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Fe magnetic moment formation and exchange interaction in Fe2P: A first-principles study
Physics Letters A 377 (2013) 731–735
Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Fe magnetic moment formation and exchange interaction in Fe2 P: A first-principles study X.B. Liu a,∗ , J. Ping Liu a , Qiming Zhang a , Z. Altounian b a b
Department of Physics, University of Texas at Arlington, Arlington, TX 76019, USA Center for the Physics of Materials and Department of Physics, McGill University, 3600 University Street, Montreal, Quebec, H3A 2T8, Canada
a r t i c l e
i n f o
Article history: Received 16 October 2012 Received in revised form 11 January 2013 Accepted 12 January 2013 Available online 18 January 2013 Communicated by R. Wu Keywords: Magnetic moment Exchange interaction Density functional
a b s t r a c t Electronic structure and magnetic properties of Fe2 P have been studied by a first-principles density functional theory calculation. The ground state is ferromagnetic and the calculated magnetic moments for Fe1 (3 f ) and Fe2 (3g) are 0.83 and 2.30 μ B , respectively. The nearest neighbor inter-site magnetic exchange coupling parameter at the Fe1 layer (0.02 mRy) is much smaller than that at the Fe2 layer (1.29 mRy). The Fe moment at the 3 f site is metastable and sensitive to the inter-site exchange interaction with its magnetic neighbors, which is responsible for the first order magnetic transition and large magneto-caloric effect around T C . © 2013 Elsevier B.V. All rights reserved.
1. Introduction Recently, more research attentions are paid to the Fe (Mn) based compound with the hexagonal Fe2 P-type structure since the discovery of the giant magneto-caloric effect (MCE) in MnFeP1−x Asx with a Fe2 P-type structure [1]. Further, large or moderate MCEs are also reported in the other Fe2 P-type compounds such as MnFeP1−x (Ge, Si)x , and (Fe, M)2 P with M = Ni, Ru, Rh, Pd, and Pt [2–9]. The large MCEs originate from the strong first order magnetic transition at T C and the field induced meta-magnetic transition above T C , which is related to the competing exchange coupling in the compounds. The knowledge on the structure, magnetic states and exchange interaction of the prototype compound Fe2 P is very helpful in understanding the magnetic transition and magneto-caloric effect in these compounds. In the past 40 years, many experimental and theoretical research works have been done on the hexagonal Fe2 P [10–17]. The hexagonal Fe2 P is a ferromagnet with a strong c-axis magnetocrystalline anisotropy and a Curie temperature T C = 217 K [10]. Fe2 P shows generally a first order magnetic phase transition from paramagnetic to ferromagnetic states at T C with a discontinuous change of lattice constants, where a increases while c decreases [13,14]. Fe atoms are positioned at two inequivalent sites: a tetragonal site Fe1 (3 f ) and a pyramidal site Fe2 (3g). Neutron diffraction studies [18] on powder sample at 77 K show Fe magnetic moments of 0.69 μ B and 2.31 μ B at the 3 f and 3g sites, respec-
*
Corresponding author. E-mail address: [email protected] (X.B. Liu).
0375-9601/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2013.01.019
tively. However, polarized neutron diffraction study on a single crystal sample gave Fe moments of 0.92 μ B and 1.7 μ B for the 3 f and 3g sites, respectively, at the same temperature [19]. Temperature and magnetic field dependence of magnetization results indicate that the first order magnetic transition could be resolved into two sequential transitions in a weak external field: from the paramagnetic state to the intermediate meta-magnetic phase, then to the low temperature ferromagnetic phase [20]. The magnetic states of Fe2 P are sensitive to the external pressure, Fe vacancy, and impurities. The external pressure will reduce the lattice volume and the axial ratio a/c, and drive the magnetic structure into non-collinear and anti-ferromagnetic states [21]. Similarly, the lattice constants decrease slightly while T C decreases rapidly and the magnetic state transforms into a meta-magnetic and/or anti-ferromagnetic state with increasing x in Fe2−x P [22]. On the other hand, a very small amount of impurity such as B, As, and Si will increase the a parameter while decreasing the c parameter and improve substantially T C in Fe2 P compound [14]. The change of magnetic ordering due to non-metallic impurities are mainly ascribed to the variation of lattice constant [14,23]. Spin polarized electronic structure calculations have been performed for Fe2 P using Korringa–Kohn–Rostoker (KKR) and linear muffin-tin orbital (LMTO) method by several groups [15–17]. The calculated Fe moments are 0.8–0.96 μ B for Fe1 (3 f ) and 1.9–2.4 μ B for Fe2 (3g), in fair agreement with the experimental values. Discussed by means of the Landau phenomenological theory [24], the Fe atoms at the 3 f site show meta-magnetic behavior, which is responsible for the first order magnetic transition. In this work, the electronic structure, Fe magnetic moment and exchange interaction have been investigated in Fe2 P within the
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X.B. Liu et al. / Physics Letters A 377 (2013) 731–735
Table 1 Calculated and experimental [35] crystallographic data in Fe2 P. Atomic position: Fe1 (3 f ): (x3 f , 0, 0) and Fe2 (3g): (x3g , 0, 0.5). a (Å)
c (Å)
x3 f
x3g
5.868 5.813
3.458 3.425
0.257 0.257
0.595 0.592
experiment calculation
