motion of the domain walls under the effect of an external field. As a result of this resistance the wall motion is jumpy and, in energetic terms, the system evolution can be regarded as a sequence of jumps between different metastable states. The number of these states is very large, due to the complexity of the energy landscape, and therefore a statistical description is mandatory.
The statistical properties of these jumps have been largely investigated at room temperature, but no work has ever been published at low temperature. This is the scenario of my thesis work.
6 The most relevant result we obtained is the observation that the probability distribution for the Barkhausen jumps dramatically changes by decreasing temperature from the ambient value down to 10 K. It is now recognized that Barkhausen noise is fractal since the probability distribution of the jumps amplitude is a power law. We observed that this behavior is preserved at low temperature, but the critical exponent of the power law increases by a factor of two at low temperature, and this is a new result which contradicts the accepted view which consider the critical exponent as a number which depends only on the dimensionality of the system. The most successful paradigm which bring to this conclusion is based on renormalization group theories. Needless to say we do not have an alternative model capable to go beyond the deepest statistical theory, the phase transitions theory developed by K. Wilson. We simply present our experimental results along with a few sporadic attempts to understand them.
The key problem is to develop a reasonable physical model for the physical phenomena taking place in our thin Fe films. As shown in this work a consideration of the so-called lifetime of the system metastable states allows to view the jump process as a two steps event. The first event is the energy barrier overcome, and this is temperature independent. The second step is the chaotic avalanche which takes place after the energy barrier has been overcome. This step turns out to be strongly temperature dependent, but not physical picture for this is now available.
Clearly, in order to obtain a better understanding on this topic, it will be necessary to formulate some hypothesis on the mutual interactions of the different domains of the sample. This task is too complex and, for this reason, we decided to attack the problem in a different way. Our approach is based on the idea that the system can be regarded as an Ising-like matrix with a quenched-in disorder (models as a randomly distributed anisotropy field). A system like this evolves, under an external field, via a series of domino-like avalanches involving the matrix spins. It is however difficult to introduce temperature in this model. This difficulty is testified by the fact that the most popular complexity generator available theory (SOC, Self Organized Criticality) has not been tested versus temperature with the only exception of a couple of contradictory works. Our model, at present, does not yet include temperature but the work is in progress and we are confident that in the close future we will be able to include this parameter. The preliminary results we obtained are quite encouraging.
7 1. Magnetism
1.1 Magnetic domains
Weiss domains
At the base of comprehension of the property of magnetic materials and relative technological applications there's the observation and measurement of magnetic domains, macroscopic regions of the sample uniformly magnetized cause the alignment of the single magnetic moments of the atoms constituting the sample; the magnetization direction is not the same for each domain, so the total magnetization is less than the one of the single domains. Magnetic domains are the elements that links the basic physical properties of a material with its macroscopic properties and application1. A rising interest in domain analysis has been developed in recent years, due to the increasing miniaturization of the devices in electronics.
Why a piece of iron, that is ferromagnetic2, can appear non magnetic? There was no progress in the understanding of this behavior until 1905, when Langevin3 developed a theory by using statistical thermodynamics in which he shown that, at room temperature, independent molecular magnets lead to a weak magnetic phenomena: the strong magnetism shown in saturation must be due to some interaction among the magnets. Two years after Weiss4 tried to model the average effect of this magnetic interaction by introducing a molecular field (that later Heisenberg5 identified in the quantum-mechanical exchange effect), and saying that ferromagnetic materials are not spontaneously magnetized to saturation because different macroscopic regions (domains) takes different magnetization direction. The analysis of the dynamics of a magnetization process led to the conclusion that the evolution occur by the propagation of a boundary between domains of different magnetization: the wall. It was Bloch6 that analyzed theoretically the transition between domains in a magnetization process. A solution about the problem of a theory of magnetization process was presented in 1935 by Landau and Lifshitz7, studying the dispersion of magnetic permeability in ferromagnetic bodies: magnetic domains and domains walls are formed to minimize the total energy, the prevalent term being that magneto-static.
The energetic balance drive the system to the possibilities to obtain various domains configurations, taking into accounts the various magnetic energy density8:
8 G G • magneto-static energy Ems = -M•B
is the potential energy of a magnetic moment in a magnetic field: M is the magnetization, B is the external magnetic field;
2JS2 • exchange energy Eex = - cosθ a3
tends to keep adjacent magnetic moments parallel to each other: J is the exchange energy, S the spin, a is the lattice step, θ is the angle between directions of adjacent spins;
G G µ • demagnetizing-field energy E = -µM•H = -0 NMcos22θ d 0 2
arises from having a discontinuity in the normal component of magnetization across an interface: M is the magnetization in saturation, θ is the angle between directions of adjacent spins, N is the demagnetizing factor, H = NM cosθ is the demagnetizing field;
24 • magneto-crystalline anisotropy Ea = Ksin24θ + K sin θ + ... for uniaxial anisotropy
describes the preferences for the magnetization to be oriented along certain crystallographic
directions: K2 and K4 are anisotropy constants, θ is the angle between directions of magnetization and easy axis;
22 • magneto-elastic energy Eme = B ecos( θ + υ sin θ) for uniaxial anisotropy
is a term proportional to strain: B is a constant, e is the strain caused by external stress, ν is the Poisson's ratio, θ is the angle between directions of stress and easy axis;
Simple models have been elaborated that show how to obtain various configurations of domains.
9 Bloch walls
For the case of uniaxial anisotropy, the magnetization vectors in adjacent domains are antiparallel to each other (for reduce the magnetostatic energy), that is a 180° domain wall exists, with the plane of the wall parallel to the easy axis in order to satisfy the boundary condition for the continuity of the normal component of magnetization. Such wall is called Bloch wall in honor or describing work of Felix Bloch about this structure. If the magnetization orientation change abruptly from 0° to 180° there is no cost in anisotropy energy, but a relevant cost in exchange energy from adjacent sites. Thus the material search another less costly way to make the transition in magnetization from one domain to the adjacent one; the exchange energy can be reduced by distributing the overall 180° rotation over several N lattice steps, in order to change the angle in adjacent spins from 180° to 180°/N, like is schematically shown in Fig. 1-1.
Fig. 1-1: magnetic domains with 180° domain wall (left) with reversion of magnetization in one atomic distance (right) with reversion of magnetization over N atomic distance
Increasing N more spins are oriented in directions of higher anisotropy energy, increases it. The correct number N is established minimizing the sum of exchange energy and anisotropy energy: in the real materials N range from 100 to 10 000. The 180° domain wall will have an internal structure like shown in Fig. 1-1: the atomic magnetic moments will make a gradual transition in orientation from one domain to the other. In Fig. 1-2 is shown an expanded version of this wall: the angle of magnetic moments is shown, with value θ = 0° at the beginning of the wall and θ = 180° at its end. The figure represents a possible form for this reorientation, the exact functional form θ(z) must be derived from micromagnetics knowing exchange energy and magnetic anisotropy.
10
Fig. 1-2: (up) spin reorientation within a 180° domain wall (down) angle of magnetic moments vs distance
Neel walls
In material of cubic anisotropy 90° walls are possible. In this case such wall has an orientation that maintains continuity of the normal component of magnetization across the wall, and the magnetization rotate within the wall in order to minimize the exchange and anisotropy energy. From micromagnetics calculations can be seen that the spin reorientation in a 90° domain wall follows the same form of 180° domain wall, like shown in Fig. 1-2 but with the vertical axis range from 0° to 90°. But also 180° domain can occur in a cubic material: the difference from this last case to that of a sequence of two 90° walls is shown in Fig. 1-3.
Fig. 1-3: (left) magnetic domains with 90° domain wall (right) spin reorientation and angle of magnetic moments vs distance
11 Another example of 90° domain walls can be taken from thin magnetic films. From Fig. 1-1, is clear that as sample thickness decreases the magnetostatic energy of the wall that extends through the thickness of the sample increases, due to the free poles at the top and the bottom of the wall. To reduce this magnetostatic energy the spins inside the wall can rotate in the plane of the sample: in this way a smaller magnetostatic energy at the internal (vertical) face of the wall is accepted as the price for removing the larger magnetostatic energy at the external (horizontal) surface of the sample. This type of wall, sketched in Fig. 1-4, is called Néel wall.
Fig. 1-4: (left) Bloch wall (right) Néel wall
Making a comparison between two walls versus thickness, we can see that • Bloch wall energy increases with decreasing film thickness, because of increasing of the magnetostatic energy due to the free surface charges above and below the wall; • Néel wall energy decreases with decreasing film thickness, because magnetostatic energy is proportional to the area of the free charged surface inside the wall.
Fig. 1-5: calculated thickness dependence for unit area of the wall energy for Bloch and Néel wall
Recently, experimental measurements and micromagnetic calculations of the surface magnetization distribution in materials with thickness greater than the Néel limit have shown that a Block wall can transform into a Néel wall near a surface9.
12 A third example of how magnetostatic energy can influence the geometry of domains come from vortex walls. Néel walls have an inherent charge because of their spin structure: the large magnetostatic energy associated with this magnetic charge can be reduced if the wall splits into a sequence of vortex and the sense of polarization of the wall alternates, from up to down and vice versa, as shown in Fig. 1-6. When this situation occur, the wall is called a cross-tie wall10.
A
B
Fig. 1-6: (up) Néel wall (down) Cross-tie wall
Inside this structure two type of singularity (called Bloch lines, considering when they are seen along the z direction, orthogonal at the sample) are found: • at the center of the vortex magnetization patterns (labeled with A in Fig. 1-6); • at the point where contiguous vortex touch each other (labeled with B in Fig. 1-6).
These examples are just a representation of the fascinating richness of micromagnetic structures that one can encounter when exchange, anisotropy, sample geometry and external field action are combined in different ways.
13 1.2 Magnetization process
Stoner-Wohlfarth model
The magnetization process is the result of complex and intricate rearrangements that involve a large number of degrees of freedom and characterize the magnetization of the sample11. It is possible identify elementary contributions (an example is that of a magnetic "particle" small enough to be a single domain) which reverse their magnetization by coherent rotation, and the overall behavior of the system is just the superposition of many of these contributions, which we suppose to act independently of each other. This approach shows in a simple way how certain physical concepts work in a simple form, and provides a basis for a number of generalizations. However this is not the only mechanism by which magnetization reversal takes place: also domain walls motions and domain nucleation are mechanism presents in magnetization process.
We can consider a small magnetic particle always saturated in a certain direction, so that we can describe its state giving the orientation of its magnetization vector8. We can consider a particular case in which the particle exhibits uniaxial anisotropy: if any domain walls are present in the sample, they are assumed to be parallel to the easy axis. Remembering the magnetic energy terms, the energy density describing the situation for which the external magnetic field is transverse to the easy axis of magnetization (i.e. parallel to hard axis) is
2 Ehard = K cosθ - MH cosθ where K is the anisotropy constant (positive), M is the magnetization in saturation, H is the external magnetic field and θ is the angle between M and H. In Fig. 1-7 is shown the schematic representation of a magnetic sample with uniaxial anisotropy along easy axis (EA); the application of a magnetic field transverse to easy axis results in a rotation of the domain configuration, but no domain wall motion is present.
14 EA
H M
M
Fig. 1-7: schematic representation of hard magnetization; dashed lines represents the situation for H = 0
The equilibrium condition is found minimizing the energy density relatively to θ, that is when its first derivative is equal to zero (zero torque on M)
dE hard = () -2K cosθ + MH sinθ = 0 dθ and second derivative is positive (stability condition)
dE2 hard = -2K cos2θ + MH cosθ > 0 dθ2
The zero-torque equation has two solutions; the first is
sinθ = 0 θ = 0; π i.e. with magnetization parallel or anti-parallel to field; from stability equation we find
2K 2K θ = 0 if H > - ; θ = π if H < M M
2K that is the situation at saturated state. We will define H = the field at saturation. sat M
15 The second solution is
MH cosθ = 2K stable in the range
2K 2K - < H < MM
Note that for H = 0 the solution is θ = π/2, that is with magnetization transverse to field. We can make more clear the situation if we consider the measured component of magnetization in the field direction, M cosθ, and normalize it to saturation value, and also the field normalized to its saturation value
M cosθ H m = = cosθ h = M Hsat
In this way the situation between the saturation states see the magnetization linearly increases until it reaches the saturation field: it has described by a line in m-h graph. The overall magnetization process description is well represented by Fig. 1-8. Even if unpinned domain walls are present parallel to easy axis, they don't move because there is no energy difference across the domain wall.
m
1
h -1 1
-1
Fig. 1-8: m-h loop for hard-axis magnetization
16 If external magnetic field is parallel to the easy axis, the energy density is
2 Eeasy = K sin θ - MH cosθ with the same description of parameters in the equation. In Fig. 1-9 is shown a schematic representation of the situation; the application of a magnetic field parallel to easy axis results in a domain wall motion, but is not present a rotation of domain configuration. Let show this behavior from energy calculations.
EA
M
M
H
Fig. 1-9: schematic representation of easy magnetization; dashed lines represents the situation for H = 0
Now for the first derivative (zero torque on M) of precedent energy density equation we have
dE easy = () 2K cosθ + MH sinθ = 0 dθ whereas for the second derivative (stability condition)
dE2 easy = 2K cos2θ + MH cosθ > 0 dθ2
We have got again two solutions: the first solution from first derivative is
sinθ = 0 θ = 0; π
17 that is stable for
2K 2K θ = 0 if H > - ; θ = π if H < M M that is magnetization parallel or anti-parallel to field. The second solution is
MH cosθ = - 2K that requires like stability condition
2K 2K H < - and H > M M
Also in this case we can reassume the situation in a graphic; the situation is more easy than precedent case, because magnetization immediately jump from one state parallel to field to the other state anti-parallel to field when field cross the zero, as shown in Fig. 1-10.
m
1
h -1 1
-1
Fig. 1-10: m-h loop for easy-axis magnetization
The field exerts no torque on domain magnetization, but exerts a torque on the spins that compose the wall, that rotate until align with field.
18 With the same idea of hard and easy magnetization just treated, the Stoner-Wohlfarth model12 is applied at an arbitrary angle between field and easy axis. In this case the energy density results
2 ESW = K sin(θ-θ 0 ) - MH cosθ
where θ0 is the angle between external field and easy axis, as shown in Fig. 1-11.
H
θ M
θ0 EA
Fig. 1-11: coordinate system for Stoner-Wohlfarth model;
field is applied at an angle θ0 relative to easy axis (EA)
The energy is minimized respect to θ0 when
2K sin θ-θ cos θ-θ + MH sinθ = 0 ()00( ) or, defining the normalized value of magnetization and field, when
2m 1-m22 cos 2θ + 1-2m sin 2θ ± 2h 1-m 2 = 0 ()00( ) ()
This equation can be solved for h as function of m having θ0 like parameter, and the graphic result is shown in Fig. 1-12, where is shown the magnetization measured in direction of field. If domain walls are considered inside the sample, in the first part of magnetization process there is no magnetization reversal by rotational switching, because the easy wall motion is preferred; the domain rotation is utilized by the system in the last part of hysteresis loop for complete the magnetization process (the dynamics of the process is shown in figure by circle with inside the evolution of magnetization).
19
Fig. 1-12: hysteresis loops in Stoner-Wohlfarth model for various angle; the circles show the various step of magnetization process
In Stoner-Wohlfarth model the magnetization process is reversible both for magnetization rotation and for wall motion; in this case, like shown in Fig. 1-12, the system show zero coercivity. In real materials the domain walls do not move reversibly as has been assumed until now, because grain, inclusions and other defects can lower the wall energy at a particular position in the material, pinning its motion, or can place a barrier that inhibits further wall motion through the defect. The presence of these defects and other arrangements that can take place inside the sample keep to the situation shown in Fig. 1-13, where the hysteresis loops of Stoner-Wohlfarth model are presented at various angle between field and anisotropy.
Fig. 1-13: hysteresis loops in Stoner-Wohlfarth model for various angle, taking account of coercivity
20 Barkhausen noise
”In every small crystal of a material there are strains and dislocations; there are impurities, dirt..."13: imperfections, that act as pinning site for the Weiss domain wall motion. Barkhausen14 discovered that magnetization process take place in discontinuous steps using an amplifier and a loudspeaker (as schematized in Fig. 1-14), in which the discontinuity give rise to a characteristic noise.
Fig. 1-14: (left) original Barkhausen article (right) Barkhausen apparatus for experiment, with M: permanent magnet - S: coil - E: iron rod - G: galvanometer - V: amplifier - T: telephone receiver
Three different mechanisms are involved during the magnetization process: domain nucleation, coherent rotation and domain wall motion. Barkhausen noise is mainly due to the domain wall motion: the schematization of the domain wall motion as a point moving in a random pinning15 field has successfully explained several properties of ferromagnetic materials.
When we go on microscopic scale, the hysteresis loop can appear not continuous, but made of jumps. In Fig. 1-15 is reported this situation: in the upper part is shown the hysteresis loop, with in the inset a pictorial representation of a "microscopic measurement", whereas in the lower part the first derivative of magnetization versus field clearly shows the presence of discontinuities in the magnetization process, the Barkhausen signal.
21
Hysteresis loop
Barkhausen signal
Fig. 1-15: (up) picture of hysteresis loop, and in the inset representation of microscopic measurements (down) Barkhausen noise, as first derivative of magnetization versus field
The most common method to detect Barkhausen signal is to surround the sample with a magnetizing and an inductive coil; the magnetic field is slowly changed by increasing the current through the magnetizing coil, and the magnetization changes are detected on the pickup coil (the entire experiment technique is well described in next chapter). The Barkhausen pulses (due to irreversible domain wall motion) occur much faster than the slow reversible changes in magnetization due to domain rotation, which are proportional to the rate of change of external magnetic field16. When a jump take place the magnetization change its direction by 180°, and there's also a change of magnetic flux through pickup coil; by Faraday law, the induced voltage in the coil is proportional to the rate change of flux, so that
ddMΦ V(t) =∝ dt dt
The Barkhausen discontinuity can be considered a coupled inductor with pickup coil, each one with its own time constant17; if the pickup coil has time constant much shorter than the time constant of Barkhausen discontinuity, the coil has very little effect on the voltage pulse, that appear an exponential decay.
22 Barkhausen noise is a well-known phenomenon18: it's been experimentally observed that the histogram of Barkhausen jumps sizes ∆M follows a power-law19, P(∆M) = ∆M-α, with the critical exponent α that spans a wide range of values dependence various conditions20. The same power-law behavior is followed by the Barkhausen jumps duration, with an obvious different critical coefficient.
Fig. 1-16> probabilitydistributionof sizes and duration of Barkhausen jumps in Si/Fe 11
Also stress affect the magnetic properties of ferromagnetic samples21, and one expect that stress also affects the Barkhausen behavior. Measurements on Fe, Ni and permalloy shown that the number of mechanically produced Barkhausen discontinuities increases linearly with stress22, up to elastic limit, although an investigation on amorphous ribbons demonstrated that the amplitude of the Barkhausen jumps (and its distribution) is not affected by mechanical stress23.
23 Criticality
Like Feynman said, "Barkhausen noise heard by a loudspeaker is like a whole rush of clicks that sound something like the noise of sand grains falling over each other as a can of sand is tilted"13. This citation was prophetic, because in the very last years this effect has attracted a growing interest as an example of complex system with avalanches and scaling behavior. Nature is complex: it starts with few elementary particles and end up with all the complex phenomena we can see all around us, like life. Bak24 said that complex behavior in nature reflects the tendency of large systems with many components to evolve into critical state, where little disturbances may lead to events, called avalanches, of all sizes: most of the changes take place through catastrophic events even rather than smooth gradual path. The evolution to this critical state is not guided from outside agents, but established solely by the mutual interactions among the elements of the system: the state is self-organized, originating what is called self-organized criticality (SOC).
The better definition I have found about critical system is of Sethna, Dahmen and Myers20, where they talk about the crackling noise (what we have called avalanches of all sizes): "Not all systems crackle. Some respond to external forces with many similar-sized small events (for example, popcorn popping as it is heated). Others give way in one single event (for example, chalk snapping as it is stressed). In broad terms, crackling noise is in between these limits: when the connections between parts of the system are stronger than in popcorn, but weaker than in the grains making up chalks, the yielding events can span many size scales. Crackling forms the transition between snapping and popping".
Until now, SOC is the only known mechanism to generate complexity. Systems in balance are not complex, because if a system in equilibrium is slightly disturbed not much happens. A system in equilibrium can show complex behavior, but under special circumstances: i.e. in first-order liquid- gas phase transition in thermodynamics or in second-order ferromagnetic-paramagnetic transition in magnetism or in Ising model regarding simulations there's a critical behavior separating two phases, but temperature must be fine tuned to obtain it. The critical state is reached when temperature assumes a well-defined value: the critical temperature (named Curie temperature in magnetic case).
24 In the cases described before, one o more order parameters X are introduced25, which value fluctuates in time and space, and whose: • average value vanishes on one side of the transition (high temperature side) and moves away from zero on the other side (low-temperature side): in this last side its behavior is described by a -α power-law
• fluctuations of the parameter behave in a way depends on value of temperature: when T = Tc, -β the probability distribution of fluctuations is represented by a power-law p(∆X) = k2∆X , where
k2 is another constant, ∆X is the fluctuation of the parameter X and β is another critical exponent.
Now we know what is a complex system: a system with a large numbers of variables, that may exist on a wide range of length scales, like a Chinese Box phenomenon. This explains why probability theories and statistical analysis is so important. The study of the origin of complexity is a nowadays question for science, thanks to high-speed computers that allow a big amount of calculus not available before. Although the statistical nature of the problem make it unpredictable, there are a large series of phenomenon that cannot be understood in the specific scientific context: Gutenberg- Richter's law about earthquakes26, Zip's law about city's population or ranking of words27, fractals of Mandelbrot28, 1/f noise29, sound emitted during martensitic phase transition30, and others. These are driven systems with many degrees of freedom, which respond to the driving external force in a series of discrete avalanche spanning a broad range of scales.
All these phenomenon are universal, following a similar behavior: like Barkhausen noise, the probability distribution of one of the characteristic parameters involved X is well described by a power-law, P(X) = X-α, with α named critical exponent (or fractal dimension in fractals theory). It's fascinating to see how the dynamics of all the elements can produce a law with such extreme simplicity.
Although several models are now present, the prototypical model of SOC system is the sand-pile.
25 We start whit a 2d grid of squares, in which every square can have a random value of grains of sand (we can choose to start with every square set to 0). Randomly, at each step a grain is deposited onto a square with coordinates x;y randomly chosen, increasing its height z of one:
z (x;y) = z(x;y) + 1
Now a rule is applied that allow the avalanche process: when the height of this square exceeds a critical value zc, for example 4, the avalanche start. One grain of its sand is sent to each of the four neighbors, and its height decrement of four units:
z (x±1;y) = z(x±1;y) + 1 z (x;y±1) = z(x;y±1) + 1 z (x;y) = z(x;y) - 4
These simple equations define completely the model. This last process continues until the avalanche stop, and a new grain is sent to the grid. A calculus of the distribution of avalanche sizes S shows how this critical behavior is well described by a power-law with exponent α = 1.1 (Fig. 1-17)
Fig. 1-17: power-law behavior probability distribution of avalanche in sand-pile model
Power-law says much more than there are avalanches of all sizes: it says that the phenomenon is scale-free, self-similar. This want to say that the probability distribution of the parameter involved doesn't depend on the resolution of the measurements, the observing scale is indistinguishable if not specified (in Fig. 1-18 this is shown for 1/f noise): the plot can be superimposed by simple linear change of scales of the two axes.
26
We also have used a model followed by a simulation to try to explain the physics behavior of our magnetic system. The meaning of the simulation is that is not possible to obtain the results of the model evolution by pen-and-paper, i.e. in a mathematical form: also if sand-pile model is simple, we cannot calculate the critical exponent of that (and of our) model31.
Fig. 1-18: 1/f noise with critical behavior, where statistical properties are preserved across scales
27 1.3 Magneto-optical Kerr effect
Magneto-optics
The Kerr effect is the phenomenon where linearly polarized light impinge over a magnetic sample and is reflected, obtaining a rotation and an ellipticity of its polarization: these two parameters are linearly dependent on the magnetization of the sample and independent from external magnetic field32, so that this phenomenon can be used like technique to measure hysteresis loops. At a microscopic level, the Kerr effect is due to the coupling of electric field of the light with the electron spin wave functions of the magnetic sample via spin-orbit interaction33.
