Controlling the magnetocrystalline anisotropy of ε-Fe2O3

The magnetocrystalline anisotropy of pristine and Co-substituted e-Fe2O3 is investigated by density functional calculations. The epsilon-iron oxide is the only polymorph of Fe2O3 magnetoelectric in its antiferromagnetic ground states other crystalline forms being α-Fe2O3 (hematite), β-Fe2O3, and γ-Fe2O3 (maghemite). The magnetizations of the four iron sublattices are antiferromagnetically aligned with slightly different magnetic moments resulting in a ferrimagnetic structure. Compared to the naturally occurring hematite and maghemite, bulk e-Fe2O3 is difficult to prepare, but e-Fe2O3 nanomaterials of different geometries and feature sizes have been fabricated. A coercivity of 20 kOe [2 T] was reported in nanocomposites of e-Fe2O3, and an upper bound for the magnetic anisotropy constant K at a low temperature of e-Fe2O3 is previously measured to be 0.1 MJ/m3. In the Co-substituted oxides, one octahedral or tetrahedral Fe atom per unit cell has been replaced by Co. The cobalt substitution substantially enhances magnetization and anisotropy.The magnetocrystalline anisotropy of pristine and Co-substituted e-Fe2O3 is investigated by density functional calculations. The epsilon-iron oxide is the only polymorph of Fe2O3 magnetoelectric in its antiferromagnetic ground states other crystalline forms being α-Fe2O3 (hematite), β-Fe2O3, and γ-Fe2O3 (maghemite). The magnetizations of the four iron sublattices are antiferromagnetically aligned with slightly different magnetic moments resulting in a ferrimagnetic structure. Compared to the naturally occurring hematite and maghemite, bulk e-Fe2O3 is difficult to prepare, but e-Fe2O3 nanomaterials of different geometries and feature sizes have been fabricated. A coercivity of 20 kOe [2 T] was reported in nanocomposites of e-Fe2O3, and an upper bound for the magnetic anisotropy constant K at a low temperature of e-Fe2O3 is previously measured to be 0.1 MJ/m3. In the Co-substituted oxides, one octahedral or tetrahedral Fe atom per unit cell has been replaced by Co. The cobalt substitution substantially enha...


I. INTRODUCTION
Iron sesquioxide, Fe 2 O 3 , exists in form of several polymorphs: the common α-Fe 2 O 3 (hematite), γ-Fe 2 O 3 (maghemite) and the rare polymorphs β-Fe 2 O 3 and ε-Fe 2 O 3 . 1 Epsilon-Fe 2 O 3 was first reported in 1934 by Forestier and Guiot-Guillain. Later, Schrader and Büttner 2 in 1963 and Trautmann and Forestier in 1965 studied its magnetic properties, especially its anisotropy. 3 ε-Fe 2 O 3 has been naturally found in the ancient Chinese pottery as patterns on the pots, 4 in archeological sites around Europe, 5,6 and very recently in young basaltic rocks. 7 Very recently, the mineral Luogufengite 8 has been identified by Xu et al., 7 as being Al-containing ε-Fe 2 O 3 . The laboratory-synthesized ε-Fe 2 O 3 and the mineral have the same structure and magnetic properties. The laboratory-prepared sample and the natural mineral have the lattice parameters as a = 5.095, b = 8.789 and c = 9.437 Å, 9 and a = 5.0647, b = 8.7131, c = 9.3842 Å, 7 respectively.
The crystal structure of ε-Fe 2 O 3 is orthorhombic, has the space group Pna2 1, and contains 8 formula units per unit cell. 9 Figure 1 shows that the unit cell contains four different Fe sites, namely two distorted octahedral sites (Fe A and Fe B ), a regular octahedral site (Fe C ) and a regular tetrahedral site (Fe D ). The interatomic exchange interaction is of A-type antiferromagnetic, with the spin arrangement of β, α, α, β for the Fe A , Fe B , Fe C , Fe D atoms, respectively. The spin structure of this system is not fully understood and it is reported as collinear 10-12 and noncollinear 1,9 ferrimagnetic. Recently, Xu et al., 13 predicted spin frustration of the Fe D sites, resulting in a noncollinear spin structure with the energy of 60 meV/f.u. lower than that of the experimentally suggested collinear spin structure. The ferrimagnetism in the bulk is due to the uncompensated moments of regular octahedral (Fe C ) and regular tetrahedral (Fe D ), both arranged in antiferromagnetic order and the moments of the two distorted octahedral being equal cancels each other. 14 The oxide is magnetoelectric 15 with a switchable ferroelectric polarization 13 and ferrimagnetic with a Curie temperature of 510 K. 10  There have been experimental attempts to further enhance and improve the coercivity of this particular phase of Fe 2 O 3 by substitution of Fe-atoms on different sites by non-magnetic atoms such as indium, aluminum, gallium, and rhodium, in different concentrations. Namai et al. 22 chemically prepared a series of Rh-substituted ε-Fe 2 O 3 nanoparticles and obtained enhanced coercivities of 2.7 and 3.1 T for isotropic and crystallographically aligned nanoparticles, respectively. In this case, the Rh-atom occupies the Fe-atom at C-site. Ohkoshi et al. 21 prepared In-, Ga-and Al-substituted ε-Fe 2 O 3 , with various concentrations and substitutions taking place at every Fe-site and obtained a tunability of the coercivity. In Al-substituted ε-Fe 2 O 3 (Fe 1.7 Al 0.3 ), the Al atoms preferentially occupy the Fe D sites 21,23 but reduces the coercivity.
In this work, we have studied the effect of Co substitution on different Fe sites. We have replaced a single A, C, and D type Fe atom per unit cell by Co and calculated the saturation magnetization (Ms), the effective magnetic anisotropy constant (K effective ) and the anisotropy field (HA). Since the anisotropy field is the upper bound to the coercivity, the variation of HA with the site substitution will give a good estimate of the coercivity of the system. We also identify the site-specific origin of the anisotropy change and compare the situation in ε-Fe 2 O 3 with the anisotropy contribution in α-Fe 2 O 3 and γ-Fe 2 O 3 .