framework of a density functional theory (DFT). It is found that the Fe magnetic moment at the 3 f site is metastable and depends on the exchange interaction with the neighboring magnetic atoms. The magnetic exchange interactions vary substantially among the different Fe–Fe pairs and the intra-layer magnetic exchange interaction is much stronger in the Fe2 (3g) layer than that in the Fe1 (3 f ) layer. 2. Computational methods We performed first-principles electronic structure calculations in the framework of density functional theory (DFT). The Vienna Ab-initio Simulation Package (VASP) [25,26] in the projector augmented wave (PAW) framework [27,28] was employed to perform DFT calculations using the generalized gradient approximation of Perdew–Burke–Ernzerhof (PBE) [29] for the exchange correlation functional. In all calculations the structural degrees of freedom are fully relaxed on a gamma centered k-grids of 21 × 21 × 15 k-mesh. The k-space integrations were performed with the tetrahedron method [30,31]. The inter-sites exchange coupling parameters J i j of the Heisenberg model have been calculated using a linear-response method [32] and implemented in LM code [33]. We adopt a Green’s function technique combined with the linear muffin-tin orbital (LMTO) method with atomic sphere approximation (ASA) in the calculation [33,34]. In this approach, exchange interactions are calculated as the response to small-angle fluctuations of the spin orientations. The band structure calculation provides the one-electron Green function. The energy integrals over the occupied part of the valence band are expressed as integrals over an energy variable along a closed path starting and ending at the Fermi energy. The integrals are numerically evaluated using the Gaussian quadrature method. Additional calculation details have been reported previously [8]. 3. Results and discussion 3.1. Density of states The hexagonal Fe2 P has a layer structure with a space group ¯ of P 62m. The atoms Fe1 , Fe2 , P1 , and P2 are distributed at the 3 f (0.257, 0, 0), 3g (0.595, 00.15), 2c (0.333, 0.667, 0) and 1b (0, 0, 0.5) sites, respectively [35]. The compound is consisted of alternative layers of Fe1 –P1 , and Fe2 –P2 layers along c-axis (Fig. 1). As shown in Fig. 1(a), Fe2 is centered at a distorted pyramid with four P1 and one P2 at the five vertices. Fe1 is centered at a distorted tetrahedron with two P1 and two P2 at the four vertices (Fig. 1(b), the unit cell is doubled along c-axis). As shown in Fig. 1(c), the unit cell length is doubled along the two a-axis, each Fe1 has two equivalent nearest neighbors (NNs) of P1 and two equivalent NNs of Fe1 in the Fe1 –P1 layer while each Fe2 has one nearest neighbor of P2 and four equivalent NNs of Fe2 in the Fe2 –P2 layer. The different atomic environments of Fe1 and Fe2 will affect their magnetic properties as shown below. The geometrical structure of Fe2 P has been fully relaxed in the calculations. The calculated structural data are in good agreement with the experimental values and the difference is within 1% (Table 1). Fig. 2 displays the total and partial density of states (DOS) of hexagonal Fe2 P compound. Fe2 P shows a typical metallic behavior
Fig. 1. (Colour online.) (a) Unit cell of Fe2 P, showing the Fe2 –P chemical bonds; (b) Unit cell doubled along c-axis, showing the Fe1 –P chemical bonds; (c) Unit cell doubled along a- and b-axis, showing the chemical bonds in the layers of Fe1 –P1 and Fe2 –P2 . The small (red), further small (blue), medium (cyan), and large (green) sized spheres are atoms P1 , P2 , Fe1 , and Fe2 , respectively. Table 2 Calculated and experimental magnetic moments M (μ B ) in Fe2 P. Phase
M Fe1
M Fe2
M1
M P2
Method
Reference
FM FM FM FM FM AFM FM AFM
0.69 0.92 0.92 0.89 0.83 0.30 0.95 0.06
2.31 1.7 2.03 2.24 2.30 2.11 2.00 1.77
– –
– –
−0.06 −0.07 −0.04 −0.001 −0.07 −0.02
−0.06 −0.06 −0.05 −0.02 −0.06 −0.03
experiment experiment LMTO KKR VASP VASP LMTO LMTO
[17] [18] [15] [16] this work this work this work this work
in both spin minority and spin majority components. A large energy gap (about 4 eV) separated the DOS into a lower part (around −12 eV) and a higher part ranging from about −7.5 eV to the Fermi level (E f = 0). The contribution to the lower part of the DOS is from the 3s-like states of the P atoms. The main contribution to the DOS at Fermi Level (E f = 0) is from Fe 3d electrons. Clearly, the exchange splitting of DOS for Fe1 is smaller than that of Fe2 , consistent with a smaller spin moment of Fe1 (Table 2). 3.2. Fe magnetic moments For layer compound Fe2 P, it has a ferromagnetic (FM) ground state and the calculated magnetic moments are 0.83 μ B and 2.30 μ B for Fe1 and Fe2 , respectively, via the VASP code (Table 2). The total moments is 3.08 μ B per formula unit due to the contribution of the small moments induced at P atoms. Similarly, the LMTO-ASA code gives the moments of 0.95 μ B and 2.00 μ B for Fe1 and Fe2 , respectively, and a total magnetic moment of 2.94 μ B per formula unit. Although the calculated total magnetic moment per
X.B. Liu et al. / Physics Letters A 377 (2013) 731–735
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Table 3 Atomic Wigner–Seitz cell Volume (WSV) and inter-atomic distances d in Fe2 P compounds. Fe1
Fe2
Atoms
d (Å)
Atoms
d (Å)
2 2 2 2 4
2.218 2.297 2.621 2.626 2.707
1 4 2 4 4
2.376 2.484 2.626 2.707 3.088
11.0
WSV (Å3 )
P1 P2 Fe1 Fe2 Fe2
WSV (Å3 )
P2 P1 Fe1 Fe1 Fe2
12.6
Fig. 3. (Colour online.) The virtual anti-ferromagnetic magnetic unit cell for Fe2 P. The small (red), medium (white), and large (green) sized spheres are atoms P, Fe1 , and Fe2 , respectively.
Fig. 2. (Colour online.) Calculated spin-projected (a) total density of states (DOS) and partial DOS at Fe1 site (b) and that at Fe2 site (c) in Fe2 P compound. The blue, red and green lines are for the contribution of Fe 3d, 4p and 4s-like states, respectively. The Fermi level is at E = 0.
formula unit (3.0 μ B or so) is almost same for the two methods, the different moment values at different Fe sites are expected due to the different atomic potential treatment in the PAW and LMTOASA methods. It is expected that the magnetic moment calculated by LMTO-ASA has a lower precision because the sphere approximation of potential shape and the overlapping among different atomic spheres will lower the potential precision. These calculated magnetic moments are in good agreement with the previous experimental and calculation results (Table 2).
