- Email: [email protected]
Predicting magnetostriction of MFe3N (M = Fe, Mn, Ir, Os, Pd, Rh) from ab initio calculations
Computational Materials Science 92 (2014) 464–467
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Predicting magnetostriction of MFe3N (M = Fe, Mn, Ir, Os, Pd, Rh) from ab initio calculations Yun Zhang a, Zhe Wang a, Juexian Cao a,b,⇑ a b
Department of Physics, Xiangtan University, Xiangtan 411105, Hunan, China Beijing Computational Science Research Center, Beijing 100084, Beijing, China
a r t i c l e
i n f o
Article history: Received 29 January 2014 Received in revised form 30 May 2014 Accepted 31 May 2014 Available online 9 July 2014 Keywords: Fe-alloys Ab initio calculations Magnetic properties
a b s t r a c t Using the first-principles full-potential linearized augmented plane-wave method, we investigate the magnetostriction of Fe4N. Its magnetostrictive coefficient is found to be 143 ppm. In order to enhance the magnetostriction, the derivative MnFe3N is taken into our consideration according to the rigid band model. Interestingly, we find that the magnetostrictive coefficient is enhanced to be +373 ppm. To further improve the magnetostriction, we calculate the magnetostrictive coefficient of the derivatives containing 4d and 5d metal due to their strong spin–orbit coupling. The results show that the magnetostrictive coefficient can reach to 564 ppm and +416 ppm for OsFe3N and IrFe3N. Our calculations give a guideline for designing giant magnetostriction material. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction Magnetostrictive materials are widely used for exploitations in sensors, actuators, micro electro mechanical system (MEMS), and energy harvesting devices [1,2]. The large magnetostrictive coefficients are discovered in rare-earth 3d transition metal compounds, such as Terfenol-D (k111 > 1000 ppm (106)). However, high value of magnetocrystalline anisotropy, high material costs and mechanical friability of these materials hinder their extensive industrial applications [3,4]. Thus, it is necessary to seek new class of magnetostrictive materials which are free of rare-earth metal. Strikingly, a new category of magnetostrictive materials has been developed based on Fe, mixed with nonmagnetic elements such as Ga, Al, Zn, Ge and Be [5–14]. In particular, Galfenol (Fe1xGax) alloys are regarded as promising magnetostrictive materials for the next generation. Tremendous interdisciplinary efforts have been carried during recent years to reveal the intrinsic mechanism of large magnetostrictive of Galfenol and to improve their performance for various applications, which lay a foundation to rationally design better magnetostrictive materials. Moreover, these rare-earth-free magnetostrictive materials have excellent features including high mechanical strength, good ductility, large magnetostriction values at low saturation magnetic field, high imposed-stress levels, and low associated cost. This makes them a potential alternative to
⇑ Corresponding author at: Department of Physics, Xiangtan University, Xiangtan 411105, Hunan, China. Tel.: 86 18673210632. E-mail address: [email protected] (J. Cao). http://dx.doi.org/10.1016/j.commatsci.2014.05.069 0927-0256/Ó 2014 Elsevier B.V. All rights reserved.
rare-earth-based giant magnetostrictive materials. There is still a challenge to find new materials that exhibit high magnetostriction in a relatively low reversal magnetic field and good mechanical properties for technological innovations. Fe4N has been studied extensively both experimentally [15–17] and theoretically [17,18] due to its remarkable chemical inertness and fascinating magnetic properties. Fe4N adopts a perovskite-like crystal structure as shown in Fig. 1(a), in which one Fe is located at the corners with Wyckoff site 1a, the other three iron atoms occupy the face centers with Wyckoff site 3c, while nitrogen atom occupies the body center with Wyckoff site 1b. In order to improve the magnetic properties of Fe4N, its derivatives MFe3N (M = Mn, Ni, Pd, Pt, Co, Rh, Ir, Ru, Os) have been studied both theoretically and experimentally recently, namely replacing the iron atoms with M atoms on Wyckoff site 1a and/or 3c. About half a century ago, NiFe3N, PdFe3N and PtFe3N were synthesized experimentally, which are fully ordered compounds [19]. RhFe3N was predicted to be thermodynamically stable at ambient pressure and latterly confirmed by experimental study [20,21]. By contrast, IrFe3N is stable only at pressures higher than 37 GPa [20]. The Os substitution effect has also been investigated with various experimental techniques [22]. Recent experimental study show that Mn atoms substitution occurring at both sites 1a and 3c, but mainly at site 1a [23]. These derivatives possess fascinating physical properties, such as ferromagnetism, high Curie temperature [6] and good ductility [24–26]. Besides, due to the large spin–orbit coupling (SOC) of 4d and 5d elements, they may exhibit large magnetocrystalline anisotropy. This motivates us to investigate the magnetostriction of Fe4N and its derivatives.
