Angular Preisach analysis of Hysteresis loops and FMR lineshapes of ferromagnetic nanowire arrays
AAngular Preisach analysis of Hysteresis loops and FMR lineshapes of ferromagneticnanowire arrays
C. Tannous, A. Ghaddar
Laboratoire de Magn´etisme de Bretagne - CNRS FRE 3117- Universit´e de Bretagne Occidentale - 6,Avenue le Gorgeu C.S.93837 - 29238 Brest Cedex 3 - FRANCE.
J. Gieraltowski
Laboratoire des domaines Oc´eaniques, IUEM CNRS-UMR 6538,Technopole Brest IROISE 29280 Plouzan´e, FRANCE.
Preisach analysis is applied to the study of hysteresis loops measured for different angles betweenthe applied magnetic field and the common axis of ferromagnetic Nickel nanowire arrays. Whenextended to Ferromagnetic Resonance (FMR) lineshapes, with same set of parameters extractedfrom the corresponding hysteresis loops, Preisach analysis shows that a different distribution ofinteractions or coercivities ought to be used in order to explain experimental results. Inspecting thebehavior of hysteresis loops and FMR linewidth versus field angle, we infer that angular dependencemight be exploited in angle sensing devices that could compete with Anisotropic (AMR) or GiantMagnetoresistive (GMR) based devices.
I. INTRODUCTION
Ferromagnetic nanowires possess interesting properties that might be exploited in spintronic devices such as race-track type magnetic non-volatile memory called MRAM (based on transverse domain-wall dynamics ) and magneticlogic devices . They might also be used in magnonic devices based on spin-wave excitation and propagation .Ferromagnetic nanowires have applications in microwave devices such as circulators , superconducting single-photonGHz detectors and counters , information storage (as recording media and read-write devices), Quantum transport(such as GMR circuits) as well as in Quantum computing and Telecommunication.They are simpler than nanotubes since their physical properties do not depend on chirality and they can be grownwith a variety of methods : Molecular Beam Epitaxy, Electrochemical methods (Template synthesis, AnodicAlumina filters), Chemical solution techniques (Self-assembly, Sol-Gel, emulsions...) and can be grown with a tunablenumber of monolayers and length .Ordered arrays of nanowires may be of paramount importance in areas such as high-density patterned mediainformation recording an example of which is the Quantum Magnetic Disk . They might be also of interest in novelhigh-frequency communication or signal-processing devices based on the exploitation of spin-waves (in magnoniccrystals made of magnetic superlattices or multilayers) to transfer and process information or spin-currents with nodissipative Joule effect.In this work, we explore the possibility for Nickel ferromagnetic nanowire arrays (FNA) to be of interest in anglesensitive devices. For this goal we perform field angle dependent hysteresis loops and FMR lineshape measurementsin the X-band (9.4 GHz). Preisach analysis is applied to extract from the measured effective anisotropy field H eff several angle dependent physical parameters (such as interaction and coercivity) while changing nanowire diameterfrom 15 nm to 100 nm.These findings might be exploited in angle dependent sensing devices that might compete with present AMR orGMR angle sensors.This work is organized as follows: In section 2, measured hysteresis loops versus field angle are presented andanalyzed with Preisach modeling, whereas in section 3 the same analysis is performed on the FMR lineshape measure-ments. We conclude the work in section 4. Appendix I details the FMR angular fitting procedure whereas AppendixII is a general overview of Preisach modeling. II. HYSTERESIS LOOPS VERSUS FIELD ANGLE
