A. Lattice volume expansion-induced ferromagnetism of the EuTiO₃ thin film

The molar ratio of Eu to Ti, estimated by the Rutherford backscattering, in the thin films deposited on LaAlO₃, SrTiO₃, and DyScO₃ substrates is shown in Table I. The experimental error in the molar ratio was evaluated to be 0.15–0.70%. The molar ratio of Eu to Ti is almost equal to 1 for all the thin films, indicating that the as-deposited thin films are stoichiometric as far as the molar ratio of cations is concerned. Also, it was demonstrated by the out-of-plane and in-plane XRD measurements that the thin films are composed of a single phase of EuTiO₃ (although the XRD patterns are not shown here). The data obtained from the XRD measurements, i.e., the reciprocal space mappings, are discussed below in detail. Figure 1 illustrates the AFM image for the EuTiO₃ thin films on the different types of substrate. LAO, STO, and DSO indicated in the figure denote the LaAlO₃, SrTiO₃, and DyScO₃ substrates, respectively. So-called stepped and terraced structure is observed, suggesting that the surface of the thin films is atomically flat and smooth.

FIG. 1.

AFM images of EuTiO₃ thin films grown on (a) LaAlO₃, (b) SrTiO₃, and (c) DyScO₃ substrates. LAO, STO, and DSO denote LaAlO₃, SrTiO₃, and DyScO₃, respectively.

TABLE I.

Molar ratio of Eu to Ti in the as-deposited europium titanate thin films grown on different sorts of the substrate. The molar ratio is almost equal to 1 in all the thin films, suggesting that stoichiometric EuTiO₃ was synthesized.

Figure 2 depicts the reciprocal space mappings of 103 reflections for EuTiO₃ thin films deposited on LaAlO₃, SrTiO₃, and DyScO₃ substrates. Q _x and Q _y stand for the reciprocal lattice vectors corresponding to the lattice vector parallel and perpendicular to the film surface, respectively. Bragg spots ascribed to the thin film and the substrate are observed in each of the reciprocal space mappings. The small circle shown in each of the illustrations denotes the reciprocal lattice for bulk EuTiO₃. It is possible to evaluate a strain or a change in the lattice volume for EuTiO₃ thin film by comparing the reciprocal lattice of the thin film with that of bulk EuTiO₃. In the case that LaAlO₃ is used as a substrate, Q _y of the EuTiO₃ thin film is much smaller than that of the bulk EuTiO₃, whereas Q _x is slightly larger for the EuTiO₃ thin film than for the bulk EuTiO₃. This fact indicates that the lattice of the EuTiO₃ thin film is elongated in a direction perpendicular to the surface of the thin film and slightly shrinks in a direction parallel to the surface. In this case, the lattice of the EuTiO₃ thin film is elongated by +1.89% in a direction perpendicular to the surface and shrinks by −0.3% in the plane parallel to the surface, respectively. Therefore, the change in lattice volume from the bulk EuTiO₃ is evaluated to be +1.56%. Similar estimation of variation in lattice constant can be performed for the SrTiO₃ and DyScO₃ substrates. The results are summarized in Table II, where Δl ₁ and Δl ₂ denote the variation in lattice constant in a direction parallel and perpendicular to the surface of the thin film, respectively, and ΔV stands for the change in lattice volume for the EuTiO₃ thin films. As expected from the degree of lattice mismatch, the lattice volume of the EuTiO₃ thin film is the largest for the DyScO₃ substrate and the smallest for the LaAlO₃ substrate. For the SrTiO₃ substrate, an intermediate value is obtained.

FIG. 2.

Reciprocal space mappings of 103 reflections for EuTiO₃ thin films as well as LaAlO₃, SrTiO₃, and DyScO₃ substrates. Q _x and Q _y stand for the reciprocal lattice vectors corresponding to the lattice vectors parallel and perpendicular to the surface of the EuTiO₃ thin film, respectively. The small circle shown in each of the mappings denotes the reciprocal lattice of bulk EuTiO₃.