Y. Zhang et al. / Computational Materials Science 92 (2014) 464–467
465
Fig. 1. (a) The atomic configuration used in the calculations, where cyan, blue and violet spheres denote M (M = Fe, Mn, Ir, Os, Pd, Rh), N and Fe atoms, respectively and (b) the total energy Etot (open squares) and magnetocrystalline anisotropy energy EMCA (open triangles) of Fe4N as a function of lattice strain along the z axis. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
2. Method and model In the present study, using first principle calculations, we studied the magnetostriction of Fe4N and its derivatives. The allelectron full-potential linearized augmented plane-wave (FLAPW) [27,28] method was used to solve the density functional Kohn– Sham equations, along with the spin-polarized generalized gradient approximation for the description of exchange–correlation interaction. No shape approximation was assumed in charge, potential, and wave function expansions. We treated the core electrons fully relativistically, and the spin–orbit coupling among valence electrons is dealt with second variationally [29]. Energy cutoffs of 225 and 16 Ry were chosen for the charge-potential and basis expansions in the interstitial region. Spherical harmonics with a maximum angular momentum of lmax = 8 were used in the muffin–tin region (rFe = 2.25 a.u., rM = 2.40 a.u., rN = 1.20 a.u.). We adopted 4200 k points to sample the irreducible Brillouin zone to ensure the numerical convergence. The MAE was obtained by applying the torque approach which has been proved to be an effective method for the reliable determination of magnetic anisotropy energy [30]. The tetragonal magnetostrictive coefficient k001 can be obtained from the strain dependences of magnetocrystalline anisotropy energy (EMCA) and total energy (Etot) as the following equation,
k001 ¼
2dEMCA =de 2
3d Etot =de2
:
ð1Þ
We applied different tetragonal strains e along the z-axis with the constant-volume distortion mode (i.e. ex = ey = e/2), such as the results of EMCA and Etot of Fe4N are showed in Fig. 1(b). 3. Results and discussion Firstly, we determined k001 of Fe4N with the slope of EMCA and the curvature of Etot at e = 0 according to Eq. (1). The calculated
value of k001 is 143 ppm, which is comparable to Fe87.5Ga12.5 (+128 ppm). However, it is desirous to further enhance k001 of Fe4N. Within the rigid band model, which can successfully predict the magnetostriction of intermetallic alloys [7], we calculated magnetocrystalline anisotropy energies of Fe4N along with the number of electrons Ne variation in the unit cell with tetragonal strain at ±1 as shown in Fig. 2(a). In the rigid-band model, the d-bands are assumed to be rigid in shape with changing atomic number. This simplifies modeling band filling by shifting the Fermi level over the fixed bands according to the number of electrons present. Obviously, the magnetostriction of Fe4N can be extremely enhanced when we down-shift the Fermi level to the vertical dash line as shown in Fig. 2(a). According to the rigid band model, one should take one electron out of the unit cell. Heuristically thinking, we can speculate MnFe3N may exhibit larger magnetostriction compared to that of Fe4N. Intriguingly, the calculated k001 of MnFe3N is +373 ppm as shown in Fig. 2(b) proves the speculation from the rigid band model well. In order to reveal the driving factor behind, we first split the contributions to EMCA from different spin parts, i.e. the majorityspin states EMCA(UU), the minority-spin states EMCA(DD), and the cross-spin coupling EMCA(UD + DU), under various strain (e = ±1%) as listed in Table 1 [14]. Obviously, one can find the contribution of DE(Total) is almost dominated by DE(DD) for both Fe4N and MnFe3N. Here, energy difference of EMCA under different strain is defined as DEðXÞ ¼ DEMCA ðXÞje¼þ1% DEMCA ðXÞje¼1% with X = Total, DD, UU, DU + UD. Therefore, we only have to discuss strain induced EMCA(DD), starting from examining the EMCA(DD) kz dependence, which obtained by integrating contributions in each kx–ky plane. For Fe4N, as presented in Fig. 3(a), EMCA(DD) kz curve shows an enhanced positive contributions (kz = 0.0–0.2 au) and a depressed negative contributions (kz = 0.2–0.5 au) for compressive strain, compared to tensile strain. Thus Fe4N exhibits negative magnetostriction. On the contrary, EMCA(DD) kz curve of MnFe3N (Fig. 3(b)) presents different behavior. A compressive strain depresses positive contributions but enhances negative
Fig. 2. (a) The calculated EMCA against the band filling, Ne, for Fe4N with +1% and 1% strain and (b) the total energy Etot (open squares) and magnetocrystalline anisotropy energy EMCA (open triangles) of MnFe3N as a function of lattice strain along the z axis.
466
Y. Zhang et al. / Computational Materials Science 92 (2014) 464–467
Table 1 The calculated total EMCA (meV) and contributions to total EMCA from spin–orbit couplings between different spin blocks and from different atoms under various strain (e = ±1%) for Fe4N and MnFe3N.