Our Nickel FNA are fabricated with an electrochemical deposition method similar to the one used by Kartopu etal. and the common length is 6 µ m for all diameters while the average interwire distance is about 350 nm.We have performed angle (0 ◦ , 30 ◦ , 45 ◦ , 60 ◦ and 90 ◦ ) dependent VSM (Vibrating Sample Magnetometry) and FMR onthese variable diameter (15 nm, 50 nm, 80 nm and 100 nm) arrays from liquid Helium (4.2 K) to room temperature .We have shown that the easy axis orientation for the 15 nm diameter sample is perpendicular to the wire axis in sharp a r X i v : . [ c ond - m a t . m t r l - s c i ] J un contrast with the 50 nm, 80 nm and 100 nm samples. This is a surprising result since we expect (from bulk Ni) thatthe easy axis along the wire axis by comparing the value of shape energy with respect to anisotropy energy. θ H θ a c x yMHz φ H φ FIG. 1: Magnetization M , applied field H and corresponding angles θ, φ, θ H , φ H they make with the nanowire axis that canbe considered as an ellipsoid-shaped single domain with characteristic lengths a = d/ c with d the diameter. When theaspect ratio c/a is large enough the ellipsoid becomes an infinite cylinder. Results obtained from the angular behavior of the resonance field H res versus θ H shows that H res is minimumat 90 ◦ for the 15 nm sample whereas it is minimum at 0 ◦ for the larger diameter samples agree with hysteresis loopsobtained from VSM measurements and confirm presence of the transition of easy axis direction from perpendicularat 15 nm to parallel to nanowire axis at 50 nm diameter.In this work we concentrate on room-temperature angle dependence experimental results and modeling. Preisachmodeling is used to understand the angular behavior of the hysteresis loops and the FMR lineshapes. After explainingthe shortcomings of the Classical Preisach Model (CPM) we use the Preisach Model for Patterned Media (PM2) tointerpret the static (hysteresis loops) and the dynamic (FMR) measurements for all angles and diameters.The Preisach modeling we use is based essentially on probability densities for interaction h i and coercive h c fields.If we rotate the field distribution ( h i , h c ) by 45 ◦ with respect to the reference system ( H α , H β ) (or switching fieldsystem; see Appendix I), we get the relations: h i = ( H α + H β ) / √ , h c = ( H α − H β − H ) / √ H is the distribution maximum.The CPM density is given by a product of two Gaussian densities pertaining to the interaction and coercive fieldsdegrees of freedom: p ( h i , h c ) = 12 πσ i σ c exp( − h i σ i ) exp( − h c σ c ) (2)where the standard deviation of the interaction and coercive fields are given by σ i and σ c respectively.The PM2 model is based on the following description: p ( h i , h c ) = 12 πσ i σ c exp( − h c σ c ) ×{ ( 1 + m − ( h i − h i ) σ i ) + ( 1 − m − ( h i + h i ) σ i ) } (3)where the normalized magnetization m = MM S has been introduced as well as an average interaction field h i . Themagnetization M is determined by double integration over the field distribution (see Appendix I). Moreover, coercivity −1−0.5 0 0.5 1 −4000 −2000 0 2000 4000 M / M s H (Oe)−1−0.5 0 0.5 1 −2000 −1000 0 1000 2000 M / M s H (Oe)−1−0.5 0 0.5 1 −2000 −1000 0 1000 2000 M / M s H (Oe)−1−0.5 0 0.5 1 −2000 −1000 0 1000 2000 M / M s H (Oe)
FIG. 2: (Color on-line) Room temperature VSM measured hysteresis loops (continuous blue lines)
M/M S versus H (in Oersteds)in the 15, 50, 80 and 100 nm diameter cases (displayed from top to bottom) and their Preisach fit (black dots). The field isalong the nanowire axis. Note the mismatch observed in the 50 nm case as discussed in the text. is represented by a single Gaussian density whereas interactions are represented by a superposition of two Gaussiandensities shifted to left and right with respect with respect to the average field h i .Loop inclination increases with σ i whereas loop width increases with σ c . Hence interactions between ”hysterons”(or nanowires in our case) are responsible for the inclination observed in the VSM hysteresis loops. The fit parametersare given in table I.The coercivity parameter reflects presence of pinning centers hampering domain motions.The Preisach fit is next made on the angular (between field and nanowire axis, see fig. 1) dependent hysteresis loopsand we present in Table II the detailed results, as an example, for the 100 nm diameter case.One of the great advantages of the PM2 model is that the loop width is controlled by the standard deviation of the TABLE I: Fitting parameters of the PM2 model of the hysteresis loops of Ni 15, 50, 80 and 100 nm diameter samples whenexternal field H is along nanowire axis i.e. θ H = 0 (see fig. 1). d (nm) H (Oe) h i (Oe) σ i (Oe) σ c (Oe)15 120 600 1800 55550 180 30 600 60080 170 170 200 390100 170 300 650 348TABLE II: Fitting parameters of the PM2 model of the hysteresis loops of Ni 100 nm diameter samples for several angles θ H between external field H and nanowire axis (see fig. 1).Angle θ H H (Oe) h i (Oe) σ i (Oe) σ c (Oe)0 ◦
