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Distortions of the statistical distribution of Barkhausen noise measured by magneto-optical Kerr effect
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 320 (2008) 1651–1656 www.elsevier.com/locate/jmmm
Distortions of the statistical distribution of Barkhausen noise measured by magneto-optical Kerr effect E. Pinotti, M. Brenna, E. Puppin Department of Physics, Polytechnic of Milano, piazza Leonardo da Vinci, 32–20133 Milano, Italy Received 10 September 2007; received in revised form 11 January 2008 Available online 24 January 2008
Abstract In magneto-optical Kerr measurements of the Barkhausen noise, a magnetization jump DM due to a domain reversal produces a variation DI of the intensity of a laser beam reflected by the sample, which is the physical quantity actually measured. Due to the nonuniform beam intensity profile, the magnitude of DI depends both on DM and on its position on the laser spot. This could distort the statistical distribution p(DI) of the measured DI with respect to the true distribution p(DM) of the magnetization jumps DM. In this work the exact relationship between the two distributions is derived in a general form, which will be applied to some possible beam profiles. It will be shown that in most cases the usual Gaussian beam produces a negligible statistical distortion. Moreover, for small DI the noise of the experimental setup can also distort the statistical distribution p(DI), by erroneously rejecting small DI as noise. This effect has been calculated for white noise, and it will be shown that it is relatively small but not totally negligible as the measured DI approaches the detection limit. r 2008 Elsevier B.V. All rights reserved. Keywords: Magneto-optical Kerr effect; Barkhausen noise; Noise statistics
1. Introduction In ferromagnetic materials the most important noise mechanism is the Barkhausen noise [1], produced by reversals (random in size and time of occurrence) of magnetic domains during the magnetization process. The importance of Barkhausen noise has been largely recognized [2] both as a probe of the various aspects of the magnetization process [3], and as an illustration of stochastic processes in complex systems [4]. It is often viewed as a prominent example of critical phenomena [5], whose most characteristic feature is a power-law probability distribution: pðDMÞ / DM a ,
(1)
where p(DM) is the probability of occurrence of a magnetization reversal of amplitude DM, and a is a number (typically ranging from 1 to 2) called the critical Corresponding author. Tel.: +39 02 2399 6132; fax: +39 02 2399 6126.
E-mail address: ermanno.pinotti@fisi.polimi.it (E. Pinotti). 0304-8853/$ - see front matter r 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2008.01.031
exponent. The exact value of a is thought to be of great importance [2], because it can discriminate between different underlying physical and statistical situations. Usually the Barkhausen noise is measured inductively, by winding a pick-up coil around the sample and detecting the voltage spikes induced into the coil by the magnetization reversals. However, sometimes this experimental setup cannot be employed, as happens for very thin films, microstructured samples and other cases of relevance. In these situations, a simple and powerful measurement apparatus employs the magneto-optical Kerr effect (MOKE) [6,7], where linearly polarized light (usually a laser beam) is directed onto the sample and the reflected light goes through an analyzing polarizer and is finally detected by a photodiode. Due to the Kerr effect a magnetization reversal DM produces a small rotation in the polarization plane of the reflected beam, which (after the analyzing polarizer) is measured by the photodiode as a variation DI in the beam intensity. By using lock-in modulation techniques, a high sensitivity can be reached, and very small magnetization reversals can be measured
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[8]. Moreover, by means of a beam expander and a focusing optics, the spot size can be varied from a few micrometers to about one centimeter [9], therefore probing Eq. (1) for DM varying over several orders of magnitude [10]. However, in a MOKE apparatus the physical quantities actually measured are the beam intensity variations DI, and hence their statistical distribution p(DI). But DI depends both on the corresponding magnetization variation DM and on its position in the illuminated spot. For example, laser beams have typically a Gaussian profile and the light intensity decreases with the distance r from the spot center: r2 IðrÞ ¼ I 0 exp 2 , 2s
(2)
where s is a parameter specifying the spot dimension. Therefore, a magnetization reversal DM happening at a distance r from the spot center would give an intensity variation DI given by r2 DI / DM exp 2 . 2s
(3)
This effect could in principle distort the measured probability distribution p(DI) with respect to the true p(DM). As an extreme example, consider a sample where all magnetization reversals have the same size DM0 illuminated by a Gaussian beam; it can be shown that the corresponding probability distribution of the DI would be p(DI) ¼ DI1, therefore giving a completely erroneous result. To our knowledge, the problem of non-uniform sample illumination has been issued a few years ago [11], and only recently a dedicated experiment has been carried out for the case of uniform and Gaussian beam profiles [12]. In this work we obtain (for the first time, to our knowledge) a theoretical analysis of the distortions possibly arising in MOKE measurements of Barkhausen noise statistics, because the non-uniform sample illumination. A general framework will be laid out, and the results for some possible beam profiles will be obtained in detail. Comparison between theory and experiments will be shown for the most usual case of the Gaussian profile. Another possible distortion of the measured statistical distribution arises for small values of DI, and is due to the noise in the experimental setup. Random fluctuations due to laser and detector noise are unavoidable, and a detection threshold has to be defined in some ways, in order to separate the ‘‘true’’ intensity variations DI from those due to the noise. However, in the vicinity of this threshold, noise fluctuations add to the ‘‘true’’ DI variations, and could produce either a rejection of a valid signals or acknowledgement of a false one, therefore distorting the statistical distribution of DI. This effect will be analyzed for Gaussian noise, and the results will be compared with experiments.
