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Journal of Magnetism and Magnetic Materials 320 (2008) 2149–2154 www.elsevier.com/locate/jmmm

Monte Carlo simulation of magnetic ordering in the Gd3Fe5O12 Ising ferrite with garnet structure Nicolae Stanicaa,, Fanica Cimpoesua, Gianina Dobrescua, Gabriel Munteanua, Soong-Hyuck Suhb a

Institute of Physical Chemistry, Coordination Chemistry, Splaiul Independentei 202, Bucharest 77208, Romania b Department of Chemical Engineering Keimyung University, Taegu 704-701, Republic of Korea Received 27 August 2007; received in revised form 11 March 2008 Available online 8 April 2008

Abstract The simulation of magnetic properties in extended quantum spin networks can be done in good conditions with Ising models within Monte Carlo–Metropolis algorithms, as our systematic studies, employing original computer codes, proved. The present analysis provides interesting insights into the exchange interactions governing the magnetic behavior of the Gd3Fe5O12 system, taken as a prototype for ferrites with garnet structure. Effective exchange interaction parameters are estimated by fitting the computed with experimental compensation temperature and temperature dependence of the magnetization. r 2008 Elsevier B.V. All rights reserved. Keywords: Ferrimagnet; Garnet-type lattice; Ising Hamiltonian; Monte Carlo algorithms

1. Introduction Cubic iron garnets based on various rare-earth ions form a prototypic class of ferrimagnetic oxides with applications in magneto-optical devices, waveguide optical isolators and magnetic bubble memories [1]. The study of ultrathin metallic magnetic films is an active field, many efforts being devoted to investigate the dimensionality and interfacial effects, anisotropy, giant magnetoresistance and magnetooptical properties [2]. At low dimensionality the properties depart from those of bulk materials, due to the new balance of magnetic interactions. Therefore, thin magnetic oxides are interesting study objects, both in fundamental and in application aspects. The Monte Carlo (MC) simulation of magnetic ordering is a valuable tool for explanatory and predictive insight into the properties of Ising ferrites and related materials. Due to good tractability in large-scale calculations, the MC procedure coupled with Ising approach is suitable to design various numerical experiments, e.g. the simulation of the Corresponding author. Tel./fax: +40 21 312 11 47.

E-mail address: [email protected] (N. Stanica). 0304-8853/$ - see front matter r 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2008.03.045

finite size and low-dimensionality effects. Here we present a case study on finite size portions of gadolinium iron garnet, Gd3Fe5O12, giving a systematic account of spin Hamiltonian parameters, in relationship with experimental magnetization curve details. It is worth recalling some generalities about the garnet crystal structure. Thus, for a general formula A3B3X2O12, with ideal O10 h Ia3d space group, the A ions occupy distorted cube sites, while B and X octahedral and tetrahedral ones, respectively [3,4]. In the rare-earth iron garnets (REIG) Ln3Fe5O12 and ytrium iron garnet Y3Fe5O12 (YIG) [5], the FeIII ions show octahedral or tetrahedral environment, while the rare-earth ions are in the centers of deformed cubes. Depending on the manner of counting the anion neighborhood, i.e. including the slightly more distant oxygen centers, the lanthanide environment can also be considered a dodecahedral distorted type. The ions placed in octahedral and tetrahedral positions are antiferromagnetically coupled [6,7]. In REIG, the iron sublattice is the same as in YIG systems and the magnetic moment of Ln3+ set is antiparallel to the total magnetic moment of the iron sublattices [8].

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parameters of the iron sublattice, taken transferable from previous analyses of the YIG system [10,11]. The YIG system represents the diamagnetic reference for the trivalent ions, adding here the new quantum spin of Gd (III) as a new step in the systematization of magnetism in REIG complex lattices. 2. Result and discussion

Fig. 1. Gd3Fe5O12 unit cell.