170 300 650 34830 ◦
170 200 750 36045 ◦
170 100 900 34560 ◦
100 50 1180 36590 ◦
100 20 1300 450 coercive fields σ c whereas its inclination is tunable by the standard deviation of the interaction fields σ i . Once we fitthe hysteresis loop we use the same parameters to evaluate the FMR lineshape as explained next. III. FMR LINESHAPE VERSUS ANGLE
Individual wires inside the array are aligned parallel to each other within a deviation of a few degrees. They arecharacterized by a cylindrical shape with a typical variation in diameter of less than 5% with a low-surface roughnessand a typical length of 6 microns.FMR experiments are performed with the microwave pumping field h rf operating at 9.4 GHz with a DC bias field H making a variable angle θ H with the nanowire axis (through sample rotation).Previously, several studies have considered reversal modes by domain nucleation and propagation (see for instanceHenry et al. for an extensive discussion of the statistical determination of reversal processes and distributionfunctions of domain nucleation and propagation fields). Moreover, Ferr´e et al. and Hertel showed the existenceof domains with micromagnetic simulations). We do not consider domain nucleation and propagation in this workand rather concentrate on transverse single domain case.Thus, the angular dependence of H res in the uniform mode is obtained by considering an ellipsoid with energy E comprised of a small second-order effective uniaxial anisotropy contribution ( K term in eq. 4) and (shape)demagnetization energy ( πM S term in eq. 4 with M S the saturation magnetization). Their sum is the total anisotropyenergy E A to which we add a Zeeman term E Z due to the external field H : E = E A + E Z = ( K + πM S ) sin θ − M S H [sin θ sin θ H cos( φ − φ H ) + cos θ cos θ H ] (4) θ is the angle the magnetization makes with the nanowire axis (see fig. 1).The resonance frequency is obtained from the Smit-Beljers formula that can be derived from the Landau-Lifshitzequation of motion with a damping term α : (cid:20) ωγ (cid:21) = (1 + α ) M S sin θ (cid:34) ∂ E∂θ ∂ E∂φ − (cid:18) ∂ E∂θ ∂φ (cid:19) (cid:35) (5)The frequency linewidth is given by: ∆ ω = γαM S (cid:18) ∂ E∂θ + 1sin θ ∂ E∂φ (cid:19) (6)The frequency-field dispersion relation is obtained from the Smit-Beljers equation after evaluating the angularsecond derivatives of the total energy and taking φ = φ H = π : ωγ = (cid:113) (1 + α )[ H eff cos 2 θ + H cos( θ − θ H )] × (cid:113) [ H eff cos θ + H cos( θ − θ H )] (7)At resonance, we have ω = ω r , the resonance frequency, θ = θ H and the applied field H = H res , the resonancefield, when we are dealing with the saturated case. In the unsaturated case the magnetization angle θ (cid:54) = θ H and onedetermines it directly from energy minimization.The above relation 7 provides a relationship between the effective anisotropy field H eff and the external field H at the resonance frequency.Generally, the effective anisotropy field H eff can be obtained from the vectorial functional derivative of the energy E A (eq. 4) with respect to magnetization H eff = − δE A δ M that becomes in the uniform case the gradient with respectto the magnetization components H eff = − ∂E A ∂ M .Moreover, we need to determine magnetization orientation θ at equilibrium. This is obtained from the minimumcondition by evaluating the first derivative ( ∂E∂θ ) θ = 0 and requiring positivity of the second derivative. Consequently,we get: ( K + πM S ) sin 2 θ = M S H sin ( θ H − θ ) (8)This equilibrium equation 8 and Smit-Beljers equation 7 are used simultaneously to determine the resonance field H res versus angle θ H as analyzed next. A. Analysis of the effective anisotropy field
