Structure and Curie temperature of Y 2 Fe 17 x Cr x

Vol. 46 No. 4 SCIENCE IN CHINA (Series G) August 2003 Structure and Curie temperature of Y 2 Fe 17 x Cr x HAO Shiqiang ( ) 1 & CHEN Nanxian ( ) 1,2 1. Department of Physics, Tsinghua University, Beijing 100084, China; 2. Institute of Applied Physics, University of Science and Technology of Beijing, Beijing 100083, China Correspondence should be addressed to Hao Shiqiang (email: shiqianghao@mails.tsinghua.edu.cn) Received September 30, 2002 Abstract The structures of Y 2 Fe 17 x Cr x are simulated by the ab initio potentials. The site preferenceofcratominy 2 Fe 17 is evaluated and the order is determined as 4f, 12j, which is close to the experimental result. Based on the site preference behavior, the calculated parameters and the atom sites of Y-Fe-Cr system are studied. The result corresponds well to observed data. Further, the DOS of the relaxed structures are calculated and the variation in Curie temperature is explained qualitatively by the spin-fluctuation theory. Keywords: structure determination, site preference, interatomic potentials, Curie temperature, lattice inversion. DOI: 10.1360/02yw0018 Introducing interstitial atoms and substituting ternary elements [1 2] are useful ways to improve magnetic properties of R 2 Fe 17, since the binary compounds have some defects such as lower Curie temperature and poor magnetocrystalline anisotropy at room temperature. Usually, the Bethe-Slater curve could be used to explain the improvement of Curie temperature. If the distance of Fe pair is shorter than 0.245 nm, the exchange interaction is negative. Otherwise, the interaction is positive [3]. When the interstitial atom and ternary element (except Si) are introduced into the cell, lattice parameters will increase, thus enhancing positive interaction. As a result, the T C increases [4]. Although the distance-dependent exchange is correlated with the variation of T C in a wide variety of the compounds, this model is too simple to explain the concerned facts. For some compounds T C does not increase as the volume increases. For example, for the Si-substituted compound Ce 2 Fe 17 x Si [5] x, the volume decreases with increasing x, but the Curie temperature T C increases. Jaswal [6] et al. used the spin-fluctuation theory to explain qualitatively the large increase in T C upon the nitrogenation of some Fe-rich compounds. Lately, they [7] used the Heisenberg model and finite-size scaling in Monte Carlo method to calculate the T C quantitatively. In this paper, based on the ab initio potentials, whose validity has been demonstrated by the a series of works [8,9], the site preference of Cr atom and the structure of Y 2 Fe 17 x Cr x are studied. Moreover, by means of the DOS calculation and spin-fluctuation theory, we give a qualitative explanation of the variation of T C. 1 Calculation method At the beginning of the 1980, Carlsson [10] reported that pair potential could be obtained by

No. 4 STRUCTURE & CURIE TEMPERATURE OF Y 2Fe 17 xcr x 357 ab initio method, but the expression was indigestible because infinite summations are included in it, each of them containing infinite terms. Different from the above method, Chen s lattice inversion formula based on ab initio cohesive energy [11 13] is a rigid and concise technique for obtaining parameter-free potentials, which can express the inversion coefficients of materials with identical structure concisely and uniformly, making it convenient to analyze them. In our previous works [8,9,11 13],somedetails of the method for obtaining the potentials are given. In this paper, the inverted potentials are expressed as the Morse approximation. The analytic expression of Morse function for potential versus the distance of two atoms is x γ x γ 1 i i 1 R0 2 R0 Φ ( x) = D 0 i e 2 e, where D 0, R 0,andγ are potential parameters. Some important potentials are shown in fig. 1. 2 Calculated results 2.1 Binary structure The structures of Y 2 Fe 17 and Y 2 Fe 17 x Cr x are simulated by the energy minimization method, Energy minimization is realized by conjugate gradient methods with the cutoff radius potentials 1.4 nm. To avoid statistic fluctuation, the model is a 3D3D3 supercell with 1026 atoms. In this work the structure stability was tested by many methods including global deformation, high temperature disturbance and atom random shift. The global deformations mean making some operations on the model, such as stretching, compressing, shearing and combinating of these techniques, while atom random shift means moving each atom that deviates from equilibrium position in random direction. The structure is disturbed with random atom shift of 0.05 nm. Each atom can recover its equilibrium position under the interaction of ab initio potentials. For the case of global deformation, the relaxed structure can recover its original form. The calculated results are listed in table 1. Notice that there is no symmetry restriction in the relaxation process. Moreover, the average energy of the unrelaxed structure may be very high. In the energy minimization, many samples can stabilize to a uniform structure. The above results do not reflect temperature effect, and the dynamic properties are neglected. To check the most stable Y 2 Fe 17 structure, the simulated anneal method is applied with NPT ensemble P =0.0001GPaandT = 1200 K. After 10 cycles, the stable Th 2 Ni 17 structure can be found. All this indicates that the structure disturbed to some extent can restore its identical final structure. Fig. 1. Some important interatomic potentials.

