Configuration dependent demagnetizing field in assemblies of interacting magnetic particles
Juan Manuel Martinez-Huerta, Armando Encinas, Joaquin De La Torre Medina, Luc Piraux
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Configuration dependent demagnetizing field in assemblies of interactingmagnetic particles
J. M. Mart´ınez-Huerta and A. Encinas ∗ Instituto de F´ısica, Universidad Aut´onoma de San Luis Potos´ı,Av. Manuel Nava 6, Zona Universitaria, 78290 San Luis Potos´ı, SLP, M´exico
J. De La Torre Medina
Facultad de Ciencias F´ısico Matem´aticas,Universidad Michoacana de San Nicol´as de Hidalgo,Avenida Francisco J. M´ujica S/N, 58060, Morelia, Michoac´an, Mexico
L. Piraux
Institute of Condensed Matter and Nanosciences, Universit´e Catholique de Louvain,Place Croix du Sud 1, B-1348, Louvain-la-Neuve, Belgium (Dated: September 14, 2018)
Abstract
A mean field model is presented for the configuration dependent effective demagnetizing andanisotropy fields in assemblies of exchange decoupled magnetic particles of arbitrary shape whichare expressed in terms of the demagnetizing factors of the particles and the volumetric shape con-taining the assembly. Perpendicularly magnetized 2D assemblies have been considered, for whichit is shown that the demagnetizing field is lower than the continuous thin film. As an exampleof these 2D systems, arrays of bistable cylindrical nanowires have been characterized by rema-nence curves as well as ferromagnetic resonance, which have served to show the correspondenceof these measurements with the model and also to validate the mean field approach. Linear chainsof cylinders and spheres have been analyzed leading to simple expressions to describe the easyaxis rotation induced by the interaction field in chains of low aspect ratio cylindrical particles, andthe dipolar magnetic anisotropy observed in the linear chain of spheres. These examples serve tounderline the dependence on the dipolar interaction field and effective demagnetizing factor on thecontributions that arise from the shape of the outer volume.1ssemblies or arrays of exchange decoupled magnets represent a very large class of magneticsystems which are of great interest in different subjects and problems in magnetism at both, fun-damental and technological level.
These include assemblies in which the particles are arrangedor dispersed on a substrate or embedded in different media such as liquids, polymers or other nonmagnetic materials.
Depending on their spatial disposition, and thus the shape or outer volumethat contains the assembly, they form one, two or 3D heterostructures or composites.
The magnetic properties of a single isolated particle are mainly governed by the shape andmagnetocrystalline anisotropy and eventually a magnetoelastic contribution. In an assembly ofmany of such particles, their individual properties are further modified by the interaction fieldproduced by the other particles. If the particles are not allowed to come into contact, then theinteraction is purely dipolar and has the same magnetostatic origin as the demagnetizing fieldand shape anisotropy associated to each particle. In this sense, from a mean field perspective,an effective demagnetizing field which includes the particle self demagnetizing field as well asthe dipolar interaction is a common feature of any assembly of exchange decoupled assemblyof particles.
Consequently, these considerations should apply to any assembly of particles,whether the particles are magnetically hard or soft, and regardless of the approach or level ofapproximation used to describe them. In particular, it should be possible to interpret the vastphenomenology of effects associated to the dipolar interaction or the effective demagnetizing field,within this framework.
However, this is usually complicated by the difficulty of finding eitheradequate expressions or reliable measurements for the dipolar interaction. Furthermore, extensionsto include the magnetization or configuration dependency of the dipolar interaction field addsadditional difficulties for establishing a suitable description for these systems.A coherent formulation is required for the effective demagnetizing field which allows to includethe configuration dependent interaction field for an assembly of particles of arbitrary shape andouter volume and that allows determining how the interaction field will modify the demagnetizingfield, the effective anisotropy or total energy and coercivity of the system.The present study is focused in deriving mean field expressions for the configuration depen-dent effective demagnetizing and anisotropy fields in assemblies of exchange decoupled magneticparticles, which depend on both the shape of the particles, the volume containing them and thepacking fraction. The magnetostatic properties of 2D and 1D assemblies have been analyzed andthe results show that the dipolar interaction depends not only on the interparticle distance, butalso on the outer shape of the assembly. In particular it is shown that this outer shape contri-2ution to the dipolar field plays an important role in the coercivity of the assembly, the limitingvalues of the effective demagnetizing fields and the magnetization reorientation transition due tothe dipolar interaction. Remanence curves and ferromagnetic resonance measurements done onarrays of bistable cylindrical nanowires have served first to identify the correspondence of thesemeasurements with the model, and second, to validate this mean field approach.
