Vortex core size in interacting cylindrical nanodot arrays
aa r X i v : . [ c ond - m a t . m t r l - s c i ] N ov Vortex core size in interacting cylindrical nanodot arrays
D. Altbir , J. Escrig , P. Landeros , F. S. Amaral , and M. Bahiana Departamento de F´ısica, Universidad de Santiago de Chile, USACH, Av. Ecuador 3493, Santiago, Chile Departamento de F´ısica, Universidad T´ecnica Federico Santa Mar´ıa, Av. Espa˜na 1680, Valpara´ıso, Chile Instituto de F´ısica, Universidade Federal do Rio de Janeiro, Caixa Postal 68.528, 21941-972, RJ, Brazil (Dated: October 25, 2018)The effect of dipolar interactions among cylindrical nanodots, with a vortex-core magnetic config-uration, is analyzed by means of analytical calculations. The cylinders are placed in a N × N squarearray in two configurations - cores oriented parallel to each other and with antiparallel alignmentbetween nearest neighbors. Results comprise the variation in the core radius with the number ofinteracting dots, the distance between them and dot height. The dipolar interdot coupling leads toa decrease (increase) of the core radius for parallel (antiparallel) arrays. PACS numbers: 75.75.+a, 75.10.-b
I. INTRODUCTION
Regular arrays of magnetic particles produced bynanoimprint lithography have attracted strong attentionduring the last decade. Common structures are arrays ofwires, [1, 2] cylinders, [3, 4] rings, [5, 6] and tubes. [7, 8]Such structures can be tailored to display different stablemagnetized states, depending on their geometric details.Besides the basic scientific interest in the magnetic prop-erties of these systems, there is evidence that they mightbe used in the production of new magnetic devices or asmedia for high density magnetic recording. [9] In partic-ular, two-dimensional arrays of magnetic nanoparticleshave been proposed as candidates for magnetoresistiverandom access memory (MRAM) devices. [10, 11, 12]Recent studies on such structures have been carriedout with the aim of determining the stable magnetizedstate as a function of the geometry of the particles.[3, 4, 13, 14, 15, 16] In the case of cylindrical particles,three idealized characteristic configurations have beenidentified: ferromagnetic with the magnetization paral-lel to the basis of the cylinder, ferromagnetic with themagnetization parallel to the cylinder axis, and a vor-tex state in which most of the magnetic moments lieparallel to the basis of the cylinder. The occurrenceof each of these configurations depends on geometricalfactors, such as the linear dimensions and their aspectratio τ ≡ H/R , with H the height and R the radius ofthe cylinder. [3, 4, 13, 14] Another issue to be consid-ered is the interaction between particles. In this casethe interparticle distance, D , is the important parame-ter. [17, 18, 19] Usually D is large enough to make theexchange coupling between particles negligible, makingthe dipolar interaction a fundamental point concerningthe magnetic state of the system. [16, 20]In this paper we focus on cylindrical particles with di-mensions such that, in the absence of an external field,the magnetic state is a vortex. For this magnetic config-uration the dipolar interaction between the dots is due tothe existence of a core region [21, 22] in which the mag-netic moments have a nonzero component parallel to thecylinder axis. In this case it is relevant to understand how the core magnetization is affected by the interparticle in-teraction. Recently, Porrati et al. [20] investigated anarray of dots using micromagnetic simulations. In smallarrays they observed that the dipolar interaction changesthe size of the magnetic core. However, and because ofthe use of micromagnetic simulations, larger arrays havenot been investigated. Therefore, analytical calculationsare very desirable to compare with experiments. Withthis in mind we examine the behavior of the core in ar-rays of dots in the vortex-core magnetic phase, in config-urations with parallel and antiparallel alignment betweenthe cores. We consider analytical calculations based ona continuous description of the dots.[23] II. SYSTEM & UNITS
The basic parameters and variables used in our calcula-tions are summarized in figure 1. We consider cylindricaldots with radius R and height, or thickness, H , in squarearrays with N × N dots and center-to-center lattice spac-ing D . The distance between any two dots in the arrayis denoted by S . Whenever necessary, cylindrical coordi-nates ρ and φ are defined on the dot plane normal to thecylindrical z -axis.The experimental measurement of the vortex core pro-file and core radius is not a simple task, and usually itis the full core magnetization, µ z , which is measured.[21, 24] With this in mind, the vortex core is character-ized through the calculation of an effective core radius, C eff , defined as the radius of an effective cylinder, uni-formly magnetized along its axis, and whose total mag-netic moment, µ z = M V = M πHC eff , (1)is the same as produced by the z -component of the vortexcore, as depicted in figure 1. Here V is the dot volumeand M is the saturation magnetization.Two types of magnetic ordering within the array areexamined: all cores parallel (configuration P) and anti-parallel nearest-neighbor cores (configuration AP). Thischoice is not an arbitrary one, because the P configura-tion corresponds to the saturated one and the AP con-figuration is the ground state of the array. FIG. 1: An illustration of geometrical parameters definingeach dot and the array.