All the magneto-optical effects (not only Kerr effect) can be quantitatively described by the use of dielectric tensor, and analyzing its off-diagonal elements34: in presence of a material, Maxwell equations must be written as
GG ∇•D = ρ freeG G GG G GG ∂B D = εε0 E + P = E ∇×E + = 0 G 11GG G GG ∂t with H = B - M = B ∇•B = 0 µ0 µ G GG GG∂D G J = σE ∇×H - = J free ∂t free
G G G G ik•r - ωt If we search for a plane wave harmonic solution E = E0 e in a electrically neutral material
ρfree = 0, we find
G G 22σ ω µ ε + i E = k E ω
Now if we put • µ ≈ µ0: relative magnetic permeability (magnetization cannot follow the rapid oscillations of field, approximation valid in visible light); σ • ε + i = εε : relative electric permeability (complex tensor); ω 0 r 22 • n = k/k0 : refraction index (with k000 = ω µ ε )
28 the wave equation becomes an eigen-values equation (also called crystalline optic equation):
G G 2 εrE = n E where (in a Jones representation)
εεεxx xy xz • εr = εεεyx yy yz is the operator (matrix) εεεzx zy zz
• n = nr + ini is the eigen-value (complex number) G E • E = x is the eigen-vector or eigen-mode Ey
G The first parameter εr represents the material, whereas n and E are relative to the e.m. wave in the material: now, given the dielectric tensor, we must search for the eigen-values and eigen-vectors solutions35. From the geometrical point of view, we can observe the behavior of the reflected wave or the transmitted wave: we'll focalize our attention more on the first case, that is relative to the Kerr effect, whereas the second represent the Faraday effect. The coordinate system chosen is represented in Fig. 1-19 for the reflected wave: the optical plane is constituted by the incident and reflected rays. Let make now some examples of geometrical configuration, regarding incident wave and magnetization component.
y
x
z
M
θ
P1 P2
Fig. 1-19: coordinate system in reflection:
P1 and P2 - incoming and reflected polarized wave; M - magnetization of the sample
29 Geometrical configuration
Magnetization shows various components of its vector nature: relatively to the coordinate system shown in Fig. 1-19, the name of various geometrical configuration of magnetization is shown in Fig. 1-20.
Mx My Mz
Fig. 1-20:geometrical configuration of magnetization (left) longitudinal (center) transverse (right) polar
Cubic crystal: let us examine the particular case of a medium having cubic symmetry and an incident wave orthogonal to the sample (i.e. θ = 0 relatively to Fig. 1-19), with no magnetization inside. In this case the transverse wave has only two component, Ex and Ey. Now, for symmetry, the dielectric tensor and the eigen-values equation become
εεxx xy ε0 0 0 0 0 εr = = with ε = ε r + iε i constant complex number εεyx yy 0 ε0
GG ε E = nE2 εE = n2 E 0 xx r 2 ε0Eyy = nE
Refraction index (the eigen-value) results n = ε0 = nr + ini , so that
0 00 nrr = ε nii = ε / 2 ε r ;
G E The eigen-modes of propagations are E = x , i.e. waves of every polarization (a cubic crystal Ey behaves optically as an isotropic continuum).
30 Faraday geometry: let us now assume that the same cubic sample is affected by a polar magnetization (Mz, i.e. orthogonal to the sample). The cubic symmetry is broken, we have got an overall uniaxial symmetry, and the dielectric tensor assumes form
εε ε ε xx xy xx xy εr = = εεyx yy-εε xy xx
so that the eigen-values equation result
2 G G εxxE x + ε xyE= y nE x εE = n2 E r 2 -εxyE x + ε xxE y = nE y
The refraction indexes are now n = εxx ± iε xy = n r + ini , and the two eigen-values are
n+ = εxx + iε xy n- = εxx - iε xy
G 1 The eigen-modes of propagation result E = E , i.e. waves of circular polarization36. 0 ± i G An incident wave Ei linearly polarized can be decomposed in the two above eigen-modes
G 11111 E = E = E + i 000 22 i -i
n± - 1 iϕ Both type of waves are reflected with different Fresnel coefficient r±± = = r e ± relative n± + 1 G to the refraction indexes found before, so that the reflected wave Er becomes
G 1111E 1r iϕϕ - 1 E = E r + r = 0 r + - e ()− + r+-0 22 i -i 2 + i r -i +
where the phases φ- and φ+ are relative to the different speed of the two traveling waves.
31 The reflected wave is an elliptically polarized wave. We can introduce the complex Kerr angle
θK = φK + iεK, where
ϕ - ϕ ϕ = -+ is the rotation of the vertical polarization K 2 r 1 - - r ε = + is the ellipticity of the elliptical wave K r 1 + - r+
If we want find how the magnetization influence rotation and ellipticity, we can develop each term of the dielectric tensor in Taylor series for each component of magnetization: for simplicity I'll develop only on the orthogonal component of magnetization Mz, stopping the development at the second order in magnetization
2 ∂∂εεij1 ij ε = ε02 + M + M ij ij zz2 ∂M2z ∂Mz
0 The first term ε ij is relative to the unperturbed system. Now, in the last case examined can de demonstrated that the symmetry conditions keep to the follow dielectric tensor
ε02 + αM-iQM ε = zz r 02 iQMzzε + αM where the Q-constant is named magneto-optical Voigt constant37. Without enter in complicated calculus, the rotation and the ellipticity of the polarized reflected wave introduced above result proportional to magnetization and odd in it:
φK ∝ Mz εK ∝ Mz
32 Cotton-Mouton geometry: in this configuration the magnetization is in the plane of the sample, that is can has two component, Mx and My. The calculation for arbitrary Mx and My are quite complicated, but can be instructive shows the result for the only presence of Mx; in this case the dielectric tensor assumes form
εε εε xx xy xx xy εr = = εεyx yy εε xy xx
so that the eigen-values equation result
2 G G εxxE x + ε xyE= y nE x εE = n2 E r 2 εxyE x + ε xxE y = nE y
The refraction indexes are now
nxxx = ε nyyy = ε
G 1 G 0 The eigen-modes of propagation result E = E and E = E ; this result is valid for general 0 0 0 1 orientation of magnetization in the plane of the sample (Mx and My), and shows that the eigen- modes are waves of linear polarization.
If now an incident wave with arbitrary polarization impinge on the sample, with the same procedure of the Faraday geometry can be shown that the reflected wave is elliptically polarized, and rotation and ellipticity dependence to the square of magnetization and are even in it:
2 2 φK ∝ M εK ∝ M
33 Kerr effect: general case
The general case is comprehensive of arbitrary magnetization and arbitrary incidence angle on the sample; in this case the dielectric tensor, stopping the development at the first order in magnetization, results
0 ε -iQMzx iQM 0 εr = iQMzyε -iQM 0 -iQMxy iQM ε
We can explain the Kerr effect introducing the reflectivity tensor: in a not magnetized sample, the incident, reflected (r in the equations below) and transmitted (t in the equations below) wave are connected by the Fresnel's coefficients
ncosθ - n cosθ nt - n r0 = 2112 = 21pp pp ncosθ + n cosθ n 2112 1 0 ncos1122θ - n cosθ rss = = t ss - 1 ncos112θ + n cosθ 2 where the subscripts pp and ss stand for parallel and perpendicular to the plane of incidence (the double letter is indicative of the fact that the polarization of incident wave not rotates after reflection, i.e. for example if a p-wave is incident on the sample, the reflected wave will has different amplitude and phase, but will be always a p-wave), θ1 and θ2 are the incidence and refracted angles, n1 and n2 the refraction indexes of first and second material. The quantities are connected by the Snell's law
n1sinθ1 = n2sinθ2
If now is present a magnetization, the reflected wave can has a rotation of its polarization and also an ellipticity; in this case is introduced the reflectivity tensor cited above, that take accounts of off- diagonal contributes (an incident p-wave generates a reflected wave with also s-component, and vice-versa): each component of the tensor depends on the magnetization, and particularly with different weight for each component of the magnetization38.
34 2 0 ttgppθ11tttg pp ss θ MMzx rpp + iQ M y iQ - G rrpp ps 24sinθ cosθ r = = 22 rr sp ss tttgpp ss θ1 MM iQzx + r0 ss 4sinθ22cosθ
The complex Kerr angle can be seen that is expressed by
r tgθ = tgϕ + iε = ps KKK()0 r ss
with φK is the rotation and εK the ellipticity, as just defined. It can be noted that for transverse magnetization My the off-diagonal elements of the reflectivity tensor are null, and the Kerr effect can be expressed only as a change ∆r of the reflectivity coefficient in the p-wave.
In Fig. 1-21 is shown the angular dependence of Kerr effect for the three different geometrical configurations of magnetization: longitudinal (Mx), polar (Mz) and transverse (My).
Fig. 1-21: angular dependence of longitudinal, polar and transverse Kerr effect
35 1.4 Bibliography
1 A. Hubert, R. Schäfer - Magnetic domains, ed. Springer (1998) 2 S. Chikazumi - Physics of ferromagnetism, ed. Oxford Science Publications (1997) 3 P. Langevin - Ann. Chim. Phys. (1905) 5, 70 4 P. Weiss - J. de Phys. Rad (1907) 6, 661 5 W. Heisenberg - Z. Phys. (1928) 49, 619 6 F. Bloch - Z. Phys. (1932) 74, 295 7 L. D. Landau, E. Lifshitz - Phys. Z. Sowjetunion (1935) 8, 153 8 R. O'Handley - Modern magnetic materials, ed. John Wiley & Sons (2000) 9 M. R. Scheinfein, J. Unguris, R. J. Celotta, D. T. Pierce - Phys. Rev. Lett. (1989) 63, 6, 668; Phys. Rev. B (1991) 43, 4, 3395 10 D. J. Craick, R. S. Tebble - Ferromagnetism and ferromagnetic domains, ed. North Holland (1965) 11 G. Bertotti - Hysteresis in magnetism, ed. Academic Press (1998) 12 E. C. Stoner, E. P. Wohlfarth - Phyl. Trans. Royal Soc. A (1948) 240, 599 13 R. Feynman, R.B. Leighton, M. Sands - The Feynman lectures on physics vol. II, ed. Addison-Wesley (1977) 14 H. Barkhausen - Phys. Z. (1919) 20, 401 15 L. Néel - Ann. Univ. Grenoble, Sect. Sci. Math. Phys. (1969) 40, 2828 16 J. McClure Jr., K.Schröder - CRC Crit. Rev. Solid State Sci. (1976) 6, 45 17 R. S. Tebble, I. C. Skidmore, W. D. Corner - Proc. Phys. Soc. London A (1950) 63, 739 18 S. Zapperi, P. Cizeau, G. Durin, H. E. Stanley - Phys. Rev. B (1998) 58, 10, 6353 19 L. V. Meisel, P. J. Cote - Phys. Rev. Lett. (1991) 67, 1334 20 J. P. Sethna, K. A. Dahmen, C. R. Meyers - Nature (2001) 410, 242 21 L: Callegaro, E. Puppin - Appl. Phys. Lett. (1996) 68, 9, 1279 22 Yu. Kharitonov - Izv. VUZ Fiz. (1967) 10, 12, 112 23 G. Durin, S. Zapperi - Journ. Appl. Phys. (1999) 85, 8, 5196 24 P. Bak - How nature works, ed. Copernicus (1999)
25 J. J. Binney, N. J. Dowrick, A. J. Fischer, M. E. J. Newman - The theory of critical phenomena, ed. Oxford Science (1992) 26 B. Gutenberg, C. F. Richter - Bull. Seismol. Soc. Am. (1944) 34, 185 27 G. K. Zipf - Human behavior and the principle of least effort, ed. Addison-Wesley (1949) 28 B. Mandelbrot - The fractal geometry of nature, ed. Freeman (1982) 36
29 P. Bak, C. Tang, K. Weisenfeld - Phys. Rev. Lett. (1987) 59, 4, 381 30 J. Ortin et al. - Journ. Phys. IV (1995) 5, 209 31 A. B. Chhabra, M. J. Feigenbaum, L. P. Kadanoff, A. J. Kolan, I. Procaccia - Phys. Rev. E (1993) 47, 5, 3099 32 P. Argyres - Phys. Rev. (1955) 97, 2, 334 33 S. Bader, J. Erksine - Uktrathin Magnetic Structure II (1994) 34 P. N. Argyres - Phys. Rev. (1955) 97, 2, 334 35 P. Paroli - Introduction to magneto-optics, XXI Course of GNSM (1986); F. Lucari - Effetti magneto-ottici per misure di magnetizzazione e di suscettività magnetica, XXII Course of GNSM (1990) 36 M. Freiser - Trans. Magn. (1968) 4, 4, 152 37 C. You, S. Shin - Jour. Appl. Phys. (1998) 84, 541 38 R. Hunt - Journ. Appl. Phys. (1967) 38, 4, 1652
37 2. Instrumentation
2.1 Inductive technique
History
Since 1919 of original Barkhausen measurements1, the technique used to record and analyze the magnetic avalanche that occurs during a magnetization process was been that inductive, using like probe a coil wound around the sample: "All you need is a coil of wire - with many thousands of turns - connected to an amplifier and a loudspeaker. [...] As you move the magnet nearer to the iron you will hear a whole rush of clicks that sound something like the noise of sand grains falling over each other as a can of sand is tilted. The domain walls are jumping, snapping, and jiggling as the field is increased. The phenomenon is called the Barkhausen effect."2 In time the technique has been perfected, but conceptually remained the same, and until now all the literature about Barkhausen noise is based on it; it's only in the last years that in some cases it's preferred the magneto-optical technique, without doubt advantages from the point of view of recording and analysis of data: inductive technique is integral, relative to the whole sample (not exactly so, as described in this chapter), while magneto-optical technique is local, cause the zone explored is only where impinge the laser beam. In this chapter I'll present both the instrumentation, part of which has been also realized by myself.
The original technique to observe Barkhausen noise consists in placing an inductive coil around the sample, and increment the external field: the magnetization jumps that take place inside the sample can be detected by measuring the voltage pulses inductively generated in the pickup coil. A series of problem arising from the interpretation of amplitude and shape of these jumps3, because is realistic expect that these jumps will be strongly affected by the relative position of the coil with respect to the sample region where the magnetization reversal take place. At this scope a new experimental apparatus has been realized4, consists in two pickup coils that allow detecting simultaneously Barkhausen noise registered by two coils. The position of the two coils can be changed, and this allows describing the spatial correlation of the jumps.
38 Apparatus A
F
C
D
E B
Fig. 2-1: head of the apparatus for measurements with inductive technique
In Fig. 2-1 is shown the head of our apparatus: all the various components are enclosed in a metallic box (F) for noise reduction, and the sample (E) is glued on a stiff plastic support and rigidly suspended within a magnetizing coil (B). Our coil is made of 1000 turns of copper wire, and is capable to supply a magnetic field of about 1000 Oe. Two little pickup coils (C and D) are wound around the sample and fixed on the plastic support: one of them can be moved up and down from the outside of the box by a manipulator (A). All these connections are bring out by a series of BNC connectors for external cabling (in Fig. 2-2 a photograph of the head of the apparatus).
Fig. 2-2: photograph of the head of the apparatus
39 The external field is generated by injecting a current in the coil B that comes from a waveform generator (HP 33120 A), set in order to perform a double linear ramp in 20 s (f = 0.1 Hz), and followed by a low-noise buffer amplifier (see Fig. 2-3) with unary gain and capacitor compensator circuit. This last apparatus has been realized with the scope of reduce the steps in the linear ramp generated by the 12 bits D/A converter of the waveform generator. The signals coming from the two pickup coils are send before at two low-pass filter (with a cut-off frequency of 18 kHz), and then at two amplifiers, in this case obtained using the input stage of two lock-in (EG&G 7260). A PC controls all the instruments, and the signals are acquired with a 12 bits A/D converter at a sampling rate of 50 kHz, so to obtain a number of 106 points for each channel in the overall acquisition time.
+12V 200kΩ −12V
In 1kΩ
+12V 1kΩ _ L272 +
470nF −12V
1A max
Out
1kΩ 1Fµ 100µ F
Fig. 2-3: electronic scheme of low-noise buffer
Measurements
For testing the apparatus, we have investigated an amorphous ribbon of Fe63B14Si8Ni15: various measurements have been made dependence of the relative distance of the two pickup coils. First of all, we have positioned both at the center of the sample, as close as possible to each other. In this case, like Fig. 2-4 shown, is clear the strong correlation between the two signals, since the two coils are probing the same region of the sample. There's a direct correspondence, although we can see different intensity in the signals. When we positioning the coils at a distance of 40 mm each other, the scenery changes completely (Fig. 2-4): no correspondence is found (except one case).
40 12 12
Sample Sample
2 mm 40 mm
1 1
2 2 Barkhausen noise [arb. units] [arb. noise Barkhausen units] [arb. noise Barkhausen
0 50 100 150 200 250 300 0 50 100 150 200 250 300
Time [µs ] Time [µs ]
Fig. 2-4: inductive signals with coils close (left) and with coils placed at a distance of 40 mm (right)
The different peak intensity revealed when the coils are close means that just at this distance the amplitude of the peaks is strongly affected by the relative position of the reversing region with respect to the coils. When the coils are positioned at 40 mm one to the other, the signal shown in Fig. 2-4 says that the attenuation of the peaks that take place in the sample doesn't allow to detect simultaneously the jumps.
The relation between the magnetic flux detected by the pickup coil and the change of magnetization that give rise to flux change depends on the geometry of the coil and on the permeability of the sample. From Deuling and Storm5, for a infinitely long cylindrical sample surrounded by a very short pickup coil the flux through the coil is a decreasing exponential, function of the distance of magnetization change from pickup coil. In our work is clearly shown like that the signals induced in the coils are very similar when the coils are placed nearby, whereas a progressive decrease of their correlation is observed by moving the coils apart
The analysis of data so collected will be object of future investigations, also for try to find a model that explain the attenuation of magnetic signal into the sample and give a quantitative analysis. The investigation addressed on this amorphous ribbon has also allowed us to study the negative jumps of magnetization that occur during the magnetization process6, as described in detail in appendix.
41 2.2 Magneto-optical technique
Barkhausen effect has attracted the interest of physicists since from its discovery for applications as a nondestructive technique, both with inductive technique and also with magnetic imaging techniques, in particular those based on the magneto-optical effects. In this way a large amount of information is now available on the details of the complex magnetic domain structure in a variety of materials and of its evolution under the effect of an external field7. The MOKE (Magneto-Optical Kerr Effect) technique can be used to easily measure the magnetic property: since 1800 has been used in experimental technique, and for the first time used by Bader and Moog8 to study thin films (SMOKE, Surface MOKE) of Fe epitaxially grown on Au(100).
If compared with other techniques, the MOKE apparatus has several advantages: • it allows to measure a full hysteresis loop in few minutes; • it is much less cost than other apparatus for magnetic measurements, like VSM; • it can measure magnetization in presence of external magnetic field.
Only in the last years this technique has been utilized to study Barkhausen noise9, modifying the classical MOKE configuration. In this chapter I'll describe the components used for measurements, making particular mention on those instruments that we've inserted in the optical and electronic chain. The layout of our MOKE apparatus in longitudinal configuration is shown in Fig. 2-5.
Fig. 2-5: layout of the magneto-optical apparatus
42 Summarily, the polarized (P) laser beam (L) impinge onto the sample (S) and after has been reflected is modulated with a photo-elastic birefringence modulator (M) at f = 50 kHz, and finally analyzed (A) and detected by a photodiode (D). The distinctive feature of our apparatus is the presence of an optical stage for tuning the spot size onto the sample, constituted by a beam- expander (B) followed by a focusing lens (F) with f = 200 mm. This allows to obtain a variable size beam whose diameter onto the sample, starting from 10 µm, can be tuned by moving the focusing lens.
All the optical components are collocated onto an optical table supported by four arms with pneumatic control (Fig. 2-6), that allow the vertical positioning and a vibration isolation control: this equipment is necessary to obtain the stability of job plane dependence of weight on it.
Fig. 2-6: arms of the optical table
Laser source
The light source is a He-Ne laser Melles-Griot at λ = 632.8 nm and power 5 mW; the laser is intensity stabilized, but its alimentation is supplied by a switching-supply operate at 25 kHz. This condition makes dangerous the operation of modulate light beam, because modulated light signal beats with this frequency (in our case the light is modulated by the photo-elastic modulator at a frequency of 50 kHz) and a noise is present at the output. For this scope a low pass filter has been inserted in the supply chain of the laser Fig. 2-7, in order to reduce this high frequency noise.
43
Fig. 2-7: low-pass filter for laser supply
Magnetic source
We've used two different coils like magnetic field sources: • two coils mounted in Helmholtz configuration, projected in order to adapt their dimensions to cryostat (that is inserted in the hole of the cylindrical coils), each one made of 444 turns and with a resistance R = 1.5 Ω (in order to use the full dynamic of the 36 V - 12 A current supply when the coils are connected in series)
Fig. 2-8: one of the two coils for cryogenic measurements
44 • an electro-magnet, useful for polar MOKE measurements (like those on magnetite and electrodeposited sample described in appendix) where an high field is required; in Fig. 2-9 is shown the current-field characteristic, with different distances between poles of electro-magnet.
15000
10000
5000
0 10 mm
H [Oe] 12 mm -5000 14 mm 16 mm -10000 18 mm 20 mm -15000
-15 -10 -5 0 5 10 15 I [A] Fig. 2-9: current-filed characteristic of electro-magnet
Polarizers
We use two Glan-Thompson calcite (CaCO3) polarizers, each one constituted by a pairs of birefringent triangular prisms glued together, that allow the transmission of the parallel component to optic axis and totally reflect the perpendicular component (Fig. 2-10). The first one, opportunely redirected along the original direction by the second prism, is transmitted; the second one is absorbed by a black opaque surface glued on the base of the first prism.
OA
E ϑ E//
E// ϑ OA
E
Fig. 2-10: triangular prisms constitute the polarizer
45 Photo-elastic modulator
Modulation allows to obtain a better signal-to-noise ratio, because reduce the low-frequency 1/f noise introduced by the electronic and acquisition chain. The polarization modulation is obtained in our apparatus by the use of a photo-elastic modulator Hinds PEM-90 in which, by photo-elastic effect, when the optical element is stressed it exhibits a birefringence proportional to stress. Photo- elastic modulators are resonant apparatus, and the birefringence effect oscillates at a fixed frequency (50 kHz for fused quartz): in our case a piezo-electric transducer is joined at the birefringent element, and an alternate tension (at the same frequency of the mechanical resonance) applied to the transducer generates a periodic oscillation of the element (see Fig. 2-11).
element
E y
Ex
transducer tension
Fig. 2-11: photo-elastic modulator, with optical element (on the left) and piezo-electric transducer (on the right)
If the optical element is relaxed the light passes through with the polarization unchanged, but if the it is stressed the polarization components parallel or perpendicular (Ex and Ey in Fig. 2-11) to the modulator axis travel at different speeds: the phase difference created between the two components oscillates as a function of time and is called the retardation. For example, when the maximum retardation reaches exactly one-fourth of the wavelength λ of light, the photo-elastic modulator acts as an oscillating quarter-wave (λ/4) plate (see Fig. 2-12).
Fig. 2-12: scheme of oscillating quarter-wave polarization modulation: the rectangles at the bottom are the relative positions of the modulator elements
46 In a Jones representation the modulator can be view as a compensator (phase retard), represented by a matrix (operator)
10 M = 0eiϕm
where φm is the phase of one component of incident light respect to the other; this phase, being modulated, can be expressed as φm = φ0 sin(ωt), where
• φ0 is the intensity of the modulation depends on the transducer excitation:
o in a quarter-wave plate (λ/4) we have got φ0 = π/2 = 1.57 = 90°
o in a half-wave plate (λ/2) we have got φ0 = π = 3.14 = 180° • ω is the resonance frequency (2π50 kHz in our case).
How we can detect Kerr signal from our sample? Let represent all the components listed in Fig. 2-5 by the appropriate Jones representation10, where the angles are referred to the optical plane :
cosγ • incident field: E = E 0 sinγ
cos2α sinα cosα • polarizer: P = sinα cosα sin2α
cosϕϕ -sin • sample: S = where φ = θ + iε is the complex Kerr angle sinϕϕ cos
10 • modulator: M = where φm = φ0 sin(ωt) is the modulation 0eiϕm
cos2βββ sin cos • analyzer: A = sinββ cos sin2 β
The signal revealed by the detector is simply the matrix product of the operators:
2 I = |Eout |= AMSPE
For certain angles of optical components, the equation reduces in complexity.