In order to define the topology of the generated clusters, taken as large multiples of the unit cell, the crystal data from Ref. [9] were used. Based on these crystal data for ferrimagnetic Y3Fe5O12, a useful drawing of the unit cell of Gd3Fe5O12 is obtained (Fig. 1). One of us (NS) developed a FORTRAN computer code, and initiated a systematic investigation of related prototype systems in a series of papers [10,11,26]. The Ising Spin model is appropriate for the systems with magnetic anisotropy, as the networks behaving as magnets implicitly are. Besides, it can account for transition to long-range order at a finite, non-zero temperature [12]. The MC technique , based on the Metropolis algorithm [13], generates a sampling of states following the Boltzmann distribution that preferentially contains configurations which minimize the interaction energy of the system and bring important contribution to the magnetization at temperature T [14]. The modeling with appropriate Hamiltonians such as the Ising one is expected to illuminate more complex intricacies related with magnetic anisotropy [15], spin canting [16] and other effects that determine the properties of a magnet [17,18], from macroscopic respects to nanoscale and the microscopic causalities [19,20]. Particularly, the inadequacies of the traditional molecular-field models must be noted [21]. For instance, due to the negligence of the energy associated with short-range order, such models lead to a large overestimation of the ordering temperature. We shall demonstrate that Ising–MC simulation based on the Metropolis algorithm gives the possibility to determine the Ne´el (or Curie) temperature TN (TC), the compensation temperature Tcomp [29] and the variation of the spontaneous sublattice magnetizations or of the magnetic susceptibility with temperature. The results of MC simulations on the Ising ferrimagnet represented by the Gd3Fe5O12 case (GdIG) are initiated starting from the

Following the well-known crystallographic notation of the garnet lattice sites, the octahedral FeIII ions will take the ‘‘a’’ label, while the tetrahedral FeIII centers the ‘‘d’’ one. The dodecahedrally coordinated GdIII ions will be marked by ‘‘c’’ label. For each site, of type r(r=a, d or c) and a given order number r (1prp512 or 1728, respectively, for 8 or 27 full cubic elementary cell (FCEC) [10,11,26]), having the system in general configuration, ascribed by the set Sz of all-site spin projections, one may assign a site Ising energy, as follows: E Ising rðrÞ fS z g ¼  2J ra

na X

aðiÞ SrðrÞ z S z  2J rd

i¼1

 2J rc

nc X

nd X

dðjÞ SrðrÞ z Sz

j¼1 cðkÞ S rðrÞ z Sz ,

(1)

k¼1

where the sums running over all coupling types, with a, d and c ions, of corresponding order indices i, j, k, respectively, are found in the neighborhood of the accounted r(r) center. The quantities labeled by na, nd and nc, respectively, are the coordination numbers. The magnetic field will add the Zeeman term on each ion E Zeeman fSz g ¼ grðrÞ mB H z S rðrÞ z . rðrÞ

(2)

The energy of the given configuration Sz can be written as  X 1 Ising Zeeman E rðrÞ fSz g þ E rðrÞ fS z g , EfS z g ¼ 2 r

(3)

where the 12 factor in the Ising term summation is taken in order to eliminate the double count of coupling interactions. The spin on a and d centers is 52, meaning that the Saz and Sdz spin projections take values in the (752, 732, 712) set. For GdIII, the spin 72 implies the (772, 752, 732, 712) set for the Scz spin projections. We have considered the following nearest-neighbor exchange parameters: Jad, Jdc, for antiferromagnetic interactions and Jac, Jaa, Jdd, Jcc for ferromagnetic interactions. In order to minimize the errors related to the edge perturbation in finite samples and accelerate convergence towards the infinite lattice limit, periodic boundary conditions (PBC) were adopted [22]. The calculations are performed on large clusters, multiplying in all directions the FCEC and adding the periodic boundary links, that enforce the formal topological contact between opposed