All linear dimensions will be considered in units of theexchange length L x , defined as L x = p A/µ M . Thedimensionless geometrical parameters are then defined as h ≡ HL x , r ≡ RL x , d ≡ DL x , b = BL x , c eff ≡ C eff L x , s ≡ SL x . (2) III. THEORETICAL MODEL
Large arrays can be studied if we adopt a simplifieddescription of the system in which the discrete distribu-tion of magnetic moments is replaced with a continuousone, defined by a function ~M ( ~r ) such that ~M ( ~r ) δV givesthe total magnetic moment within the element of vol-ume δV centered at ~r . Using such a description, theinternal energy of the array can be written in terms ofthe self-energies, E self , that is, the energies of the iso-lated dots, and the interaction contribution, E int , cor-responding to the dipolar coupling between the dots.Both E self and E int have a magnetostatic term given by E dip = ( µ / R ~M · ∇ U dV , with U ( ~r ) the magneto-static potential. Assuming that ~M ( ~r ) varies slowly onthe scale of the lattice parameter, the exchange term canbe approximated by E ex = A R P ( ∇ m i ) dV , where m i is the i -th component of the reduced magnetization withrespect the saturation value M , that is, m i = M i /M ,for i = x, y, z .[23] A. Vortex-core magnetization
For the vortex-core configuration we assume that themagnetization is independent of z and φ , that is, ~m ( ~r ) = m z ( ρ )ˆ z + m φ ( ρ ) ˆ φ , (3)where ρ is the radial coordinate, ˆ z and ˆ φ are unitaryvectors in cylindrical coordinates, and the normalization condition requires that m z + m φ = 1. The function m z ( ρ )specifies the core profile, for which we adopt the modelproposed by Landeros et al ,[14] given by m z ( ρ ) = (cid:2) − ( ρ/B ) (cid:3) n , < ρ < B . (4)and m z ( ρ ) = 0 if B < ρ < R . Here B is a parameterrelated to the core radius and the exponent n is a non-negative integer. Alternative expressions for m z ( ρ ) havebeen proposed in the literature. [25, 26, 27, 28] Figure 2illustrates the calculated magnetization profile m z ( ρ ) fora Fe dot ( R = 28 . H = 37 . et al [26], the (blue) thin line correspondsto the one proposed by Aharoni,[27] the (black) thickline corresponds to our model[14] with n = 4, the dashed(red) line represents m z ( ρ ) using the model proposed byFeldtkeller et al [25], and the dotted (green) line repre-sents the profile using the model presented by H¨ollinger et al [28]. m z [nm] Usov Aharoni ( n=1 ) Landeros ( n=4 ) Feldtkeller Hollinger
FIG. 2: Calculated magnetization profiles, m z ( ρ ), for differentcore models. Dimensions are R = 28 . H = 37 . L x = 3 .
327 nm.