47 Simple rotation: let consider, for example, a real rotation θ of the sample, normal incidence of light onto the sample and these particular conditions for the apparatus: • incident field: γ = 45° • polarizer: α = 45° • sample: φ = θ (only the real part of the angle)
• modulator: φm = φ0 sin(ωt) • analyzer: β = 0° from the matrix product we obtain as intensity revealed by the detector
I 2 I = 0 cosθ - eiϕm sinθ 2
If the rotation angle θ is little, we can approximate cosθ ≈ 1 and sinθ ≈ θ, and moreover
eiϕm = cosϕ + i sin ϕ = cos ϕω sin t + i sin ϕω sin t mmo ( )() o
Developing the trigonometric functions in series of Bessel functions of argument φ0
cos[φ0 sin(ωt)] = J0(φ0) + 2J2(φ0)cos(2ωt) + 2J4(φ0)cos(4ωt) + ...
sin[φ0 sin(ωt)] = 2J1(φ0)sin(ωt) + 2J3(φ0)sin(3ωt) + ... stopping the substitution at the second order (in view of the lock-in technique employed that filters the signal as a pass-band), the intensity results
I I ≈θϕθϕω 0 1 - 2 J - 2 J cos 2 t 2 00() 20 () ()
If we choose a particular value of the modulation intensity φ0 = 2.41 for which J0(φ0) = 0, we can see from the intensity equation that the only term containing the rotation is at frequency 2ω, that his can be extracted by demodulating the intensity signal with a lock-in, in our case an EG&G 7260.
48 Complex rotation: if now the rotation angle is complex, φ = θ +iε, following the same rules as before and stopping the Fourier development at the second order, the intensity equation results
I =ω I02 + Iω cos( t) + Iω cos( 2 ω t)
For the general case of oblique incidence, with slight modifications at Nederpel work11, the solutions for two important cases are obtained:
• β = 0 ⇒ s-polarization Iω = 4J1(φ0)tgε I 2ω = -4J2(φ0)tgθ
• β = π/2 ⇒ p-polarization Iω = 4J1(φ0)tgε I 2ω = +4J2(φ0)tgθ
Hence, apart from numerical factors, the normalized lock-in signals at frequency ω and 2ω yield the Kerr rotation and ellipticity respectively. Fluctuations due to the instabilities of light source can be efficiently eliminated taking ratios of the signals respect to the continuous component of intensity. If the angle of the analyzer is set to 45° (as in our apparatus), can be seen12 that both terms of frequency are proportional to the Kerr rotation angle θ, that to a first approximation is proportional to magnetization; for our measurements we have choose the second term, demodulating at 2ω, because of the better signal-to-noise ratio. Depending on the modulator retardation φ0, the Bessel functions can be maximized, and doing so maximizing the signal:
• φ0 = 108° implies J1(φ0) = 0.582, its maximum value;
• φ0 = 175° implies J2(φ0) = 0.486, its maximum value.
If we choose φ0 = 2.41 = 138° as seen before, this implies J0(φ0) = 0, and both J1(φ0) and J2(φ0) are close to their maximum value.
Cryostat
In order to obtain very low temperature it has been used a cryostat ST15 Cryomech based on Gifford-McMahon cycle: the compressor package supplies the cold head with pressurized helium through flexible metal hoses. Inside the cryostat there's the cold head (see Fig. 2-13), made of aluminum, which refrigerates the sample and allow to reach a minimal temperature of 8 K; also a temperature sensor and a heater are mounted on the sample holder. A heat shield can be mounted on the head in order to protects the sample from heat irradiation. Before starting the system, the cryostat must be evacuated with a vacuum pumping capable of reaching a pressure of 10-4 Torr, and water cooled.
49
Fig. 2-13:cold head of the cryogenic apparatus: on the left is visible the sample holder
The Gifford-McMahon cycle is complicated: briefly, the compressor system supplies compressed helium to the cold head through stainless steel flexible lines, and the cold head achieves the G-M cycle by expanding the high pressure helium to low pressure. This is repeated 120 times per minute by a rotary valve, which activates the displacers and the flow of helium through the regenerators and charging the expansion volumes with high pressure helium. Because of the big oscillations generated on the table by the valve, we have projected an electronic circuit (see Fig. 2-14) that allows to alternate the pumping to the acquisition: the circuit is connected to the acquisition board and controlled by the pc, and the number of periods for each phase are configurable by the acquisition program.
Fig. 2-14: electronic circuit that controls the timing for pumping and acquisition
50 2.3 Acquisition programs
Inductive measurements
All the acquisition and simulation programs have been written by myself using LabWindows CVI of National Instruments, a visual C program for virtual instrumentation that allow to completely control all the steps of the program. The graphics interfaces are easily generated by CVI, and the "only" work of the programmer is insert the code in C language necessary at various buttons, input, selections, calculus and so on. The programs are interfaced with an acquisition board AT-MIO- 16DE-10 of National Instruments with analogical/digital input/output. The inductive measurements program (a screen shot is presented in Fig. 2-15) controls in output (via RS232) the wave-function generator, that is connected to an amplifier and then to the coil for generating magnetic field, and in input receives the tensions coming from the two probing coils. The same program allows to analyze the acquired data and detect the jumps.
Fig. 2-15: acquisition program for inductive signal
51 Magneto-optical measurements
Also for the magneto-optical measurements program (a screen shot is presented in Fig. 2-16) the output directed to the wave function generator is regulated by RS232, and all the analysis of acquired data are performed by the program itself, that allows to generate the histogram of jumps detected in the data; analysis and acquisition happen in coincidence, so that an hysteresis loop is acquired while the precedent hysteresis loop is analyzed. A particular control for the cryostat has been added (as seen in chapter 2), that allow to set the number of cycles of activity and inactivity of the cryogenic pump.
Fig. 2-16: acquisition program for magneto-optical signal
52 Ising simulations
The simulations, describe in chapter 3, are programmed in C language, and the visual C of CVI helps to easily insert the parameters in the model; also the transfer of parameters via file is possible, and this option allows to fully program different simulation with different parameters, elaborated in sequence with no time delay between them. The same program (a screen shot is in Fig. 2-17) is utilized for simulation on Ising model and also for the avalanches (what is called random coercive field Ising model in chapter 3). Each parameter of the standard Ising model (dimension of the matrix, exchange interaction, temperature, external field, ...) can be easily changed by the visual interface of the program; an important algorithm has been introduced in the Ising model that allows to measures the dimension of domains that form during the evolution of the system. Like precedent program, also the code for analyze data is implemented in the program: this means the calculus of probability distribution of both type of jumps, positive and negative.
Fig. 2-17: simulation program for both Ising model and random coercive field Ising model
53 2.4 Bibliography
1 H. Barkhausen - Phys. Z. (1919) 20, 401 2 R. Feynman, R.B. Leighton, M. Sands - The Feynman lectures on physics vol. II, ed. Addison-Wesley (1977) 3 J. McClure Jr., K. Schröder - CRC Crit. Rev. Solid State Sci. (1976) 6, 45 4 E. Puppin, M. Zani, D. Vallaro, A. Venturi - Rev. Sci. Instr. (2001) 72, 4, 2058 5 H. Deuling, L. Storm - Z. Angew. Phys. (1966) 21, 4, 355 6 M. Zani, E. Puppin - Journ. of Appl. Phys. (2003) 94, 9 ,5901 7 A. Hubert, R. Schafer - Magnetic Domains, ed. Springer (1998) 8 E. Moog, S. Bader - Superl. Microstr. (1985) 1, 543; S. Bader - Journ. Magn. Magn. Mat. (1991) 100, 440 9 E. Puppin, P. Vavassori, L. Callegaro - Rev. Sci. Instr. (2000) 71, 1752 E. Puppin - Phys. Rev. Lett. (2000) 84, 5415 10 R. M. A. Azzam, N. M. Bashara - Ellipsometry and polarized light, ed North-Holland (1979) 11 P. Q. J. Nederpel, J. W. D. Martens - Rev. Sci. Instr. (1985) 56, 5, 687 12 K. Sato - Jpn. Journ. Appl. Phys. (1981) 20, 2403
54 3. Experimental results and simulations
3.1 Metastable states
Experiment
It's has just said that magnetization process is made of abrupt magnetization jumps connecting discrete metastable state; we have investigated the stability of a magnetic system in one of these state, what it is called the life-time, in presence of an increasing magnetic field. The usual method for Barkhausen investigation, consists in detect the induced peak voltage in an inductive coil by the magnetization jumps, doesn't has the necessary spatial resolution, although a variant at this standard apparatus has been presented1 in one of our works. The number of metastable states in a macroscopic system is enormous, and due to stochastic process the path followed by the system to reach the final state is every time different. A possibility is offered by studying a microscopic region of the sample with a technique at high spatial resolution: this is our MOKE apparatus, that with a focalizing lens system in the optical path allow to investigate a relative small portion of the sample, of the order of 10 µm or more if necessary. The observation of a small region of the sample allows to identify a limited number of well-defined magnetization states and the corresponding Barkhausen jumps: the process remains stochastic, but moves through identifiable states.
We have investigated a sample2 of Fe 90 nm thick epitaxially grown on a MgO(001) single crystal. The substrate has been heat-treated until the observation of a LEED (1x1) pattern, typical of bulk- terminated lattice; Fe has been evaporated from an electron beam source at a rate of 1 nm/min in a vacuum chamber with the base pressure of 5*10-11 Torr, and then covered with a 3 nm-thick protective Pt layer. The thickness has been measured with a quartz microbalance, and the sample purity checked with in-situ XPS and Auger spectroscopy. The size of the sample is ≈ 4x4 mm; the magnetization is parallel to the film plane, and the experiments have been conducted by applying the magnetic field along the easy axis of the film. We performed nearly 64 000 hysteresis loops measurements by using a high sensitivity magneto- optical Kerr apparatus, on a central region of the sample defined with a spot size of about 100 µm, and by applying a triangular-wave magnetic field with frequency 100 mHz and amplitude 100 Oe.
55
40 M(k+1)
20 jump k+1
0
M(k) M(k) jump k -20 M [norm. units] [norm. M M(k-1)
-40 H(k) H(k+1)
20 25 30 35 40
H [Oe]
Fig. 3-1: physical quantities used in the model shown in the upper branch of one hysteresis cycle
In Fig. 3-1 is shown the upper branch of an hysteresis cycle extracted form the recorded stream: it appear clear the presence of Barkhausen jumps that connect metastable states during the magnetization process. The physical quantities involved in the process are shown in the figure, and they can explained in a three-step process: • when a jump with index k, that occur at a field indicate with H(k), take place, the system has been driven from the magnetization state M(k-1) to M(k); the jump so defined has a value ∆M(k) = M(k) - M(k-1); • now the system meet a metastable state, and until the external magnetic field that start from H(k) not reach the value H(k+1) nothing happen, and the magnetization of the system rest at a constant value; in this state the field has incremented of a value ∆H(k) = H(k+1) - H(k), the width or the life of the metastable state; • now that the state has reach the end of his life and the external field has the value H(k+1), a new jump k+1 can occur, and the system will be driven form the magnetization state M(k) to M(k+1); this new jump has a value ∆M(k+1) = M(k+1) - M(k).
The statistical distribution of magnetization jumps ∆M (Barkhausen noise) has just been addressed in many theoretical and experimental works, originating the well known power-law distribution as described in the precedent chapters: this behavior is not related to the particular sample investigated, but has been observed in different continuous or structured ferromagnetic film when the laser spot is small enough3. Until now the various magnetization jumps have been treated like single entity, with no relation one to each other: now we want investigate the statistical behavior of ∆H, the life of metastable states.
56 If a sequence of hysteresis loops is taken from the acquired stream, appear clear that three main principle states, denoted with A, B, C in the life Fig. 3-2, are present; they appear on each loop at a similar, although not the same, value of magnetization M, and have a different value of width ∆H.
40 C
20
0
B
-20 M [norm. units] [norm. M A
-40
40 C
20
0
B
-20 units] [norm. M A
-40
40 C
20
0 B
-20 units] [norm. M A
-40
20 25 30 35 40 H [Oe]
Fig. 3-2: sequence of upper branch hysteresis loops; state A, B, C are explained in the text
57 For make it more clear, in Fig. 3-3 is shown the probability distribution of the parameter M(k), the final value of magnetization after that a jump has occurred.
A C
B
M) [arb. units] [arb. M)
∆ p(
-40 -20 0 20 40 M [norm. units]
Fig. 3-3: probability distribution of final magnetization state after a jump (shadow area is explained in the text)
As it appear, three main peaks are present in the figure, and labeled accordingly to the corresponding plateau of Fig. 3-2. The presence of these reproducible plateau is observed everywhere over the sample surface, although the number and position of the magnetization states is related to the particular region investigated. In fact, the existence of these states can be found also on other samples with different composition and size4.
If we want to investigate the dynamics involving only a particular state, we must extract from the original stream the jumps entering or leaving in that state. This explain the shadow area in Fig. 3-3: if only the jumps that enter or leave the state B, for instance, are considered we can filter the entire record of data with the condition that M(k) is in selected range (identified by the shadow line on x axis in Fig. 3-3), and show for example the behavior of ∆H relative to those jumps that terminate in state B or leave the state B. Both case are shown in Fig. 3-4, where, remembering Fig. 3-1, • p(H(k)) (solid dots in figure) is relative to jumps that enter in state B; • p(H(k+1)) (empty circles in figure) is relative to jumps that leave state B.
58
state B H(k) H(k+1)
units] [arb. p(H)
26 28 30 32 34 36 38 H [Oe]
Fig. 3-4: probability distribution of magnetic field for jumps that enter or leave in state B, and relative Gaussian best-fit
Also Gaussian best-fit of both plot are shown in figure, having standard deviations σk = 0.98 Oe and
σk+1 = 1.22 Oe respectively.
In Fig. 3-5 the normalized probability distribution of width state relative to state B is shown; it can be seen that such statistical life has a peculiar shape, asymmetric and no-crossing the origin of the axes. Il must be noted that our external magnetic field has a linear rate (obtained with a triangular waveform signal), and so we can speak of time-life or field-life indifferently.
0.40
0.35 0.30 state B 0.25 ]
-1 0.20
H) [Oe ∆ 0.15 p( 0.10 0.05 0.00 02468 ∆H [Oe]
Fig. 3-5: probability distribution of width of state B; fit-line is relative to the model explained after
59 Model
Now we want to try to identify the dynamics involved in our system relative to a magnetization sample that, inserted in an increasing magnetic field, leave a metastable state and jumps to another one. The model involved is based on the physical quantity p(∆H) shown before, and make an explanation of the shape of the curves obtained.
Reconnecting to Fig. 3-1, we start with our system prepared in the magnetization metastable state M(k), H(k), reached with a jump of index k. Barkhausen jumps are related to domain wall motion through a pinning site way, so we can consider that our system is trapped in a potential well of definite depth: the external magnetic field linearly increase, and when it has increased of a value ∆H the stability is destroyed, and the system can make another jump k+1 keeping itself in the state identified by M(k+1), H(k+1).
The model considered is relative to a magnetic dipole coupled with a local magnetic field: this last is the sum of three different contributes: • the external field H;
• the pinning field Hp, connected to the potential well in which the dipole is founded, that pins it in the metastable state;
• the interaction field Hw, which take accounts for the interaction between the sampled area and the rest of the sample.
The system start after a jump k, at the beginning of the metastable state, so that the local field which has subjected
Hloc(k) = H(k) + Hw(k) - Hp is a negative quantity. Now the external field increase and so the local field, until it reached a positive value and the next jump occur:
Hloc = H + Hw - Hp
Subtracting the precedent equation from this last we obtain
* Hloc = ∆H + Hw + Hloc(k) 60 with the position
* ∆H = H - H(k) Hw = Hw - Hw(k).
The equation summarizes the behavior of our model: the jump k+1 take place when the local field become positive for the first time, under the pressure of the external field linearly increasing and the fluctuating interaction field. A pictorial representation of this behavior is shown in Fig. 3-6.
H loc Hloc = ∆H + Hloc(k)
0 ∆H
* Hw
Hloc(k)
Fig. 3-6: pictorial representation of process of metastable state life * described by the equation Hloc = ∆H + Hw + Hloc(k)
What is the probability p(∆H) that the system goes out from the potential well, i.e. the local field become positive for the first time, with a ∆H value of external field? We must solve the precedent founded equation
* Hloc = ∆H + Hw + Hloc(k) for a probability distribution p(∆H) that satisfies the follow conditions: • the external field ∆H increases monotonically from zero; * * • the interaction field Hw is a statistical variable, and we call p(Hw ) its probability distribution;
• the system is always prepared in the same state, i.e. Hloc(k) (that is negative) is constant with k.
The solution is a mathematical problem involving the reliability problem, that allow to determine the life (or time-to-failure) probability of the system when the failure-rate of the system is given5.
61 Reliability problem
Be p(∆H) the probability distribution function (that we want find) that the system escape from potential well with an increment ∆H of external field. By definitions, the failure-rate (or hazard- rate) r(∆H) is the conditional probability that a system survived until ∆H will die (escape from well in our case) in the interval ∆H ÷ (∆H + d∆H), or
prob. escape with ∆H p(∆H) r ()∆H = = prob. not be escape until ∆H 1 - P()∆H
∆H where P()∆H = pxdx() is the distribution function relative to the density p(∆H). Inverting the ∫0 equation, with few mathematical passages is easy obtain
∆H p ∆H = r∆H exp r x dx () () () ∫0
* One component of the total field has statistical behavior (Hw ), so that in our case we can write
∞ r ()∆H = pw () xdx ∫−∆(Hloc + H)
* We make the assumption (the simplest possible) on the distribution pw(Hw ) that it is Gaussian with zero mean and standard deviation σw
1x2 pxw () = exp- 2 σw 2π 2σw
This assumption can be justified in the framework of the limit central theorem, since the interaction field Hw results a sum of interactions due to the domains constituting the sample.
62 Substituting the probability p(Hw) in the p(∆H) equation, this last can be calculated in a close form
11σw 22∆H p ∆H1 = + erfβ exp α erf α - βerf β + exp -α - exp -β - () 22() () () () () 2 π where we put for simplicity
H H+∆H α = loc β = loc σw 2 σw 2 and erf(x) is the Gaussian error function
2 x erf() x = exp -t2 dt π ∫0 ( )
Now, the function p(∆H) has a complicated form, but as it can see it's dependent on only two parameters, Hloc and σw, so that p(∆H) = p(∆H | Hloc; σw).
We have got a set of experimental value ∆Hi: the chose fit method is given by the maximum likelihood criterion (or maximum joint probability to obtain just those set ∆Hi of data), one of the best estimator to obtaining one or more parameters: the likelihood function is written as
LH()loc ; σww = ∏ p(∆H i | H loc ; σ ) i
where p(∆Hi | Hloc; σw) is the function just obtained evaluated in various experimental data. Only for numerical convenience, we have minimized the negative logarithmic likelihood function
LH* ; σ = - log L H ; σ = - log p ∆H | H ; σ ()locww( loc) ∑ ( i loc w) i
using a Matlab algorithm, and we have obtained the two parameters, Hloc and σw, that better fit our experimental data for each series of data (for each of the three states).
63 Conclusions
In Fig. 3-7 is shown the good agreement of data with the fit and relative parameters.
0.35
0.30
0.25
] -1 0.20 state A
H) [Oe 0.15 ∆ p( 0.10 0.05
0.00 0.35 0.30 state B 0.25
] -1 0.20
H) [Oe
∆ 0.15 p( 0.10
0.05
0.00 0.35
0.30
0.25 state C ] -1 0.20
H) [Oe
∆ 0.15 p( 0.10
0.05
0.00 02468 ∆H [Oe]
Fig. 3-7: probability distribution of width of states A,B,C and relative fit-line
64 For state B, we can compare the value of σw = 1.34 Oe with those obtained for σk = 0.98 Oe and
σk+1 = 1.22 Oe shown in Fig. 3-4; although the exact relation between these three values is very complicated, it can be seen a relative numerical agreement.
The magnetization process, when observed at a microscopic scale, shows that the process is discontinuous, and evolves through magnetization jumps and metastable states. Until now no work has been done in studying these states, also because are less easy to detect than jumps: the results we have reach are due to the high-spatial resolution of the measurement apparatus, that allow to observe very small portion of the sample, and to the very high number (64 000) of hysteresis loops measured.
65 3.2 Magnetization jumps vs temperature
Experiment
All the various models presented until now on Barkhausen noise refer to zero temperature situation, whereas the experimental data are all obtained at room temperature. Also in the field in the complex systems (of which Barkhausen noise makes part) there are contradicting results on the behavior of the avalanche in presence of temperature: in Vergeles6 model the introduction of temperature in a critical system breaks down criticality, in Caldarelli7 work the temperature introduced in a sand-pile model preserves criticality. So the need for experimental results at low temperature is strong. We have investigated the same sample of precedent paragraph, a thin Fe film2 90 nm thick epitaxially grown on a substrate of MgO(001), substrate heat-treated until the observation of a LEED (1x1) pattern, typical of bulk-terminated lattice. Fe has been evaporated from an electron- beam source at a rate of 1 nm/min in a vacuum chamber with the base pressure of 5*10-11 Torr. The thickness has been measured with a quartz microbalance, and the sample purity checked with in-situ XPS and Auger spectroscopy. We have used our magneto-optical Kerr apparatus with variable spatial-resolution, that has allowed us to vary the spot size of the laser beam from 20 to 500 µm. The sample has been mounted on a cryostat in order to change its temperature from 10 to 300K. The entire apparatus and all the problematic relative to the cryostat are described in precedent chapter; in the follow I'll present the results obtained on the sample versus temperature, with the extraordinary conclusion (first experimental data!) about temperature dependent criticality.
Coercive field
It is established that the coercive field Hc is strongly affected by temperature, and usually increases at lower temperature. A model which explains this effect in terms of domain wall motion in a disordered media has been proposed by Gaunt8, in which a wall is pinned by a potential well from which it can escape due to thermal motion. The Gaunt hypothesis are consistent with the magnetization process in our sample9, where magnetization dynamics is driven by domain walls motion, and that cause magnetization to jumps between randomly distributed pinning sites. Gaunt model predicts that Hc depends on T according to this law:
1/2 1/2 2/3 Hcc = H1 - CT 0
66 where C is a constant depending on the size and shape of the potential well. The macroscopic behavior of our sample is shown in Fig. 3-8: a series of loops with a spot size of 100 µm has been acquired, and an average cycle has been obtained simply adding the single cycles of the series: in figure is shown with filled dots the average cycle obtained at 300 K and with empty dots the average cycle measured at 10 K.
60 300 K 40
20 10 K
0
-20
M [norm. units] [norm. M H c -40
-60 -60 -40 -20 0 20 40 60
H [Oe]
Fig. 3-8: average hysteresis cycle at 300 K (filled dots) and 10 K (empty dots)
Clearly the coercive field increases between two cases, and in Fig. 3-9 is reported the relationship 1/2 between coercive field Hc and temperature T we have obtained; the data of Hc are plotted versus T2/3, and the linear trend of data in this scale confirm that coercive field depends on temperature according on Gaunt law.
5.8
5.6
5.4 ) 1/2
5.2
(Oe 1/2 c
H 5.0
4.8
4.6 0 1020304050 2/3 2/3 T (K ) Fig. 3-9: temperature dependence of coercive field
67 Metastable states
Magnetization process, when observed at a microscopic scale, is discontinuous and presents a series of magnetization jumps connecting metastable states. These steps are not deterministic, but randomly change by repeating the loop measurements. A statistical analysis of these magnetization fluctuations can be performed by considering the relevant parameters involved in the process, ∆H (width of metastable states) and ∆M (size of magnetization jumps), illustrated in the upper part of Fig. 3-10. A series of 5000 hysteresis cycles has been acquired with a laser spot of 100 µm on the sample.
