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2 Physics Letters A 377 (2013) Contents lists available at SciVerse ScienceDirect Physics Letters A Fe magnetic moment formation and exchange interaction in Fe 2 P: A first-principles study X.B. Liu a,, J. Ping Liu a, Qiming Zhang a,z.altounian b a Department of Physics, University of Texas at Arlington, Arlington, TX 76019, USA b Center for the Physics of Materials and Department of Physics, McGill University, 3600 University Street, Montreal, Quebec, H3A 2T8, Canada article info abstract 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 Electronic structure and magnetic properties of Fe 2 P have been studied by a first-principles density functional theory calculation. The ground state is ferromagnetic and the calculated magnetic moments for Fe 1 (3 f )andfe 2 (3g) are0.83and2.30μ B, respectively. The nearest neighbor inter-site magnetic exchange coupling parameter at the Fe 1 layer (0.02 mry) is much smaller than that at the Fe 2 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 Elsevier B.V. All rights reserved. 1. Introduction Recently, more research attentions are paid to the Fe (Mn) based compound with the hexagonal Fe 2 P-type structure since the discovery of the giant magneto-caloric effect (MCE) in MnFeP 1 x As x with a Fe 2 P-type structure [1]. Further, large or moderate MCEs are also reported in the other Fe 2 P-type compounds such as MnFeP 1 x (Ge, Si) x, and (Fe, M) 2 PwithM= 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 Fe 2 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 Fe 2 P [10 17]. The hexagonal Fe 2 P is a ferromagnet with a strong c-axis magnetocrystalline anisotropy and a Curie temperature T C = 217 K [10]. Fe 2 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 Fe 1 (3 f ) and a pyramidal site Fe 2 (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. address: liuxubo@uta.edu (X.B. Liu). 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 tothelowtemperatureferromagneticphase[20]. The magnetic states of Fe 2 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 Fe 2 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 Fe 2 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 Fe 2 P using Korringa Kohn Rostoker (KKR) and linear muffin-tin orbital (LMTO) method by several groups [15 17]. The calculated Fe moments are μ B for Fe 1 (3 f ) and μ B for Fe 2 (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 Fe 2 P within the /$ see front matter 2013 Elsevier B.V. All rights reserved.

3 732 X.B. Liu et al. / Physics Letters A 377 (2013) Table 1 Calculated and experimental [35] crystallographic data in Fe 2 P. Atomic position: Fe 1 (3 f ): (x 3 f, 0, 0) and Fe 2 (3g): (x 3g, 0, 0.5). a (Å) c (Å) x 3 f x 3g 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 Fe 2 (3g) layer than that in the Fe 1 (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 k-mesh. The k-space integrations were performed with the tetrahedron method [30,31]. The inter-sites exchange coupling parameters J ij 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 Fe 2 P has a layer structure with a space group of P 62m. The atoms Fe1, Fe 2, P 1, and P 2 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 Fe 1 P 1, and Fe 2 P 2 layers along c-axis (Fig. 1). As shown in Fig. 1(a), Fe 2 is centered at a distorted pyramid with four P 1 and one P 2 at the five vertices. Fe 1 is centered at a distorted tetrahedron with two P 1 and two P 2 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 Fe 1 has two equivalent nearest neighbors (NNs) of P 1 and two equivalent NNs of Fe 1 in the Fe 1 P 1 layer while each Fe 2 has one nearest neighbor of P 2 and four equivalent NNs of Fe 2 in the Fe 2 P 2 layer. The different atomic environments of Fe 1 and Fe 2 will affect their magnetic properties as shown below. The geometrical structure of Fe 2 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 Fe 2 P compound. Fe 2 P shows a typical metallic behavior Fig. 1. (Colour online.) (a) Unit cell of Fe 2 P, showing the Fe 2 P chemical bonds; (b) Unit cell doubled along c-axis, showing the Fe 1 P chemical bonds; (c) Unit cell doubled along a- andb-axis, showing the chemical bonds in the layers of Fe 1 P 1 