Magnetoelastic coupling and multiferroic ferroelastic/magnetic phase transitions in the perovskite KMnF${}_3$

The magnetic, elastic, and anelastic behavior of single-crystal KMnF${}_3$ have been investigated by superconducting quantum interference device (SQUID) magnetometry and resonant ultrasound spectroscopy (RUS) through the sequence of phase transitions: phase I, $Pm$$\overline{3}$$m$ $\to (T_{\mathrm{c}1}$ $=$ 185 K) → phase II, $I$4/mcm $\to (T_{\mathrm{c}2}$ $=T_{\mathrm{N}}$ $=$ 87 K) → phase III, antiferromagnetic, Cmcm → ($T_{\mathrm{c}3}$ $=$ 82 K) → phase IV, canted ferromagnet, Pnma. It is concluded that observed changes in the elastic properties can be explained simply in terms of strain/order parameter coupling for the octahedral tilting transitions. There appears to be no evidence in the present data or in data from the literature for coupling between the magnetic order parameter and shear strains. Any coupling between the magnetic and structural transitions is therefore weak, probably occurring only biquadratically through a small common volume strain. The combined data show unambiguously that, for the crystal used, the Néel point and the structural transition at 87 K are coincident. In other crystals, with slightly different stoichiometries and defect contents, this need not be the case, however, and the overlap of transition temperatures in KMnF${}_3$ is essentially accidental. Strong acoustic dissipation at ∼0.1–1 MHz in the stability field of phase II is attributed to the local mobility of transformation twin walls under externally applied stress. A Debye-like loss peak near 130 K is attributed to pinning of at least some twin walls by defects, but relatively high levels of acoustic dissipation below this freezing temperature imply that some of the twin walls remain mobile due to weak pinning or the absence of any pinning. Acoustic losses continue in the stability field of phase III (Cmcm) but diminish substantially in the stability field of phase IV (Pnma), implying quite different twin mobilities in the different structure types. Overlap of the structural and magnetic instabilities in KMnF${}_3$ opens up possibilities for manipulation of ferroelastic twinning by application of a magnetic field and for creation of materials in which the ferroelastic twin walls have quite different magnetic properties from the matrix in which they lie.

Figure 1

Single-crystal elastic constants for KMnF${}_3$ from the literature. (a) The original temperature scale of Reshchikova et al. (Ref. 67) has been adjusted so that the discontinuity in $C_{44}$ occurs at 186 K instead of 200 K, and the temperature scale of Cao and Barsch (Ref. 68) was lowered by 3 K so that the two sets of data overlap. The cubic↔tetragonal phase transition is marked by softening of $\frac{1}{2}\left(C_{11}-C_{12}\right)$ as $T$$\to T_{\mathrm{c}1}$ from above and below, and a small discontinuity in $C_{44}$ at $T_{\mathrm{c}1}$. Any change in bulk modulus, $K$, is within the scatter of the data. (b) Variation in the square of the velocity of longitudinal acoustic waves parallel to [100] (proportional to $C_{11}$) measured at 11.7 MHz. I → II and II → III transitions at 187.5 and 91.5 K are marked by softening from both sides and a net softening of phase II with respect to phase I and of phase III with respect to phase II.

Figure 2

SQUID measurements of the magnetic moment in KMnF${}_3$ under conditions of nominally zero applied field. (a) The phase transitions near $T_{\mathrm{c}2}$ $=$ 87.3 K and $T_{\mathrm{c}3}$ $=$ 80/82 K are clearly visible. (b) Detail from (a) plotted as (magnetic moment)${}^{-1}$, proportional to inverse susceptibility, showing abrupt discontinuities at 87.3 and 79.7 K during cooling or at 82.2 and 87.3 K during heating. (c) Detail from (a) plotted as (magnetic moment)${}^{-1}$, showing a subtle hysteresis in magnetic properties through the cubic↔tetragonal transition. Both phases are paramagnetic, so the variations are assumed to be due to paramagnetic anisotropy and differences in the topology of transformation twins present in the tetragonal phase. The break in slope at 200 K during heating suggests that some microstructure, probably tweed, remains in the cubic phase above $T_{\mathrm{c}1}$ $=$ 186 K.

