Journal of Alloys and Compounds 343 (2002) Nd Zr Fe. a, a,b b. Shi-qiang Hao *, Nan-xian Chen, Jiang Shen

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1 Journal of Alloys and Compounds 4 (2002) 5 59 L locate/ jallcom Phase stability and site preference of Nd2Fe172xT x (T=V, Ti, Nb) and Nd Zr Fe 22x x 17 a, a,b b Shi-qiang Hao *, Nan-xian Chen, Jiang Shen a Department of Physics, Tsinghua University, Beijing, , China b Institute of Applied Physics, University of Science and Technology, Beijing, 10008, China Received 5 December 2001; accepted 6 February 2002 Abstract The phase stability of Nd2Fe172xT x (T=V, Ti, Nb, x50 1.2) is tested by many means including random atom shift, global deformation and high temperature disturbance under the control of the ab initio potentials. The T atoms substitute for Fe without changing the crystal symmetry and preferentially occupy the 6c site, which is in good agreement with experimental results. The Zr atoms prefer to substitute for Nd atoms in Nd2Fe 17. Moreover, the introduction of ternary elements slightly enlarge the calculated Nd2Fe172xT x (T=V, Ti, Nb) volume, which is also close to experiment. All the results indicate that the potentials are valid for studying some structural properties of the intermetallics Elsevier Science B.V. All rights reserved. Keywords: Rare earth compounds; Transition metal compounds; Crystal structure; Computer simulations PACS: 4.20.Cf; f; j 1. Introduction bands, magnetic moments and Curie temperatures for Y2Fe 17, YFe 5 [2,4 and YFe 12 [5,6. The spin-polarized In binary R Fe compounds, R2Fe17 compounds have a tight bonding linear muffin-tin orbital (LMTO) method high magnetization. However, the applications of R2Fe17 was used to calculate the magnetic moments for Y Fe Mo are restricted by their low Curie temperatures and poor system with ThMn12 structure [7. Moreover, there are magnetocrystalline anisotropy at room temperature. There- several studies of rare earth compounds with 4f electrons. fore, many efforts have been made by either substituting Ab-initio calculations by Sabiryanov and Jaswal indicated ternary elements or by introducing interstitial atom to that the nitrogenation of Sm2Fe17 more than doubles the improve the magnetic properties [1. Curie temperature (T c) but lowers Tc of Sm2Co 17 [8. Because of its fundamental role in determining the They also calculated the exchange parameters (J ij) of magnetic properties, a good understanding of the electronic Sm2Fe16A (A=Ga, Si) by self-consistent spin-polarized structure of these materials is extremely important. How- LMTO, and simulated the Curie temperature using Monte ever, some aspects make it difficult to study rare earth Carlo method [9. Phenomenological studies have been compounds. First, the compounds have a very complex performed for R 2(Fe, T) 17 [10,11 on the ternary substitustructure with up to 57 atoms per unit cell. Second, the tion mechanism, in which three term contributions includimportant valence electrons that control their properties ing a chemical term, an electronic term and an elastic term possess both itinerant and localized characteristics. Several were considered. Recently, Chen et al. have some theoretfirst-principle calculations of R T compounds with simple ical studies on the phase stability, site preference and structure and Y Fe compounds with complex structure lattice parameters for Gd(Fe, T) 12 [12 and Sm(Fe, T) 12 have been reported. Coehoorn et al. have studied energy [1, suggesting the applicability of ab initio potentials to rare earth compounds. In this paper, atomistic studies on the structural prop- *Corresponding author. Tel.: ; fax: erties of Nd2Fe172xT x (T=V, Ti, Nb) and Nd22xZrxFe17 are address: haoshiqiang@yahoo.com.cn (S.Q. Hao). performed using ab initio interatomic potentials. The / 02/ $ see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S (02)