Fe4N MnFe3N
e (%)
DD
UD
1 1 1 1
0.082 0.106 0.174 0.175
0.023 0.025 0.052 0.037
UU 0.0008 0.0005 0.136 0.055
I
II
III
EMCA(tot)
0.004 0.007 0.015 0.014
0.166 0.222 0.292 0.303
0.390 0.421 0.611 0.272
0.058 0.081 0.341 0.251
Fig. 3. The distribution of EMCA(DD) along the kz axis in the 3D BZ of the (a) Fe4N and (b) MnFe3N. (Circles, squares, and triangles are for cases with e = +1%, 0%, and 1%, respectively). The insets display the distributions of EMCA(DD) in the kx–ky planes at kz = 0.01 a.u. Red and blue spots are for positive and negative contributions to EMCA(DD) from different K points and their size scales with the magnitude of EMCA(DD). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
contributions. Moreover, DEMCA(DD) of MnFe3N is larger than Fe4N, for example at kz = 0.01. As a result, MnFe3N shows a larger positive magnetostriction. To analyze exactly which part of BZ and which states are responsible for the strain dependence of EMCA (DD), we further ‘‘zoom in’’ toward the most active region, and show distributions of EMCA(DD) in the kx–ky planes at kz = 0.01 a.u. in the inset. Here, the positive contribution at a given k-point is represented by red dots, while the blue dots are for the contribution to the negative one. Note that the magnitude of EMCA(DD) is proportional to the size of the dots. For Fe4N in Fig. 3(a) inset, the most eye-catching region is rhombus area around BZ center with sides roughly around 1/2(C–M). The positive contributions in this region are significantly suppressed as e = 1% ? +1%, even changing to negative contributions. The magnitude of the remaining regions is almost unchanged. That is why a compressive strain produces a positive EMCA(DD) for Fe4N. Analyses on distributions of EMCA(DD) are also done for MnFe3N. The projections of EMCA(DD) in the kx–ky planes are shown in the inset of Fig. 3(b). One can clearly see that the most pronounced changes of EMCA(DD) happen around 1/3(C–X0 ), where the area, magnitude and sign all sensitively change with vertical strains. For instance, there are strong negative contributions to EMCA(DD) around BZ center roughly with radius of 1/3(C–X0 ) as e = 1%, while large portion of this region disappears and the remaining regions (1/3(C–X0 )) turn into positive contributions for e = +1%. We have performed decomposed calculations by turning on the spin–orbit coupling of one atom and off the others to illustrate the relative importance of different atoms contributions to the EMCA, as listed in Table 1. One has to note here that the sum of all atoms contributions to the EMCA is not simply the total EMCA of the system [31]. Due to the local symmetry of atoms strongly affects the EMCA of compounds, thus we classified the metal atoms into three types for discussed below advantageously: corner sites (I), face-center in Fe–M layer (II) and face-center in Fe–N layer (III). Although they occupy the same Wyckoff sites, Fe atom at site II has different
atomic symmetry (c4v) against Fe atoms at site III (c2v) under strain. We see that the change of total EMCA from Fe atoms at site III is primary for MnFe3N. It is convenient that we only have to consider the minority-spin states of Fe III atom. However, Fe II and Fe III atoms both provide contributions for Fe4N, which complicate the analytic process. So we only presented the band structure of MnFe3N in the minority spin channel and the k-dependence of EMCA(DD) along the (C–X0 ) direction in the BZ at kz = 0.01 for two different lattice strains. It is obvious in Fig. 4(a) that the major contributions to EMCA(DD) are from the vicinity near 1/3(C–X0 ), where was highlighted by arrow. Fig. 4(b) shows that bands around the Fermi level in the highlighted (arrow) region behave differently for two different lattice strains, which respond to the changes of EMCA(DD). To ultimately trace down to key electronic states, we display the character of local d components (m = 0, ±1, ±2) on Fe III atom (Fig. 4(c) and (d)) is assigned to band dispersions when the ratio of the component is larger than 10%. Lattice compression along the z axis causes the dxy band to move up-wards in energy above the Fermi level and the dxz,yz bands down-wards below the Fermi level in the shadowed region. According to the perturbation theory [32], lattice compression will produce a negative contribution to EMCA(DD) through hdxy|Lx|dxzi. While lattice elongation lifts the dxz,yz bands, this will lead to a positive contribution to EMCA(DD) through hdxz|Lz|dyzi. Meanwhile, the negative contribution from hdxy|Lx|dxzi is weakened since the dxy is shifted below the Fermi level in the shadowed region. Finally, we introduce 4d and 5d elements into the Fe4N to verify if we can further enhance magnetostriction with their large SOC. We adopted the preferably substituted site 1a for all 4d or 5d transition metal atoms [19,23], as shown in Fig. 1(a). The calculated lattice parameters and magnetic moments for Fe4N and its derivative are given in Table 2, which are consistent with the previous works [19,24]. The magnetostriction of other MFe3N (M = Ru, Pd, Os and Ir) were calculated. We found that k001 = 172, +177, 564 and +416 ppm for RuFe3N, PdFe3N, OsFe3N and IrFe3N. Among
Y. Zhang et al. / Computational Materials Science 92 (2014) 464–467
467
Fig. 4. (a) EMCA(DD) contribution EMCA(k), (b) minority-spin band structure calculated without SOC along the U–X0 direction of the two-dimensional Brillouin zone at a fixed kz = 0.01 for compressed (e = 1%, black) and stretched (e = +1%, red) MnFe3N, respectively and (c) a close-up of the minority-spin band structure along the U–X0 direction for e = 1% and e = +1%. The Fe 3d components (m = 0, ±1, ±2) are indicated by red, green and blue circles, respectively. See text for other details. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Table 2 The lattice constant a (Å), the local magnetic moment lM (lB) for M (M = Mn, Ru, Pd, Os and Ir) and lFe (lB) for Fe, the total magnetic moment ltot (lB) and the magnetostrictive coefficient k001 (ppm) of Fe4N and its derivative.