I. EXPERIMENTAL
Arrays of Co, Ni Fe and Co Fe nanowires having a low density (low porosity P ) andsmall diameter ( ≥ φ ≥ nm) have been grown by electrodeposition into the pores of 21 µ mthick lab-made track-etched polycarbonate (PC) membranes, in which the pores are parallel toeach other but randomly distributed. Also, two samples have been fabricated using anodizedallumina templates in order to have higher porosities (10 and 15%). Full details of the preparationprocess can be found elsewhere. For the electrodeposition a Cr/Au layer is evaporated previously on one side of the membraneto serve as a cathode and deposition is done at a constant potential using a Ag/AgCl referenceelectrode. For CoFe a 40 g/l FeSO + 80 g/l CoSO + 30 g/l H BO electrolyte was used with apotential of V =-0.9 V, while for NiFe the electrolyte contained 5.6 g/l FeSO + 131.4 g/l NiSO +30 g/l H BO and deposition is done at V =-1.1 V. Cobalt nanowires have been grown at V =-1Vusing a 238.5 g/l CoSO + 30 g/l H BO electrolyte with the pH set to 2.0 by addition of H SO tofavor a polycrystalline fcc-like Co structure with no magnetocrystalline anisotropy contribution. Table I shows the details of the samples considered.Ferromagnetic resonance (FMR) measurements have been done in the field swept mode withfrequencies ranging from 1 up to 50 GHz using a 150 µ m wide micro stripline with the DCmagnetic field applied parallel to the long axis of the wires, as detailed elsewhere. For all thesamples considered, the transmission spectra is recorded at a fixed frequency while the magneticfield is swept from 10 down to 0 kOe. These measurements are repeated for different frequenciesand from the collection of frequencies and their respective resonance field, the dispersion relationis obtained. Figure 1 (a) shows typical dispersion relations obtained on the samples. The dispersionrelation for an array of infinitely long cylindrical nanowires with the field applied parallel to thewires which are in the saturated state is f = γ ( H Res + H eff ) , (1)3 ABLE I. Co, NiFe and CoFe nanowire samples used in this study numbered as s1 to s16. For each samplethe following quantities are given: the wire diameter Φ , nominal packing fraction P N , the FMR effectivefield H eff , the dipolar interaction coefficient α z , and the determined values of the packing fraction P m andsaturation magnetization M ∗ s .Material Sample Φ( nm ) P N (%) H eff ( kOe ) α z (Oe) P m (%) M ∗ s (emu/cm )Co s1 39 4.8 7.65 400 4.63 1415Co s2 40 5.6 7.48 440 5.09 1406Co s3 30 3.9 7.57 240 2.91 1319Co s4 40 2.5 8.13 140 1.63 1360Co s5 30 3.9 7.77 362 4.08 1409Co s6 40 1.7 8.45 118 1.23 1396Co s7 30 4.2 7.68 370 2.85 1337Co s8 29 3.65 7.69 350 2.98 1343NiFe s9 30 3.46 4.64 140 3.19 816NiFe s10 40 3.4 4.45 180 3.58 792NiFe s11 35 8.8 5.62 580 7.87 1171NiFe s12 35 12 3.34 635 11.52 812CoFe s13 40 3.9 10.52 400 3.42 1868CoFe s14 29 5 10.24 640 5.23 1933CoFe s15 29 5 9.18 520 4.77 1706CoFe s16 18 10 8.45 1274 10.4 1953 where H Res and H eff are the resonance and effective anisotropy field, respectively. From thelinear fit of the dispersion relation, see Fig. 1 (a); the values of H eff have been determined foreach sample, and the corresponding values are given in Table I.Isothermal Remanence Magnetization (IRM) and DC demagnetization (DCD) remanencecurves measurements with the field applied parallel to the wires axis, have also been done inall the samples, from which the interaction field coefficient along the wire axis, α z , has beendetermined as α z = 2( H . r − H d ) , (2)4 !" ! " $ " % &’ ( ) * + , - ’$’(’)’%’ ./0%"12&( 2"34()56"- ’ ( % * +" , -’.& , /’ ! / , " ( . /0 % " , / % ./0%"12&( 2"34()56"- ’ ’.& $ $.& FIG. 1. (a) Typical dispersion relations obtained on arrays of NWs with the field applied parallel to thewires, the measurements are shown as symbols, while the continuous line is the best fit using Eq. (1). (b)IRM (blue) and DCD (red) remanence curves measured on sample S10 [NiFe] with the field applied parallelto the NWs axis. where H . r is the field value at which the normalized IRM curve is 0.5, while H d is the field valueat which the DCD curve is zero, as shown in figure 1 (b). The values determined for the interactionfield coefficient α z are given in table I. II. THE EFFECTIVE DEMAGNETIZING FIELD
Consider an assembly of exchange decoupled identical particles where each one has a de-magnetizing factor given by N , called the inner demagnetizing factor, contained within a macro-scopic volume described by an outer demagnetizing factor N + , as schematically shown in fig-ure 2. Furthermore, all the particles are aligned so their easy axis point in the same direction.The energy density of a given particle can include the magnetocrystalline ( K MC ), shape ( K S )and magneto elastic ( K ME ) anisotropies, which determine the total anisotropy of the particle,5 A = K MC + K S + K ME , as well as a contribution due to the interparticle interaction ( E int ) E = K A sin θ + E int (3)Where the shape anisotropy and the dipolar interaction are of magnetostatic origin and are relatedto the effective demagnetizing fields, which are considered in the following sections. Furthermore,in the following it will be assumed that the particles are homogeneously magnetized regardless oftheir shape. A. Saturated state