358 SCIENCE IN CHINA (Series G) Vol. 46 Table 1 Determination of the lattice parameters of Y 2Fe 17 Initial state (unrelaxed) Final state (relaxed) a b c/nm α β γ /deg Energy Energy /evcatom 1 a b c/nm α β γ /deg /evcatom 1 1.5, 1.4, 1.6 90, 90, 120 0.5341 0.848, 0.848, 0.831 90, 90, 120 6.2428 0.4, 0.35, 0.55 90, 90, 120 383.445 0.848, 0.848, 0.831 90, 90, 120 6.2428 0.848, 0.848, 0.831 80, 70, 135 12.813 0.848, 0.848, 0.831 90, 90, 120 6.2428 0.848, 0.848, 0.831 110, 68, 90 0.16869 0.848, 0.848, 0.831 90, 90, 120 6.2428 0.5, 0.6, 0.55 70, 120, 87 147.867 0.848, 0.848, 0.831 90, 90, 120 6.2428 1.3, 1.2, 1.7 65, 70, 100 1.173 0.848, 0.848, 0.831 90, 90, 120 6.2428 Calculated results show that the distorted system can restore the same final structure. The potentials can be thought of as long range for the restorations of the structure that undergoes global deformation. The process of restoration from the random shift atom model can be thought of as liquid-solid phase transformation. High temperature disturbance embodies the dynamic equilibrium properties at different temperature. It is reasonable, because the potentials inverted from the cohesive energy curves of virtual structures in a large scale reflect not only the characteristics of equilibrium state but also the non-equilibrium properties to some extent. All this indicates that it is effective using ab initio potential to simulate the kinds of rare-earth compounds. 1.2 Site preference In the process, we first substitute Cr atom for Fe in each site with different concentrations and then use energy minimization method to relax the ternary system under the interaction of the potentials. Thus the average energy of final structure can be investigated and compared. The results are shown in fig. 2. To avoid accidental errors, the arithmetic averages are taken for 50 stochastic samples. The symbol I denotes the range of error bar. Fig. 2 shows that Cr atom strongly prefers the 4f sites and medially substitutes for Fe atom in the 12j sites. The order of site preference is 4f, 12j. The behavior of Cr substitution can be explained by the average energy calculated Fig. 2. The dependence of average energy on the content of Cr from interatomic potentials. Fig. 1 shows that element in Y 2Fe 17-xCr x. the potential values are important in the range of 0.23 < r < 0.44 nm. Notice that Φ Fe-Cr (r) intersects Φ Fe-Fe (r) at about r = 0.27nm. When the interatomic distance r < 0.27, Φ Fe-Cr (r) >Φ Fe-Fe (r), so that it is unfavorable for the substitution of Cr atoms for the Fe atoms, and when the distance r >0.27nm,Φ Fe-Cr (r) <Φ Fe-Fe (r) and it is favorable for the substitution. On the other hand, Φ Y-Cr (r) intersects Φ Y-Fe (r) at about r =0.35nm.Whenthe distance r <0.35nm,Φ Y-Cr (r)>φ Y-Fe (r), so that it is unfavorable for the substitution.

