2950 IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 6, JUNE 2007

Magnetic Anisotropy of FePt Nanoparticles: Temperature-DependentFree Energy Barrier For Switching

Chenggang Zhou1, Thomas C. Schulthess1, and Oleg N. Mryasov2

Center for Nanophase Materials Science, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6493 USASeagate Research, Pittsburgh, PA 15222 USA

We report the calculation of free energy with constrained magnetization for 10 FePt nanoparticles. We employ an effective spinHamiltonian model constructed on the basis of constrained density functional theory calculations for 10 FePt. In this model, the Fespins (treated as classical spins in this paper) are coupled directly and via induced Pt moments with both isotropic and anisotropicinteractions. Interactions mediated by the Stoner-enhanced Pt moment stabilize the ferromagnetic order and lead to a pronounced co-ordination dependence and long-range interactions. The free energy of these nanoparticles, as a function of the temperature and theconstrained magnetization ( ), is calculated from the joint density of states ( ), using the extended Wang–Landau algo-rithm. The free energy barrier for magnetization reorientation is found to depend fairly linearly on the temperature in the ferromagneticphase and vanishes in the paramagnetic phase.

Index Terms—Magnetic anisotropy, magnetization reversal, Monte Carlo methods.

I. INTRODUCTION

MAGNETIC nanostructures for data storage and othermagneto-electronic applications have to be fabricated

using materials with sufficiently large magnetic anisotropyenergy (MAE) to provide long-term thermal stability. Magneticnanoparticles with large MAE are therefore of great techno-logical interest [1]. Recently, FePt nanoparticles withorder have been actively studied as promising material systemswith high uniaxial magnetic anisotropy constant and tight sizedistributions [2], [3]. For the last decade or so, FePt hasbeen considered as one of the prominent candidates for thenext generation of magnetic storage devices due to its highMAE, good corrosion resistance, and mechanical properties.This material both in thin film and nanoparticle forms hasbeen intensively studied mainly in the context of data storageapplications. The uniaxial anisotropy constant of FePtthin films was measured to show dependence on magnetizationas , where [4], [5]. It was found thateven for small FePt nanoparticles, the coercivity is closeto the bulk value, and the switching in nanoparticles smallerthan a critical size is characterized by coherent rotation ofmagnetization [6]. Theoretically, FePt is understood as an itin-erant ferromagnet [7]–[9]. The well-established theory of MAEtemperature dependence assumes the single-ion anisotropy[10]–[12], which does not account for important features ofmagnetic interactions in FePt and in general in 3-D–5-D (4-D)ferromagnets [9]. While the temperature dependence ofMAE has been recently understood for bulk FePt withinthe effective spin Hamiltonian model used in this paper [9],quantification and understanding of the free energy barrierremains a challenge. The free energy barrier is a prime quantityfor evaluating long-term thermal stability of magnetic nanos-tructure but it is not easy to calculate or measure in the general

Digital Object Identifier 10.1109/TMAG.2007.893795

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.iee.org.

case. In the case of coherent switching, the energy barriercan be simply calculated as , where is the uniaxialanisotropy constant and the volume of magnetic nanostruc-ture. It is not clear, however, if this approximation is applicablefor small nanoparticles. For the general case of a nanostructureof arbitrary shape and size, the direct calculations of the freeenergy barrier are rare, and to the best of our knowledge havebeen done only for . [13] The free energy barrier also canbe evaluated indirectly from the average escape time as it hasbeen done for small FePt nanoparticles [14] within the effectivespin model [9].

This paper is intended to fill the gap by applying a novelalgorithm for the free energy calculations with effective spinHamiltonian relevant to the class of 3-D–5-D (4-D) ferromag-nets, parameterized specifically for FePt. In this paper, weuse the extended Wang–Landau algorithm [15]–[17] to calculatethe joint density of states and the free energies with constraintmagnetization and identify directly the free energybarrier for nanoparticles with different sizes.