The effective or total demagnetizing factor can be expressed using the interpolation procedureintroduced by Netzelmann. To simplify the notation, in the following, the effective or total de-magnetizing factor ( N eff ) will be denoted as N T which can be expressed in terms of the packingfraction P as, N T = (1 − P ) N + P N + . (4)However, for the treatment of an assembly of magnetic particles it is more useful to rewrite thislast expression as, N T = N + ( N + − N ) P. (5)The first term on the right side corresponds to the self demagnetizing factor of the particles thatmakeup the assembly, this is, N self , while the second term contains all the contributions thatdepend on the packing and corresponds to the dipolar interaction contribution, N dip , then N T = N self + N dip . (6)When P = 0 the demagnetizing factor reduces to that of the isolated non-interacting particle.While on the opposite limit, P → this leads to the outer demagnetizing factor. If both the innerand outer demagnetizing factors are diagonal and their trace is Tr( N ) = 4 π , then from Eq. (5), Tr( N T ) = Tr( N ) + [Tr( N + ) − Tr( N )] P, (7)and from the form of (6), the following properties of N T are obtained, Tr( N T ) = 4 π, (8) Tr( N dip ) = 0 . (9)6 a) (b) (c) (d) z x y N + FIG. 2. Schematics of an assembly of identical particles characterized by an inner demagnetizing factor N contained within a macroscopic volume geometrically characterized by the outer demagnetizing factor N + (dashed lines). The i -th component of the effective demagnetizing field in the saturated state for an assembly ofparticles follows from equation (5), H DiT = M s N i + ( N + i − N i ) M s P, (10)where the first term is the usual self demagnetizing field of the particle, H Di = M s N i , and thesecond term is the dipolar interaction field, this is, H idip = ( N + i − N i ) M s P, (11)The total or effective magnetostatic anisotropy field in the saturated state defined as H ST = M s ( N xT − N zT ) , where the hard axis is along the x -direction and the easy axis is taken alongthe z -axis, then, H ST = M s ∆ N + (∆ N + − ∆ N ) M s P, (12)The first term on the right side is the shape anisotropy field of the individual particles ( H S ), whilethe second term corresponds to the total or effective dipolar field in the saturated state, H dip = (∆ N + − ∆ N ) M s P, (13)The total shape (magnetostatic) anisotropy energy is given by K ST = M s H TS / , which includesboth the shape anisotropy of the particle and the contribution of the dipolar interaction. Thisanisotropy constant now regroups K S and E int in equation (3).Finally, from the expressions obtained for the interaction field there are two other relationsof interest that follow when both the inner and outer volumes have in plane symmetry, so that N x = N y . Due to the symmetry, the in-plane components of the interaction field are equal,7 xdip = H ydip , and since Tr( N dip )=0, according to equation (9), the following relation between thecomponents of the dipolar fields is obtained, H zdip = − H xdip . (14)This shows that when both the inner and outer volume have rotational symmetry, the dipolarinteraction field along the symmetry axis is twice the dipolar field in the hard axis with oppositesign. This, for example, has been obtained by several authors. A consequence of this result isthat the dipolar part of the total or effective anisotropy field ( H dip = H xdip − H zdip ) is H dip = − H zdip , (15)which provides a relation between the component of the interaction field along the easy axis, withthe net contribution of the interaction field to the total anisotropy field. B. Non saturated states
To extend the previous expressions to the non-saturated case, consider that the particles arebistable, this is, they can only be magnetized along their easy axis in both the positive or negativedirection. The normalized magnetization, m = M ( H ) /M s , so that − ≤ m ≤ . Any magneticstate called hereafter m , can be written in terms of the fraction of particles magnetized in thepositive ( m + ) and negative ( m − ) directions, with ≤ m ± ≤ , and m = m + − m − . Moreover,since the number of particles is constant, m + + m − = 1 , and m ± can be obtained from the valueof m as, m ± = 1 ± m . (16)A recent FMR study done on non saturated arrays of bistable NWs lead to the expressions for theeffective field, the dipolar interaction term and the FMR dispersion relation, that include explicitlythe magnetic configuration of the system. In particular, it was shown that to extend the knownexpressions for the saturated case to the non-saturated case, it is necessary to rewrite the dipolarinteraction