TABLE II.

Variation in lattice constant and lattice volume for EuTiO₃ thin films grown on LaAlO₃, SrTiO₃, and DyScO₃ substrates. A relative difference in the values between the thin film and the bulk EuTiO₃ is listed. Δl ₁ and Δl ₂ denote the change in lattice constant in a direction parallel and perpendicular to the surface of the thin film, respectively. ΔV stands for the change in lattice volume relative to the bulk EuTiO₃.

Whereas the in-plane lattice constant of the EuTiO₃ thin film is varied by altering the kind of substrate compound, the out-of-plane lattice constant of the EuTiO₃ thin film is increased when compared with the bulk EuTiO₃ irrespective of the sort of substrate compound. As a result, the lattice volume of the EuTiO₃ thin film becomes larger than that of bulk EuTiO₃ even when LaAlO₃, whose lattice constant is smaller than that of bulk EuTiO₃, is used as a substrate. In general, when the lattice constant is larger for material of thin film than for the substrate material, the in-plane lattice constant of the thin film becomes shorter, and the out-of-plane lattice constant becomes longer because of the compressive strain at the interface between the thin film and the substrate.Reference Lee, Fang, Vlahos, Ke, Jung, Fitting Kourkoutis, Kim, Ryan, Heeg, Roeckerath, Goian, Bernhagen, Uecker, Hammel, Rabe, Kamba, Schubert, Freeland, Muller, Fennie, Schiffer, Gopalan, Johnston-Halperin and Schlom^28, Reference Takahashi, Onoda, Kawasaki, Nagaosa and Tokura³¹ For instance, the out-of-plane lattice constant of the EuTiO₃ thin film deposited on the (LaAlO₃)_0.3(Sr₂AlTaO₆)_0.7 substrate, the lattice constant of which is smaller than that of the bulk EuTiO₃, becomes longer by about 0.89%.Reference Takahashi, Onoda, Kawasaki, Nagaosa and Tokura³¹ Compared with this value, however, the elongation of the lattice in the out-of-plane direction is more significant for the present EuTiO₃ thin film deposited on the LaAlO₃ substrate (see Table II). Moreover, such elongation of lattice in the out-of-plane direction of the EuTiO₃ thin film takes place even for the DyScO₃ substrate whose lattice constant is larger than that of the bulk EuTiO₃, as shown in Table II. One possible origin of the large elongation of lattice in a direction perpendicular to the surface of the present EuTiO₃ thin films is that single atomic layers such as EuO and TiO₂ layers may be inserted into the perovskite structure during the deposition process to form a compound like the Ruddlesden–Popper phase. However, this possibility can be ruled out, considering the fact that postannealing of as-deposited EuTiO₃ thin films grown on SrTiO₃ substrate makes the lattice constant shrink to be just the same as the lattice constant of the bulk EuTiO₃.Reference Fujita, Wakasugi, Murai, Zong and Tanaka²⁷ Some oxygen-related point defects may cause the phenomenon, although experimental evidence is absent at this moment, and hence, further study is needed to solve the problem.