2. Effects of a non-uniform beam profile 2.1. Theoretical model Let us consider Fig. 1. The circle represents the sample surface illuminated with the laser spot, whose intensity I depends on the distance r from the center I ¼ I(r). Let us assume that a magnetization reversal DM takes place in a given location r: r indicates the distance between the center of the spot and the center of gravity of the reversed area. In the following we will assume that the intensity I can be regarded as constant over the whole reversed area and, therefore, the measured value of the reversion DI will be proportional both to DM and to the intensity I(r) of the beam. For simplicity, the proportionality factor will be taken as equal to one, and therefore DI ¼ DMIðrÞ.
(4)
Assuming a constant intensity I over the whole reversed area is of fundamental importance in our analysis, since it allows a simple and solvable analytical formulation. Without this hypothesis, the problem would be analytically intractable, having to take into account not only the value of DM and its position r, but also its geometric shape and orientation. In theory, this assumption is valid only for small reversed areas, of the order of a few percent of the beam spot. Fortunately, in most experiments this is just what one observes, with a great majority of reversals involving an area much smaller than the spot size. The assumption becomes less justifiable as the reversed area approaches the spot size: in this case the actual intensity variation DI can be sensibly higher or lower than expected from the uniform I hypothesis, depending on the geometric shape and orientation of the reversed area. However, only a few domain reversals are of this size, and therefore their weight on the shape of the overall probability distribution is negligible. Moreover, simple Monte Carlo simulations, not reported here for brevity, show that for large reversals
Fig. 1. Geometric outline of the circular sample of radius R, with a magnetization reversal DM occurring at a distance r from the sample center. The impinging laser beam is centered with the sample.
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the actual intensity variation DI (averaged over many possible shapes and orientations) is similar to the ideal one, calculated with the uniform I assumption. The two cases differ at most by a factor of about two, which is relatively unimportant if one remembers that this difference applies only to a few reversals, moreover happening at the end of the range of interest for the probability distribution. Hence, in our analysis we will rely on the uniform I assumption which produces Eq. (4). The function I(r) has the following properties:
I(r) is single valued, monotonically decreasing with r, i.e. I(r2)oI(r1) for r24r1. This choice excludes therefore multi-valued functions as the ones deriving from diffraction of the laser beam, multi-mode beams, and so on. These cases can be naturally of interest, but their mathematical analysis is much more difficult; moreover, these conditions should be avoided in a well-conducted experiment. The maximum value of I(r) arises for r ¼ 0, and in the following it will be assumed that I(0) ¼ 1. This choice comes naturally from the fact that in our MOKE measurements the experimental apparatus automatically normalizes the measured DI to the full amplitude of the measured hysteresis cycle, therefore losing information about the absolute value of the beam intensity. However, this choice requires a non-trivial scaling of the measured distributions, as already described in Ref. [4]. This scaling will be discussed in more detail in Section 3.
Since we are considering power-law distributions, we will assume that the ‘‘true’’ probability distribution of the magnetization reversals DM is of the form (5)
where a is the critical exponent and A is a normalization factor, whose value can be obtained by imposing the normalization on the probability distribution Z DM 1 pðDMÞ dDM ¼ 1. (6) DM 0
Here DM0 and DM1 are respectively the smallest and the greatest possible values for DM. In principle, both depend on the sample, but their actual value depends on the experimental setup. DM0 is dictated by the sensitivity and noise of the measurement apparatus, while DM1 cannot exceed the whole beam area (obviously the experiment cannot detect reversals extending beyond the illuminated spot). From the normalization requirement, A can be obtained: 1a if aa1, DM 1a 0 1 if a ¼ 1. A¼ lnðDM 1 =DM 0 Þ
A¼
As for the probability distribution of the reversal position r, we will assume that the DM take place with equal probability over the whole sample surface. Therefore, if we assume for simplicity a circular sample of radius R, it can be shown that the probability distribution for r is given by 2r . (8) R2 It is reasonable that, for an uniform sample, the amplitude DM of a reversal is statistically independent of its position r. Therefore the joint probability distribution p(DM, r) of a reversal of given magnitude DM appearing at a given position r is simply the product of the two probability distributions for DM and r:
pðrÞ ¼
pðDM; rÞ ¼ pðDMÞpðrÞ.
(9)
The problem is how to find the statistical distribution of the random variable (10)
DI ¼ DMIðrÞ
given the probability distributions of the random variables DM and r. The solution employs the usual tools of the statistical theory [13] and is outlined in Appendix A. Here only the final results will be given. 2.3. Resulting statistical distribution for a uniform beam As expected, for the case of a uniform beam the measured distribution p(DI) has the same analytical expression of the real distribution p(DM): r20 A DI a (11) R2 apart from the scaling factor r20/R2. This factor is proportional to the area of the beam, and it represents mathematically the reason of the scaling already described in Ref. [4]: in a MOKE experiment, the probability density p(DI) is obtained by counting how many jumps do have an amplitude DI. But, if the sample is uniform, the number of the jumps counted grows as the area of the beam spot, and so does the measured density probability. This effect has to be corrected in order to compare measurements made with different beam sizes, and the correction requires a division of the measured distribution by the spot area. pðDIÞ ¼
2.2. Probability distributions for DM and r
pðDMÞ ¼ A DM a ,
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2.4. Resulting statistical distribution for a Gaussian beam For a Gaussian beam, the measured distribution is instead given by ( " #) 2s2 1 DI a1 a pðDIÞ ¼ 2 A DI 1 1 a DM 1 R for aa1;
DM 1a 1
(7)
2s2 DM 1 1 pðDIÞ ¼ 2 A DI ln DI R
for a ¼ 1:
(12)
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Therefore, the measured distribution is given by the ‘‘true’’ one scaled by 2s2/R2 and multiplied by the expression in the curly brackets. The scaling factor again represents the spot area, which is proportional to s2, and implies the same scaling described for the uniform case [14]. The expression in curly brackets represents the distortion due to the non-uniform beam profile, and is the most important result of this analysis. The distortion it introduces in the measured probability distribution is small when aX1. In fact, since DM1 corresponds to the total beam area, it must always be DIpDM1; moreover, as the MOKE detects jumps varying over several orders of magnitude below DM1, for most of the measured DI it is DI5DM1. Therefore, when aX1 it is almost always (DI/ DM1)a151 and the distribution p(DI) becomes the correct power law: pðDIÞ ffi
2s2 A DI a R2 a 1
for a41:
(13)
As for the case a ¼ 1, the logarithm ln(DM1/DI) varies slowly when its argument is large, and again it introduces a relatively small correction. The situation is different for ao1, because in this case it is almost always (DI/DM1)a1b1, and the measured distribution becomes 2s2 DM 11a pðDIÞ ffi 2 (14) A DI 1 for ao1: 1a R Therefore, when the critical exponent is less than one, the measured distribution tends always to be of the form DI1 whatever the actual a. Fig. 2 shows the comparison between Eq. (12) and an experiment performed on an amorphous film of iron, 50 nm in thickness, with a Gaussian laser beam of three different sizes. By assuming a critical exponent equal to one
Fig. 2. Comparison between the experimental data (symbols) and the fitting curves (solid lines) derived from Eq. (12) for a laser beam with Gaussian profile. The distributions have been already scaled by the beam size, as described in the text, and are consistent with the power law over more than four orders of magnitude.