Note that the model proposed by A. Aharoni[27] isequivalent to our model (Eq. 4) using n = 1. We canobserve that our model with n = 4 agrees well with theones presented in [25] and [28], and has no discontinu-ities at m z ( ρ = B ) = 0, as in the models proposed byUsov et al [26] and Aharoni,[27] making the vortex coreprofile easily integrable. The form for m z ( ρ ) presentedabove (Equation (4)) allows us to obtain the magneti-zation profile by minimizing the total energy of the ar-ray. The value of B is determined in the minimizationprocess, as explained below, and the effective core radius( C eff ) can be evaluated equating the z -component of totalmagnetic moment (Equation (1)) with the core magneticmoment, given by µ z = M Z V m z ( ρ ) ρ dρ dz dφ . (5)Using the proposed model for m z ( ρ ), Equation (4), weobtain µ z = M πHB / ( n + 1) and therefore, C eff = B √ n + 1 . (6) B. Total energy calculation
We write all the energies in the dimensionless form,˜ E = E/µ M L x . The energies in the P and AP configu-rations will be denoted as ˜ E + and ˜ E − , respectively, andcan be written as˜ E ± = N ˜ E self + ˜ E ± int , (7)where the first term includes the self-energies of the N isolated dots, and the second term is the interdot mag-netostatic coupling.
1. Self-energy
We consider two contributions to the self-energy, ˜ E self ,of an isolate dot in the vortex-core configuration: thedipolar, ˜ E dip , and the exchange, ˜ E ex , contributions. An-alytical expressions for the dipolar and exchange energieshave been previously calculated by Landeros et al. [14]The dipolar energy results as˜ E dip = πα n b − πβ n b h F n ( b/h ) , (8)where α n ≡ n − Γ( n + 1) Γ( n + 3 / n + 5 / , β n ≡ n + 1) and F n ( x ) = P F Q [(1 / , , n + 3 / , ( n + 2 , n + 3) , − x ] . Here, P F Q denotes the generalized hypergeometric func-tion, Γ ( x ) is the gamma function and n corresponds tothe exponent used in the core model defined by Equation(4).For integer values of n , the exchange energy reduces to[14] ˜ E ex = πh [ln( r/b ) + γ n ] , (9)where γ n = (1 / H (2 n ) − n H ( − / n ). Here, H ( z ) = P ∞ i =1 [1 /i − / ( i + z )] is the generalized harmonic num-ber function[29] of the complex variable z .
2. Interdot magnetostatic coupling
The dipolar interaction between any two dots in thearray depends on the center-to-center distance betweenthem, S , and on the relative orientation of the magneticcores. An expression for this energy is obtained using themagnetostatic field experienced by one of the dots due tothe other. Details of these calculations are included inappendix. The resulting expression is˜ E ± ( S ) = ± πL x ∞ Z dk (1 − e − kH ) J ( kS ) R Z J ( kρ ) m z ( ρ ) ρdρ . (10)Since the size of the core will be obtained by energy min-imization, in the previous equation we have consideredthat the two interacting dots exhibit the same magneticprofile, that is, their cores are identical. Equation (10)allows us to write the interaction energy between twodots as ˜ E ± ( S ) = ± ˜ E ( S ), depending on the relative ori-entation of the cores. Note that if the core size increases,i.e. m z grows, then ˜ E ( S ) increases. Also we always have˜ E ( S ) >
0, and the interaction energy between two dotsis always greater than zero ( ˜ E + ( S ) >
0) if the cores areoriented in the same direction, which causes a shrink-ing of the vortex core, in agreement with micromagneticsimulations[20]. On the other hand, if the two cores areoriented in opposite directions, then the interaction termis negative ( ˜ E − ( S ) <
0) and the vortex core expands tolower the total energy.Substituting our expression for m z ( ρ ), Equation (4),we obtain˜ E ( s ) = 2 n +1 π Γ ( n + 1) h n +1 b n − ∞ Z dyy n +2 (1 − e − y ) J (cid:16) y sh (cid:17) J n +1 (cid:18) y bh (cid:19) . (11)Using equation (11) and adding up contributions overthe entire array we obtain the expression for the totalinteraction energy of the N × N square array as˜ E ± int ( N ) = 2 N N − X p =1 ( N − p )( ± p ˜ E ( pd )+ 2 N − X p =1 N − X q =1 ( N − p )( N − q )( ± p − q ˜ E (cid:16) d p p + q (cid:17) . (12)This equation was previously obtained by Laroze et al ,[18] and has been used to investigate the magnetostaticcoupling in arrays of magnetic nanowires.At this point we need to specify the value of the pa-rameter n for the core profile defined by equation (4).In a previous work, Landeros et al [14] showed that themagnetic vortex core can be well described for almostany value of n >