Magnetic properties are shown in Figs. 3 and 4 for the EuTiO₃ thin films deposited on LaAlO₃, SrTiO₃, and DyScO₃ substrates. Figure 3 depicts the temperature dependence of magnetization measured at an applied magnetic field of 100 Oe for EuTiO₃ thin films grown on LaAlO₃ (denoted by diamond), SrTiO₃ (circle), and DyScO₃ (square) substrates as well as for the bulk EuTiO₃ (triangle). The inset of the figure is a magnified view of the magnetization data for the bulk EuTiO₃. It is seen that the bulk EuTiO₃ manifests an antiferromagnetic transition at about 5 K, as reported in the literature.Reference McGuire, Shafer, Joenk, Alperin and Pickart^18, Reference Chien, DeBenedetti and Barros¹⁹ In contrast, different behavior is observed for the EuTiO₃ thin films; the magnetization drastically increases below about 5 K as the temperature is decreased for all the EuTiO₃ thin films. Figure 4 illustrates the variation of magnetization with magnetic field at 2 K for EuTiO₃ thin films grown on LaAlO₃ (denoted by diamond), SrTiO₃ (circle), and DyScO₃ (square) substrates as well as for the bulk EuTiO₃ (triangle). The insets show the magnetization at low magnetic fields for EuTiO₃ thin films grown on the SrTiO₃ substrate (right inset: circle) and for the bulk EuTiO₃ (left inset: triangle). It is obvious that the magnetic field dependence of magnetization is different between the EuTiO₃ thin film and the bulk EuTiO₃; the EuTiO₃ thin film grown on the SrTiO₃ substrate manifests magnetic field dependence of magnetization peculiar to a ferromagnet, whereas a spin-flip transition characteristic of an antiferromagnet is clearly observed for the bulk EuTiO₃. The ferromagnetic behavior is seen for the EuTiO₃ thin films grown on LaAlO₃ and DyScO₃ substrates as well. The results are in good agreement with the temperature dependence of magnetization illustrated in Fig. 3. It is also found in Fig. 4 that the saturation magnetization for the EuTiO₃ thin films is 6.4–6.8 μ_B, where μ_B is the Bohr magnetron. These values are almost identical to the theoretical magnetic moment of Eu²⁺, i.e., 7 μ_B, suggesting that almost all the europium ions are present as a divalent state in the thin films. The fact that the valence state of almost all the europium ions in the thin films is +2 was confirmed by conversion electron ¹⁵¹Eu Mössbauer spectroscopy.Reference Fujita, Wakasugi, Murai, Zong and Tanaka²⁷ For instance, the fraction of Eu²⁺ in the total number of europium ion is 96% and 98% for as-deposited and annealed EuTiO₃ thin films grown on the SrTiO₃ substrate, respectively. The fraction of Eu²⁺ and Eu³⁺ in EuTiO₃, or strictly speaking, the composition of which is EuTiO_3−δ, is very important because it may affect the magnetic properties. An increase in the fraction of Eu³⁺ possibly raises the concentration of conduction electron, leading to the ferromagnetic order of Eu²⁺ ions through the mechanism of magnetic polaron or Ruderman–Kittel–Kasuya–Yoshida interaction. The fact that the fraction of Eu³⁺ is fairly small in the preset EuTiO₃ thin films rules out the possibility of these mechanisms for the ferromagnetism observed in the present thin films. This is further supported by the fact that the evident antiferromagnetic transition occurs in the annealed EuTiO₃ thin film, whereas the ferromagnetism is observed in the as-deposited EuTiO₃ thin film although the fraction of Eu³⁺ in the annealed EuTiO₃ thin film is almost the same as that in the as-deposited EuTiO₃ thin film.Reference Fujita, Wakasugi, Murai, Zong and Tanaka²⁷

FIG. 3.

Dependence of magnetization on temperature for EuTiO₃ thin films grown on LaAlO₃ (diamond), SrTiO₃ (circle), and DyScO₃ (square) substrates as well as for bulk EuTiO₃ (triangle). The inset is a magnified view for bulk EuTiO₃.

FIG. 4.

Dependence of magnetization on magnetic field at 2 K for EuTiO₃ thin films grown on LaAlO₃ (diamond), SrTiO₃ (circle), and DyScO₃ (square) substrates as well as for bulk EuTiO₃ (triangle). The insets depict the magnetization at low magnetic fields for EuTiO₃ thin film grown on the SrTiO₃ substrate (right inset: circle) and for bulk EuTiO₃ (left inset: triangle).