The difference in the magnetic moments of Fe1 and Fe2 are related to their different local environment in the unit cell. The Wigner–Seitz cell Volume (WSV) calculations was performed following the method given by Koch and Fisher [36] and the results are listed in Table 3. Fe1 with a WSV volume of about 11.0 Å3 has four nearest P neighbors at the corners of a distorted tetrahedron with an average inter-atomic distance of about 2.25 Å, and eight Fe near neighbors with the atomic separation of about 2.6–2.7 Å. On the other hand, Fe2 with a WSV volume of about 12.6 Å3 has five P neighbors at the corners of a distorted pyramid with an average inter-atomic distance of about 2.46 Å, and has eight Fe neighbors with atomic distances varying from 2.6 to 3.1 Å. The larger WSV and larger atomic separation distances with Fe and P neighbors are responsible for the larger magnetic moment of Fe2 . Severin et al. [37] reported that the Fe atomic magnetic moment increases almost linearly with the average inter-atomic distance of Fe–P in Fe2 P, consistent to our present results. To gain more insight on the Fe moment formation, the total energy and magnetic moments for a virtual anti-ferromagnetic (AFM) structure have been calculated. For this layer compound, we, here, consider only a specific AFM structure (Fig. 3). In this virtual AFM structure, the unit cell is doubled along the c-axis and the Fe moments are distributed as Fe1 (+)Fe2 (−)Fe1 (−)Fe2 (+) along the c-axis. The intra-layer magnetic coupling of Fe1 –Fe1 and Fe2 –Fe2 in the ab-plane is ferromagnetic while the magnetic coupling of Fe1 –Fe1 and Fe2 –Fe2 along the c-axis is anti-ferromagnetic. In other words, each Fe1 layer is coupled in parallel with one near neighbor Fe2 layer and in anti-parallel with the other near neighbor Fe2 layer along the c-axis in this AFM structure. The calculated total energy of AFM structure is higher than that of FM ordering by 12 meV per formula unit of Fe2 P. Comparing with FM structure, the moment at Fe2 site decreases by about 12% (2.11 μ B ) and is
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relatively well localized in the AFM structure (Table 2). Similar to the FM ordering, P atoms have very small induced magnetic moments. However, the moment of Fe1 site is only 0.30 μ B while that in FM structure is 0.83 μ B (Table 2). The results indicate that the moment of Fe1 is metastable and depends on the exchange interaction with its neighboring magnetic atoms. For the FM structure, each Fe1 layer is magnetically coupled in parallel with the two nearest-neighbor-layers of Fe2 along the c-axis and the moment of Fe1 show moderate value. However, for the AFM structure, the inter-layer exchange interaction between Fe1 with the two nearest-neighbor-layers of Fe2 along the c-axis are compensated with each other due to the opposite magnetic coupling orientations. The sharp reduction of Fe1 moment in AFM structure implies that the moment formation at Fe1 is related to the exchange interaction with its neighboring Fe2 atoms. 3.3. Exchange interaction To understand the relationship between exchange interaction and the moment formation at the Fe1 site, the inter-sites exchange parameters have been calculated by a linear-response method [32, 33]. Fig. 4 shows the inter-sites exchange coupling parameters J i j as a function of inter-site distance in the Fe2 P layer compound. As expected, the nearest inter-site exchange interaction is the most important. With increasing inter-site distance, exchange parameters decrease rapidly. The Fe–Fe exchange interaction shows clear anisotropy and varies substantially for the different type of Fe–Fe pairs. In the Fe1 sublattice (Fig. 4(a)), the intra-layer (ab-plane) nearest Fe1 – Fe1 exchange coupling parameter has a very small and positive ab value ( J Fe = 0.02 mRy) while the exchange coupling param1 –Fe1 eter of the inter-layer (c-axis) nearest Fe1 –Fe1 pair has a negac tive value ( J Fe = −0.10 mRy). This implies that a weak fer1 –Fe1 romagnetic ordering will be preferred in the Fe1 –P1 layer while an anti-ferromagnetic ordering between different Fe1 layers will gain lower energy in the Fe1 sublattice. However, in the Fe2 sublattice (Fig. 4(b)), the nearest intra-layer exchange parameter of ab Fe2 –Fe2 has a very large and positive value ( J Fe = 1.29 mRy) 2 –Fe2 while that of Fe2 –Fe2 interlayer exchange interaction is very weak c ( J Fe = 0.08 mRy). The exchange interactions promote a ferro2 –Fe2 magnetic ordering in the Fe2 sublattice. It is clear that the nearest intra-layer exchange parameters of Fe2 –Fe2 pair are much stronger than that of Fe1 –Fe1 pairs. As shown in Fig. 4(c), the nearest inter-layer exchange coupling c constants of the Fe1 –Fe2 pair are positive ( J Fe = 0.66 mRy), 1 –Fe2 which are much larger than the inter-layer exchange interaction in the Fe1 and Fe2 sublattices. Acted by the positive exchange interaction between Fe1 and Fe2 layers, the weak anti-ferromagnetic order in the Fe1 sublattice along c-axis is overtaken, and a ferromagnetic order is established in the Fe2 P layer compound. 3.4. Relationship between Fe moment formation and exchange interaction in Fe2 P The inter-site exchange coupling calculation results support the conclusion that the moment at Fe1 site is metastable and its moment formation and magnetic ordering depend on the magnetic ordering of Fe2 site via the Fe1 –Fe2 inter-layer exchange coupling. In Fe2 P layer compound, the intra-layer exchange interaction of Fe1 –Fe1 is very weak and the intra-layer exchange interaction of Fe2 –Fe2 dominates the magnetic ordering. The strong Fe2 –Fe2 intra-layer exchange interaction makes the formation of a stable ferromagnetic state in the Fe2 layers and the Fe2 site (3g) shows a large and stable magnetic moment. This is confirmed by the calculated magnetic moments at the Fe2 site in the FM and AFM structures (Table 2). However, the moment at the Fe1 site (3 f )
Fig. 4. (Colour online.) Distance dependence of the inter-site exchange interaction parameters for Fe1 –Fe1 pairs (a), Fe2 –Fe2 pairs (b), and Fe1 –Fe2 pairs (c) in Fe2 P compound. a is the lattice constant, given in Table 1.