No. 4 STRUCTURE & CURIE TEMPERATURE OF Y 2Fe 17 xcr x 359 Site No. of Fe (< 0.27 nm) Table 2 No. of Fe (0.27 0.44 nm) Preference factors for distinct sites for Fe atoms No. of Y (<0.35 nm) No. of Y (0.35 0.44 nm) Number of benefit factors 4f 4 23 1 0 4+23 1=18 12j 8 18 2 0 8+18 2=8 12k 8 16 3 0 8+16 3=5 6g 10 13 2 0 10+13 2=1 The site preference occupation of the ternary atoms may also be analyzed by the affecting factors in table 2. The first column in table 2 gives the crystal sites in the Th 2 Ni 17 structure, and the second column shows the number of Fe atoms within the sphere centered at the Cr atom and with radius of 0.27nm. Notice that Φ Fe-Cr (r) >Φ Fe-Fe (r) in this range; the more Fe atoms in this range, the more unfavorable energy it has, so there is a negative sign. The third column shows the number of Fe atoms within the range of 0.27 0.44 nm. Here Φ Fe-Cr (r) <Φ Fe-Fe (r); the more Fe atoms in this range, the more favorable energy it has, so there is a positive sign. The fourth column shows the number of Y atoms. Because Φ Y-Cr (r) >Φ Y-Fe (r), it is unfavorable for energy to decrease, so there is a negative sign. With table 2, it is easy to have the preferential occupation sequence. 2.3 Ternary structure of Y 2 Fe 17 x Cr x Now the ab initio potentials are used to check the structure of ternary compound Y 2 Fe 17 x Cr x. For x=1, Y 2 Fe 16 Cr with Cr atom at 4f site is adopted to check the structure stability in the same way. Results show that the structure can be stabilized to Th 2 Ni 17 -type even if it undergoes either global deformation or atom random shift by 0.05 nm. The atom sites of Y 2 Fe 15 Cr 2 are evaluated and compared with different data. The results are listed in table 3. Table 3 shows that the structure of Y 2 Fe 17 x Cr x does not change with the introduction of Cr atom, and the lattice parameters are in good agreement with observed values [14]. So the validity of the ab inito potentials is verified again. Table 3 Comparison between calculated and experimental structure parameters of Y 2 Fe 15 Cr 2 Y 2Fe 17 xcr x Calc. Expt. [14] a/nm 0.839 0.841 c/nm 0.816 0.835 Y (2b) 0,0,0.246 0, 0, 0.25 Y (2d) 0.37,0.33, 0.751 1/3, 2/3, 0.75 Fe/Cr (4f) 0.333,0.666,0.098 1/3, 2/3, 0.1179 Fe (12j) 0.329,0.959,0.249 0.3299, -0.026, 0.25 Fe (12k) 0.166,0.21,0.978 0.1648, 0.3296, 0.989 Fe (6g) 0.5,0,0 0.5, 0, 0 2.4 Curie temperature of Y 2 Fe 17 x Cr x Based on the site preference of the ternary element of Cr, the densities of states of

360 SCIENCE IN CHINA (Series G) Vol. 46 Y 2 Fe 17 x Cr x are calculated by the LMTO (ASA) method [15]. In the process, the s, p, d orbitals are for Y, Fe, Cr. The atom sphere radii are chosen according to Vegard s law. The atomic positions are scaled by the relaxed structure under the ab initio potentials. For the 3d band of Fe atom, the spin-polarized DOS curve has a two-hump structure. The Fermi level E F lies in a valley between the two humps in N - (E F ) curve or near the upper boundary in the 3d + band. In fig. 3, the value of N - (E F ) are almost the same at different Cr contents, but the N + (E F )ofy 2 Fe 16 Cr is 16.79(1/eV), much lower than those of Y 2 Fe 15 Cr 2 and Y 2 Fe 17. Experimental results [14] show that when Cr is substituted in Y 2 Fe 17, the Curie temperature increases from 319 up to 415 K at x =1 and then decreases to 395 K at x =2.Inorder Fig. 3. Spin-polarized density of states for Y 2Fe 17 xcr x.the to explain qualitatively the variation behavior zero of energy represents the Fermi energy. of Y 2 Fe 17 x Cr x, we use the relation of the iron-rich alloys in spin-fluctuation theory of Mohn and Wohlfarth [16] 2 M 0 TC, (1) χ0 where M 0 is the magnetic moment of Fe ion at 0 K and χ 0 is the exchange-enhancement susceptibility 1 1 1 1 χ0 = + 2 I. (2) 2 4µ + B N ( EF) N ( EF) I is the Stoner parameter and µ B is the Bohr magneton. In this paper, as Y 2 Fe 17 x Cr x M 0 are close to each other with x concentration, χ 0 is the dominating factor for the Curie temperature variation according to eq. (2). Furthermore, χ 0 is determined mainly by the quantitative N + (E F )andn - (E F ), since the parameter I depends weakly on the local environment of Fe in the Y-Fe-Cr system. AccordingtotheDOSoftheY 2 Fe 17 x Cr x with different ternary concentrations the total DOS of Y 2 Fe 16 Cr in Fermi level are lower than those of both Y 2 Fe 17 and Y 2 Fe 15 Cr 2 at the corresponding Fermi level. So the Curie temperature of Y 2 Fe 16 Cr must be higher than those of both Y 2 Fe 17 and Y 2 Fe 15 Cr 2, and is close to the experimental value.