In order to evaluate the total effective anisotropy field H eff , we include in the energy E A , demagnetization,magnetocrystalline anisotropy, and interactions among nanowires, with the corresponding fields H dem = 2 πM S , H K = K M S and H i , thus: H eff = H dem + H i + H K (9)The interaction field H i comprises dipolar interactions between nanowires that depend on porosity P (filling factor)and additional interactions as described in the CPM and PM2 models (see Appendix II). For example, if we considerthe simplest case, demagnetization and dipolar fields are of the same form and may be written as a single term2 πM S (1 − P ).Experimentally, the resonance field H res versus field angle θ H peaks at ω r /γ , hence it is possible to extract theeffective anisotropy field H eff through the use of eq. 7. Thus the Land´e g -factor, saturation magnetization M S andcubic anisotropy constant K can be determined with a least-squares fitting method similar to the one used inAppendix I.This yields the following table III containing fitting parameters K and M S (Anisotropy and saturation magneti-zation) versus diameter.From table III, one infers that as the diameter increases the Ni bulk values are steadily approached which is a goodtest of the FMR fit. B. Preisach modeling of FMR lineshape and transverse susceptibility
The FMR lineshape is obtained from the field derivative d<χ (cid:48)(cid:48) xx >dH of the average transverse susceptibility imaginarypart < χ (cid:48)(cid:48) xx > given by: < χ (cid:48)(cid:48) xx > = (cid:120) S p ( H α , H β ) χ (cid:48)(cid:48) xx dH α dH β (10) TABLE III: Room temperature fitting parameters K and M S with corresponding Nickel nanowire diameter d and averageseparation D . Effective H eff and anisotropy H K fields are determined with Smit-Beljers. Comparing with bulk Nickelanisotropy coefficient at room temperature: K = − . × erg/cm and saturation magnetization M S =485 emu/cm weinfer that as the diameter increases we get closer to the bulk values as expected with K changing by about two orders ofmagnitude. d D K M S H eff H K (nm) (nm) (erg/cm ) (emu/cm ) (Oe) (Oe)15 256 -1.909 × × × × The fields H α , H β are the switching fields that define the Preisach plane (see Appendix II) over which the doubleintegration above is performed in order to estimate the average.The expression of the transverse susceptibility imaginary part is derived directly from the energy and given by: χ (cid:48)(cid:48) xx = ω ( ω r − ω ) + ω ∆ ω r × (cid:20) − γ (1 + α ) (cid:18) ∂ E∂θ (cid:19) ∆ ω r + αγM S ( ω r − ω ) (cid:21) (11)Performing the above double integral over the Preisach plane we spline the values obtained and take the derivativewith respect to H from the splined value (see for instance Numerical Recipes ). The results are displayed in fig. 3.The FMR derivative spectrum is easily found to be asymmetric in contrast to what is normally obtained withLandau-Lifshitz-Gilbert modeling. The Preisach PM2 results agree with lineshapes previously obtained in the liter-ature by Ebels et al. as well as Dumitru et al. but not with our measurements (see fig. 4) that display a smallfield shift with angle θ H .In table IV we display results we obtain for the PM2 parameters that fitted the hysteresis loops and FMR mea-surements of χ (cid:48)(cid:48) xx versus field in the Ni2 and Ni6 sample cases . Note that some values of Table IV are different fromthose given in Table I of ref. , nevertheless it shows that the PM2 model is capable of achieving hysteresis loop andFMR results for the samples Ni2 and Ni6. TABLE IV: Results obtained for PM2 model parameters belonging to samples Ni2 and Ni6 studied by Dumitru et al. . Wediffer from some of the parameters displayed in their Table I.Fields (Oe) Field orientation Ni2 Ni6 H in wire plane 120 180perpendicular to wire plane 125 170 h i in wire plane 1430 180perpendicular to wire plane 70 580 σ i in wire plane 260 720perpendicular to wire plane 250 610 σ c in wire plane 40 240perpendicular to wire plane 60 265 Turning to the calculated linewidths concerning our FMR measurements, we infer that they are smaller than theexperimental values which implies that we have to include additional interactions in the dynamic (FMR) calculation.This is due to the fact we concentrate on the PM2 