1. We choose n = 4, as explained in[14], and finally obtain the following expression for theself-energy,˜ E self = 0 . πb − πb hF ( b/h ) + πh (cid:16) ln rb + 2 . (cid:17) , (13)with F ( x ) = − x (256 + 384 x + 576 x +600 x + 350 x − F ( − / , / , , − x )) , where F ( a, b, c, z ) is a hypergeometric function.Using n = 4 in Eq. (11) we obtain˜ E ( s ) = 294912 πh b ∞ Z dyy (1 − e − y ) J (cid:16) y sh (cid:17) J (cid:18) y bh (cid:19) . (14)Finally, the total energy of the array is calculated as˜ E ± = N ˜ E self + ˜ E ± int , with ˜ E self given by equation (13)and ˜ E ± int given by equations (12) and (14).To determine the vortex core magnetization, we haveto minimize ˜ E ± with respect to b . Note that the onlyterm in the expression for the energy which depends onthe radius r of the dot is ˜ E self (see equation (13)). How-ever, the derivative of ˜ E self with respect to b is indepen-dent of r , leading to a core size that is independent of thedot radius. [14, 21] This follows from the fact that theexternal region of the dot (a perfect vortex) does not in-teract with the core (apart from the exchange interactionacross the interface between the two regions). That is tosay, the equation for b that minimizes the total energy ofthe vortex configuration is independent of r . IV. RESULTS AND DISCUSSION
We are now in position to investigate how the core ra-dius is affected by the interdot magnetostatic coupling.In order to obtain the value of b , the energy ˜ E ± must beminimized for fixed h , d and N . Since the dipolar inter-action is long-ranged, an increment in N leads to an in-crease in the dipolar field felt by each dot and, therefore,to a change in the core radius until a certain asymptoticvalue is reached. Figure 3 illustrates this effect for arraysof dots with r = 6, d = 12 . h = 12 (figure3(a)) and h = 8 (figure 3(b)), in configurations P andAP. For both values of h we observe that for N = 12the core radius is almost at its limiting size. From ourresults we observe an increase in the core radius with thenumber of dots in the AP configuration, and the oppo-site behavior for the P configuration, in agreement withmicromagnetic simulations.[20] For a given value of N ,AP arrays with smaller values of d present larger coreradius due to the preference for the AP ordering. Theantiparallel alignment has the lowest dipolar energy, sothe core radius increases with the number of dots in this d = 12.5 AP d = 12.5 P d = 14 AP d = 14 P N c e ff r = h = d = 12.5 AP d = 12.5 P d = 14 AP d = 14 P b ) a ) r = h = FIG. 3: Effective core radius c eff vs N for r = 6: (a) h = 12and (b) h = 8. Open symbols correspond to an antiparallel(AP) ordering and full symbols to a parallel (P) ordering.Circles correspond to d = 12 . d = 14. configuration. On the other hand, in the P configurationthe parallel coupling between the cores increases the in-teraction energy, and then the core region is reduced inorder to decrease the interaction.The dipolar energy may also be varied if the interdotdistance is changed. This effect is depicted in figure 4for the AP and P configurations. While in the antipar-allel arrays the size of the core rapidly reaches the valuefor isolated dots; in the parallel configuration, the effectof the interdot interaction is relevant for longer interdotdistances.Now, the influence of the dot height is more subtle.Figure 5 shows the behavior of c eff vs h for arrays ofdots with r = 6, d = 12, with N = 4 and 64, in the Pand AP configurations. For the isolated dot (blue solidline), the effective core follows approximately the relation
12 16 20 24 282.02.12.22.3
Isolated dot b ) r = 6 h = 8 N=2 AP N=2 P N=8 AP N=8 P c e ff d Isolated dot a ) r = 6 h = 12 N=2 AP N=2 P N=8 AP N=8 P
FIG. 4: Effective core radius c eff vs d for r = 6: (a) h = 12and (b) h = 8. Open symbols correspond to an antiparallel(AP) ordering and full symbols to a parallel (P) ordering. c eff ≈ .