and Fe 2 P 2. The small (red), further small (blue), medium (cyan), and large (green) sized spheres are atoms P 1,P 2,Fe 1,andFe 2, respectively. Table 2 Calculated and experimental magnetic moments M (μ B )infe 2 P. Phase M Fe1 M Fe2 M 1 M P2 Method Reference FM experiment [17] FM experiment [18] FM LMTO [15] FM KKR [16] FM VASP this work AFM VASP this work FM LMTO this work AFM LMTO 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 DOSisfromthe3s-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 Fe 1 is smaller than that of Fe 2, consistent with a smaller spin moment of Fe 1 (Table 2) Fe magnetic moments For layer compound Fe 2 P, it has a ferromagnetic (FM) ground state and the calculated magnetic moments are 0.83 μ B and 2.30 μ B for Fe 1 and Fe 2, respectively, via thevaspcode(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 Fe 1 and Fe 2, respectively, and a total magnetic moment of 2.94 μ B per formula unit. Although the calculated total magnetic moment per

4 X.B. Liu et al. / Physics Letters A 377 (2013) Table 3 Atomic Wigner Seitz cell Volume (WSV) and inter-atomic distances d in Fe 2 Pcompounds. Fe 1 Fe 2 Atoms d (Å) Atoms d (Å) 2P P P P Fe Fe Fe Fe Fe Fe WSV (Å 3 ) 11.0 WSV (Å 3 ) 12.6 Fig. 3. (Colour online.) The virtual anti-ferromagnetic magnetic unit cell for Fe 2 P. The small (red), medium (white), and large (green) sized spheres are atoms P, Fe 1, and Fe 2, respectively. Fig. 2. (Colour online.) Calculated spin-projected (a) total density of states (DOS) and partial DOS at Fe 1 site (b) and that at Fe 2 site (c) in Fe 2 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 (3.0μ 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 LMTO- ASA 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 Fe 1 and Fe 2 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. Fe 1 with a WSV volume of about 11.0 Å 3 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 Å. On the other hand, Fe 2 with a WSV volume of about 12.6 Å 3 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 Fe 2. Severin et al. [37] reported that the Fe atomic magnetic moment increases almost linearly with the average inter-atomic distance of Fe P in Fe 2 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 Fe 1 (+)Fe 2 ( )Fe 1 ( )Fe 2 (+) alongthe c-axis. The intra-layer magnetic coupling of Fe 1 Fe 1 and Fe 2 Fe 2 in the ab-plane is ferromagnetic while the magnetic coupling of Fe 1 Fe 1 and Fe 2 Fe 2 along the c-axis is anti-ferromagnetic. In other words, each Fe 1 layer is coupled in parallel with one near neighbor Fe 2 layer and in anti-parallel with the other near neighbor Fe 2 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 Fe 2 P. Comparing with FM structure, the moment at Fe 2 site decreases by about 12% (2.11 μ B ) and is

5 734 X.B. Liu et al. / Physics Letters A 377 (2013) 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 Fe 1 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 Fe 1 is metastable and depends on the exchange interaction with its neighboring magnetic atoms. For the FM structure, each Fe 1 layer is magnetically coupled in parallel with the two nearest-neighbor-layers of Fe 2 along the c-axis and the moment of Fe 1 show moderate value. However, for the AFM structure, the inter-layer exchange interaction between Fe 1 with the two nearest-neighbor-layers of Fe 2 along the c-axis are compensated with each other due to the opposite magnetic coupling orientations. The sharp reduction of Fe 1 moment in AFM structure implies that the moment formation at Fe 1 is related to the exchange interaction with its neighboring Fe 2 atoms Exchange interaction To understand the relationship between exchange interaction and the moment formation at the Fe 1 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 ij as a function of inter-site distance in the Fe 2 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 Fe 1 sublattice (Fig. 4(a)), the intra-layer (ab-plane) nearest Fe 1 Fe 1 exchange coupling parameter has a very small and positive value ( J ab Fe 1 Fe 1 = 0.02 mry) while the exchange coupling parameter of the inter-layer (c-axis) nearest Fe 1 Fe 1 pair has a negative value ( J c Fe 1 Fe = 0.10 mry). This implies that a weak ferromagnetic