the agreement between experiment and Eq. (12) is very good. It is worthwhile to note that Eq. (12) also introduces the correct expression for the cutoff, i.e. the decay to zero of the measured p(DI) as the maximum DM1 is approached. Only for the smallest DM the agreement between Eq. (12) and the experimental data is unsatisfactory. This is due to the noise effect of the experimental setup, which will be described in Section 3. 3. Effects of noise in the measurement apparatus An ideal noiseless MOKE apparatus could detect an intensity fluctuation DI of any amplitude, down to the smallest value physically possible. Actually, any real system has an intrinsic detection limit, dictated mainly by the noise of the laser [15] and of the photodiode which measures the amplitude fluctuations DI. The detection scheme employed in our system measures the signal fluctuations in the saturated part of the hysteresis cycle, where there are no Barkhausen jumps [9]. These fluctuations are due only to the system noise, which can be therefore characterized as having a zero mean (an offset can be easily corrected) and a standard deviation sN. Then, a detection threshold NT somewhat larger than sN is set, and the signal processing accepts as legitimate signals only the fluctuations DI4NT. However, when a fluctuation DI is slightly above the detection limit, a negative instantaneous noise could bring it below the limit, and therefore produces a rejection of a true signal. Conversely, a positive instantaneous noise could add to a fluctuation DI smaller than NT and bring it above the detection limit. It appears therefore that near this limit, the actual probability distribution p(DI) could be distorted by this phenomenon. The distortion can be calculated analytically for the case of white noise, having a Gaussian amplitude with zero mean and standard deviation sN. Let us suppose that a large number of fluctuations occur in the system, all of them having the same amplitude DI4NT. An ideal system would accept all these fluctuations. After adding the noise, the measured fluctuations will have a Gaussian distribution as shown in Fig. 3, and now only the ones corresponding to the shaded area A will be accepted. This corresponds to a ‘‘detection probability’’ PA given by the shaded area under a standard Gaussian curve: Z 1 1 ðx DIÞ2 pffiffiffiffiffiffi exp PA ðDIÞ ¼ dx 2s2 2psN NT N 1 DI N T 1 þ erf pffiffiffi , (15) ¼ 2 2sN where erf( ) is the error function [16]. Therefore, the complete expression for the probability distribution p(DI) is obtained multiplying Eq. (13) by the above calculated detection probability (only the case aX1 is reported): pðDIÞ ffi PA ðDIÞ
2s2 A DI a . R2 a 1
(16)
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sets and curves have been shifted vertically, and the effect of the cutoff as DI approaches DM1 has been neglected. The agreement at low values of DI improves, suggesting the validity of this approach. In general, the distortion of the probability distribution is of some importance only for the smallest DI, and (as expected) is greatly reduced if one chooses a detection threshold NT which is substantially greater than the noise level sN, with the acceptable drawback of a reduced range of useful data. 4. Conclusions Fig. 3. Schematic effect of apparatus noise on the measured intensity variations DI in the vicinity of the detection threshold NT. Upon addition of Gaussian noise, an ensemble of given fluctuations DI is broadened, and only the ones above the threshold (shaded area A) are correctly detected.
The non-uniform intensity profile of the sampling laser beam and the apparatus noise indeed introduce some distortions into the statistical distribution of the Barkhausen noise measured by means of MOKE. However, these distortions are of small importance for most cases of practical relevance, thus confirming the reliability of MOKE as a tool for statistical analysis of noise in ferromagnetic materials. The mathematical model developed in this paper has shown a good agreement with the experimental results for the important case of power-law statistical distributions; the analytical framework has demonstrated its soundness, and could in principle be extended with similar validity to other statistical distributions of scientific interest. Appendix A
Fig. 4. Effect of system noise on the lower tail of the statistical distribution p(DI). Symbols are the experimental data, dashed lines are the ideal power law distributions, continuous lines are the distributions corrected for the effect of noise. Different detection threshold (listed in the legend in percentage of the amplitude of hysteresis cycle) have been employed in the data extraction. The noise was the same in all cases, and was about 1% of the amplitude of hysteresis cycle. For clarity, each data set has been shifted vertically.