is metastable and the moment formation and ordering depend on the exchange coupling with the Fe2 layers. For the FM state, the weak negative inter-layer exchange coupling for the Fe1 –Fe1 pairs are overtaken by the strong positive Fe1 –Fe2 inter-layer exchange interaction. The ferromagnetic ordering at the Fe1 layer is formed and the Fe1 site has a moderate moment (0.83 μ B ). For the AFM state (Fig. 3), the inter-layer exchange interaction between Fe1 with the two nearest-neighbor-layers of Fe2 along the c-axis are compensated with each other due to the opposite magnetic coupling
X.B. Liu et al. / Physics Letters A 377 (2013) 731–735
orientations. The weak exchange coupling for the Fe1 –Fe1 pairs is not enough for the large or moderate moment formation in the Fe1 sublattice. So the Fe1 site has only a very small induced moment (0.3 μ B ). Based on the above results, the first order magnetic transition in Fe2 P could be understood as follows. The magnetic state of Fe1 site (3 f ) is metastable and sensitive to the external field, temperature and pressure. As the temperature decreases to around T C , the magnetic state of Fe2 sublattice first changes from a paramagnetic state (PM) to a ferromagnetic state (FM) due to the strong Fe2 –Fe2 exchange interaction. Upon further cooling or upon the action of an external field, the enhanced inter-layer exchange interaction of Fe1 –Fe2 overtakes the weak negative Fe1 –Fe1 inter-layer exchange coupling. The magnetic moment of Fe1 site jumps immediately from zero to a finite value via a meta-magnetic-like transition, where the discontinuous change of the magnetic moment around T C signifies the occurrence of a first order magnetic transition. This explanation conforms to the experimental fact that the magnetic transition around T C in Fe2 P has actually two sequential transitions in a weak external magnetic field: from the paramagnetic state to the intermediate meta-magnetic phase, then to the low temperature ferromagnetic phase [20]. It is also experimentally observed that the two transitions are combined together and display a first order magnetic transition under a strong external field [20]. The experimental results support the idea that one Fe sublattice (Fe2 ) first achieves a ferromagnetic order and the other sublattice (Fe1 ) subsequently reaches a ferromagnetic order with decreasing temperature across T C . This is just the situation for the occurrence of a first order magnetic transition at T C in Fe2 P based compound, contributing to a large magneto-caloric effect. 4. Conclusion Fe2 P has a ferromagnetic ground state and has magnetic moments for Fe1 (3 f ) and Fe2 (3g) of 0.83 and 2.3 μ B , respectively. The exchange interaction varies substantially for the different Fe–Fe pairs in the compound. The intra-layer exchange interaction of Fe2 –Fe2 dominates the total exchange interaction in this compound. The Fe moment at the 3 f site is metastable and sensitive to the inter-site exchange interaction with its magnetic neighbors, which is responsible for the first order magnetic transition and large magneto-caloric effect around T C . Acknowledgements The work at University of Texas at Arlington was partly supported by the DARPA/ARO under grant W911NF-08-1-0249 and ARO under grant W911NF-11-1-0507. The work at McGill University was supported by the Natural and Engineering Research
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Council of Canada and Fonds pour la Formation de Chercheurs et l’Aide à la Recherche, Quebéc. References [1] O. Tegus, E. Brück, K.H.J. Buschow, F.R. de Boer, Nature (London) 415 (2002) 150. [2] E. Brück, J. Phys. D: Appl. Phys. 38 (2005) R381. [3] D.T. Cam Thanh, E. Brück, O. Tegus, J.C.P. Klaase, T.J. Gortenmulder, K.H.J. Buschow, J. Appl. Phys. 99 (2006) 08Q107. [4] A. Yan, K.-H. Müller, L. Schultz, O. Gutfleisch, J. Appl. Phys. 99 (2006) 08K903. [5] W. Dagula, O. Tegus, X.W. Li, L. Song, E. Brück, D.T. Cam Thanh, F.R. de Boer, K.H.J. Buschow, J. Appl. Phys. 99 (2006) 08Q105. [6] X.B. Liu, Z. Altounian, D.H. Ryan, M. Yue, Z.Q. Li, D.M. Liu, J.X. Zhang, J. Appl. Phys. 105 (2009) 07A920. [7] D. Liu, M. Yue, J. Zhang, T.M. McQueen, J.W. Lynn, X. Wang, Y. Chen, J. Li, R.J. Cava, X.B. Liu, Z. Altounian, Q. Huang, Phys. Rev. B 79 (2009) 014435. [8] X.B. Liu, Z. Altounian, J. Appl. Phys. 105 (2009) 07A902. [9] D. Fruchart, F. Allab, M. Balli, D. Gignoux, E.K. Hlil, A. Koumina, N. Skyryabina, J. Tobola, P. Wolfers, R. Zach, Physica A 358 (2005) 123. [10] B. Carlsson, M. Golin, S. Rundqvist, J. Solid State Chem. 8 (57) (1973). [11] H. Fujii, T. Hokabe, T. Kamigaichi, T. Okamoto, J. Phys. Soc. Japan 43 (41) (1977). [12] R. Fruchart, A. Roger, J.P. Senateur, J. Appl. Phys. 40 (1969) 1250. [13] R. Wappling, L. Häggström, T. Eriksson, S. Devanarayanan, E. Karlsson, B. Carlsson, S. Rundqvist, J. Solid State Chem. 13 (1975) 258. [14] O. Beckman, L. Lundgren, in: K.H.J. Buschow (Ed.), Handbook of Magnetic Materials, vol. 6, North-Holland, Amsterdam, 1991, p. 181, and references therein. [15] O. Eriksson, J. Sjöström, B. Johansson, L. Häggström, H.L. Skriver, J. Magn. Magn. Mater. 