ARTICLE IN PRESS N. Stanica et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 2149–2154

margins of the whole cube. The simulations were performed on such periodic boundary ‘‘garnet lattice’’ finite samples with 512 (8 FCEC) or 1728 (27 FCEC) sites. To obtain stable results, the optimal sizes of the samples were determined by carrying out simulations on a range of different sample sizes. The minimum size that showed a finite-size effect for the studied reduced temperature range kT/|Jab| was even for one FCEC only. The results presented in this work are obtained using 8 FCEC samples. In order to check the size consistency statistics, we also tried the 27 FCEC sample in the same run-time (about 100 h for each case). The results are found to be practically the same in both cases, certifying the reliability of the obtained parameters and data (critical and compensation temperatures, magnetizations or susceptibilities). For the accuracy of the present study and its aims using the 8 FCEC samples and PBC seems to be a good choice as far as the results obtained using large samples, in the same run-time, are not different. The determination of efficient size sample is important since CPU time increases significantly with the size of the MC problem. For each site, at least 104 MC steps were performed while the first 5  103 ones were discarded as the initial transient stage [22]. To avoid freezing of the spin configuration, we have used a low cooling rate, according to the following equation: ðP0 Þiþ1 ¼ 0:99  ðP0 Þi ,

(4)

where P0 ¼ kT/|Jad| is the reduced temperature parameter. The FORTRAN code allows terms accounting for noncollinear configurations. However, it was reasonable to fit the data on YIG and GdIG garnets assuming collinear spin arrangement, because GdIII has no intrinsic anisotropy. At the same time, one must foresee that in the case of other rare-earth garnets the saturation magnetization calculated at 0 K from the Ne`el model, for example, is different from that observed experimentally. This discrepancy has been attributed [20] to canting within the ‘‘c’’ sublattice, due to the strong anisotropy field of LnIII ions. In order to avoid the over-parameterization, we systematically considered parameter transferability from previous results [5,17,23] imposing also Jcc=0. For the Y3F5O12 case, Jad, Jaa and Jdd have been computed [11] from comparison between simulated and experimental data [24]. Since amongst the rare-earth iron garnets Tc is approximately constant, it can be assumed that the FeIII–FeIII interaction exchange parameter Jad is dominant; thus, the value obtained from the Y3Fe5O12 case remains constant for Gd3Fe5O12 and for all the REIGs. For each MC step, one site of the lattice is randomly picked and the spin state is changed. If this event produces a lower energy, the change is accepted automatically; if not, the change is accepted with the probability [13] p ¼ eDE=kT ,

(5)

where DE is the energy difference between the new and the old spin state. Configurations were generated by randomly

2151

sweeping through the lattice and flipping the spins one at a time, according to the heat-bath algorithm. To do one sweep implies to visit randomly all the spins, every spin at least once. The parameters Jad, Jaa, Jdd, Jac, Jdc were replaced by following reduced parameters: P0 

kT J aa J dd J ac jJ dc j ; P1  ; P2  ; P3 ¼ ; P4 ¼ . jJ ad j jJ ad j jJ ad j jJ ad j jJ ad j (6)

For Gd3Fe5O12 case, parameters P1 and P2 were kept frozen at the respective 0.05 and 0.11 values [11], obtained from comparison between simulated and experimental data [24]. We present the results of 183 runs for a sample with 8 FCEC, where P3 and P4 are fixed for each run at one of the values presented in Table 1 for the first 153 sets; more 30 runs for the compensation temperature parameter (P0)comp in the proximity of 5.6 value along the linear curve between P3 and P4 reduced exchange parameters are also included in Fig. 2. P0 is varied by Eq. (4), which gives ‘‘the cooling rate’’. The critical temperatures, Tc or (P0)critkTc/|Jad|, were located at the inflexion point in M versus T curve [25], yielding practically the same value, 11.2 [11], for all the 183 runs. In Gd3Fe5O12 case, there is another significant material data, (P0)compkTcomp/|Jad|9, where Tcomp is the compensation temperature (TcompoTc); at Tcomp, the three sublattice magnetizations cancel out to other [29]. The internal magnetic interaction energy, the specific magnetic heat, the sublattice magnetizations and their sum, as well as the associated susceptibilities, are calculated with equations given elsewhere [26]. Variation of (P0)compkTcomp/|Jad| with P3 ¼ Jac/|Jad| and P4 ¼ |Jdc|/ |Jad| is shown in Figs. 2 and 3. These results are confirmed for layered, bimetallic ferrimagnets [27] that show both compensate and non-compensate behavior at low temperatures. The results represented in Fig. 2 are well fitted with a plane equation kT comp J ac jJ dc j ¼ 100:7  þ 137:3   3:1, jJ ad j jJ ad j jJ ad j