a
lFe lM ltot k001
Fe4N
MnFe3N
RuFe3N
PdFe3N
OsFe3N
IrFe3N
3.794 2.99 / 9.97 143
3.780 2.23 3.55 10.24 +373
3.828 2.29 0.83 7.66 172
3.872 2.65 0.41 8.26 +177
3.818 2.08 0.51 6.74 564
3.843 2.42 0.65 7.88 +416
these compounds, OsFe3N and IrFe3N have large magnetostrictive coefficients, which are potentially useful for practical applications. 4. Conclusion In summary, we performed extensive density functional calculations on the magnetostriction of Fe4N and its derivative. Furthermore, using the rigid band model, we predicted MnFe3N has an enhanced magnetostrictive coefficient compared to Fe4N. Detailed analyses on the electronic origins demonstrated that the large magnetostrictive coefficient of MnFe3N is due to bands shift under strain. We also studied the magnetostriction of MFe3N (M = Ru, Pd, Os and Ir) and found OsFe3N and IrFe3N are promising magnetostrictive materials for practical applications. Acknowledgments This work is supported by National Natural Science Foundation of China (Nos. 11074212, 11204259, 11374252), the Program for New Century Excellent Talents in University (Grant No. NCET-120722) and Changjiang Scholars and Innovative Research Team in University (Grant No. IRT1080).
References [1] A.E. Clark, K.B. Hathaway, in: G. Engdahl (Ed.), Handbook of Giant Magnetostrictive Materials, Academic, San Diego, 2002. [2] R.Q. Wu, in: M.R.J. Gibbs (Ed.), Modern Trends in Magnetostriction Study and Application, Kluwer Academic, London, 2001. [3] J. Atulasimha, A.B. Flatau, Smart Mater. Struct. 20 (2011) 043001. [4] E.M. Summers, T.A. Lograsso, M. Wun-Fogle, J. Mater. Sci. 42 (2007) 9582. [5] A.E. Clark, J.-H. Yoo, J.R. Cullen, M. Wun-Fogle, G. Petculescu, A. Flatau, J. Appl. Phys. 105 (2009) 07A913. [6] G. Petculescu, K.B. Hathaway, T.A. Lograsso, M. Wun-Fogle, A.E. Clark, J. Appl. Phys. 97 (2005) 10M315. [7] Y.N. Zhang, J.X. Cao, R.Q. Wu, Appl. Phys. Lett. 96 (2010) 062508. [8] A.E. Clark, J.B. Restorff, M. Wun-Fogle, D. Wu, T.A. Lograsso, J. Appl. Phys. 103 (2008) 07B310. [9] D. Wu, Q. Xing, R.W. McCallum, T.A. Lograsso, J. Appl. Phys. 103 (2008) 07B307. [10] J.B. Restorff, M. Wun-Fogle, K.B. Hathaway, A.E. Clark, T.A. Lograsso, G. Petculescu, J. Appl. Phys. 111 (2012) 023905. [11] R.C. Hall, J. Appl. Phys. 30 (1959) 816. [12] J.X. Cao, Y.N. Zhang, W.J. Ouyang, R.Q. Wu, Phys. Rev. B 80 (2009) 104414. [13] Y.N. Zhang, R.Q. Wu, Phys. Rev. B 82 (2010) 224415. [14] S.C. Hong, W.S. Yun, R. Wu, Phys. Rev. B 79 (2009) 054419. [15] H. Jacobs, D. Rechenbach, U. Zachwieja, J. Alloys Compd. 227 (1995) 10. [16] S.K. Chen, S. Jin, T.H. Tiefel, Y.F. Hsieh, E.M. Gyorgy, D.W. Johnson Jr., J. Appl. Phys. 70 (1991) 6247. [17] C.A. Kuhnen, R.S. de Figueiredo, V. Drago, E.Z. da Silva, J. Magn. Magn. Mater. 111 (1992) 95. [18] S. Matar, P. Mohn, G. Demazeau, B. Siberchicot, J. Phys. (Paris) 49 (1988) 1761. [19] H.H. Stadelmaier, A.C. Franker, Trans. Metall. Soc. AIME 218 (1960) 571. [20] J. von Appen, R. Dronskowski, Angew. Chem. Int. Ed. 44 (2005) 1205. [21] A. Houben, P. Muller, J. von Appen, H. Lueken, R. Niewa, R. Dronskowski, Angew. Chem. Int. Ed. 44 (2005) 7212. [22] D. Andriamandroso, S. Matar, G. Demazeau, L. Fournes, IEEE Trans. Magn. 29 (1993) 2. [23] J. Martínez, L. Lopardo, J. Desimoni, J. Alloys Compd. 557 (2013) 218. [24] E. Zhao, H. Xiang, J. Meng, Z. Wu, Chem. Phys. Lett. 96 (2007) 449. [25] Z.J. Wu, J. Meng, Appl. Phys. Lett. 90 (2007) 241901. [26] D. Music, J.M. Schneider, Appl. Phys. Lett. 88 (2006) 031914. [27] M. Weinert, E. Wimmer, A.J. Freeman, Phys. Rev. B 26 (1982) 4571. [28] E. Wimmer, H. Krakauer, M. Weinert, A.J. Freeman, Phys. Rev. B 24 (1981) 864. [29] R. Wu, A.J. Freeman, J. Magn. Magn. Mater. 200 (1999) 498. [30] X. Wang, R. Wu, D.-S. Wang, A.J. Freeman, Phys. Rev. B 54 (1996) 61. [31] S. Baud, C. Ramseyer, G. Bihlmayer, S. Blügel, Phys. Rev. B 73 (2006) 104427. [32] D.S. Wang, R.Q. Wu, A.J. Freeman, Phys. Rev. B 47 (1993) 14932.