II. MODEL AND SIMULATION

We consider chemically ordered FePt in the structure,in which the Fe and Pt atoms occupy adjacent [001] planesof a face-centered tetragonal lattice. The effect of theratio enters the model through the exchange integrals fromprevious electronic structure calculations [9]. The physicalpicture that emerges from the constrained local spin-densitycalculations [9] includes: 1) localized magnetic moments onFe atoms, which can be modeled by vector spins of unitlength, and 2) induced magnetic moments on Pt atoms ,which linearly depend on the superposition of neighboring Fespins, i.e., , where the summation isover eight nearest neighbor Fe spins and is the maximummagnetization of an Pt atom in a ferromagnetically alignedenvironment. The electronic structure calculations also showthat an effective spin Hamiltonian for FePt must contain ex-change interactions between Fe spins, easy-plane anisotropyon Fe spins ( meV) , Stoner enhancement onPt atoms [18], and an easy-axis anisotropy terms due to the Pt

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ZHOU et al.: MAGNETIC ANISOTROPY OF FEPT NANOPARTICLES: TEMPERATURE-DEPENDENT FREE ENERGY BARRIER FOR SWITCHING 2951

magnetic moments. Since the Fe–Fe exchange interactions aremediated by itinerant electrons, the exchange constants decayslowly with the distance between Fe spins and can be antifer-romagnetic between certain pairs of atoms. We have included76 neighbors for each of the Fe spins. Each Pt atom contributesa term to the Hamiltonian [9]. The Ptspins do not add additional degrees of freedom to the systembut lead to the following effective spin Hamiltonian , whichhas anisotropic exchanges due to the anisotropy on Pt atoms:

constant (1)

Since detailed electronic structure calculations for the surface,corners, and edges, are not yet available, we simply omit thesecomplexities in the present stage.

We construct our nanoparticles by cutting a truncated octa-hedron from a periodic lattice. During the switching process,the magnetization of the nanoparticle overcomes a barrier in theBragg–Williams (BW) free energy , where is thenormalized magnetization in the direction. Here, we assumebefore and after the switching process, the magnetization of thenanoparticle is in the direction, respectively. Theis calculated from the joint density of states accordingto the formula

(2)

where is calculated with the extended Wang–Landaualgorithm in [16]. At a temperature below the ordering tem-perature of the nanoparticle, is expected to havetwo local minima at , separated by a free energybarrier. Although we do not specify the dynamics of theswitching process, no matter if it is coherent or incoherent, itis a process in which changes continuously fromto . The vector magnetization of the nanoparticlemay travel through a complex path, but our algorithm aver-ages over all possible paths with the Boltzmann weight, and

is dominated by the leastcostly path in free energy.

III. RESULTS

Ground state electronic structure calculations of FePt alloys[19] show that the fully ordered systems could even be anti-ferromagnetic, but that small amounts of anti-site disorder willlead to a stabilization of the ferromagnetic state. In our model ofFePt, if the effects of Pt atoms are ignored, the exchange inter-actions between Fe atoms do predict such an antiferromagnet.However, once the effects of Pt atoms are included in the Hamil-tonian, the ferromagnetic configuration wins the ground stateby gaining 1.403 mRy (221.4 K) per spin. Although this gainin the ground state energy is not large, we do observe the fer-romagnetic phase transition at about K. Fig. 1shows the finite-size scaling plot of the magnetization, whichis consistent with a three-dimensional Ising ferromagnet ratherthan a Heisenberg ferromagnet due to the uniaxial anisotropyin the Hamiltonian. The magnetization is here defined as

Fig. 1. Finite-size scaling of magnetization with 3-D Ising exponents.Inset shows same data points scaled with 3-D Heisenberg exponents� = 0:6930; � = 0:794. Error bars are smaller than symbols.

Fig. 2. Magnetization temperature dependence calculated for FePt nanoparti-cles of radius 6 and 8 (in unit of lattice constant) and compared with bulk sim-ulation with L = 30 and periodic boundary conditions.

, where are Fe spins on a tetragonal lattice withunit cells in each direction and periodic boundary conditions.