2 54 S.Q. Hao et al. / Journal of Alloys and Compounds 4 (2002) 5 59 structure stability of Nd2Fe172xTx is studied by energy minimization method and the site preference of the system Table 1 The parameters of some important potential R A D ev g is evaluated Calculation method parameters of the 2:17 and 1:5 structures can also be described. Based on these, the magnetocrystalline aniso- tropy [19 and the easy magnetization direction [20 have been studied. However, only the geometrical transforma- tion between these structures can be embodied because these relations are derived from real compound structures. In this part, the transformation from 1:5 to 2:17 is studied by Fe Fe dumbbells substitution and crystal relaxation under the control of the calculated interatomic potentials. Since the paper focuses on the structure of Nd Fe T system, the relationship between the CaCu5-type and the At the beginning of the 1980s, Carlsson [14 reported that the cohesive energy of a metal could be obtained by an ab initio method, and the pair potentials could be inverted from the calculated cohesive energy curves, giving a method to construct interatomic potentials. In that work, the techniques were presented for constructing parameter-free pair potentials based on: (i) the assumption that the cohesive energy can be written as a sum of pair interaction energies; and (ii) electronic structure calculations of the cohesive energy as a function of volume. However, the expression of the pair potential is indigestible for including infinite summations as each of them Fe Fe Fe Nd Fe V Nd V contains infinite terms. Based on the same assumptions, Chen inverted the potentials from the ab initio cohesive Th2Zn17-type structure is concerned. energy [ It is distinct in several aspects from According to the following expression, Carlsson s method. Firstly, the calculation of the cohesive (NdFe ) Nd NdFe Nd Fe Fe Nd Fe, the energy is based on the virtual simple structure and deduced Nd Fe of Th Zn structure can be obtained from the from the total energy of the sublattice. Secondly, the NdFe cell with CaCu structure, although the NdFe cell inversion method used by Chen is a number theory is metastable. First, we build a supercell from a inversion technique, which is a rigid and concise way to NdFe5 crystal cell, then we select special rare earth atoms obtain interatomic potentials. Thirdly, the interatomic (1/ ) to be replaced by Fe Fe dumbbells. As shown in potential between the distinct atoms can also be obtained Fig. 2(c e), in the first layer, dumbbells substitute for Nd by the procedure. Lastly, it is convenient for analysis since atom in the corner. In the second layer, dumbbells replace the inversion coefficient of materials with identical struc- Nd atoms in the downright center and in the third layer ture is concise and uniform. dumbbells replace Nd atoms in the up-left center. Thus the In our previous works [12,1, the methods of obtaining composition of the supercell is 2:17. Now we relax the the cohesive energy curves and the lattice inversion supercell under the control of the potentials with energy technique have been reported in detail. In this paper, only minimization. Then the relaxed structure can be identified some parameters of the potentials are listed in Table 1 as Th2Zn17 structure with Rm space group. The atom (Fig. 1). The potentials are fitted by the Morse function, sites are close to the neutron diffraction data [21. In that which is expressed as: process, assuming the target structure of Th2Zn17 is uncertain, temporarily we set the orientation and length of x g x 2g? 21 2? 21 sd f s dg f s dg 0 S R 0 D 2 R 0 F x 5 D? e 2 2? e. dumbbells arbitrarily to some extent. After the relaxation under the potentials, the dumbbell atoms are exactly occupying the 6c site, oriented along the c axis and the. Calculated results for the binary structure Nd Fe average length is 2.46 A. To sum up, this process suggests 2 17 that the transformation can be considered not only as.1. The relationship between the CaCu5-type and geometrical but also as a physical process, since the Th Zn -type relaxation is the manner of searching energy minimal point 2 17 Just like the RFe M phase, the R Fe phase can be 122x x 2 17 derived from the CaCu structure. Each of these represents 5 in phase space (Table 2, Fig. 2). a modification of the RT structure and can be described.2. Validity of the calculated potentials 5 by basis vectors in the Descartes coordinate system [18. Once the expressions of the basis vectors for the unit cells The structure of Nd2Fe17 is simulated by an energy of the 1:5, 2:17, 1:12, :29 structures are known, the minimization method, which is realized by conjugate relations between 2:17 phase, :29 phase, 1:12 phase and gradient methods with a potential cut-off radius of 14 A. 1:5 phase can be given by the vector transformations in To avoid statistic fluctuation, we adopt a model reciprocal space. Thus, the relationship between the lattice supercell with 159 atoms.