II. METHOD
Density functional theory (DFT) based on the Vienna abinitio simulation package (VASP) [24][25][26] was used for the calculation. The Perdew, Burke, and Ernzerhof (PBE) 27 functional was used to incorporate semi-local exchange-correlation effects. The DFT+U 28 formalism was implemented to account for the strongly correlated nature of the Fe 3d localized electrons. We took U-J = 4 eV for ε-Fe 2 O 3 11,29 a value commonly used for the hematite. For Co-substituted ε-Fe 2 O 3 , the value of U-J are 4 eV and 3.3 eV 29 for the 3d-states of Fe-and Co-atom, respectively. Projected Augmented Wave (PAW) 26 method-based potentials were used for Fe-, O-, and Co-atoms. The valence-electron configurations for the Fe-, O-, and Co-atoms were taken to be d 7 s 1 , s 2 p 4 , and d 8 s 1 , respectively. The electronic wave functions were represented by a plane-wave basis set with an energy cutoff of 530 eV. A Monkhorst-Pack 30 k-point mesh of 5 × 3 × 3 was used for one unit cell for structural optimization of the pristine bulk as well as of the Co-substituted ε-Fe 2 O 3 . A convergence criterion of 10 −7 eV for electronic self-consistency and maximum forces of 0.005 eV/Å for each atom during structural optimization were chosen.
To calculate the effective magnetic anisotropy and the anisotropy field for the pristine as well Co-substituted ε-Fe 2 O 3 , we included the spin-orbit coupling as implemented in VASP by Kresse and Lebacq. A very dense Monkhorst-Pack k-point mesh of 15 × 9 × 9 was used to calculate the total energies for the magnetization directions fixed parallel to the x-, y-, and z-axes. Due to the orthorhombic nature of the crystal, there exists low symmetry and the lowest order anisotropy energy 31,32 is defined as Using Eq. 1, the effective magnetic anisotropy constant was calculated using the formula K eff = (E first hard axis − Eeasy aixs) V which yields the anisotropy field where E is the total ground state energy of the system, V is the volume of the unit cell of bulk ε-Fe 2 O 3 , µ 0 is the permeability of free space and Ms is the saturation magnetization of the bulk ε-Fe 2 O 3 .

III. RESULTS AND DISCUSSION
Our DFT optimized lattice parameters obtained for ε-Fe 2 O 3 are as a = 5.125, b = 8.854 and c = 9.563 Å, 33,34 which is in agreement with the experimental lattice parameters of Sect. I. Our calculated electronic structure yields an energy band-gap of 1.9 eV. 33,34 Figure 2 shows the unit-cell structures of the Co-substituted ε-Fe 2 O 3 . For the Co-substitution, we kept the volume of the unit cell constant and only the ionic positions were relaxed. Taking into account the non-uniaxial character of the orthorhombic lattice, 31 the total energies were calculated for magnetization directions along the three principal axes.
The saturation magnetization (Ms), effective magnetic anisotropy constant (K eff ) and anisotropy field (HA) for the Cofree and Co-substituted oxides are listed in Table I. The table shows that the K eff of the pristine bulk ε-Fe 2 O 3 is comparable to the previously measured K value of 0.1 MJ/m 3 , 19 which was the upper cutoff of anisotropy constant. Both theory and experiment yield a substantial anisotropy increase due to transition-metal substitution. One reason is the anisotropy of the starting compound, which is unusually low for a noncubic compound. The anisotropy constants of the Co-substituted oxides are typical for noncubic materials (several 0.1 MJ/m 3 ). The anisotropy field, which provides an upper bound to the coercivity, is an order of magnitude higher than the experimentally reported 1,10,16 coercivity. As explained in Ref. 32, such a difference is not unusual and means that the coercivity mechanism deviates from coherent rotation due to real-structure effects.
On all three Fe sites (distorted octahedra, regular octahedra, and regular tetrahedra), the Co atoms keep interacting antiferromagnetically, maintaining the ferrimagnetic order but enhancing the total magnetization (Table I). Among the doped systems, the substitution at the tetrahedral site does not contribute to the enhancement of the anisotropy field, because the anisotropy and magnetization changes cancel each other. Among the two octahedrals, the distorted one has the bigger effect on both K eff and on HA. The distorted octahedral is not present in the structures of hematite and maghemite; it occurs in the ε-Fe 2 O 3 crystal structure only, where it has a big effect on anisotropy and on the coercivity.

IV. CONCLUSIONS
In summary, we have studied the site substitution effect of Co on the magnetization, magnetic anisotropy, and anisotropy field of ε-Fe 2 O 3 . The distorted octahedron which is exclusive to the ε-Fe 2 O 3 crystal structure and not found in hematite or maghemite, are important for the understanding of the anisotropy of the oxide. On Co substitution, they yield a disproportional contribution to anisotropy and coercivity. On the other hand, if the substitution takes places by a d-states element, such as Rh and Co, the magnetic anisotropy constant as well as the coercivity increases.