When the fluctuation is p significantly above the detection ffiffiffi threshold (i.e. DI4N T þ 2sN ) the error function tends rapidly to one, PA-1 and the apparatus noise does not affect the measured distribution (as expected). Instead, as DI approaches NT the error function tends to zero, and the measured distribution flattens out. Fig. 4 compares Eq. (16) (continuous lines) to the data collected in the same experiment described in Section 2 (symbols). The spot size was of 280 mm, and the data analysis has been performed with three different detection thresholds: 1%, 3% and 5% of the amplitude of the hysteresis cycle (corresponding to the maximum possible variation DM1). The measurement noise sN was slightly lower than 1% of the same maximum amplitude DM1. For clarity, the three data
This appendix is largely based on the usual transformation theory for random variables, as described for example in Ref. [13], and adapted to our case. Let the magnetization reversal DM and its position of occurrence r be two continuous random variables. Being statistically independent, their joint probability density function is given by the product of the individual probability density functions: 2r . R2 Let X be the domain in the DM–r plane where the joint probability density is positive: pDM;r ðDM; rÞ40. Let us assume that pDM;r ðDM; rÞ ¼ pDM ðDMÞpr ðrÞ ¼ A DM a
Along with the variable DI ¼ DMIðrÞ the new variable r ¼ r2 is introduced. This aims to define a one-to-one transformation of domain X of two-dimensional DM–r space onto a domain W of DI–r space, which is also twodimensional. Any continuous function of DM and r could be chosen as this second variable, but the expression we choose turns out to be especially convenient. The first partial derivatives of the inverse functions DM ¼ DMðDI; rÞ and r ¼ rðDI; rÞ are continuous over W. This is indeed our case, as DI ; DM ¼ IðrÞ pffiffiffi r ¼ r:
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The Jacobian J of the transformation is non-zero: qDM 1 qr IðrÞ ¼ qr 0 qr
qDM qDI J¼ qr qDI ¼
I 0 ðrÞ 1 DI 2 pffiffiffi I ðrÞ 2 r 1 pffiffiffi 2 r
1 1 pffiffiffia0. IðrÞ 2 r
Then, the joint density of the new variables DI and r is given by pDI;r ðDI; rÞ ¼ jJjpDM ðDMðDI; rÞÞpr ðrðDI; rÞÞ pffiffiffi 1 1 DI a 2 r A ¼ pffiffiffi IðrÞ 2 r IðrÞ R2 A ¼ 2 DI a ½IðrÞa1 . R The probability density function of DI is then obtained by integrating the above joint probability density over all the possible values of r:
are bounded by the value DM1. Therefore, for a given DI the maximum admissible rM is the one for which
r DI ¼ DM 1 exp M2 . 2s The integration domain W is then given by the segment DM 1 0prp2s2 ln . DI Hence the final results ( " #) 2s2 1 DI a1 a 1 pDI ðDIÞ ¼ 2 A DI 1 a DM 1 R for aa1;
2s2 DM 1 1 A DI ln DI R2 for a ¼ 1:
pDI ðDIÞ ¼
Z pDI ðDIÞ ¼
jJjpDM ðDMðDI; rÞÞpr ðrðDI; rÞÞ dr Z A ¼ 2 DI a ½IðrÞa1 dr. R W W
The cases of uniform and Gaussian beam will be derived separately. If the beam is uniform and has a radius r0 it is simply IðrÞ ¼ 1
for 0prpr20 ;
IðrÞ ¼ 0
elsewhere:
The integration domain W corresponds to the segment 0prpr20 and the result is immediate: pDI ðDIÞ ¼
r20 A DI a . R2
If the beam is Gaussian, it is
r IðrÞ ¼ exp 2 . 2s In this second case, the integration domain W is a function of DI, and corresponds to the region where this DI results as a combination of all possible DM and r. This region is finite because the possible values of DM
References [1] H. Barkhausen, Z. Phys. 20 (1919) 401. [2] G. Durin, S. Zapperi, The Barkhausen Effect. 2004, arXiv preprint, cond-mat/0404512. [3] G. Bertotti, Hysteresis in Magnetism, Academic Press, San Diego, CA, 1998. [4] E. Puppin, Phys. Rev. Lett. 84 (2000) 5415. [5] M.C. Kuntz, J.P. Sethna, Phys. Rev. B 62 (2000) 11699. [6] Z.Q. Qiu, S.D. Bader, Rev. Sci. Instrum. 71 (2000) 1243. [7] D.H. Kim, S.B. Choe, S.C. Shin, Phys. Rev. Lett. 90 (2003) 087203. [8] L. Callegaro, E. Puppin, Rev. Sci. Instrum. 66 (1995) 5375. [9] E. Puppin, P. Vavassori, L. Callegaro, Rev. Sci. Instrum. 71 (2000) 1752. [10] E. Puppin, E. Pinotti, M. Brenna, J. Appl. Phys. 101 (2007) 063903. [11] B.E. Argyle, J.G. McCord, J. Appl. Phys. 87 (2000) 6487. [12] E. Puppin, M. Brenna, E. Pinotti, G. Valentini, R. Cubeddu, J. Appl. Phys. 101 (2007) 113905. [13] A.M. Mood, F.A. Graybill, D.C. Boes, Introduction to the Theory of Statistics, third ed., McGraw-Hill, New York, 1974 (Chapter 6). [14] In theory, the illuminated area of a Gaussian beam is infinite. However, the beam intensity decreases with the distance down to a point where the reflected light is too weak to be detected. Therefore, even such a Gaussian beam has an ‘‘effective’’ area, which is finite and proportional to s2. [15] The laser noise can be greatly reduced by employing a good-quality amplitude-stabilized laser, and allowing it to stabilize for several hours before the measurements. [16] For an easier calculation, the error function in Eq. (15) can be fairly good approximated with the hyperbolic tangent having the same argument. The maximum relative error is about 10%.