3 S.Q. Hao et al. / Journal of Alloys and Compounds 4 (2002) Table 2 Comparison between the calculated and experimental [21 structure parameters of Nd2Fe17 Nd2Fe17 Calc. Expt. a A c A Nd (6c) 0,0,0.8 0,0,0.429 Fe (6c) 0,0,0.98 0,0, Fe (18f ) 0.29,0, ,0,0 Fe (18h) 0.499,20.02, ,0, Fe (9d) 0.5,0,0.5 to test the structure stability by means of random atom shifts. One, the relaxed Th2Zn17 structure is disturbed by a random atom shift of 0.5 A. Each atom can recover its equilibrium position under the interaction of ab initio Fig. 1. Some important potentials. potentials. Second, the unrelaxed Nd2Fe17 configuration substituted from CaCu5-type is disturbed by a random As in the previous work [22, the structure stability was atom shift of 0.5 A and then relaxed under the interaction tested by many methods including global deformations, of the potentials. The final structure can also be identified high temperature disturbances and random atom shifts. The as Th2Zn 17 (Rm) type. global deformations mean that we allowed for some In the first step, the test starts with the determined operations in the model such as stretching, compressing, structure. After the disturbance and relaxation, we check shearing and a combination of these. The random atom the disturbed structure whether it can restore to the shifts mean to move each atom deviating from its equilib- Th2Zn17 structure under the force of the interatomic rium position in a random direction. There are two aspects potentials. Although the range of disturbance is very large, Fig. 2. Structural relationship between the CaCu -type and the Th Zn type.

4 56 S.Q. Hao et al. / Journal of Alloys and Compounds 4 (2002) 5 59 the start point corresponding to the distorted structure in the phase space is still near to the stable state. While in the second step, the process of searching the minimum starts from the random atom configuration, corresponding to the unknown point and stabilizes to final structure driven by the potentials. This process can be viewed as liquid solid phase transformation. 4. Calculated results for the ternary compounds Nd Fe T In the experiment, methods of doping ternary elements are used to improve the magnetic properties, different elements having different effects on the performance. In this section, the site preference of T (T=V, Ti, Nb) atoms and the phase stability of Nd Fe T system are studied Site preference Fig.. Dependence of the Nd Fe T (V, Ti, Nb). T average energy on the content of In the process, firstly we substitute T atoms for Fe in each site with different concentrations. Then the energy minimization method is applied to relax the ternary system under the interaction of the potentials. The average energy is taken as a criterion of the stability. Thus the average energy of the final structure can be investigated and compared. The results are shown in Fig.. To avoid accidental errors, the results represent the arithmetic average for 20 stochastic samples. The symbol I denotes the range of error bars. It can be seen in Fig. that the system s average energy decreases with the addition of the ternary element, illustrating that each of these elements can stabilize the crystal. The cohesive energy decreases most significantly when the T atoms preferentially occupy the 6c sites, less significantly if the T atoms occupy the 18f and 18h sites, and behaves abnormally when occupying the 9d sites. All of the above results correspond well to the experiment [2. The behavior of T substitution can be explained by considering the average energy calculated from interatomic potentials. Fig. 1 shows that the potential values are important in the range of 2.,r,4.4 A. Notice that F (r) intersects with F (r) at about r52.7 A. Fe V Fe Fe When the interatomic distance r,2.7 A, F Fe V(r).F Fe Fe(r), so that it is unfavorable for a substitution of V atoms for Fe atoms. When the distance r.2.7 A, F Fe V(r),F Fe Fe(r) and it is favorable for the