Strain relaxation dynamics of multiferroic orthorhombic manganites

Figure 3 shows results for f² and Q⁻¹ from fitting of resonance peaks in spectra from the single crystal of GdMnO₃ collected in a heating sequence using the helium flow cryostat. The automated sequence involved cooling in 30 K steps from 280 K down to 10 K followed by heating from 10 to 50 K in 1 K steps, from 50 to 70 K in 2 K steps and from 70 to 295 K in 5 K steps. Data for f² from seven resonance peaks have been scaled along the y-axis in an arbitrary manner in figure 3(a) so as to allow easy comparison of their temperature dependences. The clear overall trend is of increasing steepness of elastic stiffening with falling temperature, without the leveling off toward zero slope as T → 0 K expected for a normal crystalline material. Anomalies seen in the primary spectra (lower half of figure 2) are a small dip in f² at T_N1 and a small increase at T_N2. The different combinations of elastic moduli represented by f² for the different resonances vary by up to ∼10%, with a dip of ∼0.4% at T_N1 and a small increase at T_N2.

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The steepening trend of f² with falling temperature is accompanied by two broad peaks in Q⁻¹, centred at ∼80 and ∼150 K (figure 3(b)), as is typical of Debye-like freezing processes. Resonance modes with the largest values of Q⁻¹ (f ∼ 500 and 1050 kHz at room temperature) also have the largest changes in f², consistent with this interpretation. Q⁻¹ drops to low values as T_N1 is approached from above and there is no obvious anomaly at T_N2. This pattern of acoustic loss is essentially the same for all resonances, from which it is concluded that all the single crystal moduli are affected in more or less the same way. This, in turn, implies that the loss mechanisms do not have a strong dependence on the orientation of the shear strain that applies in each resonance mode.

Data collected separately in the Teslatron cryostat extended to lower temperatures, as illustrated in figure 4 for a sequence of cooling followed by heating in small steps through the magnetic transitions. At this level of detail, the transition at T_N1 is fully reversible and occurs close to where Q⁻¹ drops to its lowest values. There is then a hysteretic transition in the expected region of T_N2, with a break in slope of f² occurring at ∼16 K during cooling and ∼19 K during heating. This is followed by a second additional hysteretic transition at ∼8 K during cooling and ∼12 K during heating, below which the resonance frequencies all revert back to the trend established above T_N2. The second hysteretic transition correlates with T_N3 in the pattern shown for GdMnO₃ A in figure 1. On this basis, the A-type AFM structure stable between T_N2 and T_N3 is slightly stiffer than both the colinear-sinusoidal incommensurate structure (above T_N2) and the ab-cycloid structure (below T_N3).

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Finally, there is a small reversible break in slope of the temperature dependences of f² at ∼5 K. On the basis of studies in the literature, as summarised in figure 1, this appears to correspond to T_R. Goto et al [4] and Kimura et al [3] showed a stability field for the weakly ferromagnetic, paraelectric structure below T_R, i.e. that there is a transition from the ab-cycloid structure back to the canted antiferromagnetic structure with falling temperature. This would be consistent, in particular, with the data shown in figure 14 of Kimura et al [3] which show ferroelectric polarisation parallel to [100] only between ∼8 and ∼5 K. However, the RUS data in figure 4 do not show a return to the stiffer trend of the A-type AFM structure below T_R, so the small anomaly may be due only to the development of long-range ordering of the magnetic moments of Gd³⁺.

Figure 5 shows results for f² and Q⁻¹ from fitting of resonance peaks in spectra from TbMnO₃, crystal 1, collected in a heating sequence using the helium flow cryostat. The full sequence involved cooling in 30 K steps at nominal temperatures from 280 K down to 10 K followed by heating from 10 to 60 K in 2 K steps and from 60 to 295 K in 5 K steps. In a second run, spectra were collected during heating from 120 to 210 K in 2 K steps. The data in figure 5(a) for f² from 8 resonance peaks have again been scaled along the y-axis in an arbitrary manner to allow easy comparison of their temperature dependences.