Journal of Magnetism and Magnetic Materials 320 (2008) 1651–1656 www.elsevier.com/locate/jmmm
Distortions of the statistical distribution of Barkhausen noise measured by magneto-optical Kerr effect E. Pinotti, M. Brenna, E. Puppin Department of Physics, Polytechnic of Milano, piazza Leonardo da Vinci, 32–20133 Milano, Italy Received 10 September 2007; received in revised form 11 January 2008 Available online 24 January 2008
Abstract In magneto-optical Kerr measurements of the Barkhausen noise, a magnetization jump DM due to a domain reversal produces a variation DI of the intensity of a laser beam reflected by the sample, which is the physical quantity actually measured. Due to the nonuniform beam intensity profile, the magnitude of DI depends both on DM and on its position on the laser spot. This could distort the statistical distribution p(DI) of the measured DI with respect to the true distribution p(DM) of the magnetization jumps DM. In this work the exact relationship between the two distributions is derived in a general form, which will be applied to some possible beam profiles. It will be shown that in most cases the usual Gaussian beam produces a negligible statistical distortion. Moreover, for small DI the noise of the experimental setup can also distort the statistical distribution p(DI), by erroneously rejecting small DI as noise. This effect has been calculated for white noise, and it will be shown that it is relatively small but not totally negligible as the measured DI approaches the detection limit. r 2008 Elsevier B.V. All rights reserved. Keywords: Magneto-optical Kerr effect; Barkhausen noise; Noise statistics
1. Introduction In ferromagnetic materials the most important noise mechanism is the Barkhausen noise [1], produced by reversals (random in size and time of occurrence) of magnetic domains during the magnetization process. The importance of Barkhausen noise has been largely recognized [2] both as a probe of the various aspects of the magnetization process [3], and as an illustration of stochastic processes in complex systems [4]. It is often viewed as a prominent example of critical phenomena [5], whose most characteristic feature is a power-law probability distribution: pðDMÞ / DM a ,
(1)
where p(DM) is the probability of occurrence of a magnetization reversal of amplitude DM, and a is a number (typically ranging from 1 to 2) called the critical Corresponding author. Tel.: +39 02 2399 6132; fax: +39 02 2399 6126.
E-mail address: ermanno.pinotti@fisi.polimi.it (E. Pinotti). 0304-8853/$ - see front matter r 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2008.01.031
exponent. The exact value of a is thought to be of great importance [2], because it can discriminate between different underlying physical and statistical situations. Usually the Barkhausen noise is measured inductively, by winding a pick-up coil around the sample and detecting the voltage spikes induced into the coil by the magnetization reversals. However, sometimes this experimental setup cannot be employed, as happens for very thin films, microstructured samples and other cases of relevance. In these situations, a simple and powerful measurement apparatus employs the magneto-optical Kerr effect (MOKE) [6,7], where linearly polarized light (usually a laser beam) is directed onto the sample and the reflected light goes through an analyzing polarizer and is finally detected by a photodiode. Due to the Kerr effect a magnetization reversal DM produces a small rotation in the polarization plane of the reflected beam, which (after the analyzing polarizer) is measured by the photodiode as a variation DI in the beam intensity. By using lock-in modulation techniques, a high sensitivity can be reached, and very small magnetization reversals can be measured
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[8]. Moreover, by means of a beam expander and a focusing optics, the spot size can be varied from a few micrometers to about one centimeter [9], therefore probing Eq. (1) for DM varying over several orders of magnitude [10]. However, in a MOKE apparatus the physical quantities actually measured are the beam intensity variations DI, and hence their statistical distribution p(DI). But DI depends both on the corresponding magnetization variation DM and on its position in the illuminated spot. For example, laser beams have typically a Gaussian profile and the light intensity decreases with the distance r from the spot center: r2 IðrÞ ¼ I 0 exp 2 , 2s
(2)
where s is a parameter specifying the spot dimension. Therefore, a magnetization reversal DM happening at a distance r from the spot center would give an intensity variation DI given by r2 DI / DM exp 2 . 2s
(3)
This effect could in principle distort the measured probability distribution p(DI) with respect to the true p(DM). As an extreme example, consider a sample where all magnetization reversals have the same size DM0 illuminated by a Gaussian beam; it can be shown that the corresponding probability distribution of the DI would be p(DI) ¼ DI1, therefore giving a completely erroneous result. To our knowledge, the problem of non-uniform sample illumination has been issued a few years ago [11], and only recently a dedicated experiment has been carried out for the case of uniform and Gaussian beam profiles [12]. In this work we obtain (for the first time, to our knowledge) a theoretical analysis of the distortions possibly arising in MOKE measurements of Barkhausen noise statistics, because the non-uniform sample illumination. A general framework will be laid out, and the results for some possible beam profiles will be obtained in detail. Comparison between theory and experiments will be shown for the most usual case of the Gaussian profile. Another possible distortion of the measured statistical distribution arises for small values of DI, and is due to the noise in the experimental setup. Random fluctuations due to laser and detector noise are unavoidable, and a detection threshold has to be defined in some ways, in order to separate the ‘‘true’’ intensity variations DI from those due to the noise. However, in the vicinity of this threshold, noise fluctuations add to the ‘‘true’’ DI variations, and could produce either a rejection of a valid signals or acknowledgement of a false one, therefore distorting the statistical distribution of DI. This effect will be analyzed for Gaussian noise, and the results will be compared with experiments.