The physical meaning of these two parameters can be better understood by considering the evolution of the magnetization in an increasing field (shown in the lower part of Fig. 3-10): • initially the system start with a configuration labeled with A; this point correspond to a metastable equilibrium position reached by the system during its evolution through magnetization jumps. The ellipses∗ represents the area sampled with the laser spot; within this area a fraction of the sample is just oriented with the external field. The metastability of this state is determined by the presence of pinning sites that determine a local minimum in the energy landscape; • by increasing the external applied field the system remains in this state, and the only produced effect is a bowing of the domain wall; the entire situation doesn't change until a particular value of external field is reached (indicated as B in figure), where the energy of the system is enough to overcome the potential barrier; • now a magnetization jump can occur that brings the system into a new metastable configuration, indicated as C in figure;
The extra-field necessary to overcome the potential barrier is ∆H, whereas the amplitude of the avalanche that occur between the state B and C is ∆M. Now we focus our attention on the first of these parameters, the width of the metastable states. The statistical distribution of ∆H at 300 K (filled dots) and 10 K (empty dots) is shown in Fig. 3-11. In precedent paragraph we have presented a model in order to explain the shape of the distribution p(∆H): here we note that the two distribution present the same shape at both temperature.
∗ The Gaussian laser beam impinge on the sample at an angle of 45°, this explain the shape of the spot; the size spot in the text refers to the minor axis. 68
C
Mi + ∆M
A Mi B
Hi Hi + ∆H
A Mi
Hi
B M i
H + ∆H i
C M + M i ∆
H + ∆H i
Fig. 3-10: (up) parameters involved in magnetization process (down) evolution of the system between three states A, B, C
300 K 10 K
H) [arb. units] ∆
p(
012345
∆H [Oe]
Fig. 3-11: probability distribution of ∆H at 300 K (filled dots) and 10 K (empty dots)
69 Magnetization jumps
The attention is now oriented at the magnetization jumps that occur in the magnetization process: we can observe in Fig. 3-12 a series of three cycles measured respectively at 300 K and 10 K.
300 K 10 K Ø Ø
i j
Ø Ø
i+1 j+1
Ø Ø
i+2 j+2
Ø Ø
16 20 24 28 26 30 34 38 H (Oe)
Fig. 3-12: (left) three loops measured one after the other at 300 K (right) the same at 10 K
In the left part of this figure it's shown a series of three loops measured at 300 K one after the other (for clarity only a portion of each loop is shown corresponding to the upper branch) with a laser spot of 100 µm. In the right part of the figure it's shown another series measured at 10 K with the same spot; fluctuations are present in both series, but the amplitude of the steps appears to be in the average smaller for lower temperature with respect to the higher temperature situation. In Fig. 3-13 is shown the probability distribution of ∆M at each of the two temperature, represented in a log-log plot: in the upper part of the figure the distribution at 300 K can be interpolated by a power-law p(∆M) = ∆M-α with critical exponent α = -1, whereas in the lower part the distribution at 10 K is always represented by a power-law, but the critical exponent change its value to α = -1.8. The cut- off present in each single distribution is due to the finite size of the region defined by the laser spot on the sample surface.
70
T = 300 K -1 p(∆M) = ∆M
20 µm
50 µm
100 µm 250 µm 500 µm
T = 10 K -1.8 p(∆M) = ∆M 20 µm
M) [arb. units] [arb. M)
∆ 100 µm p(
500 µm
1 10 100 1000 10000 ∆M [arb. units]
Fig. 3-13: probability distribution of ∆M at 300 K (filled dots) and 10 K (empty dots)
For a definite value of the spot size of the laser the histogram (or probability distribution) is easily defined; the plotting process can be repeated for different value of the spot size, and we can assemble all the histogram into a unique plot by rescaling the axis. We can note that the particular procedure adopted9 allows us to obtain experimental data that spanning over several decades of
∆M: we consider two arbitrary different distribution with respective spot size of the laser beam D1 and D2, with D1 < D2. A magnetization jump ∆M1 in the first distribution represents a percentage 2 variation of magnetization within an area (the sampled area) that is proportional to D1 . The same 2 percentage measured in the second distribution where the sample area is proportional to D2 will 2 2 correspond to a magnetization, relatively to the first spot, larger by a factor R = D2 / D1 . This correction factor, that represents the ratio between the spot size, must be applied to the second distribution in order to plot the two distribution in a unique graphic: the horizontal axis must be multiplied times R. The vertical axis is simply correct by translating the distribution until to obtain the continuity of the functions.
71 Conclusions
Metastable states has a relative stability: when the system reach one of these state, an extra-field is necessary to escape from well and make another jump. In the precedent paragraph a model has been presented that explain the shape of p(∆H): in this model we have a magnetic dipole that interacts with the magnetic field, sum of the external field, the pinning field that pins the system in the metastable state and the interaction field that take accounts for the interaction with the rest of the sample. The dipole can escape from the potential well (i.e. orient itself in the direction of external field) when the total magnetic field has an enough value to overcome the barrier. The probability distribution of Fig. 3-11 shown that p(∆H) doesn't depend on temperature, and this observation can be interpreted by saying that the escape from well is not (or weakly) influenced by temperature.
The avalanche process follow a different way: on average, the size of the magnetization jumps increase with temperature or, more precisely, the occurrence of larger jumps is favored at higher temperature. At both temperature, 300 K and 10K, the probability distribution of Fig. 3-13 shown that p(∆M) is always described by a power-law, i.e. the system conserves its criticality with temperature, but the relative critical exponent has value α = -1 at 300 K and α = -1.8 at 10 K. The role of temperature in complex systems has been addressed in few theoretical investigations, and it is not completely understood: our data indicate that criticality is always observed from 10 to 300 K, but the critical exponent that nearly doubles at lower temperature. At our knowledge this is the first experimental evidence, at least in the field of magnetism, on the temperature dependence of criticality in a complex system10.
In synthesis, in the avalanche process the thermal activation doesn't play a relevant role in the initial overcoming of the potential barrier (so that p(∆H) presents the same shape with temperature), whereas the temperature dependence is present in the probability distribution of the avalanche amplitude (so that p(∆M) change with temperature, but the system remains critical), that is related to cooperative effects taking place during the avalanche itself.
72 3.3 Simulations
Ising model
The Ising model is simple, realistic enough and easy to simulate. It consists of a spin-lattice of definite dimension and scalar value (i.e. the spins can assume values s = ±1): introducing various parameters in the model, we are able to play with it. The program written and used to simulate the Ising model is described in chapter 3; it is also utilized for a simulation described in the next paragraph, when a disorder is introduced in the system. Now it will follow a brief description of the Ising model behavior, with the various parameters introduced in it, and using a 2d dimension lattice, because our samples are thin magnetic films.
Many important results about phase transitions have been derived by studying particular models: the most influential is the Ising model, invented by Lenz11 in 1920 and resolved by Ising12 for 1d case in 1925 and Onsager13 for 2d case (this case in absence of magnetic field) in 1944. Until now, no exact solution for the 3d and 2d model in a non-zero magnetic-field has been still presented.
The first parameter introduced in the model is the energy.
Energy: the more simple situation see one energy term for each site i: the exchange energy, extended over all pairs of nearest neighbors
E = -Jss exi ∑ i y i
The system will tends to minimize its energy, and if no other energy terms are present the system will has all the spins perfectly aligned. We can define the total magnetization as the sum of the spins aligned up and normalized to the total number of spins in the lattice: in this case the magnetization has value M = 1 (or M = -1, the situation is symmetric). The system is a perfect mono-domain, where exchange energy give rules.
73 Entropy: let introduce the temperature: for each site, there's a finite probability that his site reverse cause temperature in accord to Boltzmann statistics, also if not energetically favored; if we call E1 the initial energy situation, E2 the energy with the site reversed and ∆E = E2 - E1 (the steps described are a mixture of Metropolis14 and thermostat algorithm)
• if E2 < E1 the site reverse and brings the system in the new minor energy configuration;
• if E2 > E1 the probability that the site reverse is
∆E 1 - p = ekT rev 2
o if ∆E >> kT ⇒ prev ≈ 0 ⇒ energy wins! In this case each spin tends to stay aligned with its neighbors, and the system goes into a configuration of mono-domain (the average magnetization is simply M = 1)
o if ∆E << kT ⇒ prev ≈ 1/2 ⇒ entropy wins! The disorder introduced by temperature is strong, and system goes into a configuration where spins are uniformly distributed up or down (the average magnetization is M = 0)
What's happen when temperature has value between the two extreme situation above described? The average magnetization is shown in Fig. 3-14: we can see that there's a critical value of temperature Tc = 2.269, where magnetization approaches to zero with infinite slope (this is called second-order phase transition).
Fig. 3-14: average magnetization vs temperature: magnetization is symmetric regarding sign
74 But this is the average value: what about the fluctuations? The magnetization is not constant for a definite value of temperature, but continuously fluctuate, time by time (here the single step of time is defined as the complete analysis of all spins of the lattice: where we have terminated to analyze all spins, the control restart and we go in the next step-time); if we give a look at the graphic situation (shown in Fig. 3-15), we can see how in the lattice are present agglomerate of identical spin, the domains. What are the dimensions of these domains?
• at low temperature few spins reverse, and the statistical distribution of domains size ∆M is like a delta around size = (dim*dim), where dim is the dimension of the lattice matrix (i.e. the system is a mono-domain); • on the other side, at high temperature the system is highly disordered, and the distribution shape tends to a delta around size = 1;
• at the critical point, when T = Tc, we can see from Fig. 3-15 that are presents domains of all sizes, and the probability distribution of domains size that forms in time is described by a power-law
p(∆M) = ∆M-α
where the critical value of the exponent is α = 1.93 (see Fig. 3-16)
75
Fig. 3-15: spin distribution in the Ising lattice for a 100*100 matrix when T = Tc
1E9 1E8 1E7
1000000
100000 -1.93 10000 p(∆M) = ∆M 1000
100 M) [arb. units] [arb. M) ∆ 10 p( 1 0.1
0.01 0.1 1 10 100 1000 10000 ∆M [arb. units]
Fig. 3-16: probability distribution of domains size for a 500*500 matrix when T = Tc
76 Field: if the system is inserted in a magnetic field, a new energy term must be considered for each site: the magneto-static energy
E = -Hs msi i
It's possible to plot an hysteresis cycle for the system, reporting in a graph the average magnetization vs the external field, for various temperature: the results are reported in Fig. 3-17.
Fig. 3-17: hysteresis cycles at various temperature
(left) T > Tc: the system is disordered at H = 0 (paramagnetic)
(right) T < Tc: the system is ordered at H = 0 (ferromagnetic)
(low) T = Tc: the system has no remanence and vertical slope at H = 0 (critical state)
When the temperature is over Tc, with no field applied the system is disordered, and the system shows no magnetization remanence. Only with temperature under Tc the system goes in a ordered state, and shows remanence. Another time we can see the critical behavior when the temperature assumes its critical value: the system has a phase transition from ordered to disordered state increasing temperature around Tc.
77 Random coercive field Ising model
We can extend the Ising model in several ways, and doing so introducing disorder in the system: as just seen treating Barkhausen noise, this inclusion can brings to avalanches. In all these various Ising model, they act in the rate-independent limit where the sequence of energy minima is the only important feature, and the time doesn't play any role during this evolution: this implies that the external field remains constant during the occurrence of the avalanches15. Within this framework, all the simulations are performed at zero-temperature, with absence of thermal fluctuations.
Let make a brief explanation of various extended model, for then introduce our model and its relative results, that are able to explain several magnetic properties measured on our sample.
XY model: in this case the magnetic dipoles of Ising model don't point only in two directions (up or down), but each spin is capable of pointing in any direction within some given plane, i.e. each spin is characterized by a two component vector spin (the same operation can be obtained replacing the scalar spin by a complex number)
E = -Jss exi ∑ i y i
Heisenberg model: is similar to the precedent, but the vector spins can has various dimension (under this vision the XY model can be considered a special case of Heisenberg model with d = 2); when the dimension is d = 3, we would named the model as generalize classical Heisenberg model
G G E = -Jss exi ∑ i y i
Potts model: in the Potts model each spin can take one of several possible values (n and m in the below equation are the index within this rage of values), and the interaction energy between adjacent spins is zero if the corresponding spins are different and J if they happen to be the same; in this case there's no energetic reward for the spins to take similar but not identical values
E = -δ(n - m) J s s exi ∑ in ym i
78 Random field Ising model (RFIM): a new energy term16 is introduced for each spin, that takes into account the interaction with a local quenched random field; it is assumed that such random field is Gaussian distributed, with zero mean and standard deviation σ
E = -hs fiii
Random bond Ising model (RBIM): the disorder is now introduced17 by adding noise to the nearest neighbor exchange interaction; the value of J for each pair of spin is Gaussian distributed, with zero mean and standard deviation σ. In this case, for large values of σ some bonds can become anti-ferromagnetic
G G E = -Jss* exi ∑ i y i
Site diluted Ising model (SDIM): the model18 contains exchange interactions up to the n-th nearest neighbor, with different weight for each interaction term dependence the neighboring distance. Moreover, in some randomly chosen sites can be the absence of a spin (from which the name site diluted): this is indicate in the below equation by ε, that can has value 0 (absence) or 1 (presence)
n E = -J ε s ε s E = -Hε s exi ∑∑kii y y msi ii k=1 i
Random anisotropy Ising model (RAIM): in this case the local strong anisotropy axes are randomly distributed, and at each site is assigned a unit vector which account for the anisotropy axis orientation (this mean that the component spin has value, relative to its anisotropy axis, not more s but s*cosθ), that is uniformly distributed within a cone of maximum angle θ0; if θ0 = 0 nothing is changed, if θ0 = π/2 we have completely random distribution of anisotropy axis.
GG G G E = -Js•s E = -H•s exi ∑ i y msi i i
In order to explain the statistical magnetic properties we have obtained for our samples, we have introduce an "ad-hoc" model. This model has a different approach to the problem, treating the magnetization process evolution by the point of view of local magnetic field rather than energy minimization configuration. This instrument is again in study, so I can present only partial results.
79 Random coercive field Ising model: in this model at each site is attributed a randomly distributed coercive field, Gaussian distributed (but different distribution can be chosen, from linear to Weibull or by data in an external file) with mean Hc and standard deviation σc. Moreover, each spin site is responsible of an interaction field at its nearest neighbors that has a direction parallel to spin, so that in each site is present an overall local field sum of external field and this interaction field.
The system is prepared in a saturated magnetization condition, and then the external field increments until in one site the local field reaches a value enough to reverse the site itself. Now the site reverse, changing the total magnetization of the system and changing the local field at its 4 nearest neighbors: if for each of these 4 sites the local field condition is satisfied, the single site can reverse and so on, creating a possible avalanche. In the spirit of rate-independent hysteresis limit, during an avalanche the external field is constant. The magnetization process evolves, until it reach the other saturate condition where all the spins are reversed.
dim 100
dim 1000
M [arb. units] [arb. M
101.0 101.1 101.2 101.3 101.4 101.5 101.6 H [arb. units]
Fig. 3-18: hysteresis cycles with different dimension of lattice (up) matrix 100*100 (down) matrix 1000*1000
80 It is possible makes different hysteresis cycles, that can appear more or less continuous dependence the dimension of the matrix (Fig. 3-18): we can observe that when the avalanches have dimensions comparable with the dimension of the lattice, they are easily visible in the hysteresis cycle. An interesting result is obtained observing the behavior of the system for different values of the disorder parameter, i.e. the standard deviation σc of local coercive field (the first parameter of the model, the Hc mean value of coercive field, has the only effect to translate the hysteresis loop along the H axis). When the parameter has zero value, each site has the same coercive field, and the hysteresis loop is in a mono-domain situation: increasing the parameter increases the disorder, and the enlarged hysteresis loops obtained are shown in Fig. 3-19: this phenomenon is due to the enlarged range of coercive field values.
σ = 1.5 σ = 10 c c
M [arb. units]
70 75 80 85 90 95 100 105 110 115 120 125 130 H [arb. units]
Fig. 3-19: hysteresis loops versus disorder parameter;
σc = 1.5; 3; 5; 10 from the vertical loop to the more enlarged
Average jump [norm. units] [norm. jump Average
01234 σ [arb. units] c Fig. 3-20: average jump versus disorder parameter for a matrix 100*100
81 Another way to see the influence of disorder on the system is observe the mean value of the avalanche: from Fig. 3-20 is evident that there are two distinct regions dependence of disorder parameter: • at low disorder, the system is like a mono-domain, and big avalanches (at limit only one of the entire dimension of the matrix) are possible; • at high disorder, each site has so different values of coercive field that very little avalanches are present in the system (at limit the curve tends to an average jumps of a single site); • it can be seen from the figure that there's a particular value of the disorder parameter (a critical
value) that constitute the transition from one region to the other. This happen for σc ≈ 0.87.
This is the behavior of the average jump: what happens at the probability distribution of avalanches sizes (jumps) dependence from disorder parameter? Also in this situation we can observe a phase transition and moreover the existence of a critical phenomena that occur for a particular amount of the disorder, like we have just seen in the Ising model tuning the temperature (this behavior make us think to temperature as a form of disorder). When the external disorder parameter is tuned to the critical value the system presents avalanches of all sizes, and the probability distribution of avalanches shows a critical behavior, characterized by a power-law function (Fig. 3-21). A better statistics is obtained simply taking a matrix of higher dimension: in the case of Fig. 3-21 the critical exponent of the power-law function results α = 1.2 with a 1000*1000 matrix.
-1.2 p(∆M) = ∆M
units] [arb. M) ∆ p(
1 10 100 1000 10000 ∆M [arb. units] Fig. 3-21: probability distribution of avalanches for a matrix 1000*1000
The next step will be the possibility of include temperature and disorder in the magnetization process of our model, trying to explain the experimental results obtained regarding the different critical exponent of the probability distribution of avalanches size.
82 A new interesting results comes from the analysis of the dynamics of magnetization process. In the upper part of Fig. 3-22 is shown the inductive signal of the sample of Fe63B14Si8Ni15 (as just seen in chapter 2); we can observe how the first part (indicated ad zone S) is characterized by small avalanches, whereas the bigger avalanches are present in the central part (zone U). The same structure is present in our simulation (lower part of Fig. 3-22), where this time three different divisions (zone , Z and ) are indicated.
units] [arb. dM/dt
zone S zone U
200000 400000 t [10 µs] -> H
[arb. units] dM/dt
zone zone Z zone
101.1 101.2 101.3 101.4 101.5 101.6 101.7
H [arb. units]
Fig. 3-22: inductive signal versus time (up) measurement (down) simulation; symbols are explained in the text
Also to have different average size, the avalanches are differently distributed: if the probability distribution of the avalanches in the different sections are plotted, we have got a representation of the dynamics of the magnetization process from the statistical point of view.
83 The histograms are shown in Fig. 3-23: in both cases the behavior of the distribution is a power-law p(∆M) = ∆M-α, but with different critical exponent. At the beginning of the magnetization process, the avalanches are present in a power-law distribution (zone S for the measurement, and zone for the simulation) with elevate exponent (respectively α = 2.4 and α = 1.8); proceeding in the magnetization process (zone Z for the only simulation), we can observe how in the central part of the inductive signal, where the avalanches are bigger, the distribution is again a power-law (zone U for the measurement, and zone for the simulation), but the exponent is become 2/3 of the initial value (respectively α = 1.6 and α = 1.2). This is the first observation of this phenomenon of dynamics of statistical properties in magnetization process; it's again a work in progress, so that much work must be done in explain this behavior, but perhaps can be a stimulus for explaining why the exponent change its value not only in the temperature dependence measurements shown in this chapter, but also during the magnetic evolution.
-1.6 p(∆M) = ∆M
M) [arb. units] M)
∆ -2.4
p( p(∆M) = ∆M
∆M [arb. units]
-1.2 p(∆M) = ∆M
M) [arb. units] ∆ -1.8 p( p(∆M) = ∆M
-1.4 p(∆M) = ∆M
∆M [arb. units]
Fig. 3-23: probability distribution of avalanches (up) measurement (down) simulation symbols are relative to Fig. 3-22
84 3.4 Bibliography
1 E. Puppin, M. Zani, D. Vallaro, A. Venturi - Rev. Sci. Inst. (2001) 72, 4, 2058 2 R. Bertacco, S. De Rossi, F. Ciccacci - Journ. Vac. Sci. Technol. A (1998) 16, 2277 3 E. Puppin, S. Ricci, L. Callegaro - Appl. Phys. Lett. (2000) 76, 17, 2418 4 L. Callegaro, E. Puppin, S. Ricci - Journ. Appl. Phys. (2001) 90, 5, 2416 5 N. R. Mann, R. E. Schafer, N. D. Singpurwalla - Methods for statistical analysis of reliability and life data, ed. Wiley & Sons (1974) 6 M. Vergeles - Phys. Rev. Lett. (1995) 75, 10, 1969 7 G. Caldarelli - Physica A (1998) 252, 295 8 P. Gaunt - Phil. Mag. B (1976) 48, 261 9 E. Puppin - Phys. Rev. Lett. (2000) 84, 23, 5415 10 E. Puppin, M. Zani - Journ. of Phys. C - Cond. Matter (2004) 16, 8, 1183 M. Zani, E. Puppin - Journ. Magn. Magn. Mat. (2004) in press 11 W. Lenz - Phys. Zeit. (1920) 21, 613 12 E. Ising - Z. der Phys. (1925) 31, 253 13 L. Onsager - Phys. Rev. (1944) 65, 3-4, 117 14 N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller - Journ. Chem. Phys. (1953) 21, 1087 15 E. Vives, A. Planes - Journ. Magn. Magn. Mat. (2000) 221, 164 16 C. M. Coram, A. E. Jacobs, N. Heinig, K. B. Winterborn - Phys. Rev. B (1989) 40, 10, 6992 17 E. Vives, A. Planes - Phys. Rev. B (1994) 50, 6, 3839 18 B. Tadić - Phys. Rev. Lett. (1996) 77, 18, 3843
85 A. Appendix: other measurements and collaborations
A.1 Negative jumps
When observed at a macroscopic scale, the magnetization process is discontinuous and takes place through a series of sudden jumps. Normally, in the sample region interested by one of these jumps the final magnetization is aligned with the external field, but it has been also observed1 the presence of so-called negative Barkhausen jumps: after these jumps the local sample magnetization, initially aligned with the external field, points against the field itself. The physical mechanism responsible for the presence of these negative jumps is not clear and, in previous investigations, eddy currents2 have been considered as possible candidates for explaining them.
∆M
A
I S
H
M
H
S
∆θ ÷ ∆M
Fig. A-1: (up) inductive technique (low) magneto-optical technique
86 Inductive measurements
We have investigated an amorphous ribbons of Fe63B14Si8Ni15 35 mm long with the traditional inductive coil, which probes a large sample region, and also with our space resolved magneto- optical Kerr ellipsometer. The experimental setup for both cases is schematically shown in Fig. A-1; the sample (S) is placed inside a coil (M) which generates the magnetizing field H ranging from -50 to 50 Oe: • the inductive measurements (upper part of Fig. A-1) are performed by placing a smaller coil (I) around the sample and measuring the induced voltage peaks in the coil, generated by magnetization jumps taking place in a region which extends approximately 15 mm on both sides of the coil3 which means, in our case, nearly the whole sample; • in the magneto-optical measurements (lower part of Fig. A-1) the sample is placed in the same field, but the magneto-optical Kerr signal is detected by measuring the polarization rotation (∆θ) of the incident laser beam which is proportional to the magnetization variation ∆M along the field direction.
V (arb.V units)
∆M +
∆ M-
0 200 400 600 800 1000
t (µs)
Fig. A-2: presence of positive and negative jumps in portions of inductive data
87 During one branch of the hysteresis loop, the number of jumps detected with the inductive coil is the order of 104, and the fraction of negative jumps is 6%.
The magnetization variation associated with each jump detected with the inductive technique is defined as the area ∆M of the corresponding peak (Fig. A-2), and also the statistical distributions of the amplitude of positive (∆M+) and negative (∆M-) jumps is presented (Fig. A-3). As we can see, in both cases the distribution is nearly linear over several decades in a log-log plot and can be represented with a power-law: P(∆M) = (∆M)-α, with the critical exponent α = 1.6 as best fit.
1.6 p(∆M)=∆M
∆M +
∆M -
M) [arb. units] M)
∆ p(
10-5 10-4 10-3 10-2 ∆M/∆M tot
Fig. A-3: probability distribution of positive and negative jumps
88 Magneto-optical measurements
Magneto-optical data have been collected by measuring the hysteresis loops from a sample region, defined by the probing laser spot, of 100 µm: by moving the laser spot onto the sample surface it is possible to find regions where negative jumps take place (Fig. A-5 shows a series of three hysteresis loops, with the presence of positive magnetization jumps; Fig. A-6 shows another series of three loops, measured in a different region of the sample, but with the presence of negative magnetization jumps. The lower part of both figures shows the average loop, obtained simply adding all the loops of the series). Observing the Fig. A-4 it's clear the connection between optical and inductive results: in one case (optical) the signal is proportional to M, in the other case (inductive) the signal is proportional to the time derivative dM/dt.
