Nonlinear pressure dependence of T_N in almost multiferroic EuTiO₃

EuTiO₃ (ETO) has recently attracted new interest due to its possible multiferroic behavior and anomalously strong spin–lattice coupling [1–3]. At T_N = 5.5 K the compound undergoes a paramagnetic to antiferromagnetic (AFM) phase transition [4]. Over a large temperature range the transverse optic long wavelength mode softens, reminiscent of a ferroelectric phase transition [2, 3]. Complete softening is inhibited by quantum fluctuations [5, 6] as is also observed for SrTiO₃ (STO) and other perovskite oxides. Upon entering the AFM state this mode experiences an unexpected hardening in ETO demonstrating strong spin–lattice coupling [1–3]. The obvious analogy between ETO and STO has been extended recently by predicting and verifying experimentally that ETO also exhibits an oxygen octahedral rotational instability [7]; this occurs at a much higher temperature T_S (T_S = 282 K), however, than for STO (T_S = 105 K). This large difference in transition temperatures was a motivation for further studies not only of ETO but also of the mixed crystal ETO–STO [8, 9]. From these investigations a phase diagram for this series has been established with T_S depending nonlinearly on STO content [9]. In a pure ETO system novel dynamics have been observed via muon spin rotation (μSR), namely, that at temperature T_N < T < T_S a finite paramagnetic μSR signal is present which must stem from spin correlated regions with finite spatial extent [10]. This result is further evidence that an unusual spin–lattice coupling exists in ETO which is established at high temperatures. This interpretation of the data has been verified by demonstrating that T_S is strongly dependent on the magnetic field [11], a feature so far unknown in nominally paramagnetic insulators. In order to characterize this interesting system further, the low temperature Néel state is investigated by applying pressure and measuring the p dependence of T_N. Here we emphasize that our interest is in the bulk magnetic and structural properties of ETO only which are distinctly different from thin films or substrates; these have been the focus of numerous papers on strain and stress engineering of material properties and are beyond the current investigation. It is also important to mention in this context that in quasi-two-dimensional (2D) materials their physics is very different from that observed for their three-dimensional (3D) analogues, which makes it impossible to discuss phenomena observed in these 2D compounds in relation to the bulk materials. The low temperature phase has already been addressed experimentally via different approaches [12, 13], namely by investigating its magnetic and electric field dependences. From these studies it is concluded that the ground state of ETO is a multidomain state with the possibility of developing ferroelectricity if symmetry breaking can be achieved.

Here we apply pressure to ETO to test the stability range of the AFM state and explore the possibility of achieving a ferromagnetic (FM) state. This is motivated by the fact that in semiconducting cubic Eu chalcogenides the systems change their magnetic states from FM to AFM with increasing ionic radius [14]. This observation corresponds to an inverse pressure effect which offers the possibility that ETO can be transformed from AFM to FM with increasing pressure. On the other hand, pressure experiments on various AFM perovskites and spinels [15, 16] and Ce containing compounds [17–19] have provided evidence that T_N is stabilized and increases with increasing pressure. Calculations for ETO within a Landau–Ginzburg free energy expansion and ab initio computations support the possibility of achieving FM order in ETO [20, 21] which is in accordance with first principles calculations [8]. This has demonstrated that the AFM and FM ground states have almost the same energy with an energy gain of a few meV in favor of AFM ordering. As such it appears timely to establish the pressure dependence of the Néel state for ETO. The data are analyzed within the Bloch power law model [22] which has already been employed for numerous other FM and AFM systems and proven to be particularly useful [14]. A comparison of these model results with those derived from Monte Carlo studies for the Eu chalcogenides has demonstrated their outstanding qualifications especially when considering pressure effects on AFM and FM states [14].

The ETO powder samples used have been prepared as described in [7]. Measurements of the temperature dependence of the magnetic moment m for the sample EuTiO₃ were performed with a commercial SQUID magnetometer (Quantum Design MPMS-XL). Investigations were carried out at ambient as well as under applied pressures up to p = 57 kbar using a diamond anvil cell (DAC) [23] filled with Daphne oil which served as a pressure-transmitting medium. The pressure at low temperatures was determined by the pressure dependence of the superconducting transition temperature of Pb.

