Magnetostructural coupling behavior at the ferromagnetic transition in double-perovskite $\mathrm{S}{\mathrm{r}}_2\mathrm{FeMo}{\mathrm{O}}_6$

The ordered double-perovskite $\mathrm{S}{\mathrm{r}}_2\mathrm{FeMo}{\mathrm{O}}_6$ (SFMO) possesses remarkable room-temperature low-field colossal magnetoresistivity and transport properties which are related, at least in part, to combined structural and magnetic instabilities that are responsible for a cubic-tetragonal phase transition near 420 K. A formal strain analysis combined with measurements of elastic properties from resonant ultrasound spectroscopy reveal a system with weak biquadratic coupling between two order parameters belonging to ${\mathrm{\Gamma }}_4^{+}$ and $m{\mathrm{\Gamma }}_4^{+}$ of parent space group $Fm\overline{3}m$. The observed softening of the shear modulus by ∼50% is due to the classical effects of strain/order parameter coupling at an improper ferroelastic $\left({\mathrm{\Gamma }}_4^{+}\right)$ transition which is second order in character, while the ferromagnetic order parameter $\left(m{\mathrm{\Gamma }}_4^{+}\right)$ couples only with volume strain. The influence of a third order parameter, for ordering of Fe and Mo on crystallographic B sites, is to change the strength of coupling between the ${\mathrm{\Gamma }}_4^{+}$ order parameter and the tetragonal shear strain due to the influence of changes in local strain heterogeneity at a unit cell scale. High anelastic loss below the transition point reveals the presence of mobile ferroelastic twin walls which become pinned by oxygen vacancies in a temperature interval near 340 K. The twin walls must be both ferroelastic and ferromagnetic, but due to the weak coupling between the magnetic and structural order parameters it should be possible to pull them apart with a weak magnetic field. These insights into the role of strain coupling and relaxational effects in a system with only weak coupling between three order parameters allow rationalization and prediction of how static and dynamic properties of the material might be tuned in thin film form by choice of strain contrast with a substrate.

Figure 1

Summary of literature data for the magnetic phase transition temperature $T_{\mathrm{c}}$, degree of long-range cation order $Q_{\mathrm{od}}$, and saturation magnetization measured at low temperature, $M_{\mathrm{s}}$ [6, 7, 8, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. The data show two well-defined correlations of reducing $T_{\mathrm{c}}$ and $M_{\mathrm{s}}$ with increasing disorder; straight lines are fits to the combined literature data for the relationships between $T_{\mathrm{c}}$ and $M_{\mathrm{s}}$, and between $Q_{\mathrm{od}}$ and $M_{\mathrm{s}}$. Individual data sets of Huang et al. [32] and Shimada et al. [7] show essentially the same pattern as the collected literature data.

Figure 2

(a) Pseudocubic lattice parameters for $\mathrm{S}{\mathrm{r}}_2\mathrm{FeMo}{\mathrm{O}}_6$ as a function of temperature. Uncertainties from the least squares fitting are smaller than the size of the symbols. The reference parameter $a_o$ is given by extrapolation of a straight-line fit to the cubic lattice parameters at high temperatures. The results of both cubic and tetragonal fits are given in the intermediate temperature range, although the resolution of the primary spectra is not good enough to be certain of the tetragonal splitting. (b) Full width at half maximum height (FWHM) for the 004 and 220 reflections in space group $I4/m$ (004 in $Fm\overline{3}m$; 2θ ≈ 46º). (c) Variations of spontaneous strains determined for the cooling sequence using Eqs. (1, 2, 3, 4) and data in (a). Uncertainties propagated simply from the lattice parameters are ±∼0.0001.

Figure 3

(a) SQUID data showing magnetic hysteresis loops measured at 2, 295, and 400 K from the same sample as used for RUS. The inset shows small openings in the loops at all three temperatures. (b) Thermal evolution of the field-cooled (FC) and zero–field-cooled (ZFC) dc susceptibility measured in a 50 Oe field between 2 and 400 K (SQUID data). The inset shows the thermal evolution of the cooling and heating susceptibility at 50 Oe from room temperature to 520 K obtained from the VSM. (c) ac magnetic susceptibility from the Kappabridge, showing clearly that $T_{\mathrm{c}}$ is in the vicinity of 411–414 K, as defined by extrapolation of the steep part of the curves to zero.