∆M -
∆M + M (arb. units) (arb. M
dM/dt (arb.units) dM/dt
-3 -2 -1 0 1 2 3 H (Oe)
Fig. A-4: detail of optical data (upper) and its first derivative (lower)
89
M (arb. units) M
-20 -10 0 10 20 H (Oe)
Fig. A-5: (up) series of three hysteresis loops, with presence of positive jumps (down) average loop
M (arb. units) M
-6 -3 0 3 6 H (Oe)
Fig. A-6: (up) series of three hysteresis loops, with presence of negative jumps (down) average loop
90 A stream of 14.000 loops has been collected in order to obtain a good statistics, and Fig. A-7 shows the statistical distributions of both positive and negative jumps. The behavior is still a power-law, and the values of the corresponding critical exponents are very similar in the two cases with α = 1.6 both for negative and positive jumps.
-1.6 p(∆M )=∆M + +
units] [arb. M) ∆
p(
-1.6 p(∆M )=∆M - -
M) [arb. units] ∆
p(
∆M [arb. units]
Fig. A-7: probability distribution of positive and negative jumps
A possible explanation for the negative jumps can be found in eddy currents4 induced in metallic ferromagnets as a consequence of a magnetic induction flux variation caused by a positive jump: each negative jump5 should follow a corresponding positive jump with a time delay smaller than 0.1 µs. As shown in Fig. A-2, we observe different situations where negative jumps take place in any possible position, both very close to large positive jumps but also with no positive jumps within 200 ms: on the basis of the inductive data we can conclude that, in our sample, the role of eddy currents can be relevant, but they are not the direct cause of negative jumps. On the other end, magneto-optical data shown that negative jumps take place only in few regions of the sample, supporting the hypothesis that local defects are responsible for their existence.
91 As conclusion6, we observe that all our measurements, performed with different experimental technique, bring to the same power-law distribution both for positive and negative jumps, and the value of the critical exponent is the same α = 1.6 for all distributions.
The statistical distribution of the positive jump size is a consequence of the disorder inside the sample7. Results obtained are an indication that the same physics is involved in determining both kind of avalanches, with the difference that negative jumps do not take place everywhere inside the sample, but only around particular regions. The same type of distribution obtained indicate that avalanches proceeds through the disordered medium according to the same physical mechanisms of the positive avalanches.
92 A.2 Trilayer Fe/NiO/Fe
The coupling between ferromagnetic layers separated by a thin insulating anti-ferromagnetic spacer can be of great interest, both for fundamental reasons and for the possible applications. Particularly, one would expect a large contribution to the magnetic coupling coming from direct exchange across the spacer.
I have collaborated with Paolo Biagioni, Marco Finazzi, Alberto Brambilla (Dipartimento di Fisica of Politecnico di Milano), Paolo Vavassori (Dipartimento di Fisica of Università di Ferrara) and their collaborators for magneto-optical characterization of the sample; the magnetization has been studied at various angle with respect to external magnetic field (using longitudinal and transverse MOKE configuration, this last at Dipartimento di Fisica of Ferrara). The sample has been prepared at cross configuration (like sketched in Fig. A-8), i.e. depositing a strip of Fe on a substrate of MgO with an external magnetic field H applied, turning the sample of 90° and depositing 7 monolayer of NiO before and others 50 monolayer of Fe after.
In this way is possible analyze the various magnetic contribution simply pointing the laser beam for MOKE measurements in different regions of the sample: • in the central region is present the trilayer; • in two angles it is present the substrate ("subs" in next figures) of Fe; • in the other two angles the measurements interest the overlayer ("over" in next figures) of Fe.
subs H Fe
NiO
try over
Fig. A-8: sample preparation
In the next pages are shown the hysteresis loops measured in our MOKE laboratory with external field parallel (PARALL, in Fig. A-9) and perpendicular (PERP, in Fig. A-10) at the original magnetic field H (see the second figure in Fig. A-8) applied during the deposition of substrate of Fe.
93
Fig. A-9: hysteresis loops (see text for PARALL explanation): subs = substrate, over = overlayer
94
Fig. A-10: hysteresis loops (see text for PERP explanation): subs = substrate, over = overlayer
95 A.3 Magnetite Fe3O4
Considerable attention has been focused in the last years on highly spin polarized compounds in order to improve the performances of spin electronic devices. However the electronic and magnetic structure of bulk materials are strongly modified at the interface, thus there is a relevant interest in understanding the structural, electronic and magnetic properties of its free surface and of its interface with other materials.
A collaboration is born with Alberto Tagliaferri, Matteo Cantoni, Riccardo Bertacco (Dipartimento di Fisica of Politecnico di Milano) and their collaborators about measurements on magnetite Fe3O4, a promising candidate for the use in magnetic tunnel junctions; our group has been involved in MOKE measurements, for determining chemical, structural and magnetic properties of the sample. Our measurements have allowed to characterize the angular dependence of magnetization process in longitudinal (left of Fig. A-11) and polar (right of Fig. A-11) MOKE configuration (see chapter 2 about MOKE geometry) by measuring various hysteresis cycle versus angle.
M [arb. M units] units] [arb. M
-3000 -2000 -1000 0 1000 2000 3000 -15000 -10000 -5000 0 5000 10000 15000 H [Oe] H [Oe]
Fig. A-11: (left) hysteresis loop in longitudinal configuration: easy axis (right) hysteresis loop in polar configuration: hard axis
96 A.4 Electrodeposited CoPt "in-situ"
We have had a scientific collaboration for more than 1 year with Massimilano Bestetti, Silvia Franz, Balzarini Gaia, prof. Pietro Luigi Cavallotti (Dipartimento di Chimica of Politecnico di Milano) and their collaborators regarding characterization of electrodeposited thin magnetic films for data magnetic registration, especially Co based. Cobalt has a hexagonal structure, that generate the higher crystal anisotropy in transition metals along the axis of the hexagon: this is the easy magnetization axis. The magneto-optical technique is particularly adapt for the characterization of these samples, because the Kerr signal on Co is very high and easy to detect, the samples utilized have a very good surface due to the use of this type of samples, and the measure doesn't need to cut the sample (like in VSM measures), operation that can keep to magneto-elastic effect on the sample and distorted measures.
The first step of the activity has regarded the study of a relationship between magnetic properties and microstructure of electrodeposited CoZnP and CoPtZn(P) layers; this work has been presented at Materials Week 2002 Congress in Munich (Germany), reporting the dependence of crystal structure, composition and morphology of the deposits on electrolyte composition and operating conditions: in appendix the relative proceedings.
The second step of this collaboration was tried to realize an electrochemical cell, which enables "in situ" MOKE measurements (i.e. the magnetic measurements are performed during the growth of the deposit): this investigation is desirable to achieve detailed correlations between structure and magnetic properties of electrodeposited films. The MOKE apparatus is very important in studying the properties during crystal growth, because it has the advantage, respect to other techniques like SEMPA and photoemission, to be able of measure the magnetization sample in presence of a magnetic field. Because MOKE measurements imply that a variable magnetic field is applied to the sample, this could affect in principle the properties of the electrodeposits: this means that in order to avoid the effect of the magnetic field on the deposit growth, it is necessary to interrupt periodically the electrodeposition and perform the magnetic measurements by maintaining the deposit in a "stand-by" condition. This "stand-by" condition is represented by the absence of circulating currents. A series of experiments was performed by recording MOKE hysteresis curves, in polar geometrical configuration, during electrodeposition of CoPt layers in a home built cell, shown in Fig. A-12, fitted between the poles of the electromagnet.
97
Fig. A-12:(left) scheme of electrochemical cell: [1] cathode (copper 99.99% sheet); [2] counterelectrode (Pt wire); [3] cell (ring of teflon); [4]quartz optical window; [5] mirror (electropolished Al) (right) photograph of cell and electromagnet: the upper tip is the electrolyte inlet
During electrodeposition of CoPt, the electrolytic solution was continuously pumped by means of a peristaltic pump from a reservoir into the cell (having a volume of about 0.5×π×(1.5)2 = 3.5 cm3. The counterelectrode was a U-shaped platinum wire, and two mirror holders were attached with an adhesive tape on the rear side of the optical window. The mirror, made of aluminum electropolished in order to have high reflectivity, was mounted on the holders with an inclination of about 45°. Several preliminary MOKE measurements, under different conditions of electrodeposition, have been performed.
98 A.5 Bibliography
1 L. V. Kirensky, W. F. Ivlev - Dokl. Akad. Nauk SSSR (1951) 76, 389 2 A. Zentko, V. Hajko - Czech. J. Phys. B (1968) 18, 8, 1026 V. Hajko, A. Zentko, S. Filka - Czech. J. Phys. B (1969) 19, 4, 547 3 E. Puppin, M. Zani, D. Vallaro, A. Venturi - Rev. Sci. Inst. (2001) 72, 4, 2058 4 J. Kranz, A. Schauer - Ann. Phys. (1959) 7, 4, 84 5 A. Zentkova, A. Zentko, V. Hajko - Czech. J. Phys. B (1969) 19, 5, 650 6 M. Zani, E. Puppin - Journ. of Appl. Phys. (2003) 94, 9 ,5901 7 G. Bertotti - Hysteresis in magnetism, ed. Academic Press (1998)
99 B. Appendix: publications
A list and relative my publications are inserted, about the research work that has been involved me in these three years: • A double coil apparatus for Barkhausen noise measurements Ezio Puppin, Maurizio Zani, Davide Vallaro, Alberto Venturi Review of Scientific Instrument (2001) 72, 4, 2058 • Electrodeposition of CoZnP and CoPtZn(P) magnetic alloys Silvia Franz, Massimiliano Bestetti, Antonello Vicenzo, Vincenzo Basso, Maurizio Zani, Pietro Cavallotti Proceedings of Materials Week (2002 - Munich, Germany) • Barkhausen jumps and metastability Luca Callegaro, Ezio Puppin, Maurizio Zani Journal of Physics D: Applied Physics (2003) 36, 17, 2036
• Negative Barkhausen jumps in amorphous ribbons of Fe63B14Si8Ni15 Maurizio Zani, Ezio Puppin Journal of Applied Physics (2003) 94, 9, 5901 • Temperature dependent criticality of Barkhausen noise in thin Fe films Maurizio Zani, Ezio Puppin Proceedings of VII International Conference on Magnetism (2003 - Roma, Italy) & Journal of Magnetism and Magnetic Materials - (2004) in press • Magnetic hysteresis and Barkhausen noise in thin Fe films at 10 K Ezio Puppin, Maurizio Zani Journal of Physics: Condensed Matter (2004) 16, 8, 1183
100 REVIEW OF SCIENTIFIC INSTRUMENTS VOLUME 72, NUMBER 4 APRIL 2001
A double coil apparatus for Barkhausen noise measurements E. Puppin,a) M. Zani, D. Vallaro, and A. Venturi Istituto Nazionale per la Fisica della Materia, Dipartimento di Fisica, Politecnico di Milano, P.za L. da Vinci 32-20133 Milano, Italy ͑Received 4 October 2000; accepted for publication 11 January 2001͒ A pickup coil wound around the sample is the standard method for Barkhausen noise ͑BN͒ measurements. Here we describe an apparatus where two coils are used instead of one. The relative position of the coils can be changed and this allows the experimental investigation of spatial correlation effects in BN. The signals induced in the coils are very similar when the coils are placed nearby whereas a progressive decrease of their correlation is observed by moving the coils apart. We tested our system on a ribbon of amorphous Fe63B14Si8Ni15 100 mm long. For a distance between the coils of 40 mm the signal correlation is nearly vanished. © 2001 American Institute of Physics. ͓DOI: 10.1063/1.1353193͔
I. INTRODUCTION tween them. When the coils are very close to each other the induced signals should be very similar. By moving the coils Barkhausen noise ͑BN͒ is a well-known phenomenon apart a progressive differential of the two signals is expected. which occurs during magnetization reversal in ferro- The degree of correlation between the signals and its depen- 1 magnets. For an extensive review on BN see Ref. 2. The dence on the distance between the coils will give valuable details of this phenomenon are extremely complex and at- information on the spatial distribution of the magnetization tracted the interest of statistical physicists since BN belongs process. to a larger class of relevant physical systems whose common feature is a strongly nonlinear and dissipative behavior in II. EXPERIMENTAL APPARATUS presence of a structural disorder. In Fig. 1 the head of the apparatus is schematically The original technique for the observation of BN con- drawn. The various components are enclosed in a metallic sists in placing an inductive coil around the sample during box ͑F͒ for noise reduction. The sample ͑E͒ used for this the magnetization process.1 The sudden magnetization rever- preliminary work is an amorphous ribbon of Fe63B14S8Ni15 sals taking place inside the material can be detected by mea- having a length of 100 mm, a width of 3 mm, and a thickness suring the associated voltage pulses inductively generated in of 20 m. This sample has been glued on a stiff plastic the pickup coil. Until now this technique represented the support and rigidly suspended within a magnetizing coil ͑B͒. only way to obtain experimental information on BN and all Coil B is made of 1000 turns of copper wire capable to carry the literature on this subject is based on it. Recently, a mag- a current up to 10 A direct current, corresponding to a mag- netooptical apparatus for the observation of BN has been ͑ 3 netic field of approximately 1000 Oe. Two pickup coils C described. and D͒ made each of 1000 turns of shielded copper wire The conventional pickup coil method poses a series of ͑diameter 0.1 mm͒ are wound around the sample. Each coil problems in the interpretation of amplitude and shape of the has a thickness of 1 mm and a diameter of 10 mm and pulses induced in the probe coil. This topic has been exten- is fixed on a plastic support. In one case ͑coil C͒ this support sively discussed in Ref. 4. A relevant case is the one where can be moved up and down from the outside of the metallic the sample has an elongated shape and the pickup coil has a box with a screw manipulator ͑A͒. Coil D is fixed to a shaft width much shorter compared to the sample size. This setup with a screw and its position can be changed by shifting its is typical of many experimental investigations on BN. In the support along the shaft. A series of BNC connectors located ideal case a Barkhausen jump corresponding to magnetiza- on the cover of the metallic box allows for external cabling. ⌬ tion variation M would give rise to a pulse whose shape The acquisition electronics is shown in Fig. 2. The only depends on the electrical characteristics of the acquisi- whole chain is under the control of a PC. The magnetic field ⌬ tion chain and whose area is proportional to M. In practice is generated by injecting a current in coil B of Fig. 1. The it is more realistic to expect that shape and intensity will be current ramp comes from a wave function generator ͑HP strongly affected by the relative position of the coil with 33120A͒ followed by a buffer amplifier. Figure 3 shows the respect to the sample region where the magnetization rever- electronic scheme of the buffer. A power operational ampli- sal is taking place. On a qualitative ground it is safe to assess fier ͑L272͒ is used as an inverting amplifier with unary gain that the peak will be smaller for a larger distance between the ͑current buffer͒. The 200 k⍀ resistor is used for offset com- reversal region and the coil. A simple way of investigating pensation. The input and output capacitors are used as filters this effect consists in using two coils instead of one by re- for suppressing the steps in the linear ramp generated by the peating the measurement of BN at increasing distance be- 12 bits digital-to-analog converter ͑DAC͒ of the wave shape generators. a͒Electronic mail: [email protected] The signals from the two pickup coils are fed into the
0034-6748/2001/72(4)/2058/4/$18.002058 © 2001 American Institute of Physics Rev. Sci. Instrum., Vol. 72, No. 4, April 2001 Double coil apparatus 2059
FIG. 1. Head of the apparatus. The sample ͑E͒ is placed inside the magne- tizing coil ͑B͒. The pickup coils ͑C and D͒ can be moved along the sample length. Coil C can be moved from the outside with an external manipulator ͑A͒. The whole apparatus is closed in a thick metallic box ͑F͒ for noise shielding. input stages of two lock-in amplifiers ͑EG&G 7260͒.Weuse the input stage of the lock-in for convenience but no modulation-demodulation techniques are present in our ap- paratus. This amplifier has been used with a gain of 60 dB which corresponds to a cutoff frequency of the amplification stage around 60 kHz. The output of the amplifier is further filtered with a passive low-pass filter ͑cutoff frequency 18 kHz͒. This filter improves the signal to noise ratio by sup- pressing high frequency components from the signal. Fur- FIG. 2. Schematics of the acquisition electronics. thermore, this filter removes the oscillations observed in the output signal and related to the electrical characteristics of the coil-amplifier system. The final stage of the acquisition generates magnetic pulses even without an external field. chain consists in sampling the output signal with the analog This interesting point will be addressed in future investiga- input channels of a PC board ͑National Instruments ATI- tions. In the present work two different types of measure- MIO-16E͒. The electronic chain has been set in order to ments have been performed. In one case the pickup coils exploit the full dynamics of the 12 bits A/D converters of the have been positioned as close as possible each other and at multipurpose board. The electronic noise level in this con- the center of the sample. The magnetic field has been set in figuration is in the order of a single bit. Translated in voltage order to perform a double linear ramp in 20 s. In this way, units, with a full dynamics of Ϯ5 V the noise level has a root with a sampling frequency of 50 kHz, the number of points mean square of 3 mV being the average height of the peaks from a single coil is equal to 106. With this time resolution around 1 V. we observe many thousands of peaks for each scan. In order to show the results obtained with two coils let us consider the data shown in Figs. 4 and 5. The few peaks shown in these III. MEASUREMENTS figures are representative of the most significant features of The sample used in the present investigation is a ribbon the whole population. The crisp character of the detected of amorphous Fe63B14Si8Ni15 formed in a rapid quenching pulses is due to the low conductivity of our samples. Eddy process. Materials prepared with this technique contain many currents would produce a considerable broadening is con- nonequilibrium defects that relax at room temperature which ducting bulk ferromagnets. In the case of Fig. 4 both coils 2060 Rev. Sci. Instrum., Vol. 72, No. 4, April 2001 Puppin et al.
FIG. 5. Signals when the coils are placed at a distance of 40 mm. this case a completely different behavior is observed. With the exception of one peak all the other structures do not have FIG. 3. Electronic scheme of the low noise current buffer used for generat- any correspondence in the two signals. ing the magnetic field. IV. DISCUSSION A detailed discussion of the effects shown in Figs. 4 and have been positioned at the center of the sample, as close as 5 goes well beyond the purpose of this article focused on a possible to each other. In this experimental situation the most description of a double coil apparatus for the measurement of visible feature is the strong correlation existing between the Barkhausen noise. It is, however, clear, from the preliminary two signals. In fact, a one-to-one correspondence between data shown earlier, that a considerable amount of informa- the peaks is clearly observed. This result is expected since in tion can be extracted from a comparison between the signals this configuration the two coils are probing the same sample induced in the coils by tuning their relative distance. The region. The relative intensity of the peaks, however, changes development of a model for interpreting these data will be in the two signals. A different case is shown in Fig. 5. Here the subject of future investigations. Here we point out a few the coils have been displaced at a distance of 40 mm by aspects. moving each coil at 20 mm from the center of the sample. In The peaks of Fig. 4 are generated by magnetization re- versals taking place in the sample region close to the coils. The variation of the relative intensity of the various peaks indicates that already on this scale length the amplitude of the peaks is strongly affected by the relative position of the reversing region with respect to the coils. On the other hand, the signals shown in Fig. 5 indicates that most of the peaks detected by one coil cannot be observed in a second coil placed at a distance of 40 mm. These two considerations allow to state that a magnetization reversal taking place in- side the sample induces a peak only in a coil placed at a short distance. A more detailed analysis of our data, not reported here for space reasons, indicates that the intensity of the signal decreases with a characteristic length of 10 mm in our sample. Another consideration is that also when the coils are placed at a distance of 40 mm it is possible to observe the same peak in both coils. Since the attenuation length is in the order of a few millimeters, as discussed earlier, the presence of a simultaneous peak indicates that the magnetization re- versal is taking place in an extended region of the sample and that this region is sufficiently close to both coils. In order FIG. 4. Signals when the coils are positioned nearby. to develop a quantitative model for the interpretation of our Rev. Sci. Instrum., Vol. 72, No. 4, April 2001 Double coil apparatus 2061 data it would be therefore necessary to consider also the ACKNOWLEDGMENT spatial extension of the magnetization reversal. A proper fit- ting of our data therefore will give information on the spatial We are indebted with Luca Callegaro at the Italian Insti- ͑ ͒ properties of the magnetization process. tute of Metrology IEN for valuable discussions. As a final remark we note that the amount of experimen- tal information can be further increased by using a larger number of coils. In order to avoid redundancies the number 1 H. Barkhausen, Phys. Z. 20, 401 ͑1919͒. N of coils should not exceed the ratio between the sample 2 S. Zapperi, P. Cizeau, G. Durin, and H. E. Stanley, Phys. Rev. B 58, 6353 length L and the characteristic attenuation length .Inour ͑1998͒; G. Bertotti, Hysteresis in Magnetism ͑Academic, New York, case, being Lϭ100 mm and ϭ10 mm, Nϭ10. The most 1998͒. 3 serious problem in using a large number of coils is to balance E. Puppin, P. Vavassori, and L. Callegaro, Rev. Sci. Instrum. 71,1752 ͑2000͒; E. Puppin, Phys. Rev. Lett. 84, 5415 ͑2000͒. the amplification stages in order to match their transfer 4 J. McClure, Jr. and K. Schro¨der, CRC Crit. Rev. Solid State Sci. 6,45 functions. ͑1976͒. 1
Electrodeposition of CoZnP and CoPtZn(P) magnetic alloys
S. Franz¨, M. Bestetti¨, A.Vicenzo ¨, V.Bassoª , M.Zani§ & P.L. Cavallotti¨ ¨ Politecnico di Milano, Dipartimento di Chimica, Materiali e Ingegneria Chimica “G.Natta” 20131 Milano (Italy) ªIstituto Nazionale Elettrotecnico “Galileo Ferraris”, 10135 Torino (Italy) §Politecnico di Milano, Dipartimento di Fisica, 20133 Milano (Italy)
Abstract In this paper the relationship between magnetic properties and microstructure of electrodeposited CoZnP and CoPtZn(P) layers is presented. The dependence of crystal structure, composition and morphology of the deposits on electrolyte composition and operating conditions was also considered. CoZnP layers electrodeposited from alkaline chloride solutions were characterised by [00.1] preferred orientation (P.O.) and perpendicular magnetic anisotropy, with perpendicular coercivities (about 80 kA m-1) higher than parallel ones (about 20 kA m-1). Cobalt content was in the range 84-95 at%, zinc 3-9 at% and phosphorus 2-7 at%. Further improvement of magnetic properties was obtained by electrodepositing CoPtZn(P) alloys from alkaline sulphamate electrolytes. CoPtZn(P) layers with [00.1] P.O., perpendicular coercivities of about 140 kA m-1 and parallel coercivities of about 40 kA m-1 were obtained. Cobalt content was in the range 78-90 at%, phosphorus 6-15 at% and zinc 0.5-8 at%. By modifying operating conditions, electrodeposited CoZnP and CoPtZn(P) alloys with (10.0) and (10.1) XRD reflections with parallel coercivity higher than perpendicular one were also obtained.
Introduction Magnetic materials are used to manufacture a wide range of miniaturised devices (MEMS). In this respect electrodeposited cobalt alloys are important materials for the possibility of tailoring the magnetic properties and the microstructure at high thickness of the deposited films by controlling the deposition conditions. Matsuda showed that CoZnP films, containing Zn 1.7 wt% and P 4.6 wt%, obtained by electroless deposition, and with thickness below 1 mm, have coercivity of 90 kA m-1; by comparison CoP films, containing P 3.9 wt%, have coercivity decreasing from 60 kA m-1 at about 50 nm to 20 kA m-1 at about 0.8 mm. (1) Moreover, a CoPt alloy with Co 50at% can reach coercivities as high as 400 kA m-1 (2), and a CoPt sputtered thin film alloy with Co 80at% about 160 kA m-1 (3-4). Coercivity of about 80 kA m-1 for electrodeposited CoPt layers with Co 80at% were measured, while CoPt(P) electrodeposits of similar Co content can have coercivities higher than 400 kA m-1. The thickness of these samples was lower than 1 mm, and the coercivity decreased by increasing the layer thickness. CoPt(P) samples loose about 25% coercivity when thickness increases from 0.2 to 1.1 mm. (5) In this paper we discuss the magnetic properties of electrodeposited CoZnP and CoPtZn(P) alloys, with particular concern to crystallographic structure and its influence on magnetic properties. The dependence of crystal structure, composition and morphology of the deposits on electrolyte composition and operating conditions was also considered.
