Scaling relations for magnetic nanoparticles - APS Link Manager

PHYSICAL REVIEW B 71, 094435 共2005兲

Scaling relations for magnetic nanoparticles P. Landeros, J. Escrig, and D. Altbir Departamento de Física, Universidad de Santiago de Chile, USACH, Av. Ecuador 3493, Santiago, Chile

D. Laroze Instituto de Física, Pontificia Universidad Católica de Valparaíso, Casilla 4059, Valparaíso, Chile

J. d’Albuquerque e Castro Instituto de Física, Universidade Federal do Rio de Janeiro, Cx.Postal 68.528, 21941-972, RJ, Brazil

P. Vargas Departamento de Física, Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, Chile 共Received 2 July 2004; revised manuscript received 1 November 2004; published 31 March 2005兲 A detailed investigation of the scaling relations recently proposed 关J. d’Albuquerque e Castro, D. Altbir, J. C. Retamal, and P. Vargas, Phys. Rev. Lett. 88, 237202 共2002兲兴 to study the magnetic properties of nanoparticles is presented. Analytical expressions for the total energy of three characteristic internal configurations of the particles are obtained, in terms of which the behavior of the magnetic phase diagram for those particles upon scaling of the exchange interaction is discussed. The exponent ␩ in scaling relations is shown to be dependent on the geometry of the vortex core, and results for specific cases are presented. DOI: 10.1103/PhysRevB.71.094435

PACS number共s兲: 75.75.⫹a, 75.60.⫺d

I. INTRODUCTION

In recent years, a great deal of attention has been focused on the study of regular arrays of magnetic particles produced by nanoimprint lithography. Besides the basic scientific interest in the magnetic properties of these systems, there is evidence that they might be used in the production of new magnetic devices, or as media for high density magnetic recording.1 One of the main points in the study of such systems concerns the internal magnetic structure of the nanoparticles as a function of their shape and size. For example, in the case of cylindrically shaped particles produced by electrodeposition, the internal arrangements of the magnetic moments have been identified as being close to one of the following three 共idealized兲 characteristic configurations: ferromagnetic with the magnetization parallel to the basis of the cylinder 共F1兲, ferromagnetic with the magnetization parallel to the cylinder axis 共F2兲, and a vortex state, in which most of the magnetic moments lie parallel to the basis of the cylinder 共V兲.2,3 The occurrence of each of these configurations depends on geometrical factors, such as the linear dimensions of the cylinders and their aspect ratio. Clearly, for the development of magnetic devices based on those arrays, knowledge of the internal magnetic structure of the particles is of fundamental importance. Experimentally, attempts have been made to determine, from the analysis of hysteresis curves,4,5 the range of values of diameter D and height H of cylindrically shaped particles for which the internal arrangement of the magnetic moments is close to either one of the two ferromagnetic configurations 共F1 or F2兲 or to the vortex one 共V兲. However, such an approach does not allow a clear description of the magnetic structure of individual cylinders, since in many cases the internal magnetic configurations are not readily identifiable from magnetization curves. 1098-0121/2005/71共9兲/094435共6兲/$23.00

On the other hand, theoretical determination of the configuration of lowest energy of particles in the size range of those currently produced, based on a microscopic approach and using present standard computational facilities, is out of reach. The reason is the exceedingly large number of magnetic moments within such particles, which may exceed 109. Recently, d’Albuquerque e Castro et al.6 have proposed a scaling technique for determining the phase diagram giving the configuration of lowest energy among the three abovementioned characteristic magnetic configurations. They have shown that such a diagram can be obtained from those for much smaller particles, in which the exchange interaction J has been scaled down by a factor x ⬍ 1; i.e., for J⬘ = xJ. The diagram for the full strength of the exchange interaction is then obtained by scaling up the D⬘ and H⬘ axes in the phase diagram for J⬘ by a factor 1 / x␩. In their work, the exponent ␩ has been determined numerically from the position, as a function of x, of a triple point 共Dt , Ht兲 in the phase diagram where the three configurations have equal energy. The scaling technique has been applied to the determination of the phase diagram of cylindrically shaped6 and truncated conical7 particles. In both cases, ␩ turned out to be approximately equal to 0.55. We recall that the vortex configuration exhibits a core region within which the magnetic moments have a nonzero component parallel to the axis of either the cylinder or the truncated cone. We remark that the determination of the geometry of the core 共i.e., its shape and size兲, on the basis of a microscopic model in which the individual magnetic moments are considered, would require a prohibitively large computational effort. For this reason, d’Albuquerque e Castro et al.6 and Escrig et al.7 adopted a simplified representation of the vortex core, consisting of a single line of magnetic moments along the axis of either the cylinders or the truncated cones. The phase diagrams thus obtained are in

