Introduction

In recent years europium monoxide, EuO, has received considerable attention as a potential material for spintronics, because of its special electronic and magnetic properties. The compound has a rock salt structure and it is a ferromagnetic insulator below the Curie temperature of TC = 69 K1,2,3,4. The divalent Eu ions possess a large magnetic moment of 7 μB, originating from the half-filled 4f states, which are separated by an energy gap of 1.12 eV from the Eu 5d conduction band5. EuO is suitable as spin filter due to its spin polarization of almost 100%, as demonstrated both by experiment4,6,7,8 and theory9. Spin filter tunneling junctions (metal/EuO/metal heterojunctions) based on polycrystalline EuO have been studied in various experiments10,11,12,13,14,15,16. Integration of EuO on semiconducting GaAs17, GaN7 and Si7,18 has been demonstrated. Particularly, the possibility of growth on graphene19 and topological insulators20 is interesting for spintronics devices.

The ferromagnetism of EuO, in general, originates from indirect exchange, J1 and superexchange, J2. It is widely accepted that the indirect exchange is governed by the Eu 4f and 5d orbitals21,22, whereas the superexchange involves the hybridized Eu 4f and O 2p orbitals23, where mediation by 6s and 5d states appears to be important24,25. It has been suggested by Ingle and Elfimov21 that TC can be enhanced most effectively by reducing the gap between the Eu 4f and 5d states and by minimizing the hybridization between the Eu 4f and O 2p states. In this context, rare earth doping with La, Lu and Gd, has been studied experimentally26,27,28,29,30,31,32,33,34,35,36, by model approaches1,37,38,39 and by first-principles calculations21,22,23,31,33,40,41,42. The effects of rare earth doping, which is efficient only for low dopant concentrations, have been explained by modifications of the on-site and RKKY interactions22. On the other hand, enhancement of TC can also be achieved in O deficient EuO43,44,45 and by the application of tensile strain25. In fact, epitaxial growth of EuO on appropriate substrates can result in a tetragonal or an orthorhombic structure36.

Commonly used methods for modeling doping effects on the electronic structure of EuO are the virtual crystal approximation31,40 and the rigid band approximation33. First-principles calculations for Gd-doped EuO by the supercell approach (partial substitution of Eu by Gd) have been reported in Ref. 41 without addressing the exchange interaction. Insight into the magnetism has been accomplished in Refs. 22 and 42 for a restricted set of configurations.

In contrast, the purpose of our work is to investigate the effect of Gd doping and O deficiency in the entire concentration range relevant for experiment, focusing on the electronic structure, exchange interaction and TC. We will first analyze the effect of Gd doping and afterwards that of O deficiency.

Results

In our first-principles calculations we use a linear combination of atomic orbitals and Troullier-Martins norm-conserving relativistic pseudopotentials (as implemented in the SIESTA code)46. The wave functions are expanded in a ζ+polarization basis, except for the Eu 4f states for which we use a single-ζ basis. A cutoff of 600 Ry is employed together with 4 × 4 × 4 and 6 × 6 × 4 uniform meshes, respectively, for sampling the Brillouin zones of the cubic and tetragonal phases. To achieve an accurate description of the EuO band gap, we use the local density approximation with on-site Coulomb repulsions (U) and exchange parameters (J)47,48. Note that these parameters refer to the microscopic interacting-electron problem; in particular, these J should not be confused with the exchange interactions of the effective Heisenberg model, to be discussed in connection with Eqs. (1)–(3).

For the Eu 4f states we set Jf = 0.77 eV21, but we vary Uf between 8 and 9 eV since the band structure depends critically on the on-site potential of the Eu 4f states. A value of Uf = 8.8 eV gives the best agreement with the experimental situation (band gap of 1.1 eV and band splitting of 0.6 eV)4,6. Following Ref. 21 we use Jp = 1.2 eV and Up = 4.6 eV for the O 2p orbitals. Moreover, for the Gd 4f orbitals we set Jf = 0.7 eV and Uf = 6.7 eV49,50.

The experimental lattice constant of 5.144 Å is used for the rocksalt structure (cubic phase), with four Eu and four O atoms per unit cell. For the tetragonal phase we start from the lattice parameters a = 3.65 Å (inter-planar spacing) and c = 5.12 Å (out-of-plane spacing). For both phases, 2 × 2 × 2 supercells are built, which are shown in Fig. 1. These supercells are fully relaxed by means of the conjugate gradient method until the atomic forces have declined below 0.01 eV/Å. We obtain for the cubic phase a = 5.097 Å and for the tetragonal phase a = 3.635 Å and c = 5.080 Å. These values are kept fixed when building the respective structures under Gd doping and O deficiency. Under this constraint, we have carefully relaxed all atomic positions which, of course, is mandatory in order to be able to obtain reliable results. Gd concentrations between 6.25% and 25% are considered, by substituting Eu atoms by Gd; O vacancy concentrations in the same range are achieved by removing O atoms from the supercell. In both cases we distribute the impurities homogeneously in the supercell.