Experimental Electrolytes for deposition of CoZnP and CoPtZn(P) layers were prepared from chemicals of analytical grade and distilled water (see Table 1 for composition). Two other electrolytes were also 2
tested by addition of 0.05-0.1M Na4BO7 to the electrolyte of Table 1 and by substituting 0.05M KH2PO3 for 0.1M NaH2PO2. Table 1: Composition of electrolytes for CoZnP and CoPtZn(P) electrodeposition Reagent CoZnP CoPtZn(P) CoCl2 cobalt chloride 0.1 M ZnCl2 zinc chloride 0.005 M C5H9O2N proline 0.3 M NaH2PO2 sodium hypophosphite 0.1 M 0.1 M Co(NH2SO3)2 cobalt sulphamate 0.1 M Zn(NH2SO3)2 zinc sulphamate 0.005 M Pt(NH3)2(NO2)2 platinum p-salt 0.01 M (NH4)2C6H6O7 ammonium citrate 0.1 M NH2CH2COOH glycine 0.1 M
The pH was varied in the range 3÷9 in CoZnP plating, while it was about 8 in CoPtZn(P) plating; pH was adjusted by addition of either sodium hydroxide, ammonia or sulphamic acid. The temperature was in the range 50 ÷ 80°C and current density in the range 10 ÷ 60 mA cm-2. The substrates used for deposition were either brass, brass coated with electroless NiP or silicon wafers coated with gold or copper. Current was supplied to the cell by means of a galvanostat (AMEL 550). CoPtZn(P) films were deposited in a two compartments cell with glass-frit diaphragm. Phase structure and texture of electrodeposits were determined by X-ray (Philips PW 1830); a and c in the following tables refer to lattice parameters of the hcp structure. The composition was measured by EDS analysis; thickness and morphology were assessed by laser profilometry (UBM Microfocus) and SEM (Cambridge Stereoscan 360), respectively. Coercivity (Hc), saturation magnetisation (Ms) and remanence (Mr) were determined from room temperature hysteresis cycles measured by VSM (Vibrating Sample Magnetometer) in magnetic field both perpendicular and parallel to the substrate and MOKE (Magneto-Optical Kerr Effect) technique.
Results and discussion CoZnP – Composition and structure. CoZnP and CoPtZn(P) layers were electrodeposited with hcp structure. CoZnP alloys with [10.0] P.O., i.e. with close packed planes perpendicular to the surface, were obtained from electrolytes containing 0.2 M proline. Deposits with close-packed planes parallel to the surface, i.e. [00.2] P.O., were observed with proline concentration of 0.3 M. A further increase of proline concentration to 0.7 M resulted in [10.0] P.O.. By varying the pH (initial) from 3 to 9, at proline concentration 0.3 M, no significant changes in the crystallographic structure were observed, while composition of the CoZnP alloy changes and, correspondingly, faradaic efficiency (F) has apparently a maximum at about 7.5. Table 2 reports the results of electrodeposition tests carried out at 10 mA cm-2, 60°C on brass.
Table 2: Properties of CoZnP layers obtained at different pH.
pH Co Zn P FCo FCo a c (%at) (%at) (%at) (Å) (Å) 3 74.59 1.78 26.6 - - - - 5 71.66 11.86 16.3 25.8 4.3 - - 7 84.4 9.22 6.38 76.2 8.3 2.5299 4.0840 8 94.51 3.14 2.34 80.1 2.7 2.5109 4.0792 9 96.7 2.2 1.1 40.0 0.9 2.5448 4.0711 3
Cobalt content increases with pH and phosphorus is strongly codeposited at low pH according to the reaction
- + - H 2PO2 + 2H + e ® P + 2H 2O E° = -0.382 V at 60°C
At low pH the hydrogen gas evolution is more intense and the overall faradaic efficiency is very low. The morphology of the deposits is characterised by large grains at acidic as well as at alkaline pH, and by finer grains at pH 7-8 (Fig. 1), where the deposits were bright. Deposits obtained at high pH have hcp crystal structure and the XRD (00.2) reflection is more intense at pH around 7-8, while at pH 9 the more intense peak is (11.0). The higher content of zinc in the deposit obtained at pH 5 could be explained by the amphoteric behaviour of the metal. The increase of hypophosphite in the electrolyte favoured zinc deposition (Table 3) and increases (00.2) peak intensity (Fig. 2).
Table 3: Properties of CoZnP layers obtained at different sodium hypophosphite concentration.
NaH2PO2 Co Zn P a c (M) (%at) (%at) (%at) (Å) (Å) 0 96.73 1.89 - 2.5097 4.0961 0.05 89.56 5.59 4.01 2.5458 4.0813 0.1 89.04 6.35 4.61 2.5213 4.0841 0.15 88.82 6.21 4.97 2.5219 4.0784
Fig. 1: Surface morphologies of CoZnP alloys Fig. 2: XRD patterns of CoZnP layers at electrodeposited at pH 3, 5, 7 and 9 (10 mA different Na hypophosphite concentrations cm-2, 60°C, 60 min, brass). (10 mA cm-2, 60°C, pH 7, 45 min).
The effect of temperature and current density on CoZnP layer properties was also investigated. At 10 mA cm-2, electrodepositing on brass coated with NiP, for 60 minutes, the temperature increase from 30 to 50°C caused a change in the hcp crystallographic P.O. from [10.0] to [00.2]. A further increase of temperature resulted in a stronger [00.2] reflection (Fig. 3). At 10 and 20 mA cm-2 [00.2] P.O. was observed, while at higher current densities the crystallographic structure showed [10.0] P.O.(Fig. 4). 4
Fig. 3: XRD patterns of CoZnP layers at Fig. 4: XRD patterns of CoZnP layers at -2 different temperatures (10 mA cm , pH 7, 60 different current densities (60°C, pH 7, t 10 ¸ min). 60 minute).
CoPtZn(P) – Composition and structure. Zinc has an inhibitory effect on cobalt deposition; Fig. 5 shows a decrease of zinc content in the deposit with increasing film thickness, while platinum content decreases. As shown in Fig. 6, deposition starts with an incubation period, ti, of about 12 minutes.
Fig. 5 - Composition of CoPtZn(P) alloys as a Fig. 6: Thickness of CoPtZn(P) alloys as a function of thickness (20 mA cm-2, 80°C, function of electrodeposition time (20 mA wafer Si/Au). cm-2, 80°C, pH 8, wafer Si/Au).
Potentiodynamic curves (not reported) recorded for Co, CoPt and CoPtZn(P) systems show that zinc in solution decreases the current density at the same potential, while platinum addition increases the deposition rate. During the initial stages of the electrodeposition process, zinc inhibition favours the growth of hcp structure with basal planes parallel to the surface, i.e. with [00.1] P.O.. As the electrodeposition proceeds further, the inhibitory effect of zinc decreases and CoPtZn(P) films grows with [11.2] P.O.. Correspondingly, surface morphology is homogeneous at low thickness, and globular at higher thickness (Fig. 7). The calculated crystallite dimension, by means of the Sherrer equation, for CoPtZn(P) coatings is in the range 20-30nm. 5
Fig. 7: Surface morphology of CoPtZn(P) layers at different thickness (20 mA cm-2, 80°C, pH 8, 30¸120 min, Si/Au wafer).
The effect of temperature and c.d. on properties of CoPtZn(P) electrodeposited alloys was also considered. At 50°C, increasing c.d. from 10 to 60 mA cm-2, a change in the P.O. was observed: at 10 mA cm-2 CoPtZn(P) deposits have weak [11.0] P.O.; at 20 mA cm-2 [10.0] P.O.; at higher c.d. both (00.2) and (11.2) reflections are observed. On the other hand, at 80°C, XRD patterns show that CoPtZn(P) films have [00.2] P.O. at 10 mA cm-2 and [11.0] P.O. at higher c.d. (up to 60 mA cm-2). By increasing temperature from 50 to 80°C, cobalt content slightly increases and zinc and platinum contents keep almost constant, while alloy composition is nearly unaffected by c.d.. Faradaic efficiency of the elements is not very affected by temperature and has a maximum at about 40 mA cm-2, with the only exception of zinc, as reported in Table 4 and Table 5.
Table 4: Composition and faradaic efficiency of CoZnPt(P) layers electrodeposited at different current densities at 50°C (60 min, Si/Au wafer).
Current density Co Pt Zn FCo FPt FZn Ftot (mA cm-2) (%at) (%at) (%at) 10 76.6 13.51 9.89 14.2 0.8 0.4 15.4 20 77.79 13.79 8.42 18.0 1.3 0.9 20.1 40 81.13 13.13 5.74 21.7 1.8 1.9 25.4 60 78.94 11.81 9.25 17.5 1.3 1.4 20.2
Table 5: Composition and faradaic efficiency of CoZnPt(P) layers electrodeposited at different current densities at 80°C (60 min, Si/Au wafer).
Current density Co Pt Zn FCo FPt FZn Ftot (mA cm-2) (%at) (%at) (%at) 10 87.48 9.54 2.78 10.3 0.9 1.3 12.5 20 84.04 11.74 4.22 18.9 1.7 2.0 22.6 40 79.69 13.43 6.88 23.0 1.9 1.6 26.5 60 81.59 11.8 6.61 19.7 1.5 2.3 23.2
Addition of Na2B4O7 to the electrolyte was carried out in order to keep the pH constant during 2- - electrodeposition. Although the couple (B4O7) /(B4O7) buffers at pH 8.8-8.9 at 60-80°C, glycine and citrate additions decrease the pH of the cathodic compartment to 8.4. At 60°C CoPtZn(P) deposits have a strong [11.0] P.O., while at 80°C electrodeposited CoPtZn(P) alloys are characterised by XRD reflections with components perpendicular to the substrate, such as (00.2) and (11.2). 6
The substitution of KH2PO3 for NaH2PO2 was considered: at 60°C it favours zinc deposition compared to NaH2PO2, cobalt content in the alloy decreases whilst platinum content is almost constant (Table 6). By increasing temperature from 60 to 80°C the composition of the electrodeposited layers is very similar to that of samples obtained from sodium hypophosphite- containing electrolytes. Moreover, at 60°C most samples have [11.0]+[10.0] crystallographic orientation, with the only exception of samples obtained at 15 mA cm-2. At 80°C electrodeposited films were characterised by [00.1] P.O. (15 mA cm-2) or at least by (00.2)+(11.1) reflections (20 mA cm-2).
Table 6: Properties of CoPtZn(P) layers obtained at different current density from electrolyte containing KH2PO3 instead of NaH2PO2 (60°C, 60 min, brass).
Current density Co Pt Zn a c (mA cm-2) (at%) (at%) ( at%) (Å) (Å) 10 74.36 14.88 10.76 2.5735 4.1977 15 74.96 13.77 11.27 2.5750 4.1878 20 72.88 14.05 13.07 2.5610 4.1977 40 71.42 14.76 13.82 - -
CoZnP e CoPtZn(P) – Magnetic properties. Three main magnetic behaviours were observed, depending on crystallographic structure and plating operating conditions. CoZnP and CoPtZn(P) samples with pronounced [00.1] P.O., i.e. with magnetic easy direction perpendicular to the substrate, have strong perpendicular magnetic anisotropy, being perpendicular coercivity (Hc,^) higher than the parallel one (Hc, //) (Fig. 8-a).
Table 7: Magnetic properties of CoZnP and CoPtZn(P) alloys.
Alloy Co Pt Zn P Hc ^ Hc // XRD P.O. Thickness (at%) (at%) (at%) (at%) (kA m-1) (kA m-1) (mm) CoZnP 91.25 - 4.6 4.15 70 36.7 [00.1] 10 CoPtZn(P) 89.4 7.3 3.3 - 142 43 [00.1] 2.8 CoZnP ------CoPtZn(P) 84.04 11.74 4.22 - 169 164 [11.0]+[11.1] 5.6 CoZnP 93.86 - 3.31 2.83 6 9 [10.0] 15 CoPtZn(P) 87.48 9.54 2.78 - 101 148 [11.0] 2.1
Samples with (00.2)+(11.2) XRD reflections can be characterised by either low perpendicular anisotropy or magnetic isotropy, being perpendicular and parallel anisotropy values very similar. Magnetic isotropy was shown also by deposits characterised by (00.1)+(11.0) or (00.1)+(10.0) or (11.0) + (11.2) (Fig. 8-b) peaks (the latter probably corresponding to a [00.1]+[11.0] structure where (00.2) planes were not differentiated by XRD analysis). Finally, films with pronounced [11.0] P.O., i.e. with magnetically easy direction parallel to the substrate, have parallel magnetic anisotropy (Fig. 8-c). 7
a b c Fig. 8: Hysteresis loops of ECD CoPtZn(P) layers (a: Co 89.4 at%, Pt 7.3 at%, Zn 3.3 at%, 10 mA cm-2, 80°C, 2.8 mm Si/Au; b: Co 87.48 at%, Pt 9.54 at%, Zn 2.78 at%, 10 mA cm-2, 80 °C, 2.1 mm, Si/Au; c: Co 84.04 at%, Pt 11.74 at%, Zn 4.2 at%, 20 mA cm-2, 80 °C, 5.6 mm, Si/Au).
Magneto-optical (MOKE) measurements, obtained with magnetic field parallel to the deposit surface, gave lower coercivity values in comparison with those obtained by VSM technique. This is most probably due to the fact that MOKE technique gives information about surface magnetic properties, while VSM measurements depend on the whole sample volume. MOKE measurements in parallel magnetic field were in agreement with the suggested relation between magnetic properties and crystallographic structure if thin film XRD analysis is considered instead of bulk XRD one. By comparing hysteresis cycles measured with VSM and MOKE magnetometer it was observed that a lower degree of orientation in a direction parallel to the substrate results in a lower coercivity value in the parallel direction. Experimental results showed that coercivity of electrodeposited CoPtZn(P) alloys lowers to a constant value for film thickness higher than about 1 mm, as observed for CoPt films [6]. In CoPtZn(P) electrodeposits coercivity is attributed to the presence of compounds included or precipitated at grain boundaries as they favour intergranular separation. Zinc is likely to be present in the deposit as oxidised compounds (oxides or hydroxides) precipitated at grain boundaries, together with phosphorus compounds (phosphates or phosphides). Hc,// value decreases by increasing current density, whilst increases with temperature (Fig. 9). No similar behaviour was observed for Hc,^.The squareness S of the hysteresis cycles, i.e. the ratio between saturation magnetisation and remanence, is generally lower than 0.5. This observation is in agreement with Quinn et al. (7) and Chikazumi (8), who calculated squareness of about 0.866 and 0.5 respectively for disordered fcc and hcp structures, and lower than 0.5 for a partly ordered hcp structure. Moreover, a relation between parallel squareness S// and intensity of [00.1] peak intensity was observed: the more pronounced is the XRD reflection, the lower S// is (Fig. 10).
Fig. 9: CoPtZn(P) layers coercivity as a Fig. 10: Parallel squareness of CoPtZn(P) function of current density at 50 and 80°C. alloys as a function of (00.1) peak intensity. 8
Measured hysteresis cycles were corrected for the demagnetising field acting in the perpendicular direction. This is done by calculating the effective field Heff with the equation Heff = Ha – (J/mz)×Nd, where Ha is the applied magnetic field, J is the saturation magnetisation, mz is the magnetic 9 permeability of vacuum and Nd=1is the demagnetisation factor (). After the correction the squareness of the cycle increases from 0.09 to 0.84, the saturation field decreases from about 1000 kA m-1 to 290 kA m-1 and the remanent magnetisation increases from 0.10T to 0.94T. It is to be underlined that corrected perpendicular squareness may be higher that the real one since in the perpendicular direction sample did not achieve magnetic saturation. Samples electrodeposited from electrolyte containing Na2B4O7 have coercivity values similar to -1 those of the electrodeposits obtained without Na2B4O7 (130-160 kA m ).
Conclusions The increasing interest for miniaturised devices, such as MEMS, and recording media with larger storage capacity stimulates the study of new magnetic materials. CoZnP and CoPtZn(P) magnetic alloys were produced by electrodeposition. It is possible to control their magnetic properties by varying their composition, morphology and crystallographic structure. By modifying current density and temperature it is possible either to favour the growth of an hcp structure with most densely packed planes parallel to the substrate (with P.O. [00.1]) or perpendicular to the substrate (with [11.0] or [10.0] P.O.). Both CoZnP and CoPtZn(P) samples showed magneto-crystalline anisotropy. In fact, samples with strong [00.1] P.O. are characterised by perpendicular magnetic anisotropy. These alloys are suitable for manufacturing closed magnetic circuits perpendicular to the substrate. Films with most densely packed planes perpendicular or tilted with respect to the substrate show parallel magnetic anisotropy and are suitable materials for manufacturing closed magnetic circuits parallel to the substrate. Finally, magnetic isotropy was detected in CoZnP and CoPtZn(P) alloys with weak [00.1] P.O. or with XRD reflections of the form [10.2]. Electrochemically deposited CoZnP alloys with 10 mm thickness showed coercivity of about 70 kA m-1, measured in magnetic field perpendicular to the substrate. Great improvement was observed in CoPtZn(P) alloys, which showed coercivities of about 170 kA m-1 at thickness of about 2.8 mm.
List of Figures
Fig. 2 Fig. 1 9
Fig. 3 Fig. 5
Fig. 4
Fig. 6
Fig. 7
a b c
Fig. 8 10
Fig. 9 Fig. 10
List of Tables
Table 1 Reagent CoZnP CoPtZn(P) CoCl2 cobalt chloride 0.1 M ZnCl2 zinc chloride 0.005 M C5H9O2N proline 0.3 M NaH2PO2 sodium hypophosphite 0.1 M 0.1 M Co(NH2SO3)2 cobalt sulphamate 0.1 M Zn(NH2SO3)2 zinc sulphamate 0.005 M Pt(NH3)2(NO2)2 platinum p-salt 0.01 M (NH4)2C6H6O7 ammonium citrate 0.1 M NH2CH2COOH glycine 0.1 M
Table 2 pH Co Zn P FCo FCo a c (%at) (%at) (%at) (Å) (Å) 3 74.59 1.78 26.6 - - - - 5 71.66 11.86 16.3 25.8 4.3 - - 7 84.4 9.22 6.38 76.2 8.3 2.5299 4.0840 8 94.51 3.14 2.34 80.1 2.7 2.5109 4.0792 9 96.7 2.2 1.1 40.0 0.9 2.5448 4.0711
Table 3 NaH2PO2 Co Zn P a c (M) (%at) (%at) (%at) (Å) (Å) 0 96.73 1.89 - 2.5097 4.0961 0.05 89.56 5.59 4.01 2.5458 4.0813 0.1 89.04 6.35 4.61 2.5213 4.0841 0.15 88.82 6.21 4.97 2.5219 4.0784 11
Table 4 Current density Co Pt Zn FCo FPt FZn Ftot (mA cm-2) (%at) (%at) (%at) 10 76.6 13.51 9.89 14.2 0.8 0.4 15.4 20 77.79 13.79 8.42 18.0 1.3 0.9 20.1 40 81.13 13.13 5.74 21.7 1.8 1.9 25.4 60 78.94 11.81 9.25 17.5 1.3 1.4 20.2
Table 5
Current density Co Pt Zn FCo FPt FZn Ftot (mA cm-2) (%at) (%at) (%at) 10 87.48 9.54 2.78 10.3 0.9 1.3 12.5 20 84.04 11.74 4.22 18.9 1.7 2.0 22.6 40 79.69 13.43 6.88 23.0 1.9 1.6 26.5 60 81.59 11.8 6.61 19.7 1.5 2.3 23.2
Table 6 Current density Co Pt Zn a c (mA cm-2) (at%) (at%) ( at%) (Å) (Å) 10 74.36 14.88 10.76 2.5735 4.1977 15 74.96 13.77 11.27 2.5750 4.1878 20 72.88 14.05 13.07 2.5610 4.1977 40 71.42 14.76 13.82 - -
Table 7
Alloy Co Pt Zn P Hc ^ Hc // XRD P.O. Thickness (at%) (at%) (at%) (at%) (kA m-1) (kA m-1) (mm) CoZnP 91.25 - 4.6 4.15 70 36.7 [00.1] 10 CoPtZn(P) 89.4 7.3 3.3 - 142 43 [00.1] 2.8 CoZnP ------CoPtZn(P) 84.04 11.74 4.22 - 169 164 [11.0]+[11.1] 5.6 CoZnP 93.86 - 3.31 2.83 6 9 [10.0] 15 CoPtZn(P) 87.48 9.54 2.78 - 101 148 [11.0] 2.1
References
(1) H.Matsuda, O.Takano: J. Japan. Inst. Metals 52(4) (1988) 414. (2) D.J.Craik: Platinum Metals Rev. 16(4) (1972) 129. (3) J.A.Aboaf, S.H.Herd, E.Klokholm: IEEE Trans. Magn. 19(4) (1983) 1514. (4) M.Yanagisawa, N.Shiota, H.Yamazuchi, Y. Suganuma: Proc.IEEE Trans. Magn., (1983) 1638. (5) P.L.Cavallotti, P. Bucher, N.Lecis, G. Zangari: PV 95-18 189th Meeting ECS, (1996) 169. (6) P.L.Cavallotti, G.Zangari, G.Fontana, P.G.Maisto: Magn. Mat. Proc. Dev., 1993 Int. Conf. edited by L.T Romankiw, D.A.Herman, ECS 1994. (7) H.F.Quinn: Adv. X-Ray Anal. 4 (1961) 151. (8) S.Chikazumi: “Physics of Magnetism”, Wiley (1964). (9) G.Bertotti: ”Hysteresis in Magnetism”, Academic Press (1998). INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS J. Phys. D: Appl. Phys. 36 (2003) 2036–2040 PII: S0022-3727(03)61427-0 Barkhausen jumps and metastability
Luca Callegaro1, Ezio Puppin2 and Maurizio Zani2
1 Istituto Elettrotecnico Nazionale Galileo Ferraris, Strada delle Cacce, 91, I-10135 Torino, Italy 2 INFM—Dipartimento di Fisica, Politecnico di Milano, Pza L da Vinci, 32, I-20133 Milano, Italy
Received 28 March 2003 Published 20 August 2003 Online at stacks.iop.org/JPhysD/36/2036 Abstract The magnetization process drives a ferromagnetic system through a sequence of discrete metastable magnetization states; the abrupt transitions taking place between these states are responsible for the Barkhausen noise. In thin films, the magnetization states of a small region of the sample surface can be identified, and experimental information can be gained on the statistical properties of the magnetization sequence. Here we investigate the stability of the system in a particular state in the presence of an increasing magnetic field, which we call the state life. A simple statistical model, based on the reliability theory, is proposed in order to explain the observed data.