substitution. On the other hand, F (r) intersect with F (r) at about r5.6 A. Nd V Nd Fe It is unfavorable for the ternary substitution since F Nd V(r). F (r) at the range of r,.6 A. Nd Fe The site preference occupation of the ternary atoms may also be analyzed in Table by the preference factors, one kind of scale to measure the variation of cohesive energy. The first column in Table includes the sites occupied by the V atoms; the second column shows the number of Fe atoms within the sphere centered at the V atom and with radius of 2.7 A. Notice that F Fe V(r).F Fe Fe(r) is in this range. The more Fe atoms in this range, the more unfavorable is the energy, so there is a negative sign. The

5 S.Q. Hao et al. / Journal of Alloys and Compounds 4 (2002) Table Preference factors for distinct sites Site No. of Fe No. of Fe No. of Nd No. of Nd Number of (,2.7 A) ( A) (,.6 A) ( A) benefit factors 6c f h d third column shows the number of Fe atoms within the Table 5 The structure of Nd2Fe172xVx with different V concentration range of A. Here F Fe V(r),F Fe Fe(r). The more Fe atoms in this range the more favorable is the substitu- Nd2Fe172xVx x50 x50.5 x51 x51.2 tion, so there is a positive sign. The fourth column shows a A the number of Nd atoms. Because of F c A Nd V(r).F Nd Fe(r) V A in the range r,.6 A, it is unfavorable for the energy to Nd (6c) z decrease, so there is a negative sign. From Table, it is Fe (6c) z easy to conclude that the V atom prefer the 6c site and Fe (18f ) x avoid the 9d site. According to the same method, the site Fe (18h) x preference behavior of other ternary elements can be easily analyzed by the shape of the potentials and the crystal structure. Fe (18h) z the literature [2, but it agrees with that in the literature 4.2. Structure of the Nd2Fe172xTx [24. The structure and lattice constants of (Nd2Fe16V) is Now the ab initio potentials are applied to check the further studied at high temperatures. The molecular dyphase stability of Nd2Fe172xT x (T=V, Ti, Nb), with x50.5, namics NPT ensemble is applied using the method of 1, 1.2, respectively. For x51 Nd2Fe16V with V atoms in 6c Berendsen et al. [25 for steps, with P51 atm, site is adopted to check the structure stability by the same t ps. The dynamic simulation for (Nd2Fe16V) 81 of method. Results show that the structure can be stabilized to 159 atoms in a rhombohedral cell with periodic boundary the Th2Zn17-type even undergo either global deformation conditions is carried out at temperatures of 00, 500 and or a random atom shift of 0.5 A. The relaxation results of 700 K. After reaching equilibrium, the crystal symmetry the globally deformed Nd2Fe172xVx are listed in Table 4. can be identified as Rm space group in a certain range, Further, the atom sites of Nd2Fe172xVx with different x and the lattice constants change little with respect to the value are evaluated and listed in Table 5. Moreover, the temperature. It is very helpful to use mean square displace- structures of Nd2Fe172xT x (T=V, Ti, Nb) are calculated. ment (MSD) to understand the phase stability, since it Table 6 suggests that the lattice parameters are all in good reflects the microscopical statistical average character of agreement with observed values [2. The unit cell volindicate the system. These results are listed in Table 7, which umes are slightly enlarged by the introduction of V and Ti. that the MSD increase with temperature and the 2 While the enlargement by Nb is different from the result in average value is small as A / atom. As a Table 4 Determination of the lattice constants of Nd2Fe16V Deformed state Average energy Final state under the calculated structure Lattice constants (ev/ atom) Lattice constants Average energy a, b, c (A) abg (8) a, b, c (A) abg (8) (ev/atom).5 A, 4 A, 6 A A, A 908, 908, , 908, A, 1.5 A, 20 A A, A, A 908, 908, , 908, A, A, A A, A, A 828, 798, , 908, A, A, A A, A, A 1228, 78, , 908, A, 6.5 A, 9 A A, A, A 1108, 808, , 908, A, 15 A, 22 A A, A, A 1088, 828, , 908, 1208