74 (1988) 347. [16] S. Ishida, S. Asano, J. Ishida, J. Phys. F: Met. Phys. 17 (1987) 475. [17] J. Tobola, M. Bacmann, D. Fruchart, S. Kaprzyk, A. Koumina, S. Niziol, J.-L. Soubeyroux, P. Wolfers, R. Zach, J. Magn. Magn. Mater. 157–158 (1996) 708. [18] D. Scheerlinck, E. Legrand, Solid State Commun. 25 (1978) 181. [19] H. Fujii, S. Komura, T. Takeda, T. Okamoto, Y. Ito, J. Akimisu, J. Phys. Soc. Japan 43 (1977) 41. [20] E.A. Zavadskii, L.I. Medvedeva, A.E. Filippov, J. Magn. Magn. Mater. 43 (1984) 53. [21] H. Fujiwara, K. Kadomatsu, K. Tohma, H. Fujii, T. Okamoto, J. Magn. Magn. Mater. 21 (1980) 80. [22] L.I. Medvedeva, E.A. Zavadskii, J. Magn. Magn. Mater. 111 (1992) 189. [23] E.K. Delczeg-Czirjak, Z. Gercsi, L. Bergqvist, O. Eriksson, L. Szunyogh, P. Nordblad, B. Johansson, L. Vito, Phys. Rev. B 85 (2012) 224435. [24] H. Yamada, K. Terao, Phase Trans. 75 (2002) 231. [25] G. Kresse, J. Hafner, Phys. Rev. B 47 (1993) 558. [26] G. Kresse, J. Furthmuller, Comput. Mater. Sci. 6 (1996) 15. [27] P.E. Blöchl, Phys. Rev. B 50 (1994) 17953. [28] G. Kresse, D. Joubert, Phys. Rev. B 59 (1999) 1758. [29] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [30] O. Jepsen, O.K. Andersen, Solid State Commun. 9 (1971) 1763. [31] P.E. Blöchl, O. Jepsen, O.K. Andersen, Phys. Rev. B 49 (1994) 16223. [32] A.I. Lichtenstein, M.I. Katsnelson, V.P. Antropov, V.A. Gubanov, J. Magn. Magn. Mater. 67 (1987) 65. [33] M. van Schilfgaarde, V.P. Antropov, J. Appl. Phys. 85 (1999) 4827. [34] O.K. Andersen, Phys. Rev. B 12 (1975) 3060. [35] P. Villars, L.D. Calvert, Pearson’s Handbook of Crystallographic Data for Intermetallic Phases, vol. 3, American Society for Metals, 1986, p. 2211. [36] E. Koch, W. Fischer, Z. Kristallogr. 211 (1996) 251. [37] L. Severin, L. Häggström, L. Nordström, Y. Andersson, B. Johansson, J. Phys.: Condens. Matter 7 (1995) 185.
Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Fe magnetic moment formation and exchange interaction in Fe2 P: A first-principles study X.B. Liu a,∗ , J. Ping Liu a , Qiming Zhang a , Z. Altounian b a b
Department of Physics, University of Texas at Arlington, Arlington, TX 76019, USA Center for the Physics of Materials and Department of Physics, McGill University, 3600 University Street, Montreal, Quebec, H3A 2T8, Canada
a r t i c l e
i n f o
Article history: Received 16 October 2012 Received in revised form 11 January 2013 Accepted 12 January 2013 Available online 18 January 2013 Communicated by R. Wu Keywords: Magnetic moment Exchange interaction Density functional
a b s t r a c t Electronic structure and magnetic properties of Fe2 P have been studied by a first-principles density functional theory calculation. The ground state is ferromagnetic and the calculated magnetic moments for Fe1 (3 f ) and Fe2 (3g) are 0.83 and 2.30 μ B , respectively. The nearest neighbor inter-site magnetic exchange coupling parameter at the Fe1 layer (0.02 mRy) is much smaller than that at the Fe2 layer (1.29 mRy). The Fe moment at the 3 f site is metastable and sensitive to the inter-site exchange interaction with its magnetic neighbors, which is responsible for the first order magnetic transition and large magneto-caloric effect around T C . © 2013 Elsevier B.V. All rights reserved.
1. Introduction Recently, more research attentions are paid to the Fe (Mn) based compound with the hexagonal Fe2 P-type structure since the discovery of the giant magneto-caloric effect (MCE) in MnFeP1−x Asx with a Fe2 P-type structure [1]. Further, large or moderate MCEs are also reported in the other Fe2 P-type compounds such as MnFeP1−x (Ge, Si)x , and (Fe, M)2 P with M = Ni, Ru, Rh, Pd, and Pt [2–9]. The large MCEs originate from the strong first order magnetic transition at T C and the field induced meta-magnetic transition above T C , which is related to the competing exchange coupling in the compounds. The knowledge on the structure, magnetic states and exchange interaction of the prototype compound Fe2 P is very helpful in understanding the magnetic transition and magneto-caloric effect in these compounds. In the past 40 years, many experimental and theoretical research works have been done on the hexagonal Fe2 P [10–17]. The hexagonal Fe2 P is a ferromagnet with a strong c-axis magnetocrystalline anisotropy and a Curie temperature T C = 217 K [10]. Fe2 P shows generally a first order magnetic phase transition from paramagnetic to ferromagnetic states at T C with a discontinuous change of lattice constants, where a increases while c decreases [13,14]. Fe atoms are positioned at two inequivalent sites: a tetragonal site Fe1 (3 f ) and a pyramidal site Fe2 (3g). Neutron diffraction studies [18] on powder sample at 77 K show Fe magnetic moments of 0.69 μ B and 2.31 μ B at the 3 f and 3g sites, respec-
*
Corresponding author. E-mail address: [email protected] (X.B. Liu).