term as a parametric expression of m ± . Assuming a dipolar interaction field of theform αm , it implies that the dependence of the interaction field on m is not as a direct product butinstead, requires using Eq. (16), this is, H dip m ± = H dip ± H dip m, (17)8hich at saturation reduces to the expected value. The ± sign is introduced to include both positive( m = 1 ) and negative ( m = − ) saturation.Using Eq. (10), the i th-component of the effective demagnetizing field is, H DiT = M s N i + ( N + i − N i ) M s P ± ( N + i − N i ) M s P m. (18)The dipolar contribution now contains two terms, the first one is constant, while the second one isproportional to m and corresponds to the magnetization dependent part of the demagnetizing field.If the magnetization dependent interaction field is taken as αm , then from the last term on the righthand side, the i -th component of the dipolar interaction field coefficient α can be identified as, α i = H idip N + i − N i ) M s P . (19)Then equation (18), can be written as, H DiT = H Di + α i ± α i m, (20)where H Di is the demagnetizing field of the isolated particle. From Eq. (12), the effective or totalmagnetostatic anisotropy field is, H ST = H S + α T ± α T m, (21)and α T = α x − α z is the total dipolar field coefficient. At saturation m = 1 , both equations (20)and (21) reduce to Eqs. (10) and (12), respectively. III. APPLICATIONSA. Two dimensional arrays with rotational symmetry
An important class of magnetic assemblies are two dimensional arrays of exchange decouplednanoparticles, ideally a monolayer of single domain particles, with perpendicular magnetization.These correspond, for example, to systems obtained by lithography, including perpendicular bitpatterned media, also granular thin films with columnar structure as those used for perpendicularrecording media, self assembled monolayers and nanowire arrays.
For these systems one can assume that the height of the particles is very small compared withthe lateral dimensions of the entire array and the volume containing the particles can be considered9s an infinite thin film, so that N + x = N + y = 0 and N + z = 4 π , as depicted in figure 2 (a) and (b).Using these values in Eq. (10) and assuming particles with in-plane symmetry so that N x = N y and using N z = 4 π − N x to express quantities in terms of N x , the effective demagnetizing fieldperpendicular to the plane ( z ) is H DzT = 4 πM s − N x (1 − P ) M s . (22)Which shows that the effective demagnetizing field of a thin film made of exchange decoupledentities is less than πM s by a quantity equal to N x (1 − P ) M s .The total shape anisotropy field [Eq. (21)] is, H ST = (3 N x − π ) M s − N x M s P ∓ N x M s P m, (23)the first term on the right side is the shape anisotropy of the particle, while the second and thirdterm correspond to the dipolar part of the total anisotropy field, for m = 1 , H dip = − N x M s P, (24)which is a particular case of Eq. (15). As seen from this expression H dip is only a function of N x and P . Since the particles have in-plane symmetry, then N z = 4 π − N x , from where it followsthat if ≤ N z ≤ π , then ≤ N x ≤ π . And from the previous expression for H dip , it ispossible to establish an upper bound for the dipolar part of the total or effective anisotropy. Taking N x = 2 π , H maxdip = − πM s P. (25)The dipolar contribution to the total anisotropy field is antiferromagnetic, inferred by the negativesign in Eq. (24), as expected for a 2D array of nanomagnets with perpendicular anisotropy. B. Cylindrical Nanowires
In the particular case of a 2D array of circular cylinders of arbitrary height aligned parallel toeach other so their axes are along the z axis, as shown in figure 2 (b), their effective demagnetizingfields along the easy and hard directions are given by equation (10). As in the previous section,the outer demagnetizing factor is taken as an infinite thin film. For a cylinder of arbitrary aspectratio (wire height divided by its diameter), N z can be determined numerically. However, for theNWs considered in this study, the wire height is typically of the order of 20 µ m so the aspect ratio10s very high, and they can be considered as infinite so their demagnetizing factors are N z = 0 and N x = N y = 2 π and from equation (10) the demagnetizing field along the easy and hard axis are, H DzT = 4 πM s P, (26) H DxT = 2 πM s − πM s P. (27)Due to the symmetry properties of both inner and outer demagnetizing factors, H DxT = H DyT andthe dipolar terms satisfy both equations (9) and (14). The effective anisotropy field is, H ST = 2 πM s − πM s P, (28)that corresponds to the known expression for the effective field for an array of infinite cylindricalnanowires in the saturated state. The effective dipolar interaction field is H dip = − πM s P , andusing Eqs. (19) and (21), the magnetization dependent part of the effective interaction field is, α T = H dip πM s P, (29)in agreement with the expression obtained in Ref. [29].On the other hand, the measurement of the interaction field using Eq. (2) and the IRM andDCD remanence curves are done solely along the wire axis, this is, the measurement is related tothe component of the interaction field along this direction, as pointed out in Ref. 23. From Eq.