2. Effects of a non-uniform beam profile 2.1. Theoretical model Let us consider Fig. 1. The circle represents the sample surface illuminated with the laser spot, whose intensity I depends on the distance r from the center I ¼ I(r). Let us assume that a magnetization reversal DM takes place in a given location r: r indicates the distance between the center of the spot and the center of gravity of the reversed area. In the following we will assume that the intensity I can be regarded as constant over the whole reversed area and, therefore, the measured value of the reversion DI will be proportional both to DM and to the intensity I(r) of the beam. For simplicity, the proportionality factor will be taken as equal to one, and therefore DI ¼ DMIðrÞ.
(4)
Assuming a constant intensity I over the whole reversed area is of fundamental importance in our analysis, since it allows a simple and solvable analytical formulation. Without this hypothesis, the problem would be analytically intractable, having to take into account not only the value of DM and its position r, but also its geometric shape and orientation. In theory, this assumption is valid only for small reversed areas, of the order of a few percent of the beam spot. Fortunately, in most experiments this is just what one observes, with a great majority of reversals involving an area much smaller than the spot size. The assumption becomes less justifiable as the reversed area approaches the spot size: in this case the actual intensity variation DI can be sensibly higher or lower than expected from the uniform I hypothesis, depending on the geometric shape and orientation of the reversed area. However, only a few domain reversals are of this size, and therefore their weight on the shape of the overall probability distribution is negligible. Moreover, simple Monte Carlo simulations, not reported here for brevity, show that for large reversals
Fig. 1. Geometric outline of the circular sample of radius R, with a magnetization reversal DM occurring at a distance r from the sample center. The impinging laser beam is centered with the sample.
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the actual intensity variation DI (averaged over many possible shapes and orientations) is similar to the ideal one, calculated with the uniform I assumption. The two cases differ at most by a factor of about two, which is relatively unimportant if one remembers that this difference applies only to a few reversals, moreover happening at the end of the range of interest for the probability distribution. Hence, in our analysis we will rely on the uniform I assumption which produces Eq. (4). The function I(r) has the following properties:
I(r) is single valued, monotonically decreasing with r, i.e. I(r2)oI(r1) for r24r1. This choice excludes therefore multi-valued functions as the ones deriving from diffraction of the laser beam, multi-mode beams, and so on. These cases can be naturally of interest, but their mathematical analysis is much more difficult; moreover, these conditions should be avoided in a well-conducted experiment. The maximum value of I(r) arises for r ¼ 0, and in the following it will be assumed that I(0) ¼ 1. This choice comes naturally from the fact that in our MOKE measurements the experimental apparatus automatically normalizes the measured DI to the full amplitude of the measured hysteresis cycle, therefore losing information about the absolute value of the beam intensity. However, this choice requires a non-trivial scaling of the measured distributions, as already described in Ref. [4]. This scaling will be discussed in more detail in Section 3.
Since we are considering power-law distributions, we will assume that the ‘‘true’’ probability distribution of the magnetization reversals DM is of the form (5)
where a is the critical exponent and A is a normalization factor, whose value can be obtained by imposing the normalization on the probability distribution Z DM 1 pðDMÞ dDM ¼ 1. (6) DM 0
Here DM0 and DM1 are respectively the smallest and the greatest possible values for DM. In principle, both depend on the sample, but their actual value depends on the experimental setup. DM0 is dictated by the sensitivity and noise of the measurement apparatus, while DM1 cannot exceed the whole beam area (obviously the experiment cannot detect reversals extending beyond the illuminated spot). From the normalization requirement, A can be obtained: 1a if aa1, DM 1a 0 1 if a ¼ 1. A¼ lnðDM 1 =DM 0 Þ
A¼
As for the probability distribution of the reversal position r, we will assume that the DM take place with equal probability over the whole sample surface. Therefore, if we assume for simplicity a circular sample of radius R, it can be shown that the probability distribution for r is given by 2r . (8) R2 It is reasonable that, for an uniform sample, the amplitude DM of a reversal is statistically independent of its position r. Therefore the joint probability distribution p(DM, r) of a reversal of given magnitude DM appearing at a given position r is simply the product of the two probability distributions for DM and r:
pðrÞ ¼
pðDM; rÞ ¼ pðDMÞpðrÞ.
(9)
The problem is how to find the statistical distribution of the random variable (10)
DI ¼ DMIðrÞ
given the probability distributions of the random variables DM and r. The solution employs the usual tools of the statistical theory [13] and is outlined in Appendix A. Here only the final results will be given. 2.3. Resulting statistical distribution for a uniform beam As expected, for the case of a uniform beam the measured distribution p(DI) has the same analytical expression of the real distribution p(DM): r20 A DI a (11) R2 apart from the scaling factor r20/R2. This factor is proportional to the area of the beam, and it represents mathematically the reason of the scaling already described in Ref. [4]: in a MOKE experiment, the probability density p(DI) is obtained by counting how many jumps do have an amplitude DI. But, if the sample is uniform, the number of the jumps counted grows as the area of the beam spot, and so does the measured density probability. This effect has to be corrected in order to compare measurements made with different beam sizes, and the correction requires a division of the measured distribution by the spot area. pðDIÞ ¼
2.2. Probability distributions for DM and r
pðDMÞ ¼ A DM a ,
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2.4. Resulting statistical distribution for a Gaussian beam For a Gaussian beam, the measured distribution is instead given by ( " #) 2s2 1 DI a1 a pðDIÞ ¼ 2 A DI 1 1 a DM 1 R for aa1;
DM 1a 1
(7)
2s2 DM 1 1 pðDIÞ ¼ 2 A DI ln DI R
for a ¼ 1:
(12)
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Therefore, the measured distribution is given by the ‘‘true’’ one scaled by 2s2/R2 and multiplied by the expression in the curly brackets. The scaling factor again represents the spot area, which is proportional to s2, and implies the same scaling described for the uniform case [14]. The expression in curly brackets represents the distortion due to the non-uniform beam profile, and is the most important result of this analysis. The distortion it introduces in the measured probability distribution is small when aX1. In fact, since DM1 corresponds to the total beam area, it must always be DIpDM1; moreover, as the MOKE detects jumps varying over several orders of magnitude below DM1, for most of the measured DI it is DI5DM1. Therefore, when aX1 it is almost always (DI/ DM1)a151 and the distribution p(DI) becomes the correct power law: pðDIÞ ffi
2s2 A DI a R2 a 1
for a41:
(13)
As for the case a ¼ 1, the logarithm ln(DM1/DI) varies slowly when its argument is large, and again it introduces a relatively small correction. The situation is different for ao1, because in this case it is almost always (DI/DM1)a1b1, and the measured distribution becomes 2s2 DM 11a pðDIÞ ffi 2 (14) A DI 1 for ao1: 1a R Therefore, when the critical exponent is less than one, the measured distribution tends always to be of the form DI1 whatever the actual a. Fig. 2 shows the comparison between Eq. (12) and an experiment performed on an amorphous film of iron, 50 nm in thickness, with a Gaussian laser beam of three different sizes. By assuming a critical exponent equal to one
Fig. 2. Comparison between the experimental data (symbols) and the fitting curves (solid lines) derived from Eq. (12) for a laser beam with Gaussian profile. The distributions have been already scaled by the beam size, as described in the text, and are consistent with the power law over more than four orders of magnitude.