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As in the case of GdMnO₃, the evolution of f² for each peak follows a steepening trend with falling temperature. However, the two resonances with frequencies near 1510 and 1560 kHz at room temperature have a flat variation between 300 and ∼150 K, which almost becomes a slight softening. The most obvious anomaly in the evolution of f² is a small dip at 41 K, consistent with the value of T_N1 ≈ 42 given by Mufti et al [40]. There is perhaps a further small change in the temperature dependence of f² at the expected value of T_N2 = 26 K, but this is only at the level of noise. Overall, the elastic moduli vary by up to ∼10%, and by a very small fraction of this at the transition temperatures. Small breaks in slope of f² near 160 K may or may not be real as they occurred in the interval where the peaks were broadest and hardest to follow.

There are again two broad peaks in Q⁻¹, centred at ∼80 and ∼150 K (figure 5(b)), consistent with Debye-like freezing processes. The resonance modes with the largest values of Q⁻¹, i.e. those with frequencies near 1025, 1325 and 1865 kHz at room temperature, also have the largest changes in f², while the resonance with frequency near 1510 kHz shows the lowest values of Q⁻¹ and the smallest overall change in f². These differences imply that there is some slight dependence of the loss mechanism on the orientation of induced strains but no obviously systematic dependence on frequency. Q⁻¹ drops to low values as T_N1 is approached from above and does not display any obvious anomaly at T_N2.

Data collected separately in the Teslatron cryostat extended to lower temperatures, as illustrated in figure 6 for a sequence of cooling followed by heating in small steps through the magnetic transitions. At this level of detail, the transition at T_N1 is reversible and again occurs essentially where Q⁻¹ drops to its lowest values. There is a very slight dip discernible in the evolution of f² at ∼26 K, corresponding to the expected position of T_N2. As in the case of GdMnO₃, there are no overt anomalies in Q⁻¹ associated with either of these two transitions. In contrast with GdMnO₃, however, Q⁻¹ values extracted from all resonance peaks show a broad peak centred at ∼13 K. This is above the value of T_R ≈6 K reported in the literature and the temperature of ∼8 K where Mufti et al [40] observed a break in slope of the electric polarisation parallel to [001].

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Figure 7 contains f² and Q⁻¹ data for a representative set of resonance peaks collected from the second TbMnO₃ crystal (crystal 2) in the Teslatron cryostat. Spectra were collected in a sequence of cooling followed by heating through the temperature interval 2–295 K. The pattern of variations of f² is closely similar to that of crystal 1 (figure 5), and is also fully reversible between cooling and heating. The pattern of variations of Q⁻¹ is different, however. Firstly, the peak in Q⁻¹ at ∼80 K of crystal 1 is still present but is not as obvious for all resonances from crystal 2. Secondly, the peak at ∼150 K from crystal 1 is either absent in the data from crystal 2 or is hidden by the steep rise of Q⁻¹. Finally, Q⁻¹ returned to low values above ∼200 K for crystal 1 but continued to increase for crystal 2. There is perhaps a peak centred at ∼260 K in figure 7(b). In addition, the maximum value of Q⁻¹ near 90 K varies more substantially between resonances, implying that the loss process is more sensitive to the orientation of the strain induced in each resonance mode. The resonance with frequency near 1290 kHz actually softens with falling temperature down to ∼100 K and has the lowest values of Q⁻¹. All the resonance peaks displayed essentially the same peak in Q⁻¹ centred at ∼10 K, a few degrees above the expected value of T_R.

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At the level of detail shown in figure 8 for the same data up to 60 K, the transition at T_N1 is reversible and, as before, occurs essentially where Q⁻¹ drops to its lowest values. There is a very slight dip discernible in the evolution of f² at ∼27 K, corresponding to the expected position of T_N2. There are, again, no overt anomalies in Q⁻¹ associated with either of these transitions. The peak in Q⁻¹ values at ∼10 K is observed for all resonances and there is a small softening step in f² values with falling temperature below ∼8 K.

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RUS spectra collected from the TbMn_0.98Fe_0.02O₃ single crystal in an automated sequence involved cooling in 30 K steps at nominal temperatures from 280 K down to 10 K followed by heating from 10 to 70 K in 2 K steps and from 70 to 295 K in 5 K steps. Figure 9 shows results for f² and Q⁻¹ from fitting of eleven resonance peaks. The overall pattern for some modes is of significant elastic softening with decreasing temperature followed by stiffening. The others show more uniform stiffening. The total variation is by up to ∼7%, with a small dip at T_N1 ≈ 38 K and an even smaller change in slope at T_N2 ≈ 22 K. There is a broad asymmetric peak in Q⁻¹ at ∼150 K. Differences in the maximum values between different resonances imply that the loss process involves coupling with some specific orientation of induced strain. Subsequent measurements in the Teslatron cryostat showed that f² variations were fully reversible between cooling and heating.