(26), and Eqs. (19) and (20), the component of the magnetization dependent part of the interactionfield along the wire axis is α z = 2 πM s P. (30)Then, Eqs. (28), (29) and (30) provide different expressions for the interaction field which havedifferent physical meaning. Moreover, depending on the measuring technique, each of these threequantities can be measured independently, so it is important to distinguish between them.Since the effective anisotropy field, H ST can be determined from the dispersion relation obtainedfrom the FMR measurements, using equation (1), then from Eq. (28), H dip = H ST − πM s . So H dip has been determined for each sample from the FMR measurements using the nominal values of M s for Co (1400 emu/cm ), Ni Fe (800 emu/cm ), and Co Fe (1900 emu/cm ) and the valuesof H eff given in Table I. These values where then divided by 3 so they correspond numerically toEq. (30) which is the value of the component of the interaction field along the wire axis measuredwith the IRM and DCD remanence curves. Figure 3 shows the values obtained using the IRM andDCD remanence curves, α z plotted as a function of the values obtained by FMR, H dip / .11 ! " " $ % " & ’ ( " ) * + , - !" . /01 ’ (")*+,- !" FIG. 3. Easy axis component of the interaction field determined by magnetometry, α z = 2 πM s P , plotted asa function of H dip / , where H dip = 6 πM s P is the total interaction field measured by FMR in the saturatedstate, for all the nanowire samples listed in Table I. These results show a very good agreement, which shows that both Eqs. (28) and (30) providecorrect results for the interaction field. Moreover, these two expressions provide expressions forquantities that can be measured independently, so its possible to find a self consistent methodto determine both M s and P . Since H ST and α z are determined from FMR and magnetometrymeasurements, respectively, then combining Eqs. (28) and (30) leads to the following expressionsfor M s and P . M s = H ST + 3 α z π , (31) P = α z H ST + 3 α z . (32)The values of H ST and α z measured by FMR and the remanence curves, respectively, that aregiven in Table I have been used as input in Eqs. (31) and (32) to obtain the corresponding valuesof M s and P . The results are given in Table I in the columns labeled P m and M ∗ s , respectively.As a first point, the values of M ∗ s show a very good agreement with the known values for Co,NiFe and CoFe. Furthermore the values for NiFe and CoFe alloys are in good agreement withthose determined solely by FMR and reported elsewhere. Regarding the values of the templateporosity, comparing the nominal values determined by scanning electron microscopy P N withthose determined by the measurements P m , a very good agreement is also found in practically allthe samples. This procedure is general and can be extended to other assemblies by solving for M s and P using Eqs. (12) and (19). 12 . Linear chain So far, only 2D systems with perpendicular magnetization have been considered. Anotherinteresting example is that of the linear chain of particles and particularly for circular cylindersof different aspect ratio and spheres, as those shown in figure 2 (c) and (d), respectively. For aninfinite number of particles, the outer demagnetizing factor is an infinite cylinder of circular crosssection whose axis is assumed along the z axis so N + x = N + y = 2 π and N + z = 0 . While forboth spheres and cylinders, the components of the inner demagnetizing factor are N x = N y and N z = 4 π − N x . From equation (12), the effective anisotropy field is, H ST = M s ( N x − N z ) + M s (6 π − N x ) P. (33)For the calculations, the demagnetizing factors and the packing fraction are required. For the chainof spheres, N x = N y = N z = 4 π/ and the packing fraction is P = (2 / φ/d where d is thecenter to center distance and φ is the diameter of the sphere. For the cylinders, N z is determinedusing known expressions as a function of the aspect ratio τ = h/φ where h is the height ofthe cylinder, and in this case, P = h/d . Figure 4 (a) shows the reduced effective anisotropy ∆ N = H ST /M s for linear chains of cylinders with different aspect ratio ( τ =0.5, 0.8, 2.5 and 5) aswell as for a chain of spheres (dashed line).Consider first the case of the linear chain of cylinders. For a single, isolated and non interactingcircular cylinder, the critical aspect ratio is τ = 0 . above this value, N x > N z and the easyaxis is along the cylinder axis and at lower values N x < N z and it is perpendicular to the axis. Asseen in Fig. 4 (a) at large distances the anisotropy tends asymptotically to the expected value ofthe isolated non-interacting