228 + 0 . h . .For an isolated dot, a transition from the vortex con-figuration to a complete ferromagnetic ordering along thedot axis is observed as h increases, as shown in the phasediagrams presented in [14]. From the point of view ofthe core radius, this transition may be seen as a slowand continuous increase in c eff with h , until c eff ≈ r . Asthe dots interact in the array this behavior may change.To illustrate this point we calculate the transition linefrom the out-of-plane uniform state to the vortex-corestate configuration in arrays of 4 and 64 dots. The self-energy for the out-of-plane uniform ( u ) state has beenpresented in [30] and reads˜ E u self = πhr (cid:18) r πh − F (cid:20) − r h (cid:21)(cid:19) , (15)where F [ x ] = F [ − / , / , , x ] is a hypergeometric h r = d = c e ff N=8 AP N=2 AP isolated dot N=2 P N=8 P
FIG. 5: Effective core radius c eff vs h for dots with d = 12, r = 6. Open symbols correspond to an antiparallel (AP)ordering and full symbols to a parallel (P) ordering. Starscorrespond to N = 4 and circles to N = 64. function.The interaction energy between two dots with full mag-netization along their axis can be obtained from equation(14) using m z ( ρ ) = 1 and gives˜ E u ( s ) = 2 πhr ∞ Z dyy (1 − e − y ) J ( y sh ) J ( y rh ) , (16)as presented by Beleggia et al. [31] The total energy of anarray with out-of-plane uniform magnetization is givenby ˜ E u ± = N ˜ E u self + ˜ E u ± int , (17)with the above expressions for the self-energy (equation(15)) and the interaction energy ( ˜ E u ± int ) given by equa-tion (12) with the magnetostatic coupling of the two dotsgiven by equation (16). The transition lines between thevortex-core and uniform out-of-plane magnetic states areobtained by equating equations (7) and (17) and are rep-resented in figure 6.As we expected, the transition line shifts to lower radii(and greater heights) in the parallel array and to biggerradii (and lower heights) in the antiparallel array. Thisbehavior can be understood by analyzing the interdotdipolar coupling between two dots, ˜ E ( s ). The absolutevalue of the interaction energy increases with the coresize, so ˜ E u ( s ) > ˜ E ( s ), as the uniform out-of-plane ( u )configuration can be seen as a vortex-core with an infinitecore radius, and the magnetostatic coupling for uniformout-of-plane magnetization is stronger than the coupling N=1
N=2 P N=8 P N=2 AP N=8 AP r h FIG. 6: Dipolar induced shifts in the transition line that sepa-rates the vortex and the out-of-plane uniform magnetic states.We have fixed the ratio r/d = 0 .
48. Open symbols correspondto an antiparallel (AP) ordering and full symbols to a parallel(P) ordering. Stars correspond to N = 2 and circles to N = 8. of two vortex-cores. Therefore, the transition line in aparallel array shifts to the left, because the interactionsin an array of vortex-core dots are less than the interac-tions in a uniformly magnetized P array. On the otherhand, in the AP arrays, the interactions are negative andthe uniformly magnetized AP configuration is favorable,so the complete ferromagnetic ordering is reached for asmaller value of h . V. CONCLUSIONS
We have examined the influence of dipolar interactionsin the effective core radius of dots in a vortex configura-tion, placed in an N × N square array. Dipolar couplingamong dots occurs via core interaction, so we considertwo types of relative alignment between the cores: paral-lel, and antiparallel nearest neighbors. Whenever a cer-tain array configuration lowers the interaction energy, thecore region expands, as occurs in AP interactions. Theopposite behavior occurs for configurations increasing theinteraction energy, as the P case. These two orderingsrepresent a demagnetized configuration, the antiparallelcase, and a saturated one, corresponding to the parallelcase. Effects of the interaction between the dots in thecore size have been investigated by varying the numberof dots in the array, the distance between them, and theirheights. In all cases, an increase of the dipolar interac-tion energy leads to a decrease of the core radius. Whenwe increase the height of the dots, in the P configuration a transition from vortex to a full ferromagnetic state ishindered, while in the AP configuration the interactionfavors the transition to a full AP ferromagnetic state. Acknowledgments
This work has been partially supported by MillenniumScience Nucleus ”Basic and Applied Magnetism” P06-022F of Chile. MECESUP USA0108 project and theprogram “Bicentenario de Ciencia y Tecnolog´ıa (PBCT)”under the project PSD-031 are also acknowledged. InBrazil the authors acknowledge CNPq, PIBIC/UFRJ,FAPERJ, CAPES, PROSUL Program, and Instituto deNanotecnologia/MCT.