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Predicting magnetostriction of MFe3N (M = Fe, Mn, Ir, Os, Pd, Rh) from ab initio calculations Yun Zhang a, Zhe Wang a, Juexian Cao a,b,⇑ a b
Department of Physics, Xiangtan University, Xiangtan 411105, Hunan, China Beijing Computational Science Research Center, Beijing 100084, Beijing, China
a r t i c l e
i n f o
Article history: Received 29 January 2014 Received in revised form 30 May 2014 Accepted 31 May 2014 Available online 9 July 2014 Keywords: Fe-alloys Ab initio calculations Magnetic properties
a b s t r a c t Using the first-principles full-potential linearized augmented plane-wave method, we investigate the magnetostriction of Fe4N. Its magnetostrictive coefficient is found to be 143 ppm. In order to enhance the magnetostriction, the derivative MnFe3N is taken into our consideration according to the rigid band model. Interestingly, we find that the magnetostrictive coefficient is enhanced to be +373 ppm. To further improve the magnetostriction, we calculate the magnetostrictive coefficient of the derivatives containing 4d and 5d metal due to their strong spin–orbit coupling. The results show that the magnetostrictive coefficient can reach to 564 ppm and +416 ppm for OsFe3N and IrFe3N. Our calculations give a guideline for designing giant magnetostriction material. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction Magnetostrictive materials are widely used for exploitations in sensors, actuators, micro electro mechanical system (MEMS), and energy harvesting devices [1,2]. The large magnetostrictive coefficients are discovered in rare-earth 3d transition metal compounds, such as Terfenol-D (k111 > 1000 ppm (106)). However, high value of magnetocrystalline anisotropy, high material costs and mechanical friability of these materials hinder their extensive industrial applications [3,4]. Thus, it is necessary to seek new class of magnetostrictive materials which are free of rare-earth metal. Strikingly, a new category of magnetostrictive materials has been developed based on Fe, mixed with nonmagnetic elements such as Ga, Al, Zn, Ge and Be [5–14]. In particular, Galfenol (Fe1xGax) alloys are regarded as promising magnetostrictive materials for the next generation. Tremendous interdisciplinary efforts have been carried during recent years to reveal the intrinsic mechanism of large magnetostrictive of Galfenol and to improve their performance for various applications, which lay a foundation to rationally design better magnetostrictive materials. Moreover, these rare-earth-free magnetostrictive materials have excellent features including high mechanical strength, good ductility, large magnetostriction values at low saturation magnetic field, high imposed-stress levels, and low associated cost. This makes them a potential alternative to
⇑ Corresponding author at: Department of Physics, Xiangtan University, Xiangtan 411105, Hunan, China. Tel.: 86 18673210632. E-mail address: [email protected] (J. Cao). http://dx.doi.org/10.1016/j.commatsci.2014.05.069 0927-0256/Ó 2014 Elsevier B.V. All rights reserved.
rare-earth-based giant magnetostrictive materials. There is still a challenge to find new materials that exhibit high magnetostriction in a relatively low reversal magnetic field and good mechanical properties for technological innovations. Fe4N has been studied extensively both experimentally [15–17] and theoretically [17,18] due to its remarkable chemical inertness and fascinating magnetic properties. Fe4N adopts a perovskite-like crystal structure as shown in Fig. 1(a), in which one Fe is located at the corners with Wyckoff site 1a, the other three iron atoms occupy the face centers with Wyckoff site 3c, while nitrogen atom occupies the body center with Wyckoff site 1b. In order to improve the magnetic properties of Fe4N, its derivatives MFe3N (M = Mn, Ni, Pd, Pt, Co, Rh, Ir, Ru, Os) have been studied both theoretically and experimentally recently, namely replacing the iron atoms with M atoms on Wyckoff site 1a and/or 3c. About half a century ago, NiFe3N, PdFe3N and PtFe3N were synthesized experimentally, which are fully ordered compounds [19]. RhFe3N was predicted to be thermodynamically stable at ambient pressure and latterly confirmed by experimental study [20,21]. By contrast, IrFe3N is stable only at pressures higher than 37 GPa [20]. The Os substitution effect has also been investigated with various experimental techniques [22]. Recent experimental study show that Mn atoms substitution occurring at both sites 1a and 3c, but mainly at site 1a [23]. These derivatives possess fascinating physical properties, such as ferromagnetism, high Curie temperature [6] and good ductility [24–26]. Besides, due to the large spin–orbit coupling (SOC) of 4d and 5d elements, they may exhibit large magnetocrystalline anisotropy. This motivates us to investigate the magnetostriction of Fe4N and its derivatives.