the agreement between experiment and Eq. (12) is very good. It is worthwhile to note that Eq. (12) also introduces the correct expression for the cutoff, i.e. the decay to zero of the measured p(DI) as the maximum DM1 is approached. Only for the smallest DM the agreement between Eq. (12) and the experimental data is unsatisfactory. This is due to the noise effect of the experimental setup, which will be described in Section 3. 3. Effects of noise in the measurement apparatus An ideal noiseless MOKE apparatus could detect an intensity fluctuation DI of any amplitude, down to the smallest value physically possible. Actually, any real system has an intrinsic detection limit, dictated mainly by the noise of the laser [15] and of the photodiode which measures the amplitude fluctuations DI. The detection scheme employed in our system measures the signal fluctuations in the saturated part of the hysteresis cycle, where there are no Barkhausen jumps [9]. These fluctuations are due only to the system noise, which can be therefore characterized as having a zero mean (an offset can be easily corrected) and a standard deviation sN. Then, a detection threshold NT somewhat larger than sN is set, and the signal processing accepts as legitimate signals only the fluctuations DI4NT. However, when a fluctuation DI is slightly above the detection limit, a negative instantaneous noise could bring it below the limit, and therefore produces a rejection of a true signal. Conversely, a positive instantaneous noise could add to a fluctuation DI smaller than NT and bring it above the detection limit. It appears therefore that near this limit, the actual probability distribution p(DI) could be distorted by this phenomenon. The distortion can be calculated analytically for the case of white noise, having a Gaussian amplitude with zero mean and standard deviation sN. Let us suppose that a large number of fluctuations occur in the system, all of them having the same amplitude DI4NT. An ideal system would accept all these fluctuations. After adding the noise, the measured fluctuations will have a Gaussian distribution as shown in Fig. 3, and now only the ones corresponding to the shaded area A will be accepted. This corresponds to a ‘‘detection probability’’ PA given by the shaded area under a standard Gaussian curve: Z 1 1 ðx DIÞ2 pffiffiffiffiffiffi exp PA ðDIÞ ¼ dx 2s2 2psN NT N 1 DI N T 1 þ erf pffiffiffi , (15) ¼ 2 2sN where erf( ) is the error function [16]. Therefore, the complete expression for the probability distribution p(DI) is obtained multiplying Eq. (13) by the above calculated detection probability (only the case aX1 is reported): pðDIÞ ffi PA ðDIÞ
2s2 A DI a . R2 a 1
(16)
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sets and curves have been shifted vertically, and the effect of the cutoff as DI approaches DM1 has been neglected. The agreement at low values of DI improves, suggesting the validity of this approach. In general, the distortion of the probability distribution is of some importance only for the smallest DI, and (as expected) is greatly reduced if one chooses a detection threshold NT which is substantially greater than the noise level sN, with the acceptable drawback of a reduced range of useful data. 4. Conclusions Fig. 3. Schematic effect of apparatus noise on the measured intensity variations DI in the vicinity of the detection threshold NT. Upon addition of Gaussian noise, an ensemble of given fluctuations DI is broadened, and only the ones above the threshold (shaded area A) are correctly detected.
The non-uniform intensity profile of the sampling laser beam and the apparatus noise indeed introduce some distortions into the statistical distribution of the Barkhausen noise measured by means of MOKE. However, these distortions are of small importance for most cases of practical relevance, thus confirming the reliability of MOKE as a tool for statistical analysis of noise in ferromagnetic materials. The mathematical model developed in this paper has shown a good agreement with the experimental results for the important case of power-law statistical distributions; the analytical framework has demonstrated its soundness, and could in principle be extended with similar validity to other statistical distributions of scientific interest. Appendix A
Fig. 4. Effect of system noise on the lower tail of the statistical distribution p(DI). Symbols are the experimental data, dashed lines are the ideal power law distributions, continuous lines are the distributions corrected for the effect of noise. Different detection threshold (listed in the legend in percentage of the amplitude of hysteresis cycle) have been employed in the data extraction. The noise was the same in all cases, and was about 1% of the amplitude of hysteresis cycle. For clarity, each data set has been shifted vertically.