Figure 3

SQUID measurements of the magnetic moment in KMnF${}_3$ under conditions of applied field. (a) Moment at fields up to $\pm$15 kOe for specific temperatures and a field of 50 Oe during heating and cooling between these temperatures. (b) Detail of hysteresis measurements showing that the crystal is paramagnetic at 220, 120, and 85 K but has a ferromagnetic moment at 60 and 40 K. (c) Detail of the hysteresis measurements made at 40 K. (d) Detail of moment measurements as a function of temperature, showing significant differences before and after application of the 15-kOe field in the stability field of phase IV.

Figure 4

Segments of RUS spectra collected during heating, stacked in proportion to the temperature at which they were collected. There are obvious minima in peak frequencies at ∼185 and 83 K but no obvious change at 87 K. Marked broadening of resonance peaks in the temperature interval ∼83–185 K is attributed to the mobility of transformation twin walls.

Figure 5

Variations of the square of the frequency, $f$, and inverse mechanical quality factor, $Q^{-1}$, for selected peaks in RUS spectra collected during heating. Values of $f^2$ for different peaks have been rescaled so as to overlap near room temperature. The step at 185 K is correct in magnitude in the sense that it was possible to follow peaks through the transition at this temperature. The magnitude of the step at 83 K is not reliable, however, since it was not possible to follow peaks through the transition at this temperature with any degree of confidence. The $f^2$ values for different peaks below 83 K were rescaled to produce an overlap near 10 K. The peak in $Q^{-1}$ near 130 K and the increase in $f^2$ below this temperature are typical of a Debye-like dissipation process and are attributed here to freezing of transformation twin-wall motion by pinning to defects.

Figure 6

(a) Details of the variations with temperature of $f^2$ and $Q^{-1}$ for a resonance peak with $f$ ≈ 453 kHz at 83 K (data collected during heating). There is clearly a change in the trend of both $f^2$ and $Q^{-1}$ at 83 K but no obvious change at 87 K. (b) There is a clear break in slope of $f^2$ as a function of temperature at 87 K; straight lines are fits to the data in two different temperature intervals.

Figure 7

Comparison of $f^2$ data (left axis) for selected resonance peaks with values of Young's modulus for [110] and [100] directions calculated Eqs. (1) and (2) using the single-crystal elastic constants of Cao and Barsch (Ref. 68; note that the temperature scale of Cao and Barsch was lowered by 3 K, as shown in Fig. 1). Young's modulus values (right axis) have been rescaled so as to match the RUS data near room temperature. The observed resonance frequencies are consistent with resonances of the single-crystal plate being influenced by motions comparable with three point bending, i.e., including the influence of both $\frac{1}{2}\left(C_{11}-C_{12}\right)$ and $C_{44}$.

Figure 8

Fits of (6) to the Debye-like peaks in $Q^{-1}$ from Fig. 5. Solid lines are baselines for the fits, and broken lines are the fits themselves.

Figure 9

Variations of single-crystal elastic constants of K${}_{0.982}$Ca${}_{0.018}$F${}_3$ measured at 15 MHz, from Schranz et al.[35] Successive phase transitions corresponding to I↔II (199 K), II↔III (131 K), and III↔IV (112 K) are apparent. The transition temperatures shown are taken from the data for $C_{11}$. Data for $C_{44}$ have changes in trend that appear to place the transition temperatures a few degrees lower than these values. Note that all three transitions are marked by discontinuities in $C_{11}$ but that the transition near 131 K gives a change in slope for $C_{44}$. This is consistent with a strain/order parameter coupling model describing a sequence $Pm$$\overline{3}$$m$↔$I$4/mcm↔Cmcm↔Pnma.

Figure 10

RUS spectra collected at 189 and 220 K during heating and cooling show a distinct hysteresis in peak positions. Peaks in the spectra collected at 189 K have markedly different frequencies, but the difference between heating and cooling is small at 220 K. This corresponds to a significant difference of the bulk elastic properties between the crystal prior to being cooled through $T_{\mathrm{c}1}$ and after being heated from ∼10 K. It is interpreted as signifying the presence of a tweed microstructure in a small temperature interval above $T_{\mathrm{c}1}$. On cooling, the tweed will develop with all possible orientations, but during heating some orientations of tweed may develop preferentially from a tetragonal crystal with preferred orientations of twin domains.