We therefore believe that the ferromagnetism in FePt is stabi-lized by the Stoner enhancement of the induced magnetizationon Pt atoms. We have also performed simulations with fewerFe–Fe exchange terms and found the critical temperature was re-duced considerably. Therefore, is sensitive to the long-rangeexchange interactions. The difference from the experimental

( 750 K) is probably due to the truncation of the Fe–Feexchange in our model, which is expected to be a long-rangeinteraction. Fig. 2 shows the magnetization of two nanoparti-cles of different sizes as a function of temperature. Comparedto the simulation of a large bulk system, the magnetization ofnanoparticles are reduced and the transition to the ordered phaseis gradual.

In order to discuss the size and temperature dependence of theBW free energy barrier of nanoparticles, we plotfor several nanoparticles as a function of temperatures in Fig. 3.The barrier linearly depends on temperature where the nanopar-ticles have a spontaneous magnetization. We can fit these curveswith , where is understood

2952 IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 6, JUNE 2007

Fig. 3. Free energy barrier per spin for four (truncated octahedron) nanopar-ticles of different sizes, averaged over 20 to 40 calculations so that error barsare smaller than symbols. Linear dependence on temperature is observed in allcases below their ordering temperatures. R is distance from center to one ofnanoparticle’s square surfaces.

as the size-dependent ordering temperature of the nanoparti-cles, and is the energy difference (free energy difference)at . is simply given by the energy difference be-tween a ferromagnetic configuration lying in the - (the groundstate if the constraint is satisfied) plane and the groundstate ferromagnetic configuration parallel to the axis. For largenanoparticles, is an extensive quantity. For small nanopar-ticles, is reduced by finite-size effect, and also con-tains a surface correction term which can be important. Hence,we expect the size dependence of is asymptotically linear involume for large nanoparticle, with the lowest order correctionterm given by the surface effect. This linear dependence on tem-perature is reminiscent of the temperature dependence of coer-civity measured by Okamoto et al.. [6]. For coherent switchingprocesses, we expect the free energy barrier to be proportionalto the coercivity, and thus the temperature dependence of thefree energy barrier implies the temperature dependence of thecoercivity. Although we have not included the complexities ofsurface exchange in our model, we do find a qualitative agree-ment between our theoretical calculations and experiment.

IV. CONCLUSION

In conclusion, we have constructed a classical spin model forFePt with order based on previous first-principles calcula-tions. The ferromagnetism in bulk FePt is found to be stabilizedby the Stoner enhancement effects on the induced moments of Ptatoms. The critical temperature K found by MonteCarlo simulations and finite-size scaling is below but close to theexperimental value. A large free energy barrier develops belowthe ordering temperature. Its height linearly depends on tem-perature. This free energy barrier is a measure of the thermalstability of the magnetization. Our numerical calculation dis-

cussed here is the first attempt to quantitatively describe the freeenergy barriers for the switching of magnetization in magneticnanoparticles.

ACKNOWLEDGMENT

This work was conducted at the Center for Nanophase Ma-terials Sciences, which was sponsored at Oak Ridge NationalLaboratory by the Division of Scientific User Facilities, U.S.Department of Energy.

REFERENCES

[1] M. Koyanagi, J. Bea, C.-K. Yin, T. Fukushima, and T. Tanaka, “Newnon-volatile memory with magnetic nano-dots,” ECS Trans., vol. 2, p.209, 2006.

[2] S. Sun, C. B. Murray, D. Weller, L. Folks, and A. Moser, “Monodis-perse fept nanoparticles and ferromagnetic fept nanocrystal superlat-tices,” Science, vol. 287, p. 1989, Mar. 2000.

[3] H. Zeng, J. Li, J. P. Liu, Z. L. wang, and S. Sun, “Exchange-couplednanocomposite magnets by nanoparticle self-assembly,” Nature, vol.420, p. 395, Nov. 2002.