A look at magnetization shown in Fig. 3 reveals that the low-temperature magnetization is the highest for the DyScO₃ substrate, intermediate for the SrTiO₃ substrate, and lowest for the LaAlO₃ substrate. Considering the above-mentioned experimental fact that the lattice volume of EuTiO₃ thin film depends on the sort of substrate compound and increases in the order of DyScO₃ > SrTiO₃ > LaAlO₃, it seems that the low-temperature magnetization monotonically increases with an increase in the lattice volume of the EuTiO₃ thin film. This tendency is coincident with the result derived by theoretical approaches.Reference Ranjan, Nabi and Pentcheva^25, Reference Akamatsu, Kumagai, Oba, Fujita, Murakami, Tanaka and Tanaka²⁶ Akamatsu et al.Reference Akamatsu, Kumagai, Oba, Fujita, Murakami, Tanaka and Tanaka²⁶ calculated energy for different spin configurations in EuTiO₃ as shown in Fig. 5 by using the hybrid Hartree–Fock density functional approach. When a prototype of spin-Hamiltonian for Heisenberg system

(1)$$H_{{\rm{spin}}} \, = \, - 2\mathop\sum\limits_{i > j} {J_{ij} {\bf{S}}_i } \cdot {\bf{S}}_j\quad,$$

FIG. 5.

Magnetic configurations and magnetic interactions in EuTiO₃ lattice. A, F, and G denote A-type antiferromagnetic, ferromagnetic, and G-type antiferromagnetic states, respectively. J ₁ and J ₂ stand for the exchange-coupling constant between nearest-neighboring and next-nearest-neighboring Eu²⁺ ions, respectively.

is assumed, the energy of spin system is expressed by

(2)$$E_{\rm{F}} \, = \,E_0 \, + \,2S(S + 1)( - 12J_1 - 24J_2 )\quad,$$

and

(3)$$E_G \, = \,E_0 \, + \,2S(S + 1)(12J_1 - 24J_2 )\quad,$$

for ferromagnetic and G-type antiferromagnetic spin configurations in EuTiO₃ lattice, respectively. Here, J ₁ and J ₂ stand for the exchange coupling constant for the nearest-neighboring and the next-nearest-neighboring Eu²⁺ ions, respectively (see Fig. 5). Equations (2) and (3) indicate that the difference in energy between ferromagnetic and G-type antiferromagnetic states depends only on the sign of J ₁. In other words, whether the ferromagnetic or G-type antiferromagnetic state is stable for 4f spins in EuTiO₃ is determined only by the exchange interaction among the nearest-neighboring Eu²⁺ ions. The hybrid Hartree–Fock density functional approach indicates that the exchange coupling constant J ₁ monotonically increases as the lattice volume of EuTiO₃ increases and that J ₁ changes its sign from negative to positive when an increase in lattice volume of EuTiO₃ exceeds about 5%.Reference Akamatsu, Kumagai, Oba, Fujita, Murakami, Tanaka and Tanaka²⁶ Subsequently, the ferromagnetic state becomes more stable than the G-type antiferromagnetic state when the lattice volume is increased beyond this critical value. The result derived from the calculations that J ₁ is positive and monotonically increases with an increase in the lattice volume above the critical value is qualitatively coincident with the present experimental result shown in Fig. 3 and Table II.