cylinder, which yield negative values for τ =0.5 and 0.8, and positivefor τ =2.5 and 5.As the particles are brought closer the interaction increases. For τ > . , the interaction isferromagnetic as it favors head to tail alignment of the magnetization, which results in an increaseof the effective anisotropy as the packing fraction increases. For low aspect ratios ( τ < . )the easy axis of a given cylinder is perpendicular to the chain axis and the first term in Eq. (33)is negative, so the interaction becomes antiferromagnetic as it has the opposite sign, and com-petes against it. As the packing fraction is increased, the value of the interaction increases andovercomes the anisotropy of the particle leading to a reorientation of the magnetization easy axis.In other words, to have a reorientation of the magnetization easy axis due to the dipolar inter-action, the interaction has to be of the demagnetizing type (antiferromagnetic) with respect to the13 !" ! " -367+5 -367+9)**:;3<=)>7? @=>7A)**:;3<=)>7? $ % &’ ( ! ) * (+ , ! - . / $0%$0!$0C$01 $ / D$ D/ %$ FIG. 4. (a) Reduced effective anisotropy as a function of the reduced center to center distance d/φ for linearchains of cylinders with aspect ratio τ =0.5, 0.8, 2.5 and 5 as well as for spheres (dashed line). (b) Zeroanisotropy curve for a linear chain of cylindrical particles of aspect ratio τ as a function of the reducedcenter to center distance. easy axis direction of the particle when considered isolated. For the linear chain of cylinders thereis a critical aspect ratio for the easy axis reversal which depends on the inter particle distance, thatcorrespond to those values for which the effective anisotropy vanishes. Setting H ST = 0 in Eq.(33) and using N x = 4 π − N z , the critical packing fraction can be expressed as, P c = 1 − (cid:18) π (cid:19) N z , (34)which is only valid for N z > π/ . Here N z = N z ( τ ) gives the dependence on the aspect ratioof the cylinder which is given by well known expressions. Figure 4 (b) shows the critical aspectratio variation with the reduced center to center distance for a linear chain of cylinders. Abovethis curve, the magnetization lies along the cylinder axis and the interaction between them isferromagnetic. Below this line, the magnetization easy axis is perpendicular to the cylinder axisand the dipolar interaction is antiferromagnetic. At large distances, this curve tends asymptoticallyto τ = 0 . (horizontal dashed line) which is the critical aspect ratio for an isolated non-interacting cylinder with N z = 4 π/ . The highest packing fraction required to reverse the easyaxis is obtained when the axial demagnetizing factor of the disk N z tends to π , which from Eq.1434), corresponds to P c = 2 / .Finally, the linear chain of spheres presents features that are interesting as seen in Fig. 4 (a), inparticular the existence of an effective anisotropy and its increase as the packing fraction increases,which in light of Eq. (33) is entirely due to the dipolar interaction. Indeed, as seen in the figure,the effective anisotropy tends to zero as the distance between particles is increased consistent witha single sphere with no shape anisotropy. Moreover, the maximum value of the anisotropy isattained when the spheres come in contact and this value is lower than π , which shows that thereis not a full equivalence between a cylindrical wire and the chain of spheres. This point has beenbrought to light recently and its important since modeling of cylindrical wires is commonly doneby considering them as a chain of spheres. From Eq. (33), the maximum value of the effectiveanisotropy for the chain of spheres is reached at P = 2 / and, H ST max = 4 π M s , (35)as indicated in Fig. 4 (a). IV. DISCUSSION
Expressions for the magnetization dependent demagnetizing field and the magnetostatic anisotropyfield (energy) have been derived. These are general as the inner and outer volume are arbitrary andapply to assemblies of identical exchange-decoupled, bistable particles with a common easy axis.Moreover, this approach is a mean field approximation which only accounts for magnetostatic ef-fects. In consequence, all the particles are equivalent which is only valid if the assembly containsa very large number of particles.Regarding the form of the effective demagnetizing field in the saturated state, Eq. (10), itcontains the self demagnetizing field of the individual particle and the component of the dipo-lar interaction field, Eq. (11) which is consistent with previous expressions obtained for nearlyspherical particles. On the other hand, when compared to other expressions, which providenon physical results for particular limiting cases, Eq. (10) provides correct limiting values when P = Drobrynin et al., derived an expression for the effective demagnetizing field which contains aterm that depends on the difference of the outer and inner demagnetizing factors as in Eq. (11).which provides support for Netzelmann’s interpolation. P < . For