Figure 1
figure 1

Structure of EuO in the cubic phase (left) and the tetragonal phase (right).

Large spheres (blue) represent Eu, small spheres (red) represent O. The front side is the xz-plane.

The nearest neighbor (NN, J1) and next-nearest neighbor (NNN, J2) exchange interactions for the cubic phase are determined by fixing three spin configurations and calculating their respective energies: the ferromagnetic one (FM), an antiferromagnetic (AFM) one with the spin direction alternating in the (001) direction (AFM1) and an AFM one with the spin direction changing every second layer in the (001) direction (AFM2). The total energies (per cation) are related to the J1 and J2 as follows51:

where S = 7/2. Given J1 and J2, an effective Heisenberg model can be defined; and similarly for the tetragonal case (next paragraph).

For the tetragonal phase (c > a), we have in-plane NN (J1||), out-of-plane NN (J1), in-plane NNN (J2||) and out-of-plane NNN (J2) interactions. Note that the term in-plane refers to the xy-plane of the tetragonal supercell, which is rotated by 45° with respect to the cubic supercell. To determine the exchange interactions in this case, we have to study five spin configurations: FM, AFM1, AFM2, AFM with the spin direction alternating in the (110) direction (AFM3) and AFM with the spin direction alternating every 2nd layer in the (100) direction (AFM4). The total energies (per cation) are given by:

In mean-field approximation this results in21

We note that the magnetic moments are very localized in all our systems, since they are mainly carried by f orbitals and thus hardly depend on the spin configuration. For example, we obtain for the pristine compound in the cubic phase values of 7.09 μB (FM), 7.03 μB (AFM1) and 7.05 μB (AFM2) and in the tetragonal phase values of 7.09 μB (FM), 7.03 μB (AFM1), 7.00 μB (AFM2), 6.98 μB (AFM3) and 7.03 μB (AFM4).

In order to clarify the effect of Gd doping for both the cubic and tetragonal phases we determine the density of states (DOS) projected on the Eu 4f, 5d, Gd 4f, 5d and O 2p orbitals, see Fig. 2. The dependences of the different exchange terms and of TC on the dopant concentration are addressed in Fig. 3. We first discuss the results for the pristine structures (0% doping), which are very similar for the cubic and tetragonal phases. For the majority spin channel we distinguish three regions: the conduction band (dominated by Eu 5d states), upper valence band (dominated by localized Eu 4f states, with some hybridization with O 2p and Eu 5d) and lower valence band (dominated by O 2p states, with significant hybridization with Eu 4f and 5d). The spin minority channel shows a similar structure but without the Eu 4f contributions.

Figure 2
figure 2

DOS projected on the Eu 4f, 5d, Gd 4f, 5d and O 2p orbitals for the cubic (left) and tetragonal (right) phases of EuO for different Gd concentrations.

Figure 3
figure 3

Exchange interaction and corresponding TC as a function of the Gd concentration for the cubic phase ((a1), (b1)) and the tetragonal phase ((a2), (b2)).

For the cubic phase, exchange interactions of J1 = 0.63 K and J2 = 0.13 K have been derived from single-crystal inelastic neutron scattering52, which by Eq. (3) corresponds to TC = 88 K, while we find J1 = 0.50 K and J2 = 0.26 K and hence a TC of 80 K, consistent with the experimental result. Note that our effective J ( = J1 + J2 = 0.76 K) agrees with the experimental value. For the tetragonal phase, we obtain J1|| = 0.54 K, J2|| = 0.19 K, J1 = 0.49 K and J2 = 0.27 K, from which a mean-field TC of 77 K is calculated. However, recent experiments on films of cubic and tetragonal EuO with 10 Å thickness have found critical temperatures of 56 K and 53 K, respectively36. While the absolute values deviate from our theoretical findings, we note that the difference between the two TC values is exactly the same (3 K). This is a strong indication for the reliability of our calculations, as far as difference quantities and dependencies (like TC vs. concentration) are concerned.

The effects of Gd doping on the DOS are similar for the cubic and tetragonal phases, see Fig. 2. The exchange splitting of the Eu 5d states at the conduction band edge essentially remains the same as in the pristine system. For increasing Gd doping, the Gd 5d and Eu 5d majority spin states shift to lower energy, increasing the system's metallicity. Since there are many more Gd 5d than Eu 5d conduction states occupied, mainly the Gd 5d states determine J1 (combination of on-site and RKKY exchange). The hybridization between the Eu 4f and O 2p states decreases for increasing Gd doping, which reduces the value of J2 (superexchange).