The calculations by Akamatsu et al. also suggest that the overlap between the 4f orbital of Eu²⁺ and the 3d orbital of Ti⁴⁺ in the EuTiO₃ lattice becomes more significant as the lattice volume is decreased.Reference Akamatsu, Kumagai, Oba, Fujita, Murakami, Tanaka and Tanaka²⁶ It should be noted that an overlap between the Eu 4f and the O 2p orbitals does not take place even when the lattice volume is decreased. The overlap between the 4f orbital of Eu²⁺ and the 3d orbital of Ti⁴⁺ can bring about the superexchange interaction between the nearest-neighboring Eu²⁺ ions, leading to antiferromagnetic configuration of magnetic moments of Eu²⁺ ions. This idea stems from the fact that the Eu 4f orbital is nonorthogonal to the Ti 3d orbital. According to Anderson's theory of superexchange interaction,Reference Anderson³² antiferromagnetic coupling is stabilized by the overlap between magnetic orbitals through intervening orbital, such as p state of nonmagnetic anion, which is nonorthogonal to the magnetic orbital. Hence, the superexchange interaction between Eu²⁺ ions via the Ti 3d states is expected to be antiferromagnetic. On the other hand, the indirect exchange coupling between 4f spins via 5d state of Eu²⁺ gives rise to ferromagnetic interaction between the nearest-neighboring Eu²⁺ ions.Reference Chien, DeBenedetti and Barros^19, Reference Shafer^33–Reference Souza-Neto, Haskel, Tseng and Lapertot³⁵ The mechanism to bring about the ferromagnetic interaction is schematically illustrated in Fig. 6. The indirect exchange interaction is such a process that an electron excited from the 4f level into the vacant 5d level of an Eu²⁺ ion is transferred to the 5d level of the nearest-neighboring Eu²⁺ ion and interacts with 4f spins at the Eu²⁺ ion so that the spins in 5d and 4f states are parallel. These antiferromagnetic and ferromagnetic interactions compete with each other. The superexchange interaction via 3d state of Ti⁴⁺ becomes weaker as the lattice volume of EuTiO₃ is increased because of the less overlapping between 4f orbital of Eu²⁺ and 3d orbital of Ti⁴⁺. At the same time, the indirect exchange coupling between 4f spins via 5d state of Eu²⁺ to bring about ferromagnetic configuration becomes weaker as the lattice volume of EuTiO₃ is increased because the separation between the nearest-neighboring Eu²⁺ ions becomes longer as the lattice volume is increased.Reference Kunes, Ku and Pickett^34, Reference Souza-Neto, Haskel, Tseng and Lapertot³⁵ In EuTiO₃, the superexchange interaction via Ti⁴⁺ 3d state becomes less dominant than the indirect exchange coupling via Eu²⁺ 5d state as the lattice volume becomes larger. Consequently, ferromagnetic phase is stabilized in EuTiO₃ with expanded lattice volume.

FIG. 6.

Mechanism of indirect exchange interaction between Eu²⁺ ions to lead to ferromagnetic state. 4f spins interact ferromagnetically with each other via 5d state.

B. Ferromagnetism of amorphous EuTiO₃ and Eu₂TiO₄ thin films

An attempt was made to synthesize amorphous EuTiO₃ and Eu₂TiO₄ thin films on an SiO₂ glass substrate by using the PLD method. According to the RBS measurements, the molar ratio of Eu to Ti is estimated to be 1.00:1.00 and 2.00:1.06 for thin films prepared from the target materials with EuTiO₃ and Eu₂TiO₄ compositions, respectively; the molar ratio for the thin films is almost identical to that for the target materials. Figure 7 shows the XRD patterns for EuTiO₃ and Eu₂TiO₄ thin films as well as HRTEM image and SAED pattern for EuTiO₃. In the figure, a-ETO and a-2ETO stand for the thin films with EuTiO₃ and Eu₂TiO₄ compositions, respectively. No sharp diffraction lines ascribable to crystalline phases but halo patterns peculiar to amorphous structure are observed in the XRD diagrams, suggesting that the thin films are amorphous and include no crystalline phases. The halo patterns at around 2θ = 30° and 20° are attributable to the thin film and an SiO₂ glass substrate, respectively. The HRTEM image and the SAED pattern also indicate that the EuTiO₃ thin film contains no crystalline phases, confirming that the thin film is amorphous. Similar results were derived for the Eu₂TiO₄ thin film from its HRTEM image and SAED pattern.

FIG. 7.