6 58 S.Q. Hao et al. / Journal of Alloys and Compounds 4 (2002) 5 59 Table 6 Comparison between the experimental [2 and calculated structure of Nd Fe Compounds Lattice parameters T Expt./Calc. a (A) Expt./Calc. c (A) Expt./Calc.V (A ) Nd Fe [ / / / Nd Fe Ti 8.615/ / / Nd2Fe16.0V / / / Nd2Fe15.99Nb / / / measure of the atom deviation from the equilibrium shows that ab initio potentials are valid in simulations in position, small MSD values indicate that the structure of this kind of material. (Nd2Fe16V) does not change in this range of temperature. However, there are some special cases in the behavior of Thus, the phase stability is verified again. site preference. The Zr atoms strongly prefer Nd sites, It can be seen from calculated results that the distorted which can be concluded easily from Fig. 4(a) because the system can restore the same final structure. The potentials average energy decreases more significantly after the can be thought of as possessing the long-range characteris- substitution for Nd atoms than for Fe atoms. This can also tics for the restoration of the structure after global deformations. The process of restoration from the random atom shift model can be thought of as liquid solid phase transformation. High temperature disturbance embodies the dynamic equilibrium properties at the different temperatures. The above three aspects of the calculations suggest that the potentials possess not only the character of the equilibrium state, but also the non-equilibrium properties to some extent. All the above cases indicate that the ab initio potential can effectively simulate the structural properties of rare-earth compounds. 5. Discussion and conclusion In the present paper, we propose a series of ab initio interatomic potentials in rare earth intermetallics of the type Nd2Fe172xT x. Ab initio calculations for some simple virtual structures, together with the lattice inversion technique, provide a method to construct parameter-free potentials. These potentials have been applied to a series of examples otherwise only solved with difficulty. This includes structural stability, site preference in Nd2Fe172xTx and lattice parameters for this anisotropic system. Compared with neutron diffraction data, the lattice parameters and the atom site position are all in good agreement with the observed values. Our results, together with the structure stability and the site preference of Nd2Fe172xT x, Table 7 Lattice constants of (Nd Fe V) and MSD at different temperature 2 16 Temperature Lattice constants MSD Space (K) 2 (A / atom) group A, 8.62 A, A 0.15 Rm , , A, 8.6 A, A 0.25 Rm , 89.98, A, 8.64 A, A 0.6 Rm , , Fig. 4. Dependence of the Nd2Fe 17/Zr and Nd2Fe172xSix average energy on the content of ternary element.

7 S.Q. Hao et al. / Journal of Alloys and Compounds 4 (2002) be analyzed by the method of preference factors mentioned Acknowledgements above. This result is different from the experimental reports [2,24, in which Zr atoms were reported to prefer The authors would like to express their deep gratitude to the (Fe) 6c site rather than the Nd (6c) site. Based on our the referee and editor of J. Alloys and Compounds for their calculated results, the lattice parameters of NdZrFe17 are helpful suggestions. This work was supported by Special given as a A, c A and the volume of the Funds for Major State Basic Research of China (Grant No. unit cell V A is smaller than that of Nd2Fe17 G , No. G ) and the National Na- ( A ). Where the symmetry of the NdZrFe17 is ture Science Foundation of China (Grant No ). concerned, the crystal keeps Rm space group even undergoing the disturbance tests. In contrast to the case of transition-metal substituents (V, Ti, Nb, Zr), the substitution of nontransition-metal Si is References studied. The cohesive energy of the Nd2Fe172xSix are calculated and shown in Fig. 4(b), indicating that the [1 