(7)

obtained by a linear least-squares fit. From the intersection between this plane and an horizontal one defined by (kTcomp/|Jad|) 5.6 for Gd3Fe5O12, corresponding to experimental values [18,24] Gd T Gd comp ¼ 280 K and T Curie ¼ 560 K, !   kT Curie ¼ 11:2 , jJ ad j Gd3 Fe5 O12

(8)

one obtains the relation (9) between P4 and P3: P4 ¼ 0:73 P3 þ 0:063,

(9)

which is the line shown in Fig. 3. Fig. 3 shows the P4 vs. P3 parametric correlation, obtained as described in text from the intersection between

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Table 1 The first 153 runs with the compensation temperature (P0)comp obtained as a function of reduced exchange parameters P3 and P4 P3

P4 .000

.000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050 .055 .060 .065 .070 .075 .080 .085 .090

0.4245 0.9280 1.4320 1.9350 2.4380 2.9420 3.4460 4.0500 4.4400 4.9500 5.4500 5.9400

.005

.010

0.1040 0.6075 1.1110 1.6150 2.1180 2.6220 3.1250 3.6290 4.2200 4.6200 5.1600 5.5050 6.1800 6.8300

0.2870 0.7905 1.2940 1.7980 2.3010 2.8050 3.3080 3.8120 4.4400 4.8100 5.3150 5.8500 6.3000 6.8330

.015

0.4700 0.9735 1.4770 1.9800 2.4840 2.9880 3.4910 3.9940 4.5500 5.0500 5.4800 6.0600 6.4650 7.2200

.020

.025 0.3325 0.8360 1.3400 1.8430 2.3470 2.8500 3.3530 3.8570 4.4400 4.9250 5.4200 5.8000 6.4000 6.9700

0.1495 0.6530 1.1570 1.6600 2.1640 2.6670 3.1710 3.6740 4.1780 4.7500 5.2350 5.8000 6.1500 6.6950

.030 1.0190 1.5220 2.0260 2.5300 3.0330 3.5360 4.0400 4.6850 5.0500 5.5050 6.0600 6.5300 6.9700

.035 1.706 2.209 2.713 3.216 3.720 4.223 4.835 5.235 5.760 6.180 6.665 7.260

.040 2.392 2.896 3.399 3.903 4.406 4.950 5.450 5.940 6.370 7.075 7.630

.045 3.079 3.582 4.086 4.710 5.210 5.590 6.000 6.630 7.040 7.670

.050 3.765 4.269 4.900 5.180 5.730 6.240 6.730 7.260 7.793

.055 4.4520 4.9550 5.5000 5.9100 6.3000 6.8300 7.6700 7.9760

.060 5.160 5.650 6.060 6.630 7.260 7.790 8.270

.065 5.82

8

8

6

(P0)comp

(P0)comp

6

4

4

2 0.07 0.06 0.05 0.04 0.03 0.02 P4

2

0 0.00

0.02

0.08 0.06

0.01 0.04 P

0.04

0.00

0.06 0.08 0.10

-0.01 0.00 0.01 0.02 0.03 0.04 P4 0.05

0.100

P

3

-0.01

0.02

0.06 0.01

0.07

Fig. 2. The compensation temperature parameter (P0)comp ¼ kTcomp/|Jad| obtained as a function of reduced exchange parameters P3 ¼ Jac/|Jad| and P4 ¼ |Jdc|/|Jad|. (b) is a rotated form of (a) in order to see its planar shape.)

kTcomp/|Jad| plane and a horizontal plane described by (kTcomp/|Jad|) 5.6 for Gd3Fe5O12. Below the Ne`el temperature of a collinear ferrimagnet, there is a spontaneous magnetization, like in the ferromagnets. However, in this case the magnetization is the vector sum of the magnetizations of the three sublattices and therefore has the magnitude given by M S_Res ¼ jM S_c  jM S_d  M S_a jj.