the particular case of a perpendicularly magnetized assembly with an outer volume ofa thin film, as shown by Eq. (22), the effective demagnetizing factor is lower than the continuousfilm, N zT < π provided that P < . The extensions to include the configuration dependence are done by taking the product betweenthe interaction term for the saturated case and Eq. (16) on the component of the demagnetizingfield along the easy axis, Eq. (18), and on the total anisotropy field, Eq. (21).An important result is that the interaction field can be expressed different forms each withdifferent physical meaning, which however, are related among them. The dipolar interaction con-tributes to the components of the demagnetizing field, Eq (10), as well as to the total magnetostaticanisotropy field, Eq. (12). Then each of these contributions can be extended to non saturated statesassuming an interaction field of the form αm , where the coefficient α is given by Eq. (19). TheFMR and magnetometry measurements done on arrays of nanowires provide a validation of thisapproach, while showing that different measuring techniques can provide values that correspondto different forms of expressing of the interaction field.From the extension done to include the configuration dependency on the interaction field, thehysteresis loop shearing can be attributed to the magnetization dependent component of the effec-tive demagnetizing field along the easy axis. As discussed in Ref. 23, assuming an interaction fieldof the form αm , the shearing can be corrected once the value of the coefficient is known, which;as shown here, is given by Eq. (19).The coercive field in an assembly is defined with respect to the total or effective anisotropy ofthe entire system, regardless of the specific internal composition, and in the Stoner - Wohlfarthmodel, these quantities are equal. If we call H c (0) = M s ∆ N the coercive field of the isolatedparticle ( P = 0 ), rearranging Eq. (12), the coercive field of the assembly is given by, H c = H c (0)(1 − P ) + M s ∆ N + P, (36)which depends on the packing fraction as well as on the outer volume. The first term is the wellknown coercive field dependence on the packing fraction in assemblies where the shape anisotropyprevails. The second term represents the outer volume anisotropy contribution, which impliesthat the coercivity of the assembly is shape dependent. So for a given assembly with fixed packingfraction, the coercive field will change between samples with different outer volume, as recently16eported for granular hard magnetic films. Moreover, for the particular case when the outer vol-ume is isotropic and ∆ N + = 0 , the second term vanishes. To further emphasize the importance ofthe second term in Eq. (36), notice that if only the first term is considered, the coercivity alwaysdecreases when the packing fraction increases, regardless of how the particles are distributed. This behavior is only consistent with the case where the interaction between the particles is anti-ferromagnetic. Indeed, for example, consider the linear chain of cylinders discussed above, eachwith their easy axis parallel to the chain axis, the interaction is ferromagnetic and as the packingfraction increases, the coercivity of the entire chain should increase from the value of the isolatedcylinder of finite aspect ratio to that of the infinite cylinder. This behavior is described by Eq. (36)when the second term is taken into account.Since the total anisotropy field, Eq. (12), depends explicitly on this outer shape anisotropy M s ∆ N + P , then variations on the shape of the outer volume results in changes in the demagnetiz-ing field, the anisotropy and, as just mentioned, in the coercivity. This contribution is associatedonly to the dipolar interaction, so at low packing it is weak, while at higher packing fractions itcan induce appreciable effects, and can be used as an additional parameter to control the magneticproperties of the assembly. An interesting case is when the assembly is made by spherical parti-cles, since their shape anisotropy vanishes and the effective anisotropy field is given only by theouter shape anisotropy. Then a chain of spheres will behave close to a cylinder (its outer volume)and the entire chain has a uniaxial anisotropy whose origin is the interparticle interaction field, while the same particles forming an spherical assembly will be isotropic. Although this is a wellknown result, it shows that this effect follows from the key role played by the outer demagnetizingfactor on the value of the interaction field, which recently has been related to novel effects in gran-ular hard magnets, composites, and dipolar ferromagnetic order in superparamagnetic particlemonolayers. Another effect related to this contribution is the magnetization reorientation transition inducedby the interaction field, which was discussed for the linear chain of cylinders, but which has alsobeen reported for other assemblies.