model with the same values of the parameters that previouslyfitted the hysteresis loops for all field angles. This contrasts with Dumitru et al. who did the fit for two orientationsof the field only ( θ H = 0 ◦ and 90 ◦ ).Moreover, let us point out from the H res versus θ H fit (in Table III) that in the dynamic case, several values, suchas the anisotropy constant K changes significantly, when diameter is reduced from 100 nm to 15 nm because of −1.2−1−0.8−0.6−0.4−0.2 0 0.2 0.4 0 2000 4000 6000 8000 A b s o r p ti on d e r i v a ti v e ( a . u . ) H (Oe) 0°30°45°60°90°−1.6−1.4−1.2−1−0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0 2000 4000 6000 8000 A b s o r p ti on d e r i v a ti v e ( a . u . ) H (Oe) 0°30°45°60°90°−1.6−1.4−1.2−1−0.8−0.6−0.4−0.2 0 0.2 0.4 0 2000 4000 6000 8000 A b s o r p ti on d e r i v a ti v e ( a . u . ) H (Oe) 0°30°45°60°90°−1.6−1.4−1.2−1−0.8−0.6−0.4−0.2 0 0.2 0.4 0 2000 4000 6000 8000 A b s o r p ti on d e r i v a ti v e ( a . u . ) H (Oe) 0°30°45°60°90°
FIG. 3: (Color on-line) Calculated derivative of the absorption d<χ (cid:48)(cid:48) xx >dH versus field for several field angles θ H and all diameters.The Preisach parameters are the same used in the hysteresis loop fit. The lineshape is in arbitrary units and drawn for fieldangles of 0, 30, 45, 60 and 90 degrees, in all diameter cases: 15, 50, 80 and 100 nm (from top to bottom). the appearance of surface anisotropy . Therefore a more complex Preisach model needed in order to fit static anddynamic results with the same sets of parameters for all angles. C. Angular Analysis of FMR lineshape width results
Measured FMR lineshapes (absorption derivative spectra ) for different angles are displayed in fig. 4 for 15 nm,50 nm, 80 nm and 100 nm diameter cases. Generally the lineshapes behave versus magnetic field as the derivative ofa Lorentzian.We use a least-squares algorithm to extract the values of the Lorentzian widths as explained in Appendix I.The width results displayed in Table V show that it is possible to relate unambiguously the value of the Lorentzderivative width to the angle θ H for a given diameter, hence the possibility to build angle sensors on the basis of thatobservation. TABLE V: Lorentz derivative widths (in Oe), versus field angle θ H , fitted with respect to experimental angular FMR lineshapespertaining to 15, 50, 80 and 100 nm diameter samples. d (nm) 0 ◦ ◦ ◦ ◦ ◦
15 3557 2742 1806 1791 136450 1020 1120 1251 1495 206980 839 1092 1278 1349 2215100 638 777 909 1100 1227
Lorentz derivative widths generally decrease as we increase nanowire diameter for all angles. For a fixed diameter,they increase with angle in the 50, 80 and 100 nm whereas in the 15 nm case they decrease. This originates fromthe fact, the 15 nm case possesses a large surface anisotropy as analysed previously in ref. , besides the lineshapebehavior in this case is more complicated than the larger diameter cases. IV. DISCUSSION AND CONCLUSION
We have performed angular Preisach analysis for the static (VSM hysteresis loops) and dynamic measurements(FMR lineshape widths) and shown that many results can be deeply understood and might be further developed inorder to be embedded in applications such as angle detection sensors.The transition at 50 nm in VSM and FMR measurements is extremely promising because of several potentialapplications in race-track MRAM devices. Yan et al. predicted that in Permalloy nanowires of 50 nm and less,moving zero-mass domain walls may attain a velocity of several 100 m/s beating Walker limit obeyed in Permalloystrips with same lateral size. Hence, nanowire cylindrical geometry in contrast to prismatic geometry of stripes bearsimportant consequences on current injection in nanowires that applies Slonczewski type torques on magnetizationaffecting domain wall motion with reduced Ohmic losses .Ordered arrays of nanowires are good candidates for patterned media and may also be used in plasmonic applicationssuch as nano-antenna arrays or nanophotonic waveguides in integrated optics . Recently , heat assisted magneticperpendicular recording using plasmonic aperture nano-antenna has been tested on patterned media in order to processlarge storage densities starting at 1 Tbits/in and scalable up to 100 Tbits/in .While the Preisach PM2 model can explain separately the static or dynamic results, one might extend it throughthe use of other distributions of interaction and coercivity in order to explain the VSM and FMR measurementssimultaneously for all angles.Nonetheless, the angular behavior of the linewidth is interesting enough to consider its use in angle sensors thatmight compete with present technology based on AMR or GMR effects. Acknowledgments