No. 4 STRUCTURE & CURIE TEMPERATURE OF Y 2Fe 17 xcr x 361 3 Conclusion Using the ab initio potentials we studied the site preference of Cr atom in Y 2 Fe 17. It is found that Cr atom prefers 4f and 12j sites. Lattice parameters and the atomic sites of ternary compounds are corresponding well to the experimental results. All this validates the ab initio potentials. As a further verification, the DOS of the relaxed structures are calculated and Curie temperature is explained in terms of the spin-fluctuation theory. The ab initio potentials and spin-fluctuation prove to be useful methods for studying this kind of materials and predicting their structures and properties. Acknowledgements This work was supported by Special Funds for Major State Basic Research of China (Grant Nos. G2000067101, and G2000067106) and the National Natural Science Foundation of China (Grant No.59971006). References 1. Sun Hong, Coey, J. M. D., Otani, Y. et al., Magnetic properties of a new series of rare-earth iron nitrides: R 2Fe 17N y (y~2.6), J. Phys.: Condens. Matter., 1990, 2: 6465 6470. 2. Jacobs, T. H., Buschow, K. H. J., Zhou, G. F. et al., Magnetic interactions in R 2Fe 17-xAl x compounds (R=Ho, Y), J. Magn. Magn. Mater., 1992,116: 220 230. 3. Gvord, D., Lemaire, R., Magnetic transition and anomalous thermal expansion in R 2Fe 17 compounds, IEEE Trans. Magn., 1974, 10: 109 113. 4. Gubbens, P. C. M., van der Kraan, A. M., Jacobs, T. N. et al., 57 Fe and 169 Tm Mössbauer effect and magnetic properties of Tm 2Fe 15M 2 (M=Al, Ga, Si), J. Less-Common Met., 1990,159: 173. 5. Middleton, D. P., Buschow, K. H. J., Magnetic properties of Ce 2Fe 17-xSi x compounds, J. Alloys of Compounds, 1994, 206: L1 L2. 6. Jaswal, S. S., Yelon, W. B., Hadjipanayis, G. C. et al., Electronic and magnetic structures of the rare-earth compounds: R 2Fe 17N x, Phys. Rev. Lett., 1991,67(5): 644 647. 7. Sabiryanov,R.F.,Jaswal,S.S.,Ab initio calculations of the Curie temperature of complex permanent-magnet materials, Phys. Rev. Lett., 1997, 79: 155 158. 8. Chen, N. X., Shen, J., Su, X. P., Theoretical study on the phase stability, site preference, and lattice parameters for Gd(Fe,T) 12, J. Phys.: Condens. Matter., 2001, 13: 2727 2736. 9. Chen, N. X., Hao, S. Q., Wu, Y. et al., Phase stability and site preference of Sm(Fe,T) 12, J. Magn. Magn. Mater., 2001, 233: 169 180. 10. Carlsson, A. E., Gelatt, C. D., Ehrenreich, H., An ab initio pair potential applied to metals, Philos. Mag. A, 1980, 41: 241 250. 11. Chen, N. X., Chen, Z. D., Wei, Y. C., Multidimensional inverse lattice problem and a uniformly sampled arithmetic Fourier transform, Phys. Rev. E, 1997, 55: R5 R8. 12. Chen, N. X., Ren, G. B., Carlsson-Gelatt-Ehrenreich technique and the Möbius inversion theorem, Phys. Rev. B, 1992, 45: 8177 8180. 13. Chen, N. X., Ge, X. J., Zhang, W. Q., Atomistic analysis of the field-ion microscopy image Fe 3Al, Phys. Rev. B, 1998, 57: 14203 14208. 14. Hao, Y. M., Zhang, P. L., Zhang, J. X. et al., A high-resolution neutron study of Y 2Fe 15Cr 2 at 77K including magnetic properties, J. Phys.: Condens. Matter, 1996, 8: 1321 1324. 15. http://www.accelrys.com/cerius2/cerius246/index.html. 16. Mohn, P., Wohlfarth, E. P., The Curie temperature of the ferromagnetic transition metals and their compounds, J. Phys. F: Metal Phys., 1987, 17: 2421 2430.