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Data collected separately in the Teslatron cryostat extended to lower temperatures, as illustrated in figure 10 for a sequence of cooling followed by heating in small steps through the magnetic transitions. As for all the other samples, the transition at T_N1 is reversible. There is a very slight, rounded dip discernible in the evolution of f² for the lowest frequency peak (∼270 kHz) at ∼22 K, corresponding to the expected position of T_N2. In contrast with the other samples, there is a frequency dependent increase in Q⁻¹ starting at ∼30 K for the 1130 kHz peak and at ∼22 K for the 270 kHz peak. There is also an asymmetric peak in Q⁻¹ values with a maximum at ∼7–10 K, which correlates with smooth but slight increases in f². There is no obvious anomaly in the evolution of f² at the expected value of T_R ∼ 6.5 K.

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A thermally activated Debye freezing process observed in measurements made as a function of temperature at constant frequency can be described by (following [33, 61, 62])

$$\begin{equation}{Q}^{-1}\left(T\right)={Q}_{\mathrm{m}}^{-1}{\left[\mathrm{cosh}\left\{\frac{{E}_{\mathrm{a}}}{R{r}_{2}\left(\beta \right)}\left(\frac{1}{T}-\frac{1}{{T}_{\mathrm{m}}}\right)\right\}\right]}^{-1}.\end{equation} \tag{ 1 }$$

The temperature, T_m, at which Q⁻¹ has its maximum value, Q_m, is determined by the condition ωτ = 1, where τ is the relaxation time for the loss mechanism and ω is the angular frequency (=2πf) at which the measurement is made. E_a is an activation energy, R is the gas constant and r₂(β) is a width parameter which defines a spread of relaxation times for the dissipation process.

Raw data for Q⁻¹ from four of the samples described above show evidence of overlapping peaks which, in lowest order, can be represented in terms of a single loss process with T_m in the vicinity of 10 K and two loss processes with T_m in the vicinities of 80 and 150 K, respectively. Fits of equation (1) to the higher temperature data for selected resonances from the four samples investigated in the present study and from the single crystal of Sm_0.6Y_0.4MnO₃ described elsewhere [30] are shown in figures 11(a)–(e). Reasonable fits to the separate peaks were generally obtained, though it was necessary to constrain some of the parameters due to overlaps and noise. The overall result is that a plausible description of the variations of Q⁻¹ for all five samples in the higher temperature range is in terms of two peaks with E_a/r₂(β) ≈ 0.03–0.05 and 0.08–0.1 eV. By somewhat arbitrarily combining data for the different samples in a single Arrhenius plot (figure 11(f)), values of T_m for the loss peak near 80 K can be represented by $\tau ={\tau }_{\mathrm{o}}\enspace \mathrm{exp}\left({E}_{\mathrm{a}}/\mathrm{R}T\right)$, with τ_o = 2.5 × 10⁻¹⁰ s, E_a = 0.044 ± 0.007 eV. In combination, the two fits would constrain r₂(β) to be ∼1 for the 80 K loss peak, implying that the relaxation mechanism involves a single relaxation time. Values of T_m from fits of equation (1) in the vicinity of 150 K do not produce the same quality of correlation and are therefore not shown. This is not surprizing given that the data indicate more than one loss process in different samples.

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The steep increase in Q⁻¹ with increasing temperature above ∼250 K is present only in the data from TbMnO₃ crystal 2 and Sm_0.6Y_0.4MnO₃ which were grown under argon rather than in air. The loss mechanisms in this temperature interval are evidently sensitive to the oxygen content of the sample.

Figure 12(a) contains fits of equation (1) to representative peaks in Q⁻¹ which have T_m near 10 K. In reality there are probably two frequency dependent peaks in the vicinity of 8 and 15 K but the latter is not readily resolved in data from many of the other resonances. Average values of E_a/r₂(β) are ∼0.003 eV for TbMnO₃ crystal 1 and TbMn_0.98Fe_0.02O₃, and ∼0.004 eV for TbMnO₃ crystal 2. Figure 12(b) is an Arrhenius plot of results from fits to different resonance peaks of TbMnO₃ crystal 2 and TbMn_0.98Fe_0.02O₃. The straight lines fit to the data give values for E_a of 0.0017 ± 0.0003 and 0.0021 ± 0.0007 eV respectively. Corresponding values of τ_o are 1.4 × 10⁻⁸ s and 1.1 × 10⁻⁸ s. From the evidence of overlapping peaks in Q⁻¹ it is likely that there are two loss mechanisms with each having E_a in the vicinity of 0.002 eV.