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P. LANDEROS et al.

good agreement with experimental data, provided appropriate values of the exchange are considered. The scaling technique represents a useful tool for studying the magnetic properties of nanosized particles. It is conceptually simple and rather interesting from the theoretical point of view. Its implementation depends on the determination of the exponent ␩ in the scaling factor, which so far has been done numerically. The agreement, within error bars, between the values of ␩ for cylinders and truncated conical particles suggests that this parameter does not depend on the shape of the particles. However, there still remains the question regarding the possible dependence of ␩ on the geometry of the vortex core. The present work aims precisely at clarifying this point. We focus on cylindrically shaped particles, for which a large amount of experimental data is available. We adopt a continuous model for the internal magnetic structure of the particles, on the basis of which analytical results for the total energy in each configuration can be obtained. We use these results to investigate the behavior of the phase diagrams under scaling transformation, from which the value of ␩ can be determined. We find that the value of ␩ does depend on the geometry of the vortex core. This point is discussed at length below.

produced by electrodeposition, the crystalline anisotropy term is much smaller than the other two, 10 so that its inclusion has little effect on the phase diagram. In view of this, it will be neglected in our calculations.

II. CONTINUOUS MAGNETIZATION MODEL

where ␣ = F1, F2, and N␣ are the demagnetizing factors, given in SI units by12

We adopt a simplified description of the system, in which the discrete distribution of magnetic moments is replaced ជ 共rជ兲 such that with a continuous one, defined by a function M ជ 共rជ兲␦V gives the total magnetization within the element of M volume ␦V centered at rជ. This model provides a fairly good basis for the discussion of the magnetic properties of nanosized particles. For cylindrically shaped particles, the magជ 共rជ兲 in the two ferromagnetic configuranetization density M tions, F1 and F2, is given by M 0xˆ and M 0zˆ, respectively. Here M 0 is the saturation magnetization density, and xˆ and zˆ are unit vectors parallel to the basis and to the axis of the cylinders, respectively. For the vortex configuration, we assume that the magnetization density has the form

ជ 共rជ兲 = M 共␳兲zˆ + M 共␳兲␸ˆ , M z ␸

共1兲

where zˆ and ␸ˆ are unit vectors in cylindrical coordinates, and M z and M ␸ satisfy the relation M z2 + M ␸2 = M 20. Thus, the profile of the vortex core is fully specified just by giving the function M z共␳兲. It is worth pointing out that the functional form in Eq. 共1兲 does not take into account the possibility of a dependence of the core shape on coordinate z. We then look at the total energy of the three configurations under consideration, from which the magnetic phase diagram can be obtained and its behavior under scaling investigated. We restrict our discussion to arrays in which the separation between cylinders is sufficiently large for the interaction between them to be ignored.8,9 The internal energy per unit of volume 共Etot兲 of a single cylinder is given by the sum of three terms corresponding to the magnetostatic 共Edip兲, the exchange 共Eex兲, and the anisotropy 共EK兲 contributions. However, in the case of particles

A. Ferromagnetic configurations

Since the exchange term depends only on the relative ori共F兲 entation of the magnetic moments, it has the same value Eex in the two ferromagnetic configurations. Since it also appears as an additive term in the expression for exchange energy in the vortex configuration, it can be simply left out in our calculations. The magnetostatic term is generally given by11 Edip =