Some of the FMR measurements were kindly made by Dr. R. Zuberek at the Institute of Physics of the PolishAcademy of Science, Warsaw (Poland). −10−8−6−4−2 0 2 4 6 8 10 12 0 2000 4000 6000 8000 A b s o r p ti on d e r i v a ti v e ( a . u . ) H (Oe) 0°30°45°60°90°−1−0.5 0 0.5 1 1.5 0 2000 4000 6000 8000 A b s o r p ti on d e r i v a ti v e ( a . u . ) H (Oe) 0°30°45°60°90°−4−3−2−1 0 1 2 3 4 5 6 7 0 2000 4000 6000 8000 A b s o r p ti on d e r i v a ti v e ( a . u . ) H (Oe) 0°30°45°60°90°−40−30−20−10 0 10 20 30 40 0 2000 4000 6000 8000 A b s o r p ti on d e r i v a ti v e ( a . u . ) H (Oe) 0°30°45°60°90°
FIG. 4: (Color on-line) Measured FMR lineshape versus field for different angles with respect to nanowire axis at a frequencyof 9.4 GHz and at room temperature. The lineshape d<χ (cid:48)(cid:48) xx >dH in arbitrary units is drawn for various angles θ H = 0, 30, 45, 60and 90 degrees, for 15, 50, 80 and 100 nm nanowire diameters (from top to bottom). Appendix A: Angular FMR linewidth evaluation procedure
We have developed a procedure based on a least squares minimization procedure of the curve dχ (cid:48)(cid:48) dH versus H tothe set of n experimental measurements [ x i , y i ] i =1 ,n where x i = H i and y i = dχ (cid:48)(cid:48) ( β ; x i ) dH . β represents a set of fittingparameters.One of the parameters is the width ∆ H obtained from a fitting procedure to the derivative of a Lorentzian whoseexpression is given by: (cid:34) dχ (cid:48)(cid:48) dH (cid:35) L = A ( H − H )(∆ H + ( H − H ) ) (A1) H is not a fitting parameter since it can be determined by the intersection of the lineshape with the H axis. Theparameters A, ∆ H are determined with a fitting procedure. Writing the set of minima equations to be satisfied atthe data points: 1 n n (cid:88) i =1 (cid:40)(cid:34) dχ (cid:48)(cid:48) ( β ; x i ) dH (cid:35) L − y i (cid:41) minimum , (A2)The fitting method is based on the Broyden algorithm, a generalization to higher dimension of the one-dimensionalsecant method that allows us to determine in a least-squares fashion, the set of unknowns A, ∆ H . Broyden methodis selected because it can handle over or under-determined numerical problems and that it works from a singular valuedecomposition point of view . This means it is able to circumvent singularities and deliver a practical solution tothe problem at hand as an optimal set within a minimal distance from the real one. Appendix B: Preisach formalism overview
Preisach model is based on a statistical approach towards magnetization processes , .Comparing Preisach formalism and micromagnetics is akin to understanding the link between thermodynamics andstatistical mechanics.The nanowire array is viewed as made of single domain interacting entities each nanowire being represented by aswitching field and a local interaction field. The local interaction field in each nanowire is assumed to be constant.A system of interacting nanowires is represented by a probability density function (PDF) p ( h i , h c ) depending oninteraction and coercive fields H i and H c respectively.The main objective in Preisach modeling is to find the best p ( h i , h c ) such that the best possible agreement withsystem behavior is obtained.Preisach formalism is an energy-based description of hysteresis, and does not require that the material underinvestigation be decomposable into discrete physical entities such as magnetic particles.It assumes that the magnetic system free energy functional can be decomposed into an ensemble of elementary twolevel (double well) subsystems (TLS) .It represents the magnetic material into a collection of microscopic bistable units ”hysterons” having