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Figure 13 presents a comparison of representative acoustic loss data, Q⁻¹, for GdMnO₃ and TbMn_0.98Fe_0.02O₃ from this study with dielectric loss, tan δ (GdMnO₃ data from Vilarinho et al [25], for a ceramic sample; TbMn_0.98Fe_0.02O₃ unpublished data of A Maia, measured parallel to [001] of a single crystal from the same boule as the crystal used for RUS). The most prominent feature of the tan δ data shown for GdMnO₃ (figure 13(a)), a frequency-dependent peak in the temperature interval ∼200–250 K, does not appear to correlate with any of the data for Q⁻¹. Arrhenius treatment of the temperature at which ωτ = 1 gave an activation energy of 0.28 ± 0.03 eV. The pattern is similar for TbMn_0.98Fe_0.02O₃, though the peaks are shifted to higher temperatures (figure 13(b)). On the other hand, there is close correlation between Q⁻¹ variations and a very much weaker peak in tan δ at ∼60–70 K (figures 13(c) and (d)). Arrhenius treatment using the condition ωτ = 1 for this peak at 500 kHz and 1 MHz gave E_a = 0.019, τ_o = 7 × 10⁻⁹ s and 0.026 eV, 3 × 10⁻⁹ s, respectively for GdMnO₃ and TbMn_0.98Fe_0.02O₃. Ferreira et al [21] obtained E_a = 0.018 eV from their more comprehensive data set for GdMnO₃.

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Fitting the results for tan δ collected as a function of temperature at 500 kHz with the equivalent expression to equation (1) gave E_a/r₂(β) ∼ 0.016 and 0.019 eV, consistent with r₂(β) being close to 1. The value of E_a/r₂(β) from fitting of the peak in Q⁻¹ near 80 K is ∼0.04 eV. Although the acoustic data are too noisy to resolve a frequency dependence of the loss peak, this is at least permissive of it being due to a loss mechanism which involves freezing of local electric dipole motion coupled with strain. Values of both Q⁻¹ and tan δ reduce steeply as T → T_N1 for both samples represented in figure 13. There are anomalies in tan δ at T_N2, but their form is quite different for GdMnO₃ and TbMn_0.98Fe_0.02O₃ (figures 13(c) and (d)), implying that the dielectric relaxation mechanism involves only very weak coupling with strain in this case.

The dielectric constant of TbMnO₃ shows the same steep frequency-dependent increase with increasing temperature above ∼120–150 K [18, 19, 26]. Cui et al [18] reported activation energies of 0.15 and 0.31 eV for peaks in tan δ at T < 160 K, and T > 170 K, respectively, for their polycrystalline sample. The latter is most likely to be due to the same loss process as accounts for the peak in tan δ at ∼200–250 K in GdMnO₃. The former is perhaps due to the same loss mechanism as would account for the acoustic loss peak seen at ∼150 K in all four samples.

A formal analysis of spontaneous strains associated with phase transitions in these multiferroic perovskites, based on lattice parameter and high resolution thermal expansion data from the literature, is presented in the appendix A. There are potentially five discrete order parameters, relating to cooperative Jahn–Teller distortions, octahedral tilting, two for magnetic structures based on ordering of Mn³⁺ and magnetic ordering of Gd³⁺/Tb³⁺, each of which will be coupled to some extent with strain. The combination of octahedral tilting and cooperative Jahn–Teller distortions gives rise to two non-zero shear strains, e₄ and e_tx , with respect to the parent cubic structure [63]. e_tx is a tetragonal strain with its unique axis parallel to the [001] in the Pbnm setting and e₄ is a shear strain in the plane perpendicular to this, i.e. containing a and b of Pbnm. Values of e₄ ∼ 10% and e_tx ∼ −7% listed in table A1 for GdMnO₃, TbMnO₃ and Sm_0.6Y_0.4MnO₃ at room temperature are a reflection of the large contribution from the Jahn–Teller distortion of Mn³⁺. By way of comparison, pure tilting transitions in GdAlO₃, TbAlO₃ and CaTiO₃ give values of e₄ ∼ 1% and e_tx ∼ −0.5% at room temperature (table A1). The strains at room temperature are also significantly greater than from the combined contributions of tilting and Jahn–Teller distortions in LaMnO₃, e₄ ∼ 3%, e_tx ∼ −4% (table A1). Thus the low temperature magnetic transitions take place in crystals which have a high degree of anisotropic shear strain [1].