The temperature dependence of the magnetic susceptibility χ for EuTiO₃ recorded at ambient pressure is shown in figure 1(a) with the background signal of the empty pressure cell being subtracted. T_N is clearly visible as a distinct peak in the susceptibility data marked by the arrow and T_N = 5.5 K. Below T_N a slight increase in χ takes place which might be caused by insufficient background subtraction and small amounts of paramagnetic impurities. From the inverse susceptibility (inset to figure 1(a)) the Weiss temperature Θ_W = 3.4 K is obtained, in agreement with previous data [4]. The field dependence of the magnetization confirms a gradual change from AFM to FM with increasing magnetic field and saturation is achieved for fields larger than 2 T with a saturation magnetization of 6.73 μ_B which is very close to the spin magnetic moment of Eu²⁺ (7 μ_B). Both of these results are in agreement with data for EuZrO₃ which becomes AFM at a slightly lower temperature T_N = 4.1 K [24].

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In order to highlight the corresponding peak in the pressure dependent magnetization data more clearly, we subtract from the magnetic moment m(T) the Curie–Weiss type temperature dependence m_CW ≈ 1/χ = C/(T − Θ_W) which confirms a well defined peak at T_N = 5.5 K in m(T)–m_CW(T). In figure 1(b) this difference is shown at ambient and selected applied pressures.

The pressure dependence of T_N is shown in figure 2. As is obvious from figure 2 T_N increases nonlinearly with pressure for p > 10 kbar. Below this pressure a linear increase in T_N with pressure appears (inset to figure 2). Such behavior has been observed in various other perovskites and also in linear chain antiferromagnets [15–19]. The interpretation of those data was based on the fact that the 4f–4f overlap increases with increasing pressure and stabilizes the AFM order, whereas superexchange via the bridging oxygen ions is considered to be less effective. Figure 1(b) illustrates that with increasing pressure the peak height at T_N diminishes nonlinearly (inset to figure 1(b)) and the peak itself broadens. A similar observation has been made in CeFe₂ alloys and has been interpreted as an enhancement of AFM correlations [19]. In the case of ETO this corresponds to an increasing Eu 4f hybridization effect stabilizing the AFM nearest neighbor exchange.

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Interestingly, in most cases it has been possible to explain the data in terms of the Bloch power law, where the exchange constants adopt an interatomic distance dependence in terms of the magnetic Grüneisen power laws. In particular, for the Eu chalcogenides a convincing agreement with the pressure data could be achieved and the validity of this law was confirmed by Monte Carlo studies [14]. This has been taken as the motivation to analyze our data within the same framework. Since the bulk modulus of ETO is unknown, we first established the pressure dependence of T_N in ETO using the Heisenberg Hamiltonian to calculate T_N:

$$\begin{eqnarray} &&{T}_{\mathrm{N}}=\frac{2 S(S+1)}{3{k}_{\mathrm{B}}}(-6{J}_{1}+1 2{J}_{2}) \end{eqnarray} \tag{ 1 }$$

where J₁ is the nearest neighbor direct Eu–Eu AFM exchange interaction and J₂ the second nearest neighbor indirect FM exchange interaction, S = 7/2 being the Eu spin. By adopting the following power laws for the exchange interactions:

$$\begin{eqnarray} &&{\tilde {J}}_{1}(p)={J}_{1}(a/{a}_{0})^{-{n}_{1}}\tqs \text{and}\tqs \nonumber\\ &&{\tilde {J}}_{2}(p)={J}_{2}(a/{a}_{0})^{-{n}_{2}} \end{eqnarray} \tag{ 2 }$$

with a being the pressure dependent lattice constant and a₀ = 3.904 Å the pseudo-cubic lattice constant at ambient pressure, with n₁ = 20.9 and n₂ = 10.8 being consistent with the Grüneisen exponents, and J₁/k_B =− 0.0167 K and J₂/k_B = 0.0355 K, the pressure dependence of T_N is correctly reproduced (black line in figure 2). Note, that similar values for the exponents of J₁ and J₂ have been derived in [25]. From this methodology the pressure dependence of the lattice constant can be derived as shown in figure 3. Since the data have been taken at low temperatures, ETO is in the tetragonal phase. However, with the tetragonal distortion being very small [12, 13, 26], the pseudo-cubic lattice constant is used and plotted as a function of pressure; this shrinks linearly with pressure. Its p dependence is comparable to that for STO [27, 28], however it is slightly steeper. From this dependence the spontaneous strain e₁ = e₂ = (a(p) − a₀)a₀ is calculated (lower inset in figure 3). As compared to STO, ETO develops a larger strain with increasing p and a similar evolution of the relative volume change (upper inset in figure 3). But the general behavior for all three quantities is qualitatively the same as for STO [28].