␮0 2V

冕

ជ 共rជ兲 · 共ⵜជ U兲dV, M

共2兲

where U共rជ兲 is the magnetostatic potential. In the above expression, an additive term independent of the configuration has been left out. For the ferromagnetic configurations, we find that 1 共␣兲 = N␣␮0M 20 , Edip 2

NF1 =

共3兲

冉冊冋

冉冊册

冋

冉冊册

1 2

1 1 D F1 − , ,2,− 2 2 H 2

and

1 1 D NF2 = 1 − 2F1 − , ,2,− 2 2 H

2

−

2D , 3␲H

共4兲

+

4D . 3␲H

共5兲

2

In the above two equations, 2F1共a , b , c , d兲 is a hypergeometric function. Notice that demagnetizing factors depend on just the ratio D / H. B. Vortex configuration

ជ 共rជ兲 varies slowly on the scale of the latAssuming that M tice parameter, the exchange term for this configuration can be approximated by11 共V兲 Eex =

A V

冕关

共ⵜជ mx兲2 + 共ⵜជ my兲2 + 共ⵜជ mz兲2兴dV,

where A is the exchange stiffness constant, and mi = M i / M 0, for i = x , y , z. We recall that A is proportional to the exchange interaction energy J between the magnetic moments.11 Makជ 共rជ兲 in Eq. 共1兲, we find ing use of the expression for M 共V兲 = Eex

2A R2

冕

R

f共␳兲␳ d␳ ,

共6兲

0

where R = D / 2, and f共␳兲 = 共⳵mz / ⳵␳兲2 / 共1 − mz2兲 + 共1 − mz2兲 / ␳2, 共F兲 on the rightwith mz共␳兲 = M z共␳兲 / M 0. The additive term Eex hand side of the above equation has been omitted.

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The magnetostatic term can be also written in terms of ជ 共rជ兲. In the vortex configuration, the magnetostatic potential M is given by 1 U共rជ兲 = 4␲

冕

1 M z共 ␳ 1兲 dS1 − 4␲ 兩rជ − rជ1兩

S1

冕

S2

M z共 ␳ 2兲 dS2 , 兩rជ − rជ2兩

where S1 and S2 are the surfaces of the top and bottom basis of the cylinder, respectively. After some manipulations, the expression for U共rជ兲 reduces to U共␳,z兲 =

1 2

冕

R

␳ ⬘d ␳ ⬘ M z共 ␳ ⬘兲

0

⫻共− e

冕

⬁

+e

−k共H−z兲

兲,

where J0共x兲 is the cylindrical Bessel function of order zero. Taking this result into Eq. 共2兲, we find13 共V兲 Edip =

␲␮0 V

冕冉冕 ⬁

0

冊

R

2

␳J0共k␳兲M z共␳兲d␳ 共1 − e−Hk兲. 共7兲

dk

0

At this point, it is necessary to specify the function M z共␳兲. Since no rigorous result regarding the shape of the vortex core is available, we resort to a simple but physically plausible approximation, given by

再

M 0关1 − 共␳/␳c兲2兴n , for

␳ 艋 ␳c , otherwise

0,

共8兲

where ␳c 艋 R and n is a non-negative constant. Alternative expressions for M z共␳兲 have been proposed in the literature.14 The above functional form for M z共␳兲 allows us to evaluate the energy integrals in Eqs. 共6兲 and 共7兲 analytically. For in共V兲 in Eq. 共6兲 then teger values of n, the expression for Eex reduces to 共V兲 Eex =