Calculating linear strains with respect to the crystallographic axes of the Pbnm structure (table A1) shows that the preferred orientation of magnetic moments in the colinear-sinusoidal antiferromagnetic structure and the propagation direction of both the ab- and bc-cycloidal magnetic structures ([010]), is parallel to the direction in which the Jahn–Teller distortions produce elongation by ∼7% with respect to the reference cubic structure. Linear strains in the other two directions, e₁ and e₃, are both negative (contraction).

High resolution thermal expansion data of Meier [64] (see also [47, 65, 66]), show that spontaneous strains arising from coupling with the magnetic order parameters in GdMnO₃ and TbMnO₃ are about two orders of magnitude smaller than those due to the combination of tilting and Jahn–Teller distortions. They vary in the range ±∼0.0002 and, in combination, give a volume strain, V_s, of up to ∼−0.0004 (figure A2(b)). Linear strains occur ahead of T_N1 in both GdMnO₃ and TbMnO₃ and, although the temperature of onset and the magnitudes are not well constrained by the data owing to the need to define baselines for the reference structure (figures A2 and A3), these correlate with a well-defined change in trend of the temperature dependence of phonon-frequencies below ∼100 K [21, 67].

The sign and trend of each of the linear strains GdMnO₃ and TbMnO₃ (figures A3(b) and (d)) is similar above and below T_N1, suggesting that the local structural and magnetic changes responsible for the precursor effects are closely related to the static colinear-sinusoidal incommensurate magnetic structure. Linear strains coupled to the magnetic order parameter(s) then display differences between the two materials, reflecting differences in details of the evolution of the magnetic structures at lower temperatures. Any additional anomalies in linear strain variations through and below T_N2 are barely detectable in TbMnO₃ (figures A3(e)–(g)). In GdMnO₃, which has a different orientation of cycloid plane, there is a reversal of the trends of e₁ and e₂ with falling temperature (figure A3(b)). The final small adjustments below T_R, accompanying ordering on both the Tb and Gd sublattices, are positive for e₃ and negative for e₁ and e₂, (figures A3(b), (e)–(g)).

Volume strains are, in effect, an integration of the contribution of all ordering of moments irrespective of the actual ordering scheme and are essentially indistinguishable between GdMnO₃ and TbMnO₃ (figure A2).

Variations of spontaneous strains due to magnetic ordering in NdMnO₃ are shown for comparison in the appendix A, using original data of Meier [64] (and see also [66]). The paramagnetic to A-type antiferromagneic transition at ∼85 K, with slight canting to give a weak ferromagnetic moment parallel to [001] [1, 2, 68, 69] is accompanied by linear strains with a temperature dependence which is typical of a co-elastic phase transition. There are small contributions from precursor effects, but the magnitudes of the strains that develop below the Néel point are comparable with the combined contributions below and above ∼40 K in GdMnO₃ and TbMnO₃. Moreover, the onset temperatures for the precursor strains in the latter are more or less the same as in NdMnO₃ (compare figures A3(b) and (d), A4). The volume strains for all three materials have closely similar temperature dependences (figures A2(b), A4(b)), and there is only a slight break in slope at T_N1 of GdMnO₃ and TbMnO₃ in comparison with the sharper and more classical break in slope at the transition point of NdMnO₃. The implication is that the total strain coupling is the same for all three materials but that a substantial proportion of this arises from local magnetic ordering ahead of the phase transitions in GdMnO₃ and TbMnO₃. The small change in strain due to ordering of Nd moments below ∼20 K in NdMnO₃ (figure A4(b)) is comparable in magnitude with the small additional strains that occur below T_R in GdMnO₃ and TbMnO₃.