0375-9601/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2013.01.019
tively. However, polarized neutron diffraction study on a single crystal sample gave Fe moments of 0.92 μ B and 1.7 μ B for the 3 f and 3g sites, respectively, at the same temperature [19]. Temperature and magnetic field dependence of magnetization results indicate that the first order magnetic transition could be resolved into two sequential transitions in a weak external field: from the paramagnetic state to the intermediate meta-magnetic phase, then to the low temperature ferromagnetic phase [20]. The magnetic states of Fe2 P are sensitive to the external pressure, Fe vacancy, and impurities. The external pressure will reduce the lattice volume and the axial ratio a/c, and drive the magnetic structure into non-collinear and anti-ferromagnetic states [21]. Similarly, the lattice constants decrease slightly while T C decreases rapidly and the magnetic state transforms into a meta-magnetic and/or anti-ferromagnetic state with increasing x in Fe2−x P [22]. On the other hand, a very small amount of impurity such as B, As, and Si will increase the a parameter while decreasing the c parameter and improve substantially T C in Fe2 P compound [14]. The change of magnetic ordering due to non-metallic impurities are mainly ascribed to the variation of lattice constant [14,23]. Spin polarized electronic structure calculations have been performed for Fe2 P using Korringa–Kohn–Rostoker (KKR) and linear muffin-tin orbital (LMTO) method by several groups [15–17]. The calculated Fe moments are 0.8–0.96 μ B for Fe1 (3 f ) and 1.9–2.4 μ B for Fe2 (3g), in fair agreement with the experimental values. Discussed by means of the Landau phenomenological theory [24], the Fe atoms at the 3 f site show meta-magnetic behavior, which is responsible for the first order magnetic transition. In this work, the electronic structure, Fe magnetic moment and exchange interaction have been investigated in Fe2 P within the
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Table 1 Calculated and experimental [35] crystallographic data in Fe2 P. Atomic position: Fe1 (3 f ): (x3 f , 0, 0) and Fe2 (3g): (x3g , 0, 0.5). a (Å)
c (Å)
x3 f
x3g
5.868 5.813
3.458 3.425
0.257 0.257
0.595 0.592
experiment calculation
framework of a density functional theory (DFT). It is found that the Fe magnetic moment at the 3 f site is metastable and depends on the exchange interaction with the neighboring magnetic atoms. The magnetic exchange interactions vary substantially among the different Fe–Fe pairs and the intra-layer magnetic exchange interaction is much stronger in the Fe2 (3g) layer than that in the Fe1 (3 f ) layer. 2. Computational methods We performed first-principles electronic structure calculations in the framework of density functional theory (DFT). The Vienna Ab-initio Simulation Package (VASP) [25,26] in the projector augmented wave (PAW) framework [27,28] was employed to perform DFT calculations using the generalized gradient approximation of Perdew–Burke–Ernzerhof (PBE) [29] for the exchange correlation functional. In all calculations the structural degrees of freedom are fully relaxed on a gamma centered k-grids of 21 × 21 × 15 k-mesh. The k-space integrations were performed with the tetrahedron method [30,31]. The inter-sites exchange coupling parameters J i j of the Heisenberg model have been calculated using a linear-response method [32] and implemented in LM code [33]. We adopt a Green’s function technique combined with the linear muffin-tin orbital (LMTO) method with atomic sphere approximation (ASA) in the calculation [33,34]. In this approach, exchange interactions are calculated as the response to small-angle fluctuations of the spin orientations. The band structure calculation provides the one-electron Green function. The energy integrals over the occupied part of the valence band are expressed as integrals over an energy variable along a closed path starting and ending at the Fermi energy. The integrals are numerically evaluated using the Gaussian quadrature method. Additional calculation details have been reported previously [8]. 3. Results and discussion 3.1. Density of states The hexagonal Fe2 P has a layer structure with a space group ¯ of P 62m. The atoms Fe1 , Fe2 , P1 , and P2 are distributed at the 3 f (0.257, 0, 0), 3g (0.595, 00.15), 2c (0.333, 0.667, 0) and 1b (0, 0, 0.5) sites, respectively [35]. The compound is consisted of alternative layers of Fe1 –P1 , and Fe2 –P2 layers along c-axis (Fig. 1). As shown in Fig. 1(a), Fe2 is centered at a distorted pyramid with four P1 and one P2 at the five vertices. Fe1 is centered at a distorted tetrahedron with two P1 and two P2 at the four vertices (Fig. 1(b), the unit cell is doubled along c-axis). As shown in Fig. 1(c), the unit cell length is doubled along the two a-axis, each Fe1 has two equivalent nearest neighbors (NNs) of P1 and two equivalent NNs of Fe1 in the Fe1 –P1 layer while each Fe2 has one nearest neighbor of P2 and four equivalent NNs of Fe2 in the Fe2 –P2 layer. The different atomic environments of Fe1 and Fe2 will affect their magnetic properties as shown below. The geometrical structure of Fe2 P has been fully relaxed in the calculations. The calculated structural data are in good agreement with the experimental values and the difference is within 1% (Table 1). Fig. 2 displays the total and partial density of states (DOS) of hexagonal Fe2 P compound. Fe2 P shows a typical metallic behavior
Fig. 1. (Colour online.) (a) Unit cell of Fe2 P, showing the Fe2 –P chemical bonds; (b) Unit cell doubled along c-axis, showing the Fe1 –P chemical bonds; (c) Unit cell doubled along a- and b-axis, showing the chemical bonds in the layers of Fe1 –P1 and Fe2 –P2 . The small (red), further small (blue), medium (cyan), and large (green) sized spheres are atoms P1 , P2 , Fe1 , and Fe2 , respectively. Table 2 Calculated and experimental magnetic moments M (μ B ) in Fe2 P. Phase
M Fe1
M Fe2
M1
M P2
Method
Reference
FM FM FM FM FM AFM FM AFM
0.69 0.92 0.92 0.89 0.83 0.30 0.95 0.06
2.31 1.7 2.03 2.24 2.30 2.11 2.00 1.77