ordering will be preferred in the Fe 1 P 1 layer while 1 layers will an anti-ferromagnetic ordering between different Fe 1 gain lower energy in the Fe 1 sublattice. However, in the Fe 2 sublattice (Fig. 4(b)), the nearest intra-layer exchange parameter of Fe 2 Fe 2 has a very large and positive value ( J ab Fe 2 Fe 2 = 1.29 mry) while that of Fe 2 Fe 2 interlayer exchange interaction is very weak ( J c Fe 2 Fe 2 = 0.08 mry). The exchange interactions promote a ferromagnetic ordering in the Fe 2 sublattice. It is clear that the nearest intra-layer exchange parameters of Fe 2 Fe 2 pair are much stronger than that of Fe 1 Fe 1 pairs. As shown in Fig. 4(c), the nearest inter-layer exchange coupling constants of the Fe 1 Fe 2 pair are positive ( J c Fe 1 Fe 2 = 0.66 mry), which are much larger than the inter-layer exchange interaction in the Fe 1 and Fe 2 sublattices. Acted by the positive exchange interaction between Fe 1 and Fe 2 layers, the weak anti-ferromagnetic order in the Fe 1 sublattice along c-axis is overtaken, and a ferromagnetic order is established in the Fe 2 P layer compound Relationship between Fe moment formation and exchange interaction in Fe 2 P The inter-site exchange coupling calculation results support the conclusion that the moment at Fe 1 site is metastable and its moment formation and magnetic ordering depend on the magnetic ordering of Fe 2 site via the Fe 1 Fe 2 inter-layer exchange coupling. In Fe 2 P layer compound, the intra-layer exchange interaction of Fe 1 Fe 1 is very weak and the intra-layer exchange interaction of Fe 2 Fe 2 dominates the magnetic ordering. The strong Fe 2 Fe 2 intra-layer exchange interaction makes the formation of a stable ferromagnetic state in the Fe 2 layers and the Fe 2 site (3g) showsa large and stable magnetic moment. This is confirmed by the calculated magnetic moments at the Fe 2 site in the FM and AFM structures (Table 2). However, the moment at the Fe 1 site (3 f ) Fig. 4. (Colour online.) Distance dependence of the inter-site exchange interaction parameters for Fe 1 Fe 1 pairs (a), Fe 2 Fe 2 pairs (b), and Fe 1 Fe 2 pairs (c) in Fe 2 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 Fe 2 layers. For the FM state, the weak negative inter-layer exchange coupling for the Fe 1 Fe 1 pairs areovertakenbythestrongpositivefe 1 Fe 2 inter-layer exchange interaction. The ferromagnetic ordering at the Fe 1 layer is formed and the Fe 1 site has a moderate moment (0.83 μ B ). For the AFM state (Fig. 3), the inter-layer exchange interaction between Fe 1 with the two nearest-neighbor-layers of Fe 2 along the c-axis are compensated with each other due to the opposite magnetic coupling

6 X.B. Liu et al. / Physics Letters A 377 (2013) orientations. The weak exchange coupling for the Fe 1 Fe 1 pairs is not enough for the large or moderate moment formation in the Fe 1 sublattice. So the Fe 1 site has only a very small induced moment (0.3μ B ). Based on the above results, the first order magnetic transition in Fe 2 P could be understood as follows. The magnetic state of Fe 1 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 Fe 2 sublattice first changes from a paramagnetic state (PM) to a ferromagnetic state (FM) due to the strong Fe 2 Fe 2 exchange interaction. Upon further cooling or upon the action of an external field, the enhanced inter-layer exchange interaction of Fe 1 Fe 2 overtakes the weak negative Fe 1 Fe 1 inter-layer exchange coupling. The magnetic moment of Fe 1 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 Fe 2 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 (Fe 2 ) first achieves a ferromagnetic order and the other sublattice (Fe 1 ) 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 Fe 2 Pbased compound, contributing to a large magneto-caloric effect. 4. Conclusion Fe 2 P has a ferromagnetic ground state and has magnetic moments for Fe 1 (3 f ) and Fe 2 (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 Fe 2 Fe 2 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 and ARO under grant W911NF The work at McGill University was supported by the Natural and Engineering Research 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. 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