Y. Zhang et al. / Computational Materials Science 92 (2014) 464–467
465
Fig. 1. (a) The atomic configuration used in the calculations, where cyan, blue and violet spheres denote M (M = Fe, Mn, Ir, Os, Pd, Rh), N and Fe atoms, respectively and (b) the total energy Etot (open squares) and magnetocrystalline anisotropy energy EMCA (open triangles) of Fe4N as a function of lattice strain along the z axis. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
2. Method and model In the present study, using first principle calculations, we studied the magnetostriction of Fe4N and its derivatives. The allelectron full-potential linearized augmented plane-wave (FLAPW) [27,28] method was used to solve the density functional Kohn– Sham equations, along with the spin-polarized generalized gradient approximation for the description of exchange–correlation interaction. No shape approximation was assumed in charge, potential, and wave function expansions. We treated the core electrons fully relativistically, and the spin–orbit coupling among valence electrons is dealt with second variationally [29]. Energy cutoffs of 225 and 16 Ry were chosen for the charge-potential and basis expansions in the interstitial region. Spherical harmonics with a maximum angular momentum of lmax = 8 were used in the muffin–tin region (rFe = 2.25 a.u., rM = 2.40 a.u., rN = 1.20 a.u.). We adopted 4200 k points to sample the irreducible Brillouin zone to ensure the numerical convergence. The MAE was obtained by applying the torque approach which has been proved to be an effective method for the reliable determination of magnetic anisotropy energy [30]. The tetragonal magnetostrictive coefficient k001 can be obtained from the strain dependences of magnetocrystalline anisotropy energy (EMCA) and total energy (Etot) as the following equation,
k001 ¼
2dEMCA =de 2
3d Etot =de2
:
ð1Þ
We applied different tetragonal strains e along the z-axis with the constant-volume distortion mode (i.e. ex = ey = e/2), such as the results of EMCA and Etot of Fe4N are showed in Fig. 1(b). 3. Results and discussion Firstly, we determined k001 of Fe4N with the slope of EMCA and the curvature of Etot at e = 0 according to Eq. (1). The calculated
value of k001 is 143 ppm, which is comparable to Fe87.5Ga12.5 (+128 ppm). However, it is desirous to further enhance k001 of Fe4N. Within the rigid band model, which can successfully predict the magnetostriction of intermetallic alloys [7], we calculated magnetocrystalline anisotropy energies of Fe4N along with the number of electrons Ne variation in the unit cell with tetragonal strain at ±1 as shown in Fig. 2(a). In the rigid-band model, the d-bands are assumed to be rigid in shape with changing atomic number. This simplifies modeling band filling by shifting the Fermi level over the fixed bands according to the number of electrons present. Obviously, the magnetostriction of Fe4N can be extremely enhanced when we down-shift the Fermi level to the vertical dash line as shown in Fig. 2(a). According to the rigid band model, one should take one electron out of the unit cell. Heuristically thinking, we can speculate MnFe3N may exhibit larger magnetostriction compared to that of Fe4N. Intriguingly, the calculated k001 of MnFe3N is +373 ppm as shown in Fig. 2(b) proves the speculation from the rigid band model well. In order to reveal the driving factor behind, we first split the contributions to EMCA from different spin parts, i.e. the majorityspin states EMCA(UU), the minority-spin states EMCA(DD), and the cross-spin coupling EMCA(UD + DU), under various strain (e = ±1%) as listed in Table 1 [14]. Obviously, one can find the contribution of DE(Total) is almost dominated by DE(DD) for both Fe4N and MnFe3N. Here, energy difference of EMCA under different strain is defined as DEðXÞ ¼ DEMCA ðXÞje¼þ1% DEMCA ðXÞje¼1% with X = Total, DD, UU, DU + UD. Therefore, we only have to discuss strain induced EMCA(DD), starting from examining the EMCA(DD) kz dependence, which obtained by integrating contributions in each kx–ky plane. For Fe4N, as presented in Fig. 3(a), EMCA(DD) kz curve shows an enhanced positive contributions (kz = 0.0–0.2 au) and a depressed negative contributions (kz = 0.2–0.5 au) for compressive strain, compared to tensile strain. Thus Fe4N exhibits negative magnetostriction. On the contrary, EMCA(DD) kz curve of MnFe3N (Fig. 3(b)) presents different behavior. A compressive strain depresses positive contributions but enhances negative
Fig. 2. (a) The calculated EMCA against the band filling, Ne, for Fe4N with +1% and 1% strain and (b) the total energy Etot (open squares) and magnetocrystalline anisotropy energy EMCA (open triangles) of MnFe3N as a function of lattice strain along the z axis.
466
Y. Zhang et al. / Computational Materials Science 92 (2014) 464–467
Table 1 The calculated total EMCA (meV) and contributions to total EMCA from spin–orbit couplings between different spin blocks and from different atoms under various strain (e = ±1%) for Fe4N and MnFe3N.