– –
– –
−0.06 −0.07 −0.04 −0.001 −0.07 −0.02
−0.06 −0.06 −0.05 −0.02 −0.06 −0.03
experiment experiment LMTO KKR VASP VASP LMTO LMTO
[17] [18] [15] [16] this work this work this work this work
in both spin minority and spin majority components. A large energy gap (about 4 eV) separated the DOS into a lower part (around −12 eV) and a higher part ranging from about −7.5 eV to the Fermi level (E f = 0). The contribution to the lower part of the DOS is from the 3s-like states of the P atoms. The main contribution to the DOS at Fermi Level (E f = 0) is from Fe 3d electrons. Clearly, the exchange splitting of DOS for Fe1 is smaller than that of Fe2 , consistent with a smaller spin moment of Fe1 (Table 2). 3.2. Fe magnetic moments For layer compound Fe2 P, it has a ferromagnetic (FM) ground state and the calculated magnetic moments are 0.83 μ B and 2.30 μ B for Fe1 and Fe2 , respectively, via the VASP code (Table 2). The total moments is 3.08 μ B per formula unit due to the contribution of the small moments induced at P atoms. Similarly, the LMTO-ASA code gives the moments of 0.95 μ B and 2.00 μ B for Fe1 and Fe2 , respectively, and a total magnetic moment of 2.94 μ B per formula unit. Although the calculated total magnetic moment per
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Table 3 Atomic Wigner–Seitz cell Volume (WSV) and inter-atomic distances d in Fe2 P compounds. Fe1
Fe2
Atoms
d (Å)
Atoms
d (Å)
2 2 2 2 4
2.218 2.297 2.621 2.626 2.707
1 4 2 4 4
2.376 2.484 2.626 2.707 3.088
11.0
WSV (Å3 )
P1 P2 Fe1 Fe2 Fe2
WSV (Å3 )
P2 P1 Fe1 Fe1 Fe2
12.6
Fig. 3. (Colour online.) The virtual anti-ferromagnetic magnetic unit cell for Fe2 P. The small (red), medium (white), and large (green) sized spheres are atoms P, Fe1 , and Fe2 , respectively.
Fig. 2. (Colour online.) Calculated spin-projected (a) total density of states (DOS) and partial DOS at Fe1 site (b) and that at Fe2 site (c) in Fe2 P compound. The blue, red and green lines are for the contribution of Fe 3d, 4p and 4s-like states, respectively. The Fermi level is at E = 0.
formula unit (3.0 μ B or so) is almost same for the two methods, the different moment values at different Fe sites are expected due to the different atomic potential treatment in the PAW and LMTOASA methods. It is expected that the magnetic moment calculated by LMTO-ASA has a lower precision because the sphere approximation of potential shape and the overlapping among different atomic spheres will lower the potential precision. These calculated magnetic moments are in good agreement with the previous experimental and calculation results (Table 2).
The difference in the magnetic moments of Fe1 and Fe2 are related to their different local environment in the unit cell. The Wigner–Seitz cell Volume (WSV) calculations was performed following the method given by Koch and Fisher [36] and the results are listed in Table 3. Fe1 with a WSV volume of about 11.0 Å3 has four nearest P neighbors at the corners of a distorted tetrahedron with an average inter-atomic distance of about 2.25 Å, and eight Fe near neighbors with the atomic separation of about 2.6–2.7 Å. On the other hand, Fe2 with a WSV volume of about 12.6 Å3 has five P neighbors at the corners of a distorted pyramid with an average inter-atomic distance of about 2.46 Å, and has eight Fe neighbors with atomic distances varying from 2.6 to 3.1 Å. The larger WSV and larger atomic separation distances with Fe and P neighbors are responsible for the larger magnetic moment of Fe2 . Severin et al. [37] reported that the Fe atomic magnetic moment increases almost linearly with the average inter-atomic distance of Fe–P in Fe2 P, consistent to our present results. To gain more insight on the Fe moment formation, the total energy and magnetic moments for a virtual anti-ferromagnetic (AFM) structure have been calculated. For this layer compound, we, here, consider only a specific AFM structure (Fig. 3). In this virtual AFM structure, the unit cell is doubled along the c-axis and the Fe moments are distributed as Fe1 (+)Fe2 (−)Fe1 (−)Fe2 (+) along the c-axis. The intra-layer magnetic coupling of Fe1 –Fe1 and Fe2 –Fe2 in the ab-plane is ferromagnetic while the magnetic coupling of Fe1 –Fe1 and Fe2 –Fe2 along the c-axis is anti-ferromagnetic. In other words, each Fe1 layer is coupled in parallel with one near neighbor Fe2 layer and in anti-parallel with the other near neighbor Fe2 layer along the c-axis in this AFM structure. The calculated total energy of AFM structure is higher than that of FM ordering by 12 meV per formula unit of Fe2 P. Comparing with FM structure, the moment at Fe2 site decreases by about 12% (2.11 μ B ) and is
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relatively well localized in the AFM structure (Table 2). Similar to the FM ordering, P atoms have very small induced magnetic moments. However, the moment of Fe1 site is only 0.30 μ B while that in FM structure is 0.83 μ B (Table 2). The results indicate that the moment of Fe1 is metastable and depends on the exchange interaction with its neighboring magnetic atoms. For the FM structure, each Fe1 layer is magnetically coupled in parallel with the two nearest-neighbor-layers of Fe2 along the c-axis and the moment of Fe1 show moderate value. However, for the AFM structure, the inter-layer exchange interaction between Fe1 with the two nearest-neighbor-layers of Fe2 along the c-axis are compensated with each other due to the opposite magnetic coupling orientations. The sharp reduction of Fe1 moment in AFM structure implies that the moment formation at Fe1 is related to the exchange interaction with its neighboring Fe2 atoms. 