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The nearest and next nearest neighbor exchange constants as derived from equation (2) are shown in figure 4 as a function of the pressure dependent lattice constant. While J₂ increases steadily with decreasing lattice constant (increasing pressure) J₁ decreases within the same range, supporting the AFM order. This trend for J₁ is supported by LDA + U calculations where a small energy gain in favor of G-type AFM order is achieved as compared to FM order [29]. If the system could, however, be artificially tensile stressed and the lattice constant enlarged by 14% a sign change of J₁ takes place enabling a transition to a FM state. Such large tensile stresses are experimentally not achievable and correspondingly a transition from AFM to FM can be excluded. The experimentally accessible range of a is highlighted in figure 4 by the shaded area. Interestingly, however, recent amorphization of the ETO samples has been shown to result in FM order at about the same temperature as AFM order is established in the crystalline sample [30, 31]. This has been attributed to a volume expansion together with the 5d magnetic polarization of Eu²⁺ with the former being consistent with the trends predicted here. Since these data have been obtained on thin films, a direct comparison to our analysis is not possible. It is important to note in addition, that for the compound Eu_0.5Ba_0.5TiO₃ the lattice constant is enlarged by 1.13% as compared to ETO [32]. Here magnetic susceptibility data suggest possible FM behavior with the Curie temperature being below 4 K. Since, as outlined above, this tensile strain is insufficient to induce FM order, the dilution of Eu moments due to Ba substitution could be the cause of such a transition. On the other hand, in more recent experiments on the same composition samples [33], lower temperatures than used in [32] could be attained and a transition to AFM order seen at T_N = 1.9 K, which is rather consistent with our conclusions from figure 3. In tensile and compressive biaxial strain engineered films of ETO a transition to FM order has been observed [34] at 1% strain only. In this case, which cannot be compared to our bulk samples, as outlined above, it is very likely that the stronger reduction of nearest neighbor spin–spin interactions as compared to the second nearest neighbors (in strictly 2D z₁ = 5, z₂ = 8) is the reason for the appearance of a FM state with a much reduced strain than that suggested from our analysis.

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From the pressure dependence of the exchange constants the pressure dependent Weiss temperature Θ_W is derived. As has been demonstrated before, Θ_W is anomalous not only in pure ETO [25] but also in STO–ETO mixed crystals since it is positive, as a consequence of the vicinity of ETO to FM order [8, 9]. By plotting Θ_W as a function of the lattice constant (exaggerating the possible scale of a) it is seen that AFM order is supported by pressure (inset to figure 4). Tensile strain which would cause an increase in the lattice constant, on the other hand, also leads to a decrease in the Weiss temperature; here, however, a sign change does not take place. Since such methodologies can only be performed on thin films, we cannot compare this with the bulk material and its hydrostatic pressure dependence. Overall, the dependence of Θ_W on a is anomalous since nonlinear behavior is observed, caused by the competing exchange interactions and their different power law dependences.

In conclusion, we have investigated the pressure dependence of the low temperature paramagnetic–antiferromagnetic phase transition for ETO. For low pressures a linear increase of T_N with p is observed which adopts a nonlinear dependence with higher p. The data have been analyzed within the Bloch power law model from which the pressure dependence of the pseudo-cubic lattice constant has been derived. Both exchange constants, namely nearest and next nearest neighbor, decrease/increase with increasing p, respectively, thus not allowing for the appearance of a FM state. On the other hand, exceedingly large tensile strain offers the possibility for a FM state, however, with values orders of magnitude beyond those experimentally achievable. The Weiss temperature remains positive in all cases demonstrating the unusual nature of the AFM state. Remarkable is ETO's highly nonlinear evolvement of T_N with p which has no comparable analogs.

This work is partly supported by the Swiss National Science Foundation, the NCCR Program MaNEP, and a SCOPES Grant No. IZ73Z0_128242.