W.B. Yelon, Z. Hu, W.J. James, G.K. Marasinghe, J. Appl. Phys. 79 energy increases when Si substitute for the 6c, 18f and 18h (8) (1996) 599. sites, and slightly decreases for the 9d site. Obviously, as [2 S.S. Jaswal, W.B. Yelon, G.C. Hadjipanayis, Y.Z. Wang, D.J. Sellmyer, Phys. Rev. Lett. 67 (5) (1991) 644. can be seen in Fig. 4(b), Si atoms strongly avoid the 6c [ O. Isnard, S. Miraglia, D. Fruchart, E. Akiba, K. Momura, J. Alloys site. But we cannot identify the order of site preference Comp. 257 (1997) 150. since Si has a strong preference for the 18h site in [4 R. Coehoorn, Phys. Rev. B9 (18) (1989) experimental reports [2,26. Similarly, the preference [5 S.S. Jaswal, Y.G. Ren, D.J. Sellmyer, J. Appl. Phys. 67 (1990) behavior of Ga atom cannot be identified by the above [6 R. Coehoorn, Phys. Rev. B41 (17) (1990) energy criterion. These results can not be explained only [7 R. Lorenz, J. Hafner, S.S. Jaswal, D.J. Sellmyer, Phys. Rev. Lett. 74 by the coordination effect. Many body effects should be (18) (1995) 688. taken into account for the strong covalent properties of Si [8 R.F. Sabiryanov, S.S. Jaswal, Phys. Rev. Lett. 79 (1) (1997) 155. and Ga. Moreover, entropy effects of the ternary element [9 R.F. Sabiryanov, S.S. Jaswal, J. Appl. Phys. 81 (8) (1997) [10 Er. Girt, Z. Altounian, Phys. Rev. B57 (1998) substitution should also be considered. [11 Er. Girt, Z. Altounian, J. Appl. Phys. 87 (9) (2000) It is well known that the process of energy minimization [12 N.X. Chen, J. Shen, X.P. Su, J. Phys. Condens. Matter. 1 (2001) is statistical dynamics. There are three disadvantages: the effect of initial configuration, local minimum and zero [1 N.X. Chen, S.Q. Hao, Y. Wu, J. Shen, J. Magn. Magn. Mater. 2 temperature. The energy minimization process is searching (2001) 169. for the local minimum value in the phase space from the [14 A.E. Carlsson, C.D. Gelatt, H. Ehrenreich, Phil. Mag. A 41 (1980) starting point. For a complex system, many local minima 241. [15 N.X. Chen, Z.D. Chen, Y.C. Wei, Phys. Rev. E55 (1997) R5. occur in phase space, so the energy minimization can not [16 N.X. Chen, G.B. Ren, Phys. Rev. B45 (1992) search the most stable structure with the lowest energy. To [17 N.X. Chen, X.J. Ge, W.Q. Zhang, F.W. Zhu, Phys. Rev. B57 (1998) find it, many cases must be considered. At zero tempera ture, the vibration properties and other dynamics properties [18 H.F. Han, H.G. Pan, H.L. Liu, F.M. Yang, Phys. Rev. B56 (1997) are neglected. Against the drawbacks, three methods are applied to deal with them. The global deformation and [19 N. Tang, X.C. Kou, F.R. de Boer, K.H.J. Buschow, J.L. Wang, F. random atom shift solve the question of the initial structure Yang, J. Phys. Condens. Matter. 11 (1999) 51. [20 D. Courtois, H.S. Li, J.M. Cadogan, Solid State Commun. 98 (6) to some extent. Moreover, molecular dynamics are applied (1996) 565. to overcome the initial configuration and the contributions [21 Er. Girt, Z. Altounian, J. Yang, J. Appl. Phys. 81 (8) (1997) of temperature are considered. [22 S.Q. Hao, N.X. Chen, J. Shen, J. Magn. Magn. Mater. (in press). Experimentally, for the Nd2Fe172xT x (T=V, Ti, Nb, Zr) [2 W.B. Yelon, Z. Hu, W.J. James, G.K. Marasinghe, J. Appl. Phys. 79 system, when the amount of T atoms is small, the main (8) (1996) 599. phase is the 2:17 structure. When the concentration of T [24 I.A. Al-Omari, Y. Yeshurun, S.S. Jaswal, J. Zhou, D.J. Sellmyer, J. atoms increases, there will be a multi-phase system [24. Magn. Magn. Mater. 217 (2000) 8. [25 H.J.C. Berendsen, J.M.P. Postma, W.F. van Gunstern et al., J. Chem. So, the ternary component concentration x is taken as Phys. 81 (1984) when the crystal structure is studied. The degree of [26 G.J. Long, G.K. Marasinghe, S. Mishra, O.A. Pringle, F. Grandjean, solubility of the ternary element is not considered in this K.H.J. Buschow, D.P. Middleton, W.B. Yelon, F. Pourarian, O. paper, which is left for a further study. Isnard, Solid State Commun. 88 (199) 761.