3.3. Exchange interaction To understand the relationship between exchange interaction and the moment formation at the Fe1 site, the inter-sites exchange parameters have been calculated by a linear-response method [32, 33]. Fig. 4 shows the inter-sites exchange coupling parameters J i j as a function of inter-site distance in the Fe2 P layer compound. As expected, the nearest inter-site exchange interaction is the most important. With increasing inter-site distance, exchange parameters decrease rapidly. The Fe–Fe exchange interaction shows clear anisotropy and varies substantially for the different type of Fe–Fe pairs. In the Fe1 sublattice (Fig. 4(a)), the intra-layer (ab-plane) nearest Fe1 – Fe1 exchange coupling parameter has a very small and positive ab value ( J Fe = 0.02 mRy) while the exchange coupling param1 –Fe1 eter of the inter-layer (c-axis) nearest Fe1 –Fe1 pair has a negac tive value ( J Fe = −0.10 mRy). This implies that a weak fer1 –Fe1 romagnetic ordering will be preferred in the Fe1 –P1 layer while an anti-ferromagnetic ordering between different Fe1 layers will gain lower energy in the Fe1 sublattice. However, in the Fe2 sublattice (Fig. 4(b)), the nearest intra-layer exchange parameter of ab Fe2 –Fe2 has a very large and positive value ( J Fe = 1.29 mRy) 2 –Fe2 while that of Fe2 –Fe2 interlayer exchange interaction is very weak c ( J Fe = 0.08 mRy). The exchange interactions promote a ferro2 –Fe2 magnetic ordering in the Fe2 sublattice. It is clear that the nearest intra-layer exchange parameters of Fe2 –Fe2 pair are much stronger than that of Fe1 –Fe1 pairs. As shown in Fig. 4(c), the nearest inter-layer exchange coupling c constants of the Fe1 –Fe2 pair are positive ( J Fe = 0.66 mRy), 1 –Fe2 which are much larger than the inter-layer exchange interaction in the Fe1 and Fe2 sublattices. Acted by the positive exchange interaction between Fe1 and Fe2 layers, the weak anti-ferromagnetic order in the Fe1 sublattice along c-axis is overtaken, and a ferromagnetic order is established in the Fe2 P layer compound. 3.4. Relationship between Fe moment formation and exchange interaction in Fe2 P The inter-site exchange coupling calculation results support the conclusion that the moment at Fe1 site is metastable and its moment formation and magnetic ordering depend on the magnetic ordering of Fe2 site via the Fe1 –Fe2 inter-layer exchange coupling. In Fe2 P layer compound, the intra-layer exchange interaction of Fe1 –Fe1 is very weak and the intra-layer exchange interaction of Fe2 –Fe2 dominates the magnetic ordering. The strong Fe2 –Fe2 intra-layer exchange interaction makes the formation of a stable ferromagnetic state in the Fe2 layers and the Fe2 site (3g) shows a large and stable magnetic moment. This is confirmed by the calculated magnetic moments at the Fe2 site in the FM and AFM structures (Table 2). However, the moment at the Fe1 site (3 f )
Fig. 4. (Colour online.) Distance dependence of the inter-site exchange interaction parameters for Fe1 –Fe1 pairs (a), Fe2 –Fe2 pairs (b), and Fe1 –Fe2 pairs (c) in Fe2 P compound. a is the lattice constant, given in Table 1.
is metastable and the moment formation and ordering depend on the exchange coupling with the Fe2 layers. For the FM state, the weak negative inter-layer exchange coupling for the Fe1 –Fe1 pairs are overtaken by the strong positive Fe1 –Fe2 inter-layer exchange interaction. The ferromagnetic ordering at the Fe1 layer is formed and the Fe1 site has a moderate moment (0.83 μ B ). For the AFM state (Fig. 3), the inter-layer exchange interaction between Fe1 with the two nearest-neighbor-layers of Fe2 along the c-axis are compensated with each other due to the opposite magnetic coupling
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orientations. The weak exchange coupling for the Fe1 –Fe1 pairs is not enough for the large or moderate moment formation in the Fe1 sublattice. So the Fe1 site has only a very small induced moment (0.3 μ B ). Based on the above results, the first order magnetic transition in Fe2 P could be understood as follows. The magnetic state of Fe1 site (3 f ) is metastable and sensitive to the external field, temperature and pressure. As the temperature decreases to around T C , the magnetic state of Fe2 sublattice first changes from a paramagnetic state (PM) to a ferromagnetic state (FM) due to the strong Fe2 –Fe2 exchange interaction. Upon further cooling or upon the action of an external field, the enhanced inter-layer exchange interaction of Fe1 –Fe2 overtakes the weak negative Fe1 –Fe1 inter-layer exchange coupling. The magnetic moment of Fe1 site jumps immediately from zero to a finite value via a meta-magnetic-like transition, where the discontinuous change of the magnetic moment around T C signifies the occurrence of a first order magnetic transition. This explanation conforms to the experimental fact that the magnetic transition around T C in Fe2 P has actually two sequential transitions in a weak external magnetic field: from the paramagnetic state to the intermediate meta-magnetic phase, then to the low temperature ferromagnetic phase [20]. It is also experimentally observed that the two transitions are combined together and display a first order magnetic transition under a strong external field [20]. The experimental results support the idea that one Fe sublattice (Fe2 ) first achieves a ferromagnetic order and the other sublattice (Fe1 ) subsequently reaches a ferromagnetic order with decreasing temperature across T C . This is just the situation for the occurrence of a first order magnetic transition at T C in Fe2 P based compound, contributing to a large magneto-caloric effect. 4. Conclusion Fe2 P has a ferromagnetic ground state and has magnetic moments for Fe1 (3 f ) and Fe2 (3g) of 0.83 and 2.3 μ B , respectively. The exchange interaction varies substantially for the different Fe–Fe pairs in the compound. The intra-layer exchange interaction of Fe2 –Fe2 dominates the total exchange interaction in this compound. The Fe moment at the 3 f site is metastable and sensitive to the inter-site exchange interaction with its magnetic neighbors, which is responsible for the first order magnetic transition and large magneto-caloric effect around T C . Acknowledgements The work at University of Texas at Arlington was partly supported by the DARPA/ARO under grant W911NF-08-1-0249 and ARO under grant W911NF-11-1-0507. The work at McGill University was supported by the Natural and Engineering Research
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