Fe4N MnFe3N
e (%)
DD
UD
1 1 1 1
0.082 0.106 0.174 0.175
0.023 0.025 0.052 0.037
UU 0.0008 0.0005 0.136 0.055
I
II
III
EMCA(tot)
0.004 0.007 0.015 0.014
0.166 0.222 0.292 0.303
0.390 0.421 0.611 0.272
0.058 0.081 0.341 0.251
Fig. 3. The distribution of EMCA(DD) along the kz axis in the 3D BZ of the (a) Fe4N and (b) MnFe3N. (Circles, squares, and triangles are for cases with e = +1%, 0%, and 1%, respectively). The insets display the distributions of EMCA(DD) in the kx–ky planes at kz = 0.01 a.u. Red and blue spots are for positive and negative contributions to EMCA(DD) from different K points and their size scales with the magnitude of EMCA(DD). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
contributions. Moreover, DEMCA(DD) of MnFe3N is larger than Fe4N, for example at kz = 0.01. As a result, MnFe3N shows a larger positive magnetostriction. To analyze exactly which part of BZ and which states are responsible for the strain dependence of EMCA (DD), we further ‘‘zoom in’’ toward the most active region, and show distributions of EMCA(DD) in the kx–ky planes at kz = 0.01 a.u. in the inset. Here, the positive contribution at a given k-point is represented by red dots, while the blue dots are for the contribution to the negative one. Note that the magnitude of EMCA(DD) is proportional to the size of the dots. For Fe4N in Fig. 3(a) inset, the most eye-catching region is rhombus area around BZ center with sides roughly around 1/2(C–M). The positive contributions in this region are significantly suppressed as e = 1% ? +1%, even changing to negative contributions. The magnitude of the remaining regions is almost unchanged. That is why a compressive strain produces a positive EMCA(DD) for Fe4N. Analyses on distributions of EMCA(DD) are also done for MnFe3N. The projections of EMCA(DD) in the kx–ky planes are shown in the inset of Fig. 3(b). One can clearly see that the most pronounced changes of EMCA(DD) happen around 1/3(C–X0 ), where the area, magnitude and sign all sensitively change with vertical strains. For instance, there are strong negative contributions to EMCA(DD) around BZ center roughly with radius of 1/3(C–X0 ) as e = 1%, while large portion of this region disappears and the remaining regions (1/3(C–X0 )) turn into positive contributions for e = +1%. We have performed decomposed calculations by turning on the spin–orbit coupling of one atom and off the others to illustrate the relative importance of different atoms contributions to the EMCA, as listed in Table 1. One has to note here that the sum of all atoms contributions to the EMCA is not simply the total EMCA of the system [31]. Due to the local symmetry of atoms strongly affects the EMCA of compounds, thus we classified the metal atoms into three types for discussed below advantageously: corner sites (I), face-center in Fe–M layer (II) and face-center in Fe–N layer (III). Although they occupy the same Wyckoff sites, Fe atom at site II has different
atomic symmetry (c4v) against Fe atoms at site III (c2v) under strain. We see that the change of total EMCA from Fe atoms at site III is primary for MnFe3N. It is convenient that we only have to consider the minority-spin states of Fe III atom. However, Fe II and Fe III atoms both provide contributions for Fe4N, which complicate the analytic process. So we only presented the band structure of MnFe3N in the minority spin channel and the k-dependence of EMCA(DD) along the (C–X0 ) direction in the BZ at kz = 0.01 for two different lattice strains. It is obvious in Fig. 4(a) that the major contributions to EMCA(DD) are from the vicinity near 1/3(C–X0 ), where was highlighted by arrow. Fig. 4(b) shows that bands around the Fermi level in the highlighted (arrow) region behave differently for two different lattice strains, which respond to the changes of EMCA(DD). To ultimately trace down to key electronic states, we display the character of local d components (m = 0, ±1, ±2) on Fe III atom (Fig. 4(c) and (d)) is assigned to band dispersions when the ratio of the component is larger than 10%. Lattice compression along the z axis causes the dxy band to move up-wards in energy above the Fermi level and the dxz,yz bands down-wards below the Fermi level in the shadowed region. According to the perturbation theory [32], lattice compression will produce a negative contribution to EMCA(DD) through hdxy|Lx|dxzi. While lattice elongation lifts the dxz,yz bands, this will lead to a positive contribution to EMCA(DD) through hdxz|Lz|dyzi. Meanwhile, the negative contribution from hdxy|Lx|dxzi is weakened since the dxy is shifted below the Fermi level in the shadowed region. Finally, we introduce 4d and 5d elements into the Fe4N to verify if we can further enhance magnetostriction with their large SOC. We adopted the preferably substituted site 1a for all 4d or 5d transition metal atoms [19,23], as shown in Fig. 1(a). The calculated lattice parameters and magnetic moments for Fe4N and its derivative are given in Table 2, which are consistent with the previous works [19,24]. The magnetostriction of other MFe3N (M = Ru, Pd, Os and Ir) were calculated. We found that k001 = 172, +177, 564 and +416 ppm for RuFe3N, PdFe3N, OsFe3N and IrFe3N. Among
Y. Zhang et al. / Computational Materials Science 92 (2014) 464–467
467
Fig. 4. (a) EMCA(DD) contribution EMCA(k), (b) minority-spin band structure calculated without SOC along the U–X0 direction of the two-dimensional Brillouin zone at a fixed kz = 0.01 for compressed (e = 1%, black) and stretched (e = +1%, red) MnFe3N, respectively and (c) a close-up of the minority-spin band structure along the U–X0 direction for e = 1% and e = +1%. The Fe 3d components (m = 0, ±1, ±2) are indicated by red, green and blue circles, respectively. See text for other details. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Table 2 The lattice constant a (Å), the local magnetic moment lM (lB) for M (M = Mn, Ru, Pd, Os and Ir) and lFe (lB) for Fe, the total magnetic moment ltot (lB) and the magnetostrictive coefficient k001 (ppm) of Fe4N and its derivative.