[4] J.-U. Thiele, K. R. Coffey, M. F. Toney, J. A. Hedstrom, and A. J.Kellock, “Temperature dependent magnetic properties of highly chem-ically ordered Fe Ni Pt L1 films,” J. Appl. Phys., vol. 91, pp.6595–6600, 2002.

[5] S. Okamoto, N. Kikuchi, O. Kitakami, T. Miyazaki, Y. Shimada, andK. Fukamichi, “Chemical-order-dependent magnetic anisotropy andexchange stiffness constant of FePt (001) epitaxial films,” Phys. Rev.B, vol. 66, p. 024413, 2002.

[6] S. Okamoto, O. Kitakami, N. Kikuchi, T. Miyazaki, Y. Shimada,and Y. K. Takahashi, “Size dependences of magnetic properties andswitching behavior in FePt L1 nanoparticles,” Phys. Rev. B, vol. 67,p. 094422, 2003.

[7] J. B. Staunton, S. Ostanin, S. S. A. Razee, B. L. Gyorffy, L. Szun-yogh, B. Ginatempo, and E. Bruno, “Temperature dependent magneticanisotropy in metallic magnets from an ab initio electronic structuretheory: L1 -ordered FePt,” Phys. Rev. Lett., vol. 93, p. 257204, 2004.

[8] T. Burkert, O. Eriksson, S. I. Simak, A. V. Ruban, B. Sanyal, L. Nord-strom, and J. M. Wills, “Magnetic anisotropy of L1 FePt and FeMn Pt,” Phys. Rev. B, vol. 71, p. 134411, 2005.

[9] O. N. Mryasov, U. Nowak, K. Y. Guslienko, and R. W. Chantrell,“Temperature-dependent magnetic properties of : Effective spin Hamil-tonian model,” Europhys. Lett., vol. 86, pp. 805–811, 2005.

[10] N. Akulov, “Zur quantentheorie der temperaturabhängigkeit der mag-netisierungskurve,” Z. Phys., vol. 100, p. 197, 1936.

[11] C. Zener, “Classical theory of the temperature dependence of magneticanisotropy energy,” Phys. Rev., vol. 96, pp. 1335–1337, 1954.

[12] H. B. Callen and E. Callen, “The present status of the temperature de-pendence of magnetocrystalline anisotropy, and the l(l + 1)=2 powerlaw,” J. Phys. Chem. Solids, vol. 27, pp. 1271–1285, 1966.

[13] D. A. Garanin and H. Kachkachi, “Surface contribution to theanisotropy of magnetic nanoparticles,” Phys. Rev. Lett., vol. 90, p.065504, 2003.

[14] U. Nowak, O. N. Mryasov, R. Weiser, and K. Guslienko, “Spin dy-namics of magnetic nano-particles: Beyond browns theory,” Phys. Rev.B, vol. 72, p. 172410, 2005.

[15] F. Wang and D. P. Landau, “Efficient, multiple-range random walkalgorithm to calculate the density of states,” Phys. Rev. Lett., vol. 86,pp. 2050–2053, 2001.

[16] C. Zhou, T. C. Schulthess, and D. P. Landau, “Monte carlo simulationsof NiFe O nanoparticles,” J. Appl. Phys., vol. 99, p. 08H906, 2006.

[17] C. Zhou, T. C. Schulthess, S. Torbrugge, and D. P. Landau,“Wang–Landau algorithm for continuous models and joint den-sity of states,” Phys. Rev. Lett., vol. 96, p. 120201, 2006.

[18] P. M. Marcus and V. L. Moruzzi, “Stoner model of ferromagnetismand total-energy band theory,” Phys. Rev. B, vol. 38, pp. 6949–6953,1988.

[19] G. Brown, B. Kraczek, A. Janotti, T. C. Schulthess, G. M. Stocks, andD. D. Johnson, “Competition between ferromagnetism and antiferro-magnetism in FEPT,” Phys. Rev. B, vol. 68, p. 052405, 2003.

Manuscript received October 31, 2007 (e-mails: zhouc@ornl.gov;tcschulthess@ornl.gov; oleg.mryasov@seagate.com).