(a) X-ray diffraction patterns for amorphous EuTiO₃ (a-ETO) and Eu₂TiO₄ (a-2ETO) thin films. (b) HRTEM image and SAED pattern for amorphous EuTiO₃.Reference Akamatsu, Fujita, Zong, Takemoto, Murai and Tanaka²⁹

Figure 8 illustrates the variation of magnetization with temperature measured at an applied DC magnetic field of 100 Oe for the amorphous EuTiO₃ and Eu₂TiO₄ thin films. The magnetization is very low at high temperatures; the temperature dependence of magnetic susceptibility at high temperatures is describable in terms of the Curie–Weiss law:

(4)$${\rm{\chi }} = {{NM_{\rm{B}} ^{\,\,\,\,2} {\rm{\mu }}_{\rm{B}} ^{\,\,\,\,2} } \over {3k_{\rm{B}} \left( {T - {\rm{\theta }}_{\rm{W}} } \right)}}\quad,$$

FIG. 8.

Dependence of magnetization on temperature at an applied dc magnetic field of 100 Oe for amorphous EuTiO₃ (a-ETO) and Eu₂TiO₄ (a-2ETO) thin films. The arrows indicate an inflection point in each of the magnetization curves. The Curie temperature is defined as the temperature giving the inflection point.Reference Akamatsu, Fujita, Zong, Takemoto, Murai and Tanaka²⁹

where χ is the magnetic susceptibility, T is the temperature, N is the number density of magnetic moment, μ_B is the Bohr magneton, M _B is the effective number of Bohr magnetons, k _B is the Boltzmann constant, and θ_W is the Weiss temperature. On the other hand, as the temperature is decreased, the magnetization increases drastically below a certain temperature. The behavior indicates that the thin films are ferromagnetic at low temperatures. The Curie temperature estimated as an inflection point (as indicated by an arrow in Fig. 8) in the magnetization as a function of temperature is T _C = 5.5 K and 14 K for the amorphous EuTiO₃ and Eu₂TiO₄, respectively. First of all, it is surprising that amorphous EuTiO₃ exhibits the ferromagnetic phase transition although the stable phase of crystalline EuTiO₃ is a G-type antiferromagnet as mentioned above. Besides, the Curie temperature, i.e., 5.5 K, is comparable with the Néel temperature of the crystalline counterpart, i.e., 5.3 K. Secondly, the ferromagnetic phase transition is observed for the amorphous Eu₂TiO₄ thin film as well as the crystalline phase of Eu₂TiO₄, and moreover, the Curie temperature of the amorphous phase, i.e., T _C = 14 K, is rather higher than that of the crystalline counterpart, i.e., T _C = 9 K.Reference Greedan and McCarthy³⁶ The Weiss temperature derived from the fit of Eq. (4) to the magnetic susceptibility data at high temperatures is θ_W = +8.0 K and +17.6 K for the amorphous EuTiO₃ and Eu₂TiO₄, respectively.Reference Akamatsu, Fujita, Zong, Takemoto, Murai and Tanaka²⁹ These values are significantly higher than the Weiss temperature of their crystalline counterparts, i.e., θ_W = +3.8 KReference Chien, DeBenedetti and Barros¹⁹ and +10 KReference Souza-Neto, Haskel, Tseng and Lapertot³⁵ for crystalline EuTiO₃ and Eu₂TiO₄, respectively, clearly indicating that the amorphization enhances the ferromagnetic interaction among Eu²⁺ ions. Considering the effect of amorphization on the magnetic properties for other transition metal oxides, these phenomena are very rare. In most of the amorphous oxides or oxide glasses containing 3d transition metal and/or trivalent rare-earth ions, antiferromagnetic rather than ferromagnetic interaction is predominant among the magnetic ions.Reference Sanchez, Friedt, Horne and Van Duyneveldt^37–Reference Akamatsu, Oku, Fujita, Murai and Tanaka⁴⁶ Furthermore, it is usual that the magnetic transition temperature is decreased by one order of magnitude when crystalline oxides are converted into amorphous state with the composition unchanged. For instance, whereas crystalline α-Fe₂O₃ is known to show weak ferromagnetism below 950 K, amorphous Fe₂O₃ manifests a cluster spin glass transition at 35.1 ± 0.1 K.Reference Mukadam, Yusuf, Sharma, Kulshreshtha and Dey⁴⁰ Also, amorphous BiFeO₃ shows a spin glass transition at 20 K,Reference Nakamura, Soeya, Ikeda and Tanaka³⁹ whereas crystalline BiFeO₃ is antiferromagnetic with the Néel temperature of 643 K.Reference Bhide and Multani⁴⁷ The ferromagnetic interaction observed even in an amorphous oxide is characteristic of the Eu²⁺ ion, as a positive Weiss temperature was observed for some oxide glasses containing a large amount of Eu²⁺ ions.Reference Schoenes, Kaldis, Thöni and Wachter^48, Reference Akamatsu, Fujita, Murai and Tanaka⁴⁹ The ferromagnetic phase transition was also observed for aluminoborosilicate glass containing 60.0 mol% of EuO, although the Curie temperature is as low as 2.2 K.Reference Akamatsu, Fujita, Murai and Tanaka⁴⁹