For 6.25% Gd doping the stronger exchange interaction between the Gd/Eu 5d and Eu 4f states (the reduced energy gap supports the f-d hopping) in combination with the RKKY exchange mediated by the conduction states22 enhances TC to around 120 K, both in the cubic and tetragonal phases, see Fig. 3(b1),(b2), in good agreement with the experimental value of 129 K for 10% Gd doping28. In Ref. 22 a maximal TC of 160 K for 10% Gd doping has been obtained on the basis of the virtual crystal approximation (using the parameters Jf = 0.6 eV and Uf = 6.1 eV for the Eu 4f states). While the validity of the virtual crystal approximation is difficult to assess in this context, we note that TC depends strongly on the parameter Uf, which in our work was chosen to be Uf = 8.8 eV, in order to reproduce the experimental band gap. With increasing Uf, the band gap opens, hence TC decreases and vice versa. On the other hand, Ref. 22 demonstrates that the thermodynamic properties of Gd doped EuO are well captured by the mean field approach.

Above 6.25% Gd doping we observe that the Gd/Eu 5d majority spin states shift further towards the Eu 4f states, which should enhance TC. However, the minority spin states start getting filled and, as a consequence, the spin polarization at the Fermi energy is reduced. This compromises the RKKY interaction and therefore effectively lowers J1 and TC. In addition, an antiferromagnetic J2 is observed for 18.75% and higher doping in both phases, which can be explained by enhanced hybridization between the Gd 5d, Eu 5d and O 2p states: see, for example, the developing joint DOS peaks close to −5 eV.

We next analyze the effects of O deficiency by means of the DOS projected on the Eu 4f, 5d and O 2p orbitals, see Fig. 4, for different O vacancy concentrations. In addition, Fig. 5 addresses the dependences of the different exchange terms and of TC on the O vacancy concentration. As expected, O deficiency in EuO causes almost rigid shifts of all states to lower energy, giving rise to the well known metal-insulator transition. With increasing O deficiency more and more of the charge donated by the O vacancies occupies the Eu 5d conduction bands. It is generally accepted that positive effects on TC due to O deficiency originate from this extra charge populating the conduction band and giving rise to enhanced RKKY exchange1,25,43,44, which corresponds to an increase in J1. However, also the gap between the majority spin Eu 5d and 4f states decreases substantially and the f-d hopping is enhanced correspondingly, see Fig. 5(a1). The band structure (not shown) demonstrates that the exchange splitting of the Eu 5d states at the conduction band edge is reduced significantly for 6.25% O vacancy concentration, as compared to the pristine case and further slightly decreases for higher O vacancy concentrations.

Figure 4
figure 4

DOS projected on the Eu 4f, Eu 5d and O 2p orbitals for the cubic (left) and tetragonal (right) phases of EuO for different O vacancy concentrations.

Figure 5
figure 5

Exchange interaction and corresponding TC as a function of the O vacancy concentration for the cubic phase ((a1), (b1)) and the tetragonal phase ((a2), (b2)).

In addition, the DOS demonstrates that hybridization between the Eu 4f, 5d and O 2p states plays a significant role for the TC value. We first focus on the cubic phase, see the left hand side of Fig. 4. Hybridization between the Eu 4f and O 2p states decreases as the O vacancy concentration increases, which enhances J2, up to 12.5% O vacancy concentration. Afterwards J2 declines rapidly. According to Fig. 5(b1), TC increases, as J1 increases, up to a maximum value of 190 K for 18.75% O vacancy concentration and decreases thereafter, as J2 decreases. The tetragonal phase, see the right hand side of Fig. 4, overall shows similar characteristics, i.e., J1||, J1, J2|| and J2, see Fig. 5(a2), first increase with O deficiency. However, now the Eu 5d minority spin states get filled for 12.5% and higher O vacancy concentrations and J1 is reduced accordingly, resulting in a maximum in TC of about 175 K, see Fig. 5(b2).

Discussion

We have performed first principles calculations for both Gd doped and O deficient EuO to clarify the mechanisms that determine the critical temperatures of the cubic and tetragonal phases. We extend previous theoretical considerations for the cubic phase to high defect concentrations and present the first comprehensive account of the role of defects in the tetragonal phase. The calculated maximum in TC, as a function of Gd concentration, is in good agreement with the experimental value. The observed behavior is explained by a complex combination of different exchange mechanisms. While both the on-site and RKKY interactions increase with increasing (but low) doping, filling of the Gd 5d minority spin states at high doping counteracts the RKKY exchange. In addition, the superexchange is modified at high doping due to growing hybridization between the Gd 4f and O 2p states. The dependence of TC on the O deficiency is controlled by a similar mechanism, though now the Eu 5d states take over the role of the Gd 5d states. As a consequence, optimal values exist both for the Gd dopant and O vacancy concentrations.