Figure 4

Differential heat flow measured in the DSC from two separate temperature intervals, at different heating rates. (a) 310–380 K at $10\mathrm{K}\mathrm{mi}{\mathrm{n}}^{-1}$, showing an irregular anomaly in the vicinity of 340 K. (b) 405–480 K at $5\mathrm{K}\mathrm{mi}{\mathrm{n}}^{-1}$ showing three distinct anomalies. The blue lines are at 338.8 (line 1), 415 (line 2), 419 (line 3), and 454 K (line 4).

Figure 5

(a) Segments of RUS spectra from a polycrystalline sample of SFMO obtained during heating and stacked in proportion to the temperature at which they were collected. (b) Segments of spectra from the high-temperature instrument obtained during heating (red) and cooling (blue). Resonance peaks which do not shift in frequency with increasing or decreasing temperature are from the alumina buffer rods while peaks which move are from the sample. Each spectrum has been offset up the $y$ axis in proportion to the temperature at which it was collected. (c) Variations of the square of the frequency $f^2$ and inverse mechanical quality factor $Q^{-1}$ for the resonance near 230 kHz from high-temperature spectra and near 199, 850, and 950 kHz from low-temperature spectra. The $f^2$ values have been scaled to be the same at room temperature and to overlap at lower temperatures. Vertical lines are shown at 338.8, 415, 419, and 454 K to indicate the temperatures of anomalies seen in the differential heat flow data from Fig. 4. The pink solid curve is a fit of Eq. (5) to a segment of the $Q^{-1}$ data.

Figure 6

Relationships between symmetry-adapted strains, octahedral tilt angle ϕ to represent $q_3$, and the refined magnetization of Fe, M, to represent $m_3$ (as defined in the Supplemental Material [60]). (a) $e_{\mathrm{t}}$ varies linearly with ${\varphi }^2$, as expected for a tilting transition in perovskites. (b) For the data with the most reliable values of $e_{\mathrm{t}}$, there are approximately linear relationships between $e_{\mathrm{t}}$ and $M^2$, but with different slopes for samples with different values of $Q_{\mathrm{od}}$. (c) The data for $\mathrm{B}{\mathrm{a}}_2\mathrm{FeMo}{\mathrm{O}}_6$ show a linear relationship between $e_{\mathrm{a}}$ and $M^2$, as expected in the absence of a tilting transition, but the most reliable $e_{\mathrm{a}}$ data for SFMO do not, due to the additional contribution from the tilting transition.

Figure 7

Effects on the degree of Fe/Mo ordering, as expressed in terms of the order parameter $Q_{\mathrm{od}}$ on transition temperatures. (a) Compiled data from the literature and specific data sets are permissive of the expected relationships $T_{\mathrm{c}},T_{\mathrm{s}}\propto Q_{\mathrm{od}}^2$, but there is significant scatter which is probably due to other factors such as sample stoichiometry. The data point at $T_{\mathrm{s}}=367\mathrm{K}$, from the ordered sample of Sánchez et al. [14], is clearly an outlier. (b) Variation of magnetic transition temperature $T_{\mathrm{c}}$ with structural transition temperature $T_{\mathrm{s}}$, where $T_{\mathrm{s}}$ is from fits to $e_{\mathrm{t}}$ shown in Fig. (A1b) in [60]. If both $T_{\mathrm{s}}$ and $T_{\mathrm{c}}$ scaled linearly with $Q_{\mathrm{od}}^2$, they would be expected to define a straight line passing through the origin. The line shown as a guide to the eye is for $T_{\mathrm{s}}=T_{\mathrm{c}}$.

Figure 8

Data from the literature showing the variations across order parameter space of the octahedral tilt angle ϕ, representing the order parameter for the tilting transition, and the refined magnetic moment of Fe, M, representing the magnetic order parameter.