a
lFe lM ltot k001
Fe4N
MnFe3N
RuFe3N
PdFe3N
OsFe3N
IrFe3N
3.794 2.99 / 9.97 143
3.780 2.23 3.55 10.24 +373
3.828 2.29 0.83 7.66 172
3.872 2.65 0.41 8.26 +177
3.818 2.08 0.51 6.74 564
3.843 2.42 0.65 7.88 +416
these compounds, OsFe3N and IrFe3N have large magnetostrictive coefficients, which are potentially useful for practical applications. 4. Conclusion In summary, we performed extensive density functional calculations on the magnetostriction of Fe4N and its derivative. Furthermore, using the rigid band model, we predicted MnFe3N has an enhanced magnetostrictive coefficient compared to Fe4N. Detailed analyses on the electronic origins demonstrated that the large magnetostrictive coefficient of MnFe3N is due to bands shift under strain. We also studied the magnetostriction of MFe3N (M = Ru, Pd, Os and Ir) and found OsFe3N and IrFe3N are promising magnetostrictive materials for practical applications. Acknowledgments This work is supported by National Natural Science Foundation of China (Nos. 11074212, 11204259, 11374252), the Program for New Century Excellent Talents in University (Grant No. NCET-120722) and Changjiang Scholars and Innovative Research Team in University (Grant No. IRT1080).
References [1] A.E. Clark, K.B. Hathaway, in: G. Engdahl (Ed.), Handbook of Giant Magnetostrictive Materials, Academic, San Diego, 2002. [2] R.Q. Wu, in: M.R.J. Gibbs (Ed.), Modern Trends in Magnetostriction Study and Application, Kluwer Academic, London, 2001. [3] J. Atulasimha, A.B. Flatau, Smart Mater. Struct. 20 (2011) 043001. [4] E.M. Summers, T.A. Lograsso, M. Wun-Fogle, J. Mater. Sci. 42 (2007) 9582. [5] A.E. Clark, J.-H. Yoo, J.R. Cullen, M. Wun-Fogle, G. Petculescu, A. Flatau, J. Appl. Phys. 105 (2009) 07A913. [6] G. Petculescu, K.B. Hathaway, T.A. Lograsso, M. Wun-Fogle, A.E. Clark, J. Appl. Phys. 97 (2005) 10M315. [7] Y.N. Zhang, J.X. Cao, R.Q. Wu, Appl. Phys. Lett. 96 (2010) 062508. [8] A.E. Clark, J.B. Restorff, M. Wun-Fogle, D. Wu, T.A. Lograsso, J. Appl. Phys. 103 (2008) 07B310. [9] D. Wu, Q. Xing, R.W. McCallum, T.A. Lograsso, J. Appl. Phys. 103 (2008) 07B307. [10] J.B. Restorff, M. Wun-Fogle, K.B. Hathaway, A.E. Clark, T.A. Lograsso, G. Petculescu, J. Appl. Phys. 111 (2012) 023905. [11] R.C. Hall, J. Appl. Phys. 30 (1959) 816. [12] J.X. Cao, Y.N. Zhang, W.J. Ouyang, R.Q. Wu, Phys. Rev. B 80 (2009) 104414. [13] Y.N. Zhang, R.Q. Wu, Phys. Rev. B 82 (2010) 224415. [14] S.C. Hong, W.S. Yun, R. Wu, Phys. Rev. B 79 (2009) 054419. [15] H. Jacobs, D. Rechenbach, U. Zachwieja, J. Alloys Compd. 227 (1995) 10. [16] S.K. Chen, S. Jin, T.H. Tiefel, Y.F. Hsieh, E.M. Gyorgy, D.W. Johnson Jr., J. Appl. Phys. 70 (1991) 6247. [17] C.A. Kuhnen, R.S. de Figueiredo, V. Drago, E.Z. da Silva, J. Magn. Magn. Mater. 111 (1992) 95. [18] S. Matar, P. Mohn, G. Demazeau, B. Siberchicot, J. Phys. (Paris) 49 (1988) 1761. [19] H.H. Stadelmaier, A.C. Franker, Trans. Metall. Soc. AIME 218 (1960) 571. [20] J. von Appen, R. Dronskowski, Angew. Chem. Int. Ed. 44 (2005) 1205. [21] A. Houben, P. Muller, J. von Appen, H. Lueken, R. Niewa, R. Dronskowski, Angew. Chem. Int. Ed. 44 (2005) 7212. [22] D. Andriamandroso, S. Matar, G. Demazeau, L. Fournes, IEEE Trans. Magn. 29 (1993) 2. [23] J. Martínez, L. Lopardo, J. Desimoni, J. Alloys Compd. 557 (2013) 218. [24] E. Zhao, H. Xiang, J. Meng, Z. Wu, Chem. Phys. Lett. 96 (2007) 449. [25] Z.J. Wu, J. Meng, Appl. Phys. Lett. 90 (2007) 241901. [26] D. Music, J.M. Schneider, Appl. Phys. Lett. 88 (2006) 031914. [27] M. Weinert, E. Wimmer, A.J. Freeman, Phys. Rev. B 26 (1982) 4571. [28] E. Wimmer, H. Krakauer, M. Weinert, A.J. Freeman, Phys. Rev. B 24 (1981) 864. [29] R. Wu, A.J. Freeman, J. Magn. Magn. Mater. 200 (1999) 498. [30] X. Wang, R. Wu, D.-S. Wang, A.J. Freeman, Phys. Rev. B 54 (1996) 61. [31] S. Baud, C. Ramseyer, G. Bihlmayer, S. Blügel, Phys. Rev. B 73 (2006) 104427. [32] D.S. Wang, R.Q. Wu, A.J. Freeman, Phys. Rev. B 47 (1993) 14932.