statisticallydistributed coercive and interaction fields. Each unit is characterized by a rectangular hysteresis loop (see fig. 5) andits status is determined by the actual field and history of the applied external fields.The classical version (CPM) of the model is based on the use of a joint distribution of normalized interactionfields H i and coercive fields H c . The interaction fields H i induce a shift in the elementary hysteresis loop (see fig. 5)whereas the coercive fields increase its width. Integrating the density over a given path in the Preisach plane yieldsa magnetization process.These fields originate from the existence of switching fields ( H α , H β ) that span the Preisach plane (see fig. 5) suchthat: H α = H i + H c , H β = H i − H c ,H i = H α + H β , H c = H α − H β has demonstrated that the necessary and sufficient conditions for a system to be rigorously describedby a CPM are the wiping-out and congruency properties . A hysteretic system will present the wiping-out property1 α β H H
HH icM H
FIG. 5: (a) Preisach ( H α , H β ) plane displaying different hysteresis loops at different locations. At each point an elementaryhysteresis loop is displayed characterized by the values of the interaction H i and coercive H c fields. (b) An elementary hysteronwith a square hysteresis loop is shown shifted by the interaction field H i and possessing a half-width given by the coercive field H c . when it returns to the same state after performing a minor loop. The second property refers to the shape of theminor loops measured in the same field range; if all these minor loops are congruent within a given field range andthis property does not depend on the actual field range used in the experiment, the system obeys the congruencyproperty.The major hysteresis loop is obtained when a path linear in the applied field is used , . Thus, in order to estimatethe the hysteresis loop, we determine the magnetization M with a double integration over the PDF as given below: M = 2 M S (cid:90) ∞ dh i (cid:90) b ( h i )0 dh c p ( h i , h c ) (B2)where the function b ( h i ) represents such path.The H i , H c PDF can be either analytically built from standard PDF (Gaussian, Lorentzian, uniform etc...) orexperimentally determined from FORC (First Order Reversal Curves) measurements. This originates from the factmagnetic interactions between nanowires are a major determinant of noise levels in magnetic media whether it is usedfor storage or processing such as in spintronics.While conventional methods of characterizing magnetic interactions utilize Isothermal Remanent Magnetization(IRM) and remanence DC Demagnetization (DCD) curves , Preisach modeling is based on some distribution whichis supposed to adequately describe the magnetic system at hand.FORC is a popular measurement leading to an appropriate Preisach model. It begins with sample saturation witha large positive field. The field is ramped down to a reversal field H α . FORC consists of a measurement of the2magnetization as the field is then increased from H α back up to saturation. The magnetization at applied field H β on the FORC with reversal point H α is denoted by M ( H α , H β ), where H β ≥ H α .The PDF is obtained from the second mixed derivative: p ( H α , H β ) = − ∂ M∂H α ∂H β (B3)Let us assume we adopt a double Lorentzian PDF given by: p ( h i , h c ) = 2 πσ i H [ π + tan − ( σ i )] × h i + h c − H σ i H ) ][1 + ( h i − h c − H σ i H ) ] (B4)with h i , h c a set of normalized fields, σ i , σ c the standard deviation of the individual Lorentzian/Gaussian distri-butions considered as independent) we find hysteresis loops that are upstraight whereas the VSM measured loopsexhibit some inclination.When one uses rather a double Gaussian PDF as in the CPM case, we get inclined hysteresis loops as observedwith the VSM measurements.Nevertheless both approaches do not agree with the hysteresis loops that we find experimentally as described byDumitru et al. . This is why we use the PM2 model as explained in section II. A Singh, S Mukhopadhyay and A Ghosh, Phys. Rev. Lett. , 067206 (2010). A Singh and A Ghosh, Phys. Rev.
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