As discussed above, this transition results with in-creasing the packing fraction when the shape anisotropy of the single particle and the dipolar termhave opposite signs. Qualitatively, this can also be interpreted as a competition between the shapeanisotropies of the inner and outer volume and how their respective easy axis are oriented withrespect to each other. At very low packing, the easy axis of a given particle in the assembly is thatof the inner volume, while as P → H c is the total shape anisotropy and H c (0) = M s ∆ N corre-sponds to the inner volume shape anisotropy, it follows that the total anisotropy will never changesign if the second term is not included.The validity of this mean field approach is determined mainly by the extent to which the indi-vidual particles in the assembly remain bistable and are able to rotate independently from otherparticles, this is, that no collective effects take place while individual reverse their magnetic state.Starting with a very diluted assembly of bistable particles, both conditions are fulfilled, howeveras the packing fraction increases, each particle feels the interaction field which increases as thedistance between particles is reduced. In this sense both bi-stability and the ability to reverse inde-pendently are susceptible to change as a consequence of any finite value of the interaction field. Onone hand, bi-stability can be related to the strength of the coercive field of the individual particle,or to the height of the energy barrier that separates the two minima configurations ( + m and − m ).On the other hand, the ability of the particles to rotate independently also depends on the value ofthe coercive field and the width of the intrinsic switching field distribution. So in both cases, thecompetition between the height of the intrinsic energy barrier of each particle and the magnitudeof the interaction field determines if these conditions are fulfilled and in this sense, there is nounique criteria but rather it will depend on the type of material and the main contributions to theenergy.In particular, the height of the energy barrier of the particles depends on the magnetocrys-talline anisotropy. Since this contribution is intrinsic to each particle, it follows that the effects ofthe dipolar interaction are expected to become less relevant as the magnetocrystalline anisotropyof the individual particles increases. In this sense, the magnetic hardness parameter κ =[ | K MC | / (2 πM s )] / , provides a useful empirical parameter of the height of the energy barrier.In particular, the higher the value of κ , the higher the packing fraction attainable without loss ofbistability or independent reversal. On the other limit, this is, low values of κ both the bi stabilityand independent reversal are expected to be limited to low and moderate packing fractions, assuggested by previous reports which show that magnetic percolation in assemblies takes place for18acking fractions between 20 and 40%. V. CONCLUSION
In conclusion, mean field expressions have been derived for the demagnetizing field and themagnetostatic anisotropy field which include the magnetization dependent part of the interactionfield for assemblies of magnetic particles with arbitrary inner and outer demagnetizing factors.Special emphasis was given to 2D arrays of nanomagnets with perpendicular magnetization, wherethe expressions for the demagnetizing field and total anisotropy field have been derived. Themodel was successfully tested using FMR and magnetometry measurements done on arrays ofcylindrical nanowires, where the equivalence between the interaction field determined by each ofthese techniques was shown. This formalism provides a simple mean field framework to describemagnetostatic effects in a wide variety of magnetic assemblies and composites. Moreover, it showsthe role played by the outer demagnetizing factor in the value and characteristics of the interactionfield which could be used to fine tune the magnetic properties of the assembly or composite.
VI. ACKNOWLEDGEMENTS
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