APPENDIX .The dipolar interaction energy between two identicaldots i and j in the vortex core configuration is given by E ± dip = µ Z V i ~M i ( ~r ) · ∇ U j ( ~r ) dV ′ where ~M i is the magnetization of the i -th dot and U j is the magnetostatic potential of the j -th dot. For thevortex core model defined by Eq. (3)we have ∇ · ~M = 0,so the magnetostatic potential reduces to[14] U j ( ~r ) = 14 π ∞ X p = −∞ π Z e ip ( φ − φ ′ ) dφ ′ R Z M z ( ρ ′ ) J p ( kρ ′ ) ρ ′ dρ ′ ∞ Z J p ( kρ j )[ e − k ( H − z ) − e − kz ] dk . (A18)In this expression we have used the expansion[32]1 | ~r − ~r ′ | = ∞ X p = −∞ e ip ( φ − φ ′ ) Z ∞ J p ( kρ ) J p ( kρ ′ ) e − k ( z > − z < ) dk . (A19)Here J p ( z ) are Bessel functions of first kind. The an-gular integration gives us R π e ip ( φ − φ ′ ) dφ ′ = 2 πe ipφ δ p, ,leading to U j ( ρ j , z ) = 12 Z R ρ ′ M z ( ρ ′ ) J ( kρ ′ ) dρ ′ Z ∞ J ( kρ j ) h e − k ( H − z ) − e − kz i dk. (A20)Since the potential has no dependence on φ , the expres-sion for the energy reduces to E ± = µ H Z π Z R Z M z ( ρ ) ∂U j ( ρ j , z ) ∂z ρdρdφ dz , (A21)and using Eq. (A20) it is straightforward to obtain E ± = µ π Z dφ ∞ Z dk R Z J ( kρ j ) M z ( ρ ) ρdρ R Z J ( kρ ′ ) M z ( ρ ′ )(1 − e − kH ) ρ ′ dρ ′ . (A22)We need to evaluate the potential due to dot j on dot i a distance S apart. Then we have to relate the radialcoordinates of both dots through the relation ρ j = p ρ + S − ρS cos( φ + β ) , with β an arbitrary angle. Using the following identity[32] J ( kρ j ) = J (cid:16) k p ρ + S − ρS cos( φ + β ) (cid:17) = ∞ X p = −∞ e ip ( φ + β ) J p ( kρ ) J p ( kS )in the expression for the energy (Eq. A22), after theangular integration, we obtain˜ E ± [ S ] = ± πL x ∞ Z dk (1 − e − kH ) J ( kS ) R Z J ( kρ ) m z ( ρ ) ρdρ . (A23) [1] M. V´azquez, Physica B , 302-313 (2001).[2] K. Nielsch, R. Hertel, R. B. Wehrspohn, J. Barthel, J.Kirschner, U. G¨osele, S. F. Fischer, and H. Kronm¨uller,IEEE Trans. Magn. , 2571 (2002).[3] R. P. Cowburn, D. K. Koltsov, A. O. Adeyeye, M. E.Welland, and D. M. Tricker, Phys. Rev. Lett. , 1042(1999).[4] C. A. Ross, M. Hwang, M. Shima, J. Y. Cheng, M.Farhoud, T. A. Savas, Henry I. Smith, W. Schwarzacher,F. M. Ross, M. Redjdal, and F. B. Humphrey, Phys. Rev.B , 144417 (2002).[5] F. J. Casta˜no, C. A. Ross, A. Eilez, W. Jung, and C.Frandsen, Phys. Rev. B , 144421 (2004).