1. Introduction the external driving force of the continuously increasing applied magnetic field; in the following this property of the During magnetization reversal in a magnetic material, the magnetization states will be called the life of metastable states. presence of the so-called Barkhausen noise indicates that the A statistical model, which explains the experimental results, system evolves through a series of sudden jumps connecting will be presented. different metastable states characterized by a particular value of the sample magnetization. The usual experimental 2. Experimental details procedure for detecting Barkhausen jumps ([1], for a review see [2], [3]) consists in sensing the related magnetic flux We performed hysteresis loop measurements by using a variations with an inductor coupled to the material: a sensitive high-sensitivity magneto-optical Kerr hysteresigraph, which method which, apart from particular arrangements [1, 4], does is based on polarization modulation of the light source with not possess spatial resolution. a photoelastic birefringence modulator. The instrument is The number of possible metastable states of a macroscopic extensively described in [6]; the signal to noise ratio has been sample is enormous; each repetition of the magnetization improved to 60 dB with a single field sweep acquisition. The process follows a different path among the energy landscape light source is an intensity-stabilized 5 mW HeNe laser beam, even though limited reproducibility has been observed in some focused on the sample with a zoom optics, which permits us to systems [5]. vary the spot size from a few millimetres down to 15 µm. The A new technique, based on magneto-optical ellipsometry instrument is under complete computer control: the hysteresis with a focused laser beam [6], permits the detection of loops are continuously measured and each one is separately Barkhausen jumps in thin films and particles [7–10]. In thin recorded for off-line processing. films, the spatial resolution obtained is sufficient to isolate The sample investigated [11] is a 90 nm thick Fe film and follow the magnetization of a small portion of the whole epitaxially grown on a MgO(001) single crystal. The sample. substrate has been heat-treated until the observation of a LEED The observation of a small region of the sample allows (1 × 1) pattern, typical of bulk-terminated lattice; Fe has us to identify a limited number of well-defined magnetization been evaporated from an electron-beam source at a rate of states and the corresponding Barkhausen jumps. The process 1nmmin−1 in a vacuum chamber with a base pressure of remains stochastic, but moves through identifiable states. 5 × 10−11 Torr, and then covered with a 3 nm thick protective This circumstance allows us to study the statistical Pt layer. The thickness has been measured with a quartz properties of the magnetization phenomenon by preparing the microbalance, and the sample purity checked with in situ sample in a well-defined magnetization state. In the following, XPS and Auger spectroscopy. The size of the sample is we will focus on the stability properties of such states against ≈4×4mm2; the magnetization is parallel to the film plane, and
0022-3727/03/172036+05$30.00 © 2003 IOP Publishing Ltd Printed in the UK 2036 Barkhausen jumps and metastability the experiments have been conducted by applying the magnetic (i.e. the histogram) p(M) of the Barkhausen jump parameter field along the easy axis of the film. M(k) evaluated on the whole stream of loops; the peaks We measured, at room temperature and on a central of p(M) have been labelled according to the corresponding region of the sample defined with a spot size of ≈100 µm, plateaux shown in figure 2. The presence of reproducible nearly 64 000 hysteresis loops, by applying a triangular-wave plateaux is observed everywhere over the sample surface, magnetic field with frequency 100 mHz and amplitude 100 Oe. although the number and position of the magnetization states Each hysteresis loop has been analysed with a computer are related to the particular region investigated. In fact, the program which identifies Barkhausen events and records their existence of these states can also be found on other samples properties; details of such a procedure can be found in [10]. with a different composition and size [9, 10]. In order to investigate the dynamics of the system 3. Experimental results involving a particular state, it is possible to extract from the entire record of Barkhausen events only those entering or Figure 1 shows the upper branch of a hysteresis loop extracted leaving that state. For instance, it is possible to select only from the recorded stream. It is apparent that magnetization jumps leaving state B by setting the filtering condition that is not a smooth function of the applied field, but evolves M(k) belongs to a window around peak B (shown in figure 3 through magnetization plateaux and Barkhausen jumps. This as a shaded area). Figure 4 plots the experimental probability feature is not related to the particular sample investigated, distributions p(H(k)) (dots) and p(H(k +1)) (open circles) of but has been observed in different continuous or structured the jumps belonging to state B, i.e. to the jumps, respectively, ferromagnetic films, when either the spot size, or the overall entering and exiting state B. The corresponding Gaussian best sample dimension, are sufficiently small [7–9]. fits are also shown, having standard deviations σk = 0.98 Oe In figure 1 are also illustrated the physical quantities and σk+1 = 1.22 Oe, respectively. measured by the analysis program: the jump with index k, taking place at the applied magnetic field H(k), carries the 20 25 30 35 40 − system from a magnetization state M(k 1) to the final state 40 = − − C M(k), and has an amplitude M(k) M(k) M(k 1). 20 After this jump, the state M(k) is maintained until a new jump 0 B with index k + 1 occurs; the state M(k) has the corresponding -20 A width H (k) = H(k +1) − H(k). -40 The statistical distribution of the jump height M, the 40 C Barkhausen ‘amplitude’ probability distribution indicated as 20 p M , shows critical behaviour and has been the subject of 0 much experimental and theoretical work [2]; the technique B -20 A described in section 2 permitted us to study p M on thin films -40 M [arb. units] and magnetic structures. In such experiments and theories, 40 C each jump is treated as a single entity having no correlation 20 with the others. 0 B Figure 2 shows a sequence of three different hysteresis -20 A loops extracted from the main stream. It is apparent that three -40 major plateaux, labelled A, B, C, appear on each loop at similar 20 25 30 35 40 values of the magnetization M, although their width H H [G] changes considerably from one loop to another. The presence of three main magnetization states is confirmed by looking Figure 2. A sequence of three hysteresis loops (upper branch) from at figure 3, which shows the experimental probability density the main stream, with marked (A, B, C) magnetization plateaux.
40 A M(k+1) C
20 jump k+1 B 0 M(k) M(k)
jump k [arb. units] [norm. units] [norm. -20 M M(k-1) p(M) ∆ H(k) -40 H(k) H(k+1)
20 22 24 26 28 30 32 34 36 38 40 -60 -40 -20 0 20 40 60 H [G] M [norm. units]
Figure 1. Upper branch of a hysteresis loop extracted from the Figure 3. Experimental probability distribution p(M); (A, B, C) recorded stream of 64 000. The physical quantities used in identify the modes of the distribution. See text for a description of developing the model are presented (see text for details). the shaded area.
2037 L Callegaro et al cannot be immediately named, but has a certain resemblance state B to distributions encountered in reliability testing, such as the H(k) H(k +1) Weibull distribution [12].
4. A model for the life of the magnetization states
[arb units.] The major issue of this paper is to identify the physics involved
p(H) when a complex system such as a macroscopic ferromagnetic sample leaves a particular metastable state and jumps to another one. More precisely, a simple quantitative model based 26 28 30 32 34 36 38 on this physical insight will be presented. This model allows H [Oe] us to explain the shape of p( H ). Let us suppose that the system has already reached, with a Figure 4. Experimental probability distributions p(H(k)) (•) and Barkhausen jump of index k occurring under the applied field p(H(k +1)) (◦) for the jumps entering and exiting state B of H(k), a metastable magnetization state M(k); we can imagine figure 3. The corresponding Gaussian best fits (——) are also that the system is trapped in a potential well of finite depth. shown. The applied magnetic field h continues to increase, until the stability of the state is destroyed and the system leaves the state with another Barkhausen jump of index k + 1 occurring at the field H(k +1). In order to model this process let us consider a one- dimensional magnetic dipole m. This dipole is coupled to a field, which in turn, is the sum of three different fields, each with a particular physical meaning. The first field, h, is the external field. The second, Hp(k), is the pinning field connected to the fact that a local potential well exists which pins the system in the metastable state. Finally, we introduce an interaction field, hw, which accounts for the interaction between the sampled area and the rest of the sample. In other words, the interaction field models the interaction between the system and the entire magnetic film, which is not observed.
4.1. Construction of the model equation The dipole m senses the local field
hloc = h + hw − Hp (1)
and has an energy E = µ0mhloc. Immediately after a jump k, the dipole is (by definition) in a stable equilibrium position; the corresponding local field
Hloc(k) = H(k)+ Hw(k) − Hp (2) is a negative quantity. After the jump, h increases and so does hloc; the stability of the dipole decreases until hloc goes positive and another jump with index k + 1 at the field h = H(k +1) occurs. By combining (1) and (2), we can rewrite the model as: = − − Figure 5. Probability distribution p( H ) for states A, B, C of hloc (h H(k))+ (hw Hw(k)) + Hloc(k). (3) figure 3; experimental data (◦) and results from the model (——). = − Model parameters Hloc and σw are reported for each continuous Introducing the quantities h h H(k), current field ∗ = − curve. increase from the last jump, and hw hw Hw(k), we can rewrite (3) as Figure 5 shows the normalized experimental probability ∗ h = h + h + H (k). (4) distributions p( H ) corresponding to each peak A, B, C of loc w loc figures 2 and 3. p( H ), the statistical distribution of the Equation (4) summarizes our model of jump occurrence: permanence of the system in a single metastable state, can jump k + 1 occurs when hloc, initially negative, under the ∗ be called the life of the state (either time life or field life, pressure of h and of the fluctuating field hw goes positive field being proportional to time for a triangular excitation). for the first time. Figure 6 shows a graphical representation of Such a peculiar shape, asymmetric and not crossing the origin, the stochastic process described by equation (4).
2038 Barkhausen jumps and metastability
H = 0 Oe σ = 1 Oe (for all curves) loc W
0.4 -2 Oe -3 Oe -4 Oe
] -1 Oe -1 ) [Oe h ∆ ( 0.2 p
0.0 0123456 ∆h [Oe] Figure 6. A pictorial representation of the stochastic process described by equation (4). H = -1 Oe (for all curves) loc 4.2. Analytical solution 10 Oe 0.4 We now want to solve the model, i.e. to determine the 3 ] probability p( h) of the occurrence of a jump k +1 -1 at a particular value of h, under the following simple 2 Oe ) [Oe assumptions: h ∆
( σ = 1 Oe 0.2 W • Immediately after the occurrence of the jump k, Hloc(k) p is a constant of the state k. That is, the observed system is ‘prepared’ always in the same initial metastable state, a well with constant depth Hloc. • h increases monotonically from zero. 0.0 • hw is a statistical variable independent of h. It follows 0123456 ∗ ∗ = ∆h [Oe] that, for the expectation value of hw, hw 0; we call ∗ pw the probability density of hw. Figure 7. Dependence of the model distribution p( h|Hloc,σw) on The solution of model equation (4) is a mathematical the parameters Hloc (upper) and σw (lower). problem strictly connected to the reliability problem, i.e. of determining the life, or time-to-failure, probability of a device, This assumption can be justified in the framework of the when the failure rate or hazard rate function is given. What limit central theorem, since the interaction field hw results from follows is simply a rewriting of the general solution to the a sum of interactions due to the several domains constituting reliability problem [13]. the magnetic thin film. By assuming that the system ‘survived’, i.e. did not jump, For this particular case equation (6) can be computed in up to a particular value of h, the ‘hazard rate’, in our closed form: language the probability of escape for that particular h,is 1 σw = | p( h) = [1 + erf(β)]exp √ α erf(α) − β erf(β) r( h) p(hloc > 0 h), i.e. 2 2 ∞ 1 h = +√ (exp(−α2) − exp(−β2)) − (8) r( h) pw(x) dx. (5) π 2 −(Hloc+ h) where we put, for simplicity: The reliability theory, given the hazard rate, allows us H H + h to calculate the ‘life’ of the system, in our language the α = √ loc ,β= loc√ (9) probability of a jump k + 1 occurrence for each h: 2σw 2σw h and erf(z) is the usual Gauss error function: p( h) = r( h) exp − r(x)dx (6) 2 z 0 erf(z) = √ exp(−t2) dt. (10) π which represents the general solution of (4). 0 Any further step in the solution of the model requires Although p( h|Hloc,σw) may have a complicated form, ∗ some assumptions on the distribution pw of hw. The it is only dependent on two parameters, Hloc and σw, which simplest possible assumption is that of a Gaussian probability change its shape. Figure 7 shows a collection of such possible distribution, with zero mean and a standard deviation σw, i.e. shapes. It is not difficult to see that in the limit Hloc σw the shape is Gaussian and roughly centred on h ≈ Hloc; in the 1 t2 p (t) = √ exp − . (7) limit Hloc σw, the shape becomes exponential as expected w 2 σw 2π 2σw for a constant hazard-rate process.
2039 L Callegaro et al 4.3. Identification of the model parameters of hysteresis must take into account the jumpy evolution of the magnetic system through these states. All previous research The identification of the parameters H and σ of the model loc w concentrated on jumps, instead of states, very likely because can be made by fitting the model solution p( h|H ,σ ) to loc w jumps are much easier to detect than states. the experimental data. Each data set is a collection of observed With a new experimental technique, by observing small H s , s = 1,...,N. The most direct fit method is given by the regions of thin-film samples, it is possible to identify the maximum likelihood criterion: we write the likelihood function metastable magnetization states; this permits classification as a product of N model probabilities evaluated at each H s of Barkhausen jumps (which in standard experiments are, N in a sense, all equivalent) as transitions between particular s L(Hloc,σw) = p( H |Hloc,σw) (11) states. s=1 We concentrated on the stability of a single state, and assumed that the state is left when the potential well stability The likelihood is a function of the parameters to be is destroyed by the continuously increasing magnetic field. identified, Hloc and σw; the maximization of L gives the values The process is completely deterministic, apart from an added s of Hloc, σw which fit in the best way the experimental data H . random magnetic field which models the interaction between For numerical calculation convenience we numerically solved the sample region observed and the rest of the sample; the equivalent problem of minimizing the negative logarithmic elementary assumptions for this randomness (Gaussian) give, likelihood by analytic calculation, a non-trivial distribution for the life of N the state itself, which can be well fitted to the experimental ∗ s data. The fit parameters have a well-defined physical L (Hloc,σw) =− log(p( H |Hloc,σw)) (12) s=1 meaning. with a simple Matlab program (using the system-defined Nelder–Mead algorithm). References
[1] Barkhausen H 1919 Z. Phys. 20 401 4.4. The model applied to experimental results [2] Bertotti G 1998 Hysteresis in Magnetism (San Diego, CA: Academic) Figure 5 shows the results of such a fit, showing a good [3] Hubert A and Schafer¨ R 1998 Magnetic Domains (Berlin: agreement between model curves and experimental data. Springer) Looking at figure 5(b), the fit gives parameter ranges the [4] Puppin E, Zani M, Vallaro D and Venturi A 2001 Rev. Sci. values Hloc =−1.33 Oe and σw = 1.34 Oe. If we Instrum. 72 2058 remember the meaning of σ , i.e. the standard deviation of [5] Urbach J S, Madison R C and Markert J T 1995 Phys. Rev. w Lett. 75 4694 the interaction magnetic field hw, we can compare its value [6] Puppin E, Vavassori P and Callegaro L 2000 Rev. Sci. Instrum. with the standard deviations σk and σk+1 of the corresponding 71 1752 p(H(k)) or p(H(k +1)) of figure 4; although the exact [7] Puppin E, Ricci S and Callegaro L 2000 Appl. Phys. Lett. 76 relation between σw, σk and σk+1 is very complicated, a rough 2418 numerical agreement between these values adds confidence to [8] Puppin E 2000 Phys. Rev. Lett. 84 5415 [9] Callegaro L, Puppin E and Ricci S 2001 J. Appl. Phys. 90 2416 the physical meaning of the model, since they originate from [10] Callegaro L, Puppin E and Ricci S 2000 IEEE Trans. Magn. 36 the same phenomenon. 3087 [11] Bertacco R, De Rossi S and Ciccacci F 1998 J. Vac. Sci. Technol. A 16 2277 5. Conclusions [12] Weibull W 1951 J. Appl. Mech. 18 293 [13] Mann N R, Schafer R E and Singpurwalla N D 1974 Methods The hysteresis process originates from the very existence of for Statistical Analysis of Reliability and Life Data metastable magnetization states. A complete understanding (New York: Wiley) chapter 4
2040 JOURNAL OF APPLIED PHYSICS VOLUME 94, NUMBER 9 1 NOVEMBER 2003
Negative Barkhausen jumps in amorphous ribbons of Fe63B14Si8Ni15 Maurizio Zania) and Ezio Puppin INFM, Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci, 32, I-20133 Milano, Italy ͑Received 8 May 2003; accepted 13 August 2003͒
Negative Barkhausen jumps in amorphous ribbons of Fe63B14Si8Ni15 have been investigated both with the traditional inductive coil, which probes a large sample region, and also with a space resolved magneto-optical Kerr ellipsometer. After these negative jumps the magnetization vector is antiparallel with respect to the external field in the sample region where the reversal took place. Magneto-optical data indicate that negative jumps take place only in selected regions of the sample. The amplitude ⌬M of both positive and negative jumps follows a power-law probability distribution: P(⌬M)ϭ⌬M Ϫ␣. The observed values of the critical exponent ␣ are very similar, within the experimental error, both for positive and negative jumps ͑␣ϭ1.6͒.©2003 American Institute of Physics. ͓DOI: 10.1063/1.1616994͔
INTRODUCTION with these techniques allows one to obtain a better physical insight on the physics of NBJs. The magnetization process in ferromagnets consists of a modification of the sample magnetization under the effect of an external field. At the end of the process the overall mag- EXPERIMENT netization switches from one saturated state to the opposite. When observed at a microscopic scale this process is discon- We investigated amorphous ribbons of Fe63B14Si8Ni15 tinuous and takes place through a series of sudden jumps, the 20 m thick, 35 mm long, and 5 mm wide, glued on a plastic so-called Barkhausen jumps.1 During each jump the system support. Magnetization was in the sample plane, and the ex- undergoes a transition between two different metastable ternal magnetic field has been applied parallel to the sample states initially separated by an energy barrier. Each jump surface along the longest side. The external field consisted of takes place when this barrier is washed out by the external a linear ramp at a frequency of 0.1 Hz ranging from Ϫ50 to field or it is overcome by thermal activation. In more visual 50 Oe. terms the domain wall separating regions with different mag- The experimental setup is schematically outlined in Fig. netization is stripped off from its pinning sites and moves 1. In both cases ͑inductive and magneto-optical measure- through the sample until it is trapped in another pinning con- ments͒ the sample ͑S͒ is placed inside a coil ͑M͒ which gen- figuration. Normally, in the sample region interested by one erates the magnetizing field H. The inductive measurements of these jumps the final magnetization is aligned with the ͑upper part of Fig. 1͒ are performed by placing a smaller coil external field. However, the presence of so-called negative ͑I͒ around the sample. The induced voltage peaks in the coil Barkhausen jumps ͑NBJs͒ has also been observed: after these are sent to an amplifier ͑A͒ and sampled at a frequency of 50 jumps the local sample magnetization, initially aligned with kHz with a general purpose input/output board driven with a the external field, points against the field itself. PC ͑see Ref. 12 for details͒. In 1930 Becker2 predicted the existence of NBJ; his In the magneto-optical measurements the sample is model, however, involved only magnetization rotation and placed in the same field ͑Fig. 1, lower part͒ but the magneto- not domain wall motion. The first experimental evidence of optical Kerr signal is detected by measuring the polarization NBJs was published by Kirensky3 in 1951 with a study on rotation ͑⌬͒ of the incident laser beam which, in turn, is polycrystalline Ni. NBJs have also been observed in poly- proportional to the magnetization variation ⌬M along the crystalline Fe–Si ͑Ref. 4͒ and in a single crystal frame of field direction. Our high sensitivity Kerr ellipsometer has a Fe–Si investigated both with the powder image technique5 signal-to-noise ratio ͑defined as the ratio of the overall width and with the inductive technique.6,7 It is interesting to ob- of the hysteresis loop with the root mean square of noise in serve that NBJs can also be observed in ferroelectrics the region of the loop where magnetization is flat͒ of the systems8 and their existence has been theoretically order of 103. See Ref. 11 for a complete description of the addressed.9 The physical mechanism responsible for the apparatus. presence of NBJs is not clear and, in previous investigations, local inclusions10 and eddy currents6,7 have been considered as possible candidates for explaining NBJs. RESULTS In the present article we compare experimental data In our sample magnetization reversal takes place through taken with two different techniques, the traditional inductive a series of discrete jumps. During one branch of the hyster- coil and the recently introduced focused magneto-optical 11 esis loop the number of jumps detected with the inductive Kerr effect. The statistical properties of NBJs measured coil is of the order of 104 and the fraction of negative jumps is 6%. The probing coil has a lateral size of 2 mm and the a͒Electronic mail: [email protected] induced peaks are generated by magnetization jumps taking
0021-8979/2003/94(9)/5901/4/$20.005901 © 2003 American Institute of Physics 5902 J. Appl. Phys., Vol. 94, No. 9, 1 November 2003 M. Zani and E. Puppin
FIG. 2. Selected portions of the inductive data showing the presence of both positive and negative jumps.
possible to directly observe the Barkhausen jumps and their statistical fluctuations.13,14 In Fig. 5 a series of three loops FIG. 1. Experimental setup for inductive ͑upper͒ and magneto-optical mea- measured one after the other is shown ͑for clarity we show surements ͑lowest͒. only a selected portion of the upper branch͒. In this case the spot size is 50 m. Clearly, the loops are made of a series of magnetization jumps whose width and field position ran- place in a region which extends approximately 15 mm on domly fluctuate by repeating the cycle. In the bottom of Fig. both sides of the coil12 which means, in our case, nearly the 5 it is also shown the average loop, simply obtained by add- whole sample. ing all the loops of a stream. This loop is different from the Figure 2 shows selected portions of the Barkhausen sig- average loop measured over a much larger sample region and nal with positive ͑pointing up͒ and negative ͑pointing down͒ shown in Fig. 4. This difference indicates the presence of jumps. The width of the observed peaks is determined by the local inhomogeneities at this length scale onto the sample time constant ͑18 kHz͒ of a low pass filter placed after the surface. signal amplifier.12 The magnetization variation associated In the loops of Fig. 5 all the magnetization jumps take with each jump is defined as the area ⌬M of the correspond- place along the field direction. By moving the laser spot onto ing peak as indicated in Fig. 1. The statistical distributions of the sample surface it is possible to find regions where nega- the amplitude of positive (⌬M ϩ) and negative (⌬M Ϫ) jumps are shown in Fig. 3. The magnetization variation as- sociated with each jump is normalized to the overall sample ⌬ magnetization M tot . In both cases the distribution is nearly linear over several decades in a log-log plot and can be there- fore represented with a power law: P(⌬M)ϭ(⌬M)Ϫ␣. The line corresponding to ␣ϭ1.6 is shown for comparison. This value of the critical exponent ␣ represents the best fit for all the experimental data of Fig. 3. Magneto-optical data have been collected by measuring the hysteresis loops from a sample region defined by the probing laser spot. This technique is therefore intrinsically space resolved. The hysteresis loop measured with a spot size of 5 mm, i.e., as large as the ribbon, is shown in Fig. 4. With this spot size, by repeating the loop measurement the FIG. 3. Statistical distributions for the amplitudes of both positive (⌬M ϩ) shape of the curve does not change within the experimental and negative (⌬M Ϫ) jumps obtained from inductive data. The amplitude is ⌬ noise. However, if the spot size is sufficiently small it is normalized to the overall sample magnetization ( M tot). J. Appl. Phys., Vol. 94, No. 9, 1 November 2003 M. Zani and E. Puppin 5903
FIG. 4. Average magneto-optical loop measured with a spot size of 5 mm.