Figure 9 depicts the magnetic field dependence of magnetization measured at 2.0 K for the amorphous EuTiO₃ and Eu₂TiO₄ thin films. In this figure, the theoretical magnetization curve calculated by using the Brillouin function for paramagnetic Eu²⁺ ion at 2.0 K is also shown. The slope of the magnetization curve at low magnetic fields is steeper for the amorphous EuTiO₃ and Eu₂TiO₄ than for the Brillouin function. This fact suggests that the ferromagnetic interaction is dominant in the amorphous thin films. We can estimate the magnetic moment per Eu ion to be 7 μ_B from the saturation magnetization at high magnetic fields for both the thin films. This value is coincident with the theoretical one for 4f Reference Jin, Tiefel, McCormack, Fastnacht, Ramesh and Chen⁷ state of Eu²⁺, indicating that almost all the europium ions are present as a divalent state in the amorphous thin films.

FIG. 9.

Dependence of magnetization on magnetic field at 2.0 K for amorphous EuTiO₃ (a-ETO) and Eu₂TiO₄ (a-2ETO) thin films. The solid curve represents the theoretical one calculated using the Brillouin function for paramagnetic Eu²⁺ ions at 2.0 K.Reference Akamatsu, Fujita, Zong, Takemoto, Murai and Tanaka²⁹