[6] J. Rothman, M. Kl¨aui, L. Lopez-Diaz, C. A. F. Vaz, A.Bleloch, J. A. C. Bland, Z. Cui, and R. Speaks, Phys.Rev. Lett. , 1098 (2001).[7] K. Nielsch, F. J. Casta˜no, C. A. Ross, and R. Krishnan,J. Appl. Phys. , 034318 (2005).[8] Kornelius Nielsch, Fernando J. Casta˜no, Sven Matthias,Woo Lee, and Caroline A. Ross, Adv. Eng. Mat. , 217-221 (2005).[9] S. Y. Chou, Proc. IEEE , 652 (1997); G. Prinz, Science , 1660 (1998).[10] J. M. Daughton, A. V. Pohm, R. T. Fayfield, and C. H.Smith, J. Phys. D , R169 (1999).[11] J. G. Zhu, Y. Zheng, and Gary A. Prinz, J. App. Phys. , 6668 (2000).[12] Stuart S. P. Parkin, Christian Kaiser, Alex Panchula,Philip M. Rice, Brian Hughes, Mahesh Samant, and See-Hun Yang, Nat. Mater. , 862 (2004).[13] J. d’Albuquerque e Castro, D. Altbir, J. C. Retamal, andP. Vargas, Phys. Rev. Lett. , 237202 (2002).[14] P. Landeros, J. Escrig, D. Altbir, D. Laroze, J.d’Albuquerque e Castro, and P. Vargas, Phys. Rev. B , 094435 (2005).[15] F. Porrati, and M. Huth, Appl. Phys. Lett. , 3157 (2004).[16] J. Escrig, P. Landeros, D. Altbir, M. Bahiana, and J.d’Albuquerque e Castro, Appl. Phys. Lett. , 132501(2006).[17] Konstantine L. Metlov, Phys. Rev. Lett. , 127205(2006).[18] D. Laroze, J. Escrig, P. Landeros, D. Altbir, M. V´azquez,and P. Vargas, Nanotechnology , 415708 (2007).[19] J. Escrig, D. Altbir, M. Jaafar, D. Navas, A. Asenjo, andM. V´azquez, Phys. Rev. B , 184429 (2007).[20] F. Porrati, and M. Huth, J. Magn. Magn. Mater. ,145-148 (2005).[21] T. Shinjo, T. Okuno, R. Hassdrof, K. Shigeto, and T.Ono, Science , 930 (2000).[22] A. Wachowiak, J. Wiebe, M. Bode, O. Pietzsch, M. Mor-genstern, and R. Wiesendanger, Science , 577 (2002).[23] A. Aharoni, Introduction to the Theory of Ferromag-netism (Oxford: Clarendon, 1996).[24] I. V. Roshchin, and I. K. Schuller, private comm.[25] E. Feldtkeller, and H. Thomas, Phys. Kondens. Mater. , 8 (1965); P. O. Jubert, and R. Allenspach, Phys. Rev.B , 144402 (2004).[26] N. A. Usov, and S. E. Peschany, J. Magn. Magn. Mater. , L290 (1993).[27] A. Aharoni, J. Appl. Phys. , 2892-2900 (1990).[28] R. H¨ollinger, A. Killinger, and U. Krey, J. Magn. Magn.Mater. , 178-189 (2003).[29] O. Espinosa, and V. H. Moll, Integral Transforms andSpecial Functions , 101-115 (2004).[30] S. Tandon, M. Beleggia, Y. Zhu, and M. De Graef, J.Magn. Magn. Mater. , 21 (2004).[31] M. Beleggia, S. Tandon, Y. Zhu, and M. De Graef, J.Magn. Magn. Mater.278