tive jumps take place. A series of three loops measured in one of such regions, and with the same spot size as in Fig. 5, is shown in Fig. 6. In this case, the presence of negative jumps, i.e., loop portions with negative derivative, can be also observed in the average loop, shown in the bottom of the figure. Figure 7 showns a portion of the loop of Fig. 6 along FIG. 6. Series of three magneto-optical loops ͑a portion of the upper branch͒ with the first derivative of the loop itself. The connection measured with a spot size of 50 m. In the lowest part of the figure is shown between optical and inductive results clearly appears by ob- the average loop simply obtained by adding all the individual loops of the stream. At variance with respect to Fig. 5 also negative jumps are observed. serving that whereas in one case ͑optical͒ the signal is pro- portional to M, in the other case ͑inductive͒ the signal is proportional to the time derivative dM/dt. In Fig. 7 the hori- amplitude ⌬M in the magneto-optical data is defined as the zontal axis represents the magnetic field which, in our ex- width of the magnetization variation associated with each periment, changes linearly in time. Therefore the curve in the jump and indicated as ⌬M in Fig. 7. lower part of this figure is proportional to dM/dt and can be In order to obtain good statistics a stream of 14.000 directly compared with the inductive signal of Fig. 1. The loops, such as those of Fig. 6, has been collected and the
FIG. 5. Series of three magneto-optical loops ͑a portion of the upper branch͒ FIG. 7. Detail of one of the loops of Fig. 6 along with its first derivative measured with a spot size of 50 m. In the lowest part of the figure is shown showing how positive and negative jumps appear in magneto-optical data the average loop simply obtained by adding all the individual loops of the and how the first derivative of the loop can be directly compared with the stream. inductive data such as those of Fig. 1. 5904 J. Appl. Phys., Vol. 94, No. 9, 1 November 2003 M. Zani and E. Puppin
jumps since, as shown in Fig. 2, a clear correlation between positive and negative jumps is not observed. In Fig. 2 we observe different situations where negative jumps take place in any possible position, both very close to large positive jumps as in the upper part but also with no positive jumps within 200 s as in the lowest part of the figure. On the basis of the inductive data we therefore conclude that, in our sample, eddy currents are not the direct cause of NBJs. On the other hand, our optical data show that NBJs take place only in a few regions of the sample surface and this supports the hypothesis that local defects are responsible for their existence. It is beyond the purpose of this work to dis- cuss the details of this process since we are more interested in the large scale statistical properties of NBJs in connection with those of positive jumps. It is quite remarkable that all our measurements, performed with a different experimental technique, bring the same power-law distribution both for positive and negative jumps. More than this, the value of the critical exponent extracted from our data is the same, ␣ϭ1.6, for all distributions. The critical behavior of Barkhausen noise is well known17 and all the available theories rely upon the idea of a
FIG. 8. Statistical distributions for the amplitudes of positive (⌬M ϩ) and domain wall moving in a randomly disordered medium. The negative (⌬M Ϫ) jumps obtained from magneto-optical data. The cutoff is statistical distribution of the jump size therefore is a direct due to the finite size of the laser spot ͑50 m͒. consequence of this disorder, at least for positive jumps. Our data show that the statistical properties are the same also for NBJs and this is an indication that the same physics is in- statistical distributions of both positive and negative jumps volved in determining both kinds of avalanches. The most are shown in Fig. 8. The observed behavior is still a power relevant difference is that NBJs do not take place everywhere law, at least over one decade, and the values of the corre- inside the sample but only around particular regions whose sponding critical exponents are very similar in the two cases size is of the order of 100 m as our magneto-optical data being ␣ϭ1.6 both for negative and positive jumps. The pres- indicate. In these regions are probably present local defects ence of a cutoff in the distributions of Fig. 8 must be related which act as seeds for the avalanches, and for this reason to the finite size of the spot.14 NBJs are not observed everywhere. However, after the initial triggering, the avalanche proceeds through the disordered DISCUSSION medium according to the same physical mechanisms of the Let us first consider in more detail the available models positive avalanches. During this process, in particular during for NBJs. In Ref. 10 the motion of a domain wall in a crystal the avalanche itself, the role of eddy currents is probably with isolated inclusions is discussed. It is shown that, if the relevant but, as discussed above, they cannot be considered size of the inclusions is of the order of the wall thickness, the the direct cause of NBJs. secondary domain structure surrounding the inclusion can generate magnetization reversals against the external field. A 1 H. Barkhausen, Z. Phys. 20, 401 ͑1919͒. true experimental proof that this is indeed the case has never 2 R. Becker, Z. Phys. 62, 253 ͑1930͒. been published. 3 L. V. Kirensky and W. F. Ivlev, Dokl. Akad. Nauk SSSR 76,389͑1951͒ Another possible explanation for NBJ can be found, ac- ͓Sov. Phys. Dokl. 1,1͑1956͔͒. 4 J. Kranz and A. Schauer, Ann. Phys. 7,84͑1959͒. cording to Ref. 4, in eddy currents induced in metallic ferro- 5 V. Kavetchansky, Doctoral thesis, Faculty of Natural Sciences, Kosˆice, magnets as a consequence of a magnetic induction flux varia- 1967. tion caused by a positive jump. Each negative jump 6 A. Zentko and V. Hajko, Czech. J. Phys., Sect. B 18, 1026 ͑1968͒. 7 therefore15 should follow a corresponding positive jump with V. Hajko, A. Zentko, and S. Filka, Czech. J. Phys., Sect. B 19,547͑1969͒. 8 ͑ ͒ B. Tadic, Eur. Phys. J. B 28,81 2002 . a time delay smaller than 0.1 s. The chief argument in favor 9 V. Ya. Shur, Phase Transitions 65,49͑1998͒. of the eddy currents conjecture comes from the observation 10 R. S. Tebble, Proc. Phys. Soc. London, Sect. B 86, 1017 ͑1955͒. that NBJs are not observed in high resistivity ferromagnetic 11 E. Puppin, P. Vavassori, and L. Callegaro, Rev. Sci. Instrum. 71, 1752 ͑ ͒ material such as ferrites.16 In our experiment on an amor- 2000 . 12 E. Puppin, M. Zani, D. Vallaro, and A. Venturi, Rev. Sci. Instrum. 72, phous alloy, NBJs are clearly observed and their amplitude 2058 ͑2001͒. distribution is the same as the one of positive jumps. This 13 E. Puppin, S. Ricci, and L. Callegaro, Appl. Phys. Lett. 76,2418͑2002͒. observation apparently supports the idea that NBJs are gen- 14 E. Puppin, Phys. Rev. Lett. 84, 5415 ͑2000͒. 15 erated by positive jumps and therefore their statistical prop- A. Zentkova, A. Zentko, and V. Hajko, Czech. J. Phys., Sect. B 19,650 ͑1969͒. erties must be the same. However, it is difficult to attribute 16 A. Zentkova, Czech. J. Phys., Sect. B 19,1454͑1969͒. NBJs simply to eddy currents induced by large positive 17 G. Bertotti, Hysteresis in Magnetism ͑Academic, San Diego, 1998͒. 3B2v7:51c MAGMA : 14903 Prod:Type:COM ED:MangalaGowri GML4:3:1 pp:123ðcol:fig::NILÞ PAGN: lmn SCAN: anand ARTICLE IN PRESS
1
3 Journal of Magnetism and Magnetic Materials ] (]]]]) ]]]–]]] 5
7 Temperature dependent criticality of Barkhausen noise in thin 9 Fe films 11 Maurizio Zani*, Ezio Puppin 13 Istituto Nazionale per la Fisica della Materia, Dipartimento di Fisica, Politecnico di Milano, P.zza Leonardo da Vinci 32, 20133 Milano, 15 Italy
17 Abstract 19 We investigated the magnetization process of a thin Fe film grown on MgO, mounted on a cryostat in order to 21 change its temperature from 10 to 300 K. Statistical distribution of magnetization jumps show a power-law at each of two temperature, with critical exponent a ¼ 1 at 300 K and a ¼ 1:8 at 10 K, showing that criticality is preserved; the 23 recent experimental technique, based on the magneto-optical Kerr effect with a variable spot size of the laser, allows to obtain probability distribution of magnetization jumps spanning over several decades. 25 r 2004 Elsevier B.V. All rights reserved.
27 PACS: 75.60.Ej; 75.70.Ak
29 Keywords: Magnetization processes; Avalanches; Barkhausen noise; Thin magnetic films; Self-organized criticality
31 The magnetization process of a ferromagnetic materi- ratio of the overall width of the hysteresis loop with the 57 33 al, when observed on a macroscopic scale, appears to be root mean square of noise in the region of the loop continuous and well represented by an hysteresis loop; where magnetization is flat) of the order of 103.A 59 35 on the other hand, if we go on a microscopic scale the focussing system of the laser beam impinging on the process is discontinuous, and the system evolves through sample allows to vary the size of the spot from 10 mmup 61 37 a series of magnetization jumps called Barkhausen noise to several mm: see Ref. [3] for a complete description of [1] connecting metastable states, due to domain walls the apparatus. This technique has been already used for 63 39 motion occurring during magnetization. The statistical room temperature investigations of Barkhausen noise in properties of such jumps have been largely investigated, thin films [4] and microstructures [5]. 65 41 in particular their size (DM) distribution, that appears to The sample [6] is an epitaxial Fe film having a be linear in a log–log plot, i.e. it can be expressed by a thickness of 900 A.The( substrate is a single crystal of 67 a ! 43 power law: PðMÞ¼DM [2], where the coefficient a is MgO (0 0 1) heat treated in vacuum in order to observe a called critical exponent. sharp (1 1) LEED pattern characteristic of the bulk- 69 45 We investigated the temperature dependence of terminated lattice. The Fe evaporation has been Barkhausen jumps in a thin Fe film grown on MgO: performed in a vacuum chamber with a base pressure 11 ( 71 47 the sample is placed at the centre of a pair of of 5 10 Torr at a rate of 10 A/min. The film magnetizing coils in Helmholtz configuration, with a thickness has been measured with a quartz microbalance 73 49 triangular waveUNCORRECTED applied at a frequency of 0.1 Hz. The and purity has beenPROOF checked with XPS and Auger magnetic signal of the sample is revealed by a Kerr spectroscopy. The sample has been mounted on a 75 51 ellipsometer in longitudinal geometry, whose sensitivity cryostat in order to change its temperature from 10 to allows to obtain a signal-to-noise ratio (defined as the 300 K, and a stream of 5000 loops has been measured at 77 53 each temperature; during the acquisition of each stream, the laser beam has been kept in a fixed position on the *Corresponding author. Tel.: +39-02-2399-6142; fax: +39- 79 sample surface, with a fixed value of the spot between 20 55 02-2399-6126. E-mail address: [email protected] (M. Zani). and 500 mm.
0304-8853/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.12.191 MAGMA : 14903 ARTICLE IN PRESS 2 M. Zani, E. Puppin / Journal of Magnetism and Magnetic Materials ] (]]]]) ]]]–]]]
1 Under the effect of the applied field, magnetization smaller for lower temperature with respect to the higher 57 evolves through a series of jumps separated by flat temperature situation. 3 regions: these steps are not deterministic, and randomly The probability distribution of DM at two different 59 change by repeating the loop measurement. temperatures is shown in Fig. 2. The experimental 5 An example is shown in Fig. 1: in the left part of this procedure adopted is described in detail in Ref. [4]: here 61 figure, a series of three loops measured at 300 K one we note that this procedure allows to obtain reliable 7 after the other (for clarity only a portion of each loop is experimental data spanning over several decades of DM; 63 shown corresponding to the upper branch) with a laser obtained by convoluting different curves. In both cases 9 spot of 100 mm is shown. In the right part of the figure is the experimental curve can be fitted with a power law, 65 shown another series measured at 10 K with the same whose critical exponent has a value a ¼ 1 at 300 K and 11 spot; fluctuations are present in both series, but the a ¼ 1:8 at 10 K, indicating that large jumps are less 67 amplitude of the steps appears to be in the average likely to occur at lower temperature. This difference is 13 relevant and is qualitatively consistent with the trend 69 observed in Fig. 1. 15 The jumpy evolution of magnetization represents an 71 example of a complex system which evolves through a 17 300 K 10 K series of avalanches whose size spans over a large range 73 of values and whose probability distribution shows scale 19 invariance, as observed in critical phenomena. The role 75 of temperature in complex systems has been addressed in 21 few theoretical investigations, and it is not completely 77 understood: in Ref. [7] a nonzero temperature breaks 23 i j down criticality, in Ref. [8] the introduction of 79 temperature in a sandpile model preserves criticality. 25 81
27 83
29 85
31 87 20µm 33 i+1 j+1 89 50µm
35 100µm 91
37 T = 300 K 250µm 93 P(∆M) = ∆M-1 500µm 39 95
41 M) (arb. units) 97 ∆ P( 43 i+2 j+2 20µm 99
45 101
µ 47 100 m 103
49 UNCORRECTEDT = 10 K PROOF 105 P(∆M) = ∆M-1.8 µ 51 500 m 107 16 20 24 28 26 30 34 38 ∆ 53 M (arb. units) 109 H (Oe) Fig. 2. Probability distribution of DM at 300 K (upper part) 55 Fig. 1. Left: stream of three loops measured one after the other and at 10 K (lower part). In both case the distribution is a 111 at a temperature of 300 K. Right: the same at 10 K. power law, but with a different value of the critical exponent. MAGMA : 14903 ARTICLE IN PRESS M. Zani, E. Puppin / Journal of Magnetism and Magnetic Materials ] (]]]]) ]]]–]]] 3
1 Our data indicate that criticality is always observed from [2] G. Bertotti, Hysteresis in Magnetism, Academic Press, San 10 to 300 K, but the critical exponent of the correspond- Diego, 1998. 13 3 ing power law nearly doubles at higher temperature. At [3] E. Puppin, P. Vavassori, L. Callegaro, Rev. Sci. Instrum. 71 our knowledge this is the first experimental evidence, at (2000) 1752. 15 5 least in the field of magnetism, on the temperature [4] E. Puppin, Phys. Rev. Lett. 84 (2000) 5415. [5] E. Puppin, L. Callegaro, S. Ricci, Appl. Phys. Lett. 76 dependence of criticality in a complex system. 17 (2000) 2418. 7 [6] R. Bertacco, S. De Rossi, F. Ciccacci, J. Vac. Sci. Technol. A 16 (1998) 2277. 19 9 References [7] M. Vergeles, Phys. Rev. Lett. 75 (1995) 1969. [8] G. Caldarelli, Physica A. 252 (1998) 295. 11 [1] H. Barkhausen, Z. Phys. 20 (1919) 401.
UNCORRECTED PROOF INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER J. Phys.: Condens. Matter 16 (2004) 1183–1188 PII: S0953-8984(04)71977-3
Magnetic hysteresis and Barkhausen noise in thin Fe films at 10 K
Ezio Puppin and Maurizio Zani
INFM—Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy
Received 13 November 2003 Published 13 February 2004 Onlineatstacks.iop.org/JPhysCM/16/1183 (DOI: 10.1088/0953-8984/16/8/004)
Abstract The role of temperature during magnetization reversal in a thin Fe film grown on MgO has been investigated. At a microscopic level the process takes place through jumps whose amplitude M follows a power-law probability distribution: P( M) = M−α,with α = 1at300 K and 1.8at10K.During each avalanche, thermal activation does not play a relevant role in the initial overcoming of the potential barrier and therefore the temperature-dependent amplitude probability is related to cooperative effects taking place during the avalanche itself.
The magnetization process in ferromagnets has been extensively investigated. At a macroscopic level the hysteresis loop describes how the overall magnetization of the sample changes under the effect of an external field. This process is normally smooth and reproducible. At amicroscopic level, however, the process is discontinuous and random [1, 2] and takes place through a series of jumps which drive the system between different metastable states. This effect has been discovered a long time ago and is called Barkhausen noise (BN) [3]. The physical reason of this random behaviour is the presence of defects, impurities, local stress, and so on, which act as pinning sites for the domain walls. Since its discovery, BN has been extensively investigated with the major aimofinterpreting its statistical properties and to create a bridge between the macroscopic and the microscopic picture of the magnetization process. In recent years the interest for BN extended well beyond the field of magnetism since it is a relevant example of complex systems (see [4] for an extended review on complexity and the role of BN in this field). One of the most striking features of BN, shared with many complex systems in the so-called ‘critical’ state, is that the amplitude M of the magnetization jumps has a probability distribution P( M) which follows a power-law behaviour over several decades. In other words, the distribution is linear in a log–log plot and can be interpolated with asimple exponential law: P( M) = M−α.Inordertoexplain the value of the critical exponent α observed in BN experiments different models, based on different approaches, have been developed. A comparison between their predictions and the experimental observations can be found in [4].
0953-8984/04/081183+06$30.00 © 2004 IOP Publishing Ltd Printed in the UK 1183 1184 EPuppin and M Zani
All the models for BN presented until now refer to zero temperature whereas all the experimental data on BN have been obtained at room temperature. More generally, in the field of complex systems we areawareonly of two works in which the role of temperature is discussed within the framework of self-organized criticality [5]. The models described in these two works bring two contradicting results [6, 7]. The need for experimental data on BN at low temperature, i.e. in a situation closer to the available theoretical predictions, is therefore strong. In the present paper we present experimental data on the statistical properties of BN in a thin Fe film at different temperatures. It will be shown that the effect of temperature is strong and the observed values of the critical exponent α change from α = 1atroom temperature to 1.8 at 10 K. Let us first summarize the available knowledge on the role of temperature in magnets when they are observed at a macroscopic scale. In this situation it is well established that the coercive force Hc is strongly affected by temperature. A model which explains this effect in terms of wall motion in a disordered medium has been proposed [8]. In this model the wall is pinned by a potential well from which it can escape due to thermal motion. The model 1/2 1/2 2/3 predicts that Hc depends on T according to the following law: Hc = Hco (1 − CT ), where C is a constant depending on the size and shape of the potential well and Hco is the coercive force at zero temperature [8]. Our experiment consisted in a measurement of BN in a thin Fe film grown on MgO. The recently introduced experimental technique is basedonthe magneto-optical Kerr effect [9] and has been already used for room temperature investigations of BN in thin films [10] and magnetic microstructures [11]. The measurements have been conducted with a magneto-optical Kerr ellipsometer whose sensitivity allows us to perform the acquisition of a hysteresis loop in 1 s or less with a noise level of the order of 10−3 times the overall width of the loop [9]. The sample has been mounted on a cryostat in order to change its temperature from 10 to 300 K. In our experiment a stream of 5000 loops has been measured at each temperature. Each loop has been taken by sweeping the field with a triangular wave at a frequency of 0.1 Hz. During the acquisition of each stream the laser beam has been kept in a fixed position on the sample surface and also the size of the laser spot has been fixed to a particular value. Different streams have been collected with a spot size ranging between 20 and 500 µm1. The sample [12] is an epitaxial Fe film 900 Å thick. The substrate is a single crystal of MgO (001) heat-treated in vacuum in order to observe a sharp (1×1) LEED pattern characteristic of the bulk-terminated lattice. Fe has been evaporated in a vacuum chamber with a base pressure of 5×10−11 Torr at a rate of 10 Å min−1.Thefilm’sthickness has been measured with a quartz microbalance and purity has been checked with XPS and Auger spectroscopy. Magnetization is in theplane of the film and the loops have been measured with the external field parallel to the sample surface. The macroscopic behaviour of our sample versus temperature is summarized in figure 1. The filled dots in the upper part show the average loop at 300 K, simply obtained by adding all the data of a stream taken with a spot diameter of 100 µm. The empty dots of figure 1 show the average loop measured on the same sample region at 10 K. Clearly, the coercive force increases as expected.Therelationship between Hc and T is shown in the lower part of 2/3 figure 1 where the square root of Hc is plotted versus T .Thislinear trend confirms that, on a macroscopic scale, the coercive force depends on temperature according to the model described in [8]. This is consistent with the microscopic picture of the magnetization process in our sample where the domain walls jump between randomly distributed pinning sites [10].
1 The Gaussian laser beam impinges onto the sample surface at an angle of 45◦ off-normal. Therefore the actual shape of the spot is elliptical and the value of 100 µmrefers to the minor axis. Magnetic hysteresis and Barkhausen noise in thin Fe films at 10 K 1185
C ∆ Mi + M
300 K A M 10 K i B ∆ Hi Hi + H arb. units)
M( H c A Mi
6.0 -40 -20 0 20 40 Hi H(Oe)
) 5.5 B M
1/2 i
(Oe H + ∆H 1/2
c i 5.0 H
∆ 4.5 C Mi + M 01020304050 2/3 2/3 ∆ T (K ) Hi + H
Figure 1. Upper panel: average hysteresis loops at Figure 2. Example of a loop showing magnetization 300 K (filled dots) and 10 K (empty dots). Lower panel: jumps. It illustrates the evolution of the systems between temperature dependence of the coercive force. The three states A, B and C (see the text for a detailed linearity in this plot indicates a functional dependence discussion). as predicted by [4].
As already stated, at variance with the average loop, each individualloop of the series presents fluctuations, as shown by the experimental points in the upper part of figure 2 [10] which illustrate the upper branch of a typical loop measured at room temperature with a spot diameter of 100 µm. Under the effect of the applied field, magnetization evolves through a series of jumps separated by flat regions. These steps are not deterministic and randomly change by repeating the loop measurement. At 10 K fluctuations are still observed, but the amplitude of the steps appears to be, on average, smaller with respect to the highest temperature situation. Astatistical analysis of these magnetization fluctuations can be performed by considering the relevant parameters of the steps. In this work our attention is focused on two of these parameters, H and M, both illustrated in figure 2. The physical meaning of these two quantities can be better understood by considering the evolution of magnetization at an increasing field. Let us start from a system configuration such as the one represented in figure 2 by point A. This point corresponds to a metastable equilibrium position reached by the system during its jumpy evolution and is pictorially represented in the lower part of figure 2. The ellipses represent the area sampled with the laser beam. Within this area a fraction of the sample magnetization is already aligned with the external field whereas the other fraction is still opposing the field. The metastability of this state is determined by the presence of pinning sites that determine a local minimum in the energy landscape. By increasing the value of the applied field the system remains in this state and the only effect produced is a bowing of 1186 EPuppin and M Zani
300 K 10 K (arb. units) H) ∆ P(
0 1 2 3 4 5 ∆H(Oe)
Figure 3. Probability distributions of H (see figure 2 and the text for details). the domain wall. This situationlasts until a particular value of the applied field is reached, as indicated by point B of figure 2. At this stage the energy of the system is sufficient to overcome the potential barrier and a jump occurs which brings the system into a new metastable state labelled by C in the figure. The extra field necessary to produce the wall depinning is H . The amplitude of the avalanche which brings the system from point B to point C is M. The statistical distributions of H and M are shown, respectively, in figures 3 and 4. Figure 3 shows P( H ),the probability distribution of H at two different temperatures, 300 K(filled dots) and 10 K (empty dots). The probability distribution of M at two different temperatures is shown in figure 4. The experimental procedure adopted for obtaining the curve of figure 4 is described in detail in [10]. Here we note that this procedure allows us to obtain reliable experimental data spanning over several decades of M.Inboth cases theexperimental curve is nearly linear in a log–log plot and therefore can be fitted with a power law: P( M) = M−α.The critical exponent has a value α = 1at300 K and 1.8 at 10 K. In other words, at both temperatures the statistics is still described by a power law, but thedifferent values of the critical exponent indicates that, on average, the size of the jumps increases with temperature. More precisely, the occurrence of large jumps is favoured at higher temperatures. In this connection we observe that the upper distribution shown in figure 4 is obtained by convoluting different curves, each having a well defined cut-off. This cut-off is due to the finite size of the region defined by the laser spot on the sample surface. In the lowest distribution this cut-off is not present and this is another indication that large jumps, i.e. whose size is comparable to or larger with respect to thespot size, are much less favoured at lower temperatures. As already stated, the accepted view of the magnetization process is schematically represented in figure 2 where the system evolves through a series of jumps between different metastable states. Each of these states has a relative stability and, in order to escape to make ajumptoanother metastable state, the system must overcome an energy barrier. In BN experimentsthe escape from the initial state is determined by the increasing applied field H . Asimplemodel has been presented [13] which explains the shape of P( H ).Inthismodel amagnetic dipole mimics the complex magnetic system initially trapped in a potential well such as case A of figure 2. This dipole senses a local field Hloc which, in turn, is the sum of three different magnetic fields. The first isthefieldexperimentally generated, H ,whichwill be assumed to be positive. Since the dipole is in a metastable configuration, a pinning field Magnetic hysteresis and Barkhausen noise in thin Fe films at 10 K 1187
P(∆M) = ∆M -1
20µm
50µm
100µm 250µm T=300 K 500µm
∆ ∆ -1.8
M) (arb. units) P( M) = M ∆
P( 20µm
100µm
T=10K 500µm
∆M(arb. units)
Figure 4. Probability distributions of M (see figure 2 and the text for details).
+ Hloc
0 ∆ - H
Jump to state C takesplace here
-Hp Hw (randominteraction field) A
Figure 5. Pictorial representation of the stochastic process which brings the system from one metastable state (such as A in figure 2) to another (such as C).
Hp is also acting. The value of this field is negative (in opposition to the external field H ). Finally, the dipole interacts with the rest of the sample via an interaction field Hw.Sincethis field depends on the configuration of the whole sample, i.e. on the orientation of all the other dipoles, it is a random field which suddenly changes any time a Barkhausen jump takes place 1188 EPuppin and M Zani within the sample. In summary:
Hloc = H + Hw − Hp. Apictorial representation of the escape from the potential well is shown in figure 5. The initial value of the local field is −Hp and the system will leave the well when the field increases above zero. With no interaction field (Hw = 0) the process is deterministic and a single value of H would be observed, the one corresponding to the intersection between the horizontal axis and the straight line which represents the experimental ramp in H .The presence of a random interaction field is responsible for the spread in the observed values of H .Thedistribution P( H ) therefore depends both on Hp and on Hw,i.e.ontheparameters whichcharacterize the escape process. The datainfigure3showthatP( H ) does not depend on temperature and this observation can be interpreted by saying that the escape of the system from the potential is weakly influenced by temperature. After the system has overcome the energy barrier a complex chain of events eventually brings the system to a new metastable state. It is hard to model this avalanche but our data indicate that it is at this stage of the process that the thermodynamic temperature plays its role by determining the statistical properties of the avalanche size. Trying to interpret the details of this behaviour in terms of the available theories is beyond our scope. Much work has to be done in this field and our hope is that the present paper will stimulate interest in this argument of the statistical physics community. As a final remark we point out that the observed variation of the critical exponent when temperature changes from 10 K to room temperature is close to the spread of the available theoretical predictions and the experimental data on BN (see, for instance, figure 9 of [4]). This is a strong reason for doing more work on the role of temperature on BN and, more generally, on complex systems.
References
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