The reason why the ferromagnetic state is stabilized in the amorphous EuTiO₃ and Eu₂TiO₄ thin films can be understood in terms of the local structure of Eu²⁺ in the amorphous phases and the above-mentioned mechanism to bring about the ferromagnetic interaction among Eu²⁺ ions. Figure 10(a) and 10(b) illustrate experimental EXAFS spectra (EXAFS oscillation curves χ(R)s as a function of radial distance) at Eu L ₃-edge (denoted by open squares) for the amorphous EuTiO₃ and Eu₂TiO₄ thin films, respectively. The insets depict the EXAFS oscillation curves in the q-space. The solid curves stand for the best-fits of theoretical ones to the first Eu–O coordination peaks. Structural parameters evaluated from the analysis, i.e., the average coordination number of Eu²⁺, n _Eu²⁺, and the nearest Eu–O bond length, d _Eu–O, are listed in Table III. For crystalline EuTiO₃, n _Eu²⁺ = 12 and d _Eu–O = 0.275 nm, and for crystalline Eu₂TiO₄, the structure of which is K₂NiF₄-type, n _Eu²⁺ = 9 and d _Eu–O = 0.270 nm (an averaged value).Reference Chien, DeBenedetti and Barros¹⁹ Hence, both the coordination number of Eu²⁺ and the Eu–O bond length are much smaller for the amorphous EuTiO₃ and Eu₂TiO₄ than for their crystalline counterparts. The coordination number and the Eu–O bond length in the amorphous phases are rather similar to those of crystalline EuO with the rock salt-type structure, for which n _Eu²⁺ = 6 and d _Eu–O = 0.257 nm.Reference McGuire, Shafer, Joenk, Alperin and Pickart¹⁸ The result of the structural analysis on the EXAFS spectra discloses the origin of the ferromagnetic interaction stronger in the amorphous thin films than in their crystalline counterparts. The indirect exchange interaction illustrated in Fig. 6 is proportional to J _intrab ²/U _fd², where J _intra is the intraatomic exchange coupling constant between the 4f and 5d levels of Eu²⁺, b is the transfer integral between the nearest-neighboring Eu²⁺ ions, and U _fd is the difference in energy between the 4f and 5d levels. As readily seen from Fig. 6, U _fd becomes smaller as the crystal field splitting of 5d level is greater, leading to stronger ferromagnetic interactions.Reference Chien, DeBenedetti and Barros¹⁹ For instance, the crystal field is much stronger in crystalline EuO than in crystalline EuTiO₃ because the coordination number of Eu²⁺ is 6 in the former and 12 in the latter. Subsequently, the ferromagnetic interaction among Eu²⁺ ions is so strong that the Curie temperature as high as 77 K is observed in crystalline EuO,Reference Matthias, Bozorth and Van Vleck⁵⁰ while antiferromagnetic transition takes place at rather low temperature like 5.3 K in crystalline EuTiO₃. Since the coordination number of Eu²⁺ and Eu–O bond length in the amorphous EuTiO₃ and Eu₂TiO₄ are rather close to those in crystalline EuO as demonstrated by the EXAFS spectra, the crystal field splitting of 5d level is larger in the amorphous EuTiO₃ and Eu₂TiO₄ than in their crystalline counterparts, resulting in the enhancement of ferromagnetic interaction in the amorphous phases.

FIG. 10.

Eu L ₃-edge EXAFS spectra in R-space (open squares) and theoretical curves (solid lines) fitted to the first-shell components of the experimental data for (a) EuTiO₃ (a-ETO) and (b) Eu₂TiO₄ (a-2ETO) thin films. R is the radial distance from Eu and χ(R) is the EXAFS oscillation curve. The insets illustrate a comparison of the real part of Fourier filtered first-shell signals in q-space (open squares) and the theoretical curves (sold lines).Reference Zong, Fujita, Akamatsu, Nakashima, Murai and Tanaka³⁰

TABLE III.

Structural parameters obtained from EXAFS for Eu²⁺ in amorphous EuTiO₃ (a-ETO) and Eu₂TiO₄ (a-2ETO) thin films. n _Eu²⁺ and d _Eu–O denote the average coordination number of Eu²⁺ and the nearest-neighboring Eu–O bond length, respectively.Reference Zong, Fujita, Akamatsu, Nakashima, Murai and Tanaka³⁰

The stabilization of ferromagnetic state by the amorphization was also observed in EuZrO₃.Reference Zong, Fujita, Akamatsu, Murai and Tanaka⁵¹ Crystalline EuZrO₃ is an antiferromagnet with the Néel temperature of 4.1 K,Reference Zong, Fujita, Akamatsu, Murai and Tanaka²² whereas the amorphous EuZrO₃ thin film prepared by the PLD method shows ferromagnetic transition at 8 K.Reference Zong, Fujita, Akamatsu, Murai and Tanaka⁵¹ It should be noted that the magnetic transition temperature is higher for the amorphous phase than for its crystalline counterpart. The mechanism based on the indirect exchange interaction leads to the ferromagnetic configuration of magnetic moments of Eu²⁺ ions in the amorphous EuZrO₃ as well as the amorphous EuTiO₃ and Eu₂TiO₄.