冉

冊

2A R ln + ␥n , R2 ␳c

where ␥n = 21 H关2n兴 − nH关−1 / 2n兴. Here, ⬁

H关z兴 =

兺 i=1

冉

1 1 − i i+z

共9兲

冊

冋

冉冊册

6W0d␳3c ␳c ␳c = ␣n − ␤n F n, HR2 4H H

22n−1⌫共n + 1兲3 , 3 5 ⌫ + n ⌫ + 2n 2 2

冉冊冉冊

册

Having evaluated all relevant contributions to the total energy in the three cases of interest, we are in a position to investigate the magnetic phase diagram for cylinders. In particular, we can look at the position of the triple point 共Dt , Ht兲 as a function of the factor x, which scales the stiffness constant A 共or exchange interaction J兲. We notice that since the 共F1兲 and energy of the two ferromagnetic configurations Etot 共F2兲 Etot are equal at the triple point, we immediately get the equation whose solution is ␰t = Dt / Ht = 1.103 17. . . 共independent of A or J兲.12 As a consequence, Dt and Ht are proportional and must exhibit the same functional dependence on x 共or equivalently, on A兲. We proceed in our analysis by looking at the case considered by d’Albuquerque e Castro et al.,6 in which the core radius is independent of x, and of the order of the lattice spacing 共first core model兲. This corresponds to taking the limit ␳c Ⰶ Rt = Dt / 2 in the expressions for the total energy. In this limit, ln共R / ␳c兲 becomes much larger in modulus than ␥n, so that the latter can be safely neglected in Eq. 共9兲. Equations. 共3兲, 共9兲, and 共10兲 then give the following equation for R t: 2A Rt 1 N␣␮0M 20 = 2 ln , 2 Rt ␳c

共11兲

共14兲

where ␣ is either F1 or F2. Now, if we scale down the exchange interaction by a factor x ⬍ 1; that is to say, if we consider a reduced exchange stiffness A⬘ = xA, and assume that Rt and the new radius at the triple point R⬘t are related according to R⬘t = x␩Rt, we find ln

2A x␩Rt Rt = x1–2␩ 2 ln . ␳c ␳c Rt

This expression gives us the following equation for ␩: ln

共10兲

where

␣n =

冎

Here, 3F2 denotes the generalized hypergeometric function.

R2t

,

共13兲

4␳2c 1 3 ,1, + n ,兵n + 2,2n + 3其,− 2 . 2 2 H

␳c F n, = 3F 2 H

2A

is the generalized harmonic number function of the complex variable z.15 For the dipolar energy term in Eq. 共7兲, we obtain 共V兲 Edip

1 W0d = ␮0M 20 , 6

NF1共␰t兲 = NF2共␰t兲,

III. TOTAL ENERGY CALCULATION AND SCALING TRANSFORMATION

M z共 ␳ 兲 =

共12兲

IV. RESULTS

dk J0共k␳兲J0共k␳⬘兲

0

−kz

冉冊冋再

␤n = 1/共1 + n兲2 ,

␩ Rt ln x. = ␳c x2␩−1 − 1

共15兲

It is clear from this equation that ␩ must in all cases be greater than 0.5. It approaches this lower bound only when Rt is much larger than the lattice spacing 共i.e., Rt Ⰷ ␳c兲, in other words, when the particles have macroscopic sizes. The behavior of ␩ in Eq. 共15兲 is presented in Fig. 1. Figure 1共a兲 shows ␩ as a function of Rt, for 20 nm艋 Rt

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P. LANDEROS et al.

FIG. 2. Reduced magnetization mz = M z / M 0 as a function of ␳, for n = 2 共dotted line兲, 4 共dashed line兲, and 10 共solid line兲. The two sets of curves correspond to H = 20 nm and 100 nm. Values of A and M 0 have been taken from Ref. 16, and correspond to those for Co.

␩ results in inaccuracies ␦H and ␦D in the coordinates in the scaled diagram. Indeed, if we write ␩ = ␩0 ± ␦␩, with ␦␩ / ␩0 Ⰶ 1, we immediately get

冏冏 ␦H H0

FIG. 1. Exponent ␩ given by Eq. 共15兲 plotted as a function of Rt, for x = 0.1 共solid line兲 and 0.2 共dashed line兲共a兲, and as a function of x, for Rt = 44 nm 共b兲.