(10)

Because the sublattice magnetizations have quite different temperature dependences, the MS vs. T curves are not restricted to a Brillouin-type shape, as in the case of ferromagnets. It was shown [28] that, using the molecular field approximation, the MS vs. T data could be fitted with more than one set of exchange constants. Consequently, it was necessarily to note that based on the actual Ising model and MC–Metropolis procedure, the computed MS

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P4= - 0.728∗P3 + 0.063

P4 = | Jdc | / | Jad |

0.06 0.05 0.04 0.03 0.02 0.01 0.00 -0.01 0.00

0.02

0.04 0.06 P3 = | Jac | / | Jad |

0.08

0.10

15 10 5 0 -5 -10 resultant magnetization dodecahedral Gd(24c) sublattice magnetization tetrahedral Fe2(24d) sublattice magnetization

-15 -20

octahedral Fe1(16a) sublattice magnetization

0

Fig. 3. The linear dependence of P4 vs. P3 parameters yielding the same compensation temperature (P0)comp ¼ kTComp/|Jad| ¼ 5.6.

18

100

200

300

400 T/K

500

600

700

Fig. 5. Calculated resultant and sublattice spontaneous magnetizations, respectively, MS_res, MS_a, MS_d, MS_c, as functions of temperature.

magnetization of the unsaturated sites decreases with T faster than that of the saturated sites, so that MS_Res decreases if it is parallel to the unsaturated sites (Fig. 5).

16 calculated experimental data for G3Fe5O12d

14

Resultant and sublattices magnetizations / μB

20

0.07

Magnetization / chemical formula / μB

2153

12

3. Conclusions

10 8 6 4 2 0 -2 -100

0

100

200

300 400 T/K

500

600

700

Fig. 4. Calculated and experimental [24] resultant spontaneous magnetization in Bohr magnetons as a function of temperature, MS_res [mB] vs. T [K].

vs. T curves show distinct dependences on the set of exchange constants. Thus, the fit (Fig. 4) of simulated data with the experimental ones [24] becomes unique for the corresponding reduced parameters kT J aa J dd P0  ¼ 11:2; P1  ¼ 0:05; P2  ¼ 0:11, jJ ad j jJ ad j jJ ad j

(1) We have demonstrated that a simple model can reproduce the ferrimagnetic behavior of garnets, particularly for Gd3Fe5O12 case. Stable results explaining the behavior of Gd3Fe5O12 garnet were obtained using an Ising model with MC procedure. (2) The parameters corresponding to iron network subsystem Jad, Jaa, and Jdd were taken by transferability from previous [11] analyzed magnetic data on Y3Fe5O12 [24]. The new parameters for Gd3Fe5O12 are Jac and Jdc, considering the spins in collinear arrangement. (3) Further related work will consider the next intricacies of modeling the REIG magnetism, such as the anisotropy effects in magnetization vs. temperature dependence. With the outlined procedure, it is also possible to study the magnetic properties of 1D or 2D samples, in order to understand dimensionality and anisotropy effects.

Acknowledgments

and for the corresponding absolute values:

This work was supported by the CNCSIS-UEFISCU ‘‘IDEI’’ 874/2007 research grant.

J ad ¼ 34:7 cm1 ; J aa ¼ 1:7 cm1 ; J dd ¼ 3:9 cm1 .

References

If only one sublattice is saturated (or near saturation), then it is apparent that the interaction involving the unsaturated paramagnetic ions is smaller than those acting on the saturated paramagnetic ions. Therefore, the

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