When the fluctuation is p significantly above the detection ffiffiffi threshold (i.e. DI4N T þ 2sN ) the error function tends rapidly to one, PA-1 and the apparatus noise does not affect the measured distribution (as expected). Instead, as DI approaches NT the error function tends to zero, and the measured distribution flattens out. Fig. 4 compares Eq. (16) (continuous lines) to the data collected in the same experiment described in Section 2 (symbols). The spot size was of 280 mm, and the data analysis has been performed with three different detection thresholds: 1%, 3% and 5% of the amplitude of the hysteresis cycle (corresponding to the maximum possible variation DM1). The measurement noise sN was slightly lower than 1% of the same maximum amplitude DM1. For clarity, the three data
This appendix is largely based on the usual transformation theory for random variables, as described for example in Ref. [13], and adapted to our case. Let the magnetization reversal DM and its position of occurrence r be two continuous random variables. Being statistically independent, their joint probability density function is given by the product of the individual probability density functions: 2r . R2 Let X be the domain in the DM–r plane where the joint probability density is positive: pDM;r ðDM; rÞ40. Let us assume that pDM;r ðDM; rÞ ¼ pDM ðDMÞpr ðrÞ ¼ A DM a
Along with the variable DI ¼ DMIðrÞ the new variable r ¼ r2 is introduced. This aims to define a one-to-one transformation of domain X of two-dimensional DM–r space onto a domain W of DI–r space, which is also twodimensional. Any continuous function of DM and r could be chosen as this second variable, but the expression we choose turns out to be especially convenient. The first partial derivatives of the inverse functions DM ¼ DMðDI; rÞ and r ¼ rðDI; rÞ are continuous over W. This is indeed our case, as DI ; DM ¼ IðrÞ pffiffiffi r ¼ r:
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The Jacobian J of the transformation is non-zero: qDM 1 qr IðrÞ ¼ qr 0 qr
qDM qDI J¼ qr qDI ¼
I 0 ðrÞ 1 DI 2 pffiffiffi I ðrÞ 2 r 1 pffiffiffi 2 r
1 1 pffiffiffia0. IðrÞ 2 r
Then, the joint density of the new variables DI and r is given by pDI;r ðDI; rÞ ¼ jJjpDM ðDMðDI; rÞÞpr ðrðDI; rÞÞ pffiffiffi 1 1 DI a 2 r A ¼ pffiffiffi IðrÞ 2 r IðrÞ R2 A ¼ 2 DI a ½IðrÞa1 . R The probability density function of DI is then obtained by integrating the above joint probability density over all the possible values of r:
are bounded by the value DM1. Therefore, for a given DI the maximum admissible rM is the one for which
r DI ¼ DM 1 exp M2 . 2s The integration domain W is then given by the segment DM 1 0prp2s2 ln . DI Hence the final results ( " #) 2s2 1 DI a1 a 1 pDI ðDIÞ ¼ 2 A DI 1 a DM 1 R for aa1;
2s2 DM 1 1 A DI ln DI R2 for a ¼ 1:
pDI ðDIÞ ¼
Z pDI ðDIÞ ¼
jJjpDM ðDMðDI; rÞÞpr ðrðDI; rÞÞ dr Z A ¼ 2 DI a ½IðrÞa1 dr. R W W
The cases of uniform and Gaussian beam will be derived separately. If the beam is uniform and has a radius r0 it is simply IðrÞ ¼ 1
for 0prpr20 ;
IðrÞ ¼ 0
elsewhere:
The integration domain W corresponds to the segment 0prpr20 and the result is immediate: pDI ðDIÞ ¼
r20 A DI a . R2
If the beam is Gaussian, it is
r IðrÞ ¼ exp 2 . 2s In this second case, the integration domain W is a function of DI, and corresponds to the region where this DI results as a combination of all possible DM and r. This region is finite because the possible values of DM
References [1] H. Barkhausen, Z. Phys. 20 (1919) 401. [2] G. Durin, S. Zapperi, The Barkhausen Effect. 2004, arXiv preprint, cond-mat/0404512. [3] G. Bertotti, Hysteresis in Magnetism, Academic Press, San Diego, CA, 1998. [4] E. Puppin, Phys. Rev. Lett. 84 (2000) 5415. [5] M.C. Kuntz, J.P. Sethna, Phys. Rev. B 62 (2000) 11699. [6] Z.Q. Qiu, S.D. Bader, Rev. Sci. Instrum. 71 (2000) 1243. [7] D.H. Kim, S.B. Choe, S.C. Shin, Phys. Rev. Lett. 90 (2003) 087203. [8] L. Callegaro, E. Puppin, Rev. Sci. Instrum. 66 (1995) 5375. [9] E. Puppin, P. Vavassori, L. Callegaro, Rev. Sci. Instrum. 71 (2000) 1752. [10] E. Puppin, E. Pinotti, M. Brenna, J. Appl. Phys. 101 (2007) 063903. [11] B.E. Argyle, J.G. McCord, J. Appl. Phys. 87 (2000) 6487. [12] E. Puppin, M. Brenna, E. Pinotti, G. Valentini, R. Cubeddu, J. Appl. Phys. 101 (2007) 113905. [13] A.M. Mood, F.A. Graybill, D.C. Boes, Introduction to the Theory of Statistics, third ed., McGraw-Hill, New York, 1974 (Chapter 6). [14] In theory, the illuminated area of a Gaussian beam is infinite. However, the beam intensity decreases with the distance down to a point where the reflected light is too weak to be detected. Therefore, even such a Gaussian beam has an ‘‘effective’’ area, which is finite and proportional to s2. [15] The laser noise can be greatly reduced by employing a good-quality amplitude-stabilized laser, and allowing it to stabilize for several hours before the measurements. [16] For an easier calculation, the error function in Eq. (15) can be fairly good approximated with the hyperbolic tangent having the same argument. The maximum relative error is about 10%.