艋 100 nm, and ␳c = 0.2 nm. We notice that in this range of Rt, 0.54⬍ ␩ ⬍ 0.58. It is also interesting to look at the behavior of ␩ as a function of x. Figure 1共b兲 shows ␩ as a function of x, for 0.01艋 x 艋 1, Rt = 44 nm, and ␳c = 0.2 nm. From the curves in Figs. 1共a兲 and 1共b兲, we find that for x 艌 0.05, ␩ turns out to be close to 0.55, as numerically obtained by d’Albuquerque e Castro et al.6 It is worth commenting on the effect of using a single value of ␩, say 0.55, to scale phase diagrams for the core model considered just above. As already pointed out, the diagram for the full strength of the exchange interaction can be obtained from the one corresponding to a reduced interaction J⬘ = xJ 共with x ⬍ 1兲 by multiplying the axes H⬘ and D⬘ of the latter by 1 / x␩. Thus, an inaccuracy ␦␩ in the value of

= − 共␩0 ln x兲

冏冏

␦␩ , ␩0

where H0 = x␩0H⬘. Since ␩0 ⬇ 0.55 and ␦␩ / ␩0 ⬇ 0.01 关estimated from Fig. 1共a兲兴, we find that, even for x as small as 0.05, the relative error ␦H / H0 is smaller than 2%. Thus, we do not expect large discrepancies between the calculated phase diagram and the experimental data resulting from such inaccuracy in ␩ since a relative error of 2% should not exceed the experimental error. We remark that the above results for ␩ hold also when the core radius corresponds to several interatomic distances and is kept fixed as the exchange interaction is scaled up or down. We next consider the case in which ␳c is adjusted so as to minimize the energy of the vortex configuration 共second core model兲. From Eqs. 共9兲 and 共10兲, we obtain the following equation for ␳c: 3␣n

␳3c ␳4c ␤n ␳6c − ␤ F共n, ␳ /H兲 + G共n, ␳c/H兲 n c H3 H4 2共n + 2兲 H6

=

2A

␮0M 20H2

where G共n, ␳c/H兲 = 3F2

,

冋再

冎

册

4␳2 3 5 ,2, + n ,兵3 + n,4 + 2n其,− 2c . 2 2 H

Equation 共15兲 can be solved numerically for ␳c in terms of H, A, and n. We remark that for the core model under consideration, ␳c does not depend on the radius R. This follows from the fact that the outer region of the cylinder does not interact with the core 共apart from the exchange interaction across the interface between the two regions兲. As a conse-

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We remark that these results holds for any other integer values of n, due to the fact that since ␳c is adjusted so as to minimize the energy in the vortex configuration, the effective radius of the core turns out to be independent of n, as clearly shown in Fig. 2.

V. CONCLUSIONS

FIG. 3. Phase diagram for Co cylinders corresponding to x = 0.12 共dashed lines兲, x = 0.24 共dotted lines兲, and x = 1 共solid lines兲 obtained for core model 2 共see text兲.

quence, for a given value of ␳c, the difference between the total energy of two cylinders of the same height but different radii does not depend on ␳c; hence, it does not contribute to 共V兲 the derivative of Etot with respect to ␳c. That is to say, the equation for ␳c that minimizes the total energy of the vortex configuration is independent of R. Figure 2 illustrates M z共␳兲 for A = 87.39 meV/ nm, M 0 = 1.4⫻ 106 A / m, and two values of H: 20 and 100 nm. For each H, results are presented for n = 2 共dotted line兲, 4 共dashed line兲, and 10 共solid line兲. The values of A and M 0 correspond to those for Co, and have been taken from Ref. 16. The value of ␳c in each case has been obtained from Eq. 共15兲. In order to investigate the behavior of magnetic phase diagram upon scaling of the exchange interaction for this second core model, we take n = 4, which according to Fig. 2 provides a physically sound description of the core profile, and calculate the phase diagrams for distinct values of x. Figure 3 shows results for cylinders of Co 共A = 87.39 meV/ nm and M 0 = 1.4⫻ 106 A / m兲 corresponding to x = 0.12 共dashed lines兲 and x = 24 共dotted lines兲. We then find that, for the present core model, the coordinates 共Dt , Ht兲 of the triple point follow the relations Dt共x兲 = 25.61x0.5 ,

共16兲

Ht共x兲 = 23.22x0.5 ,

共17兲

in which ␩ = 0.5. The diagram for the the full strength of the exchange interaction, x = 1, is represented by solid line.

Y. Chou, Proc. IEEE 85, 652 共1997兲; G. Prinz, Science 282, 1660 共1998兲. 2 J. N. Chapman, P. R. Aitchison, K. J. Kirk, S. McVitie, J. C. S. Kools, and M. F. Gillies, J. Appl. Phys. 83, 5321 共1998兲. 3 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 65, 144417 共2002兲. 1 S.

We have carried out a detailed analysis of scaling technique recently proposed by d’Albuquerque e Castro et al.6 to investigate the magnetic phase diagram of nanoparticles. As already pointed out, this technique enables us to obtain the phase diagram for particles in the nanometer size range from those corresponding to much smaller particles, in which the exchange interaction has been reduced. The scaling technique is easily implemented and represents a rather useful tool for dealing with nanoparticle systems. In addition, the existence of scaling relations and their connection with the model adopted to describe the magnetic particles bring about interesting theoretical considerations. The present work sheds light on a very interesting feature of the scaling relations; namely, the dependence of the exponent ␩ on the model adopted for describing the core of the vortex configuration. Based on a continuous magnetization model, we were able to derive analytical expressions for the total energy in each configuration, which allowed us to determine the exponent ␩. We found that in the case of nanoparticles for which the core dimensions, and consequently its contribution to the total energy, can be either neglected or do not change much upon scaling of A, ␩ turns out to be weakly dependent on x and quite close to 0.55. Nevertheless, when the contribution from the core is relevant and its size upon scaling of A changes so as to minimize the total energy in the vortex configuration, then ␩ becomes exactly equal to 0.5.

ACKNOWLEDGMENTS

This work has been partially supported by Fondo Nacional de Investigaciones Científicas y Tecnológicas 共FONDECYT, Chile兲 under Grant Nos. 1040354, 1020071, 1010127, and 7010127, and Millennium Science Nucleus “Condensed Matter Physics” P02-054F of Chile, and CNPq, FAPERJ, and Instituto de Nanociências/MCT of Brazil. CONICYT Ph.D. Program Fellowships, as well as MECESUP USA0108 and FSM-990 projects are gratefully acknowledged.

4 R.

P. Cowburn, D. K. Koltsov, A. O. Adeyeye, and M. E. Welland, D. M. Tricker, Phys. Rev. Lett. 83, 1042 共1999兲. 5 A. Lebib, S. P. Li, M. Natali, and Y. Chen, J. Appl. Phys. 89, 3892 共2001兲. 6 J. d’Albuquerque e Castro, D. Altbir, J. C. Retamal, and P. Vargas, Phys. Rev. Lett. 88, 237202 共2002兲. 7 J. Escrig, P. Landeros, J. C. Retamal, D. Altbir, and J.

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P. LANDEROS et al. d’Albuquerque e Castro, Appl. Phys. Lett. 82, 3478 共2003兲. Grimsditch, Y. Jaccard, and Ivan K. Schuller, Phys. Rev. B 58, 11 539 共1998兲. 9 K. Yu Guslienko, Sug-Bong Choe, and Sung-Chul Shin, Appl. Phys. Lett. 76, 3609 共2000兲. 10 C. A. Ross, Henry I. Smith, T. Savas, M. Schattenburg, M. Farhoud, M. Hwang, M. Walsh, M. C. Abraham, and R. J. Ram, J. Vac. Sci. Technol. B 17, 3168 共1999兲. 11 A. Aharoni, Introduction to the Theory of Ferromagnetism 共Clarendon, Oxford, 1996兲. 8 M.

12 S.

Tandom, M. Beleggia, Y. Zhu, and M. De Graef, J. Magn. Magn. Mater. 271, 9 共2004兲. 13 See, e.g., A. Hubert and R. Schäfer, Magnetic Domains: The Analysis of Magnetic Microstructures 共Springer, Berlin, 1998兲, and references therein. 14 See, e.g., N. A. Usov and S. E. Peschany, J. Magn. Magn. Mater. 118, L290 共1993兲. 15 O. Espinosa and V. H. Moll, math.CA/0304079 共unpublished兲. 16 http://magnet.atp.tuwien.ac.at/scholz/projects/diss/html/node86.html

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