The fact that changes in resonance frequencies at T_N1 and T_N2 described above confirms that coupling of the magnetic order parameters with strain is weak. Lowest order coupling terms of linear strains, e₁, e₂, e₃, with the driving order parameter, M, for magnetic transitions in an orthorhombic crystal will have the form λeM². These would be expected to give rise to the classic pattern of elastic softening below a second order transition shown first for the displacive transition in SrTiO₃ [70]. However, such softening requires that the order parameter can relax on the same time scale as the variations of strain in the acoustic resonance modes. This is seen at the antiferromagnetic ordering transition in CoF₂, for example, where the magnitude of the coupled strains is also less than ∼0.001 [71]. Accompanying the elastic anomaly in CoF₂ is a classic peak in Q⁻¹ due to critical slowing down as the transition point is approached from above and below. The absence of both effects through the magnetic transitions in any of the crystals investigated here implies that relaxation of the magnetic order parameter in response to an induced strain is slower than ∼10⁻⁶ s. Evidence of the Debye and dielectric loss peaks at ∼80 K is that switching of local electric dipoles is already reduced to less than this timescale before the transition temperatures are reached. If there is coupling between local magnetic and electric dipole moments, the relaxation time of electric dipoles would provide the rate limiting step for relaxation of the magnetic order parameter.

Coupling of the form λe² M² is always allowed and does not require dynamical relaxation. It will give stiffening or softening in proportion to M², depending on the sign and magnitude of the coupling coefficient, λ, (as set out in detail in [72], for example). The variation of f² at T_N1 suggests slight stiffening (λ positive). Any anomaly at T_N2 is barely detectable in TbMnO₃ (figures A3(d)–(g)), consistent with the colinear-sinusoidal to cycloidal magnetic transition being accompanied by changes in the magnitudes of the strain coupling coefficients that are negligibly small. Anomalies in f² at T_N2 and T_N3 in GdMnO₃ indicate that there are small differences in the strength of coupling with the magnetic order parameters of the colinear-sinusoidal incommensurate, antiferromagnetic and cycloidal structures.

Given that there are small changes in e₁, e₂ and e₃ near T_R for both GdMnO₃ and TbMnO₃ (figures A3(b)–(g)), ordering of Gd³⁺ and Tb³⁺ moments might to be expected to give rise to closely similar changes of elastic properties. Rounding of the peak in heat capacity (e.g. [42, 45, 52, 55, 73, 74] for GdMnO₃) is indicative of ordering in a field created by the ordered Mn³⁺ moments rather than at a discrete phase transition, but there are indeed small anomalies in the temperature dependence of f² close to the expected values of T_R (figures 4 and 8). However, there is a distinct Debye-like peak in Q⁻¹ ahead of T_R in the three crystals containing Tb but not in data from the crystal containing Gd. It appears, therefore, that there is a defect pinning or freezing process which is unique to the cycloid structure.

An increase in the imaginary part of the ac magnetic susceptibility of TbMnO₃, measured at frequencies of 0.5–3 kHz, has also been observed below ∼10 K [29]. The form of this is consistent with there being a peak in magnetic loss below the lowest temperature data point of 5 K. Extrapolation to a measuring frequency of 3 kHz of the straight line fit to the acoustic loss data for crystal 2 in figure 12(b) would give ωτ = 1 at ∼3 K. In other words the data are indicative of a magnetoelastic relaxation mechanism, with dynamics constrained by an activation energy of ∼0.002 eV. The loss mechanism is, as yet, undefined.

Two separate loss peaks in figure 12(a) imply two separate processes with slightly different relaxation times. One possibility for the loss mechanisms relates to freezing of dynamical motion(s) of Tb³⁺ moments which have some alignment with Mn³⁺ moments at 15 K in TbMnO₃ [38], i.e. ahead of T_R. A second possibility is suggested by low temperature dielectric spectroscopy data between 5 and 16 K for DyMnO₃ which were interpreted as being due to motion of boundaries between ab- and bc-cycloid domains on a timescale of ∼10⁻⁷ s [75]. However, there should not be any shear strain contrast across domain walls of the bc-cycloid so that an externally applied shear stress would not be expected to cause them to move. The significant point is that the activation energy barrier is substantially smaller than for the loss process at ∼80 K which is interpreted as being due to coupling of local electric dipoles with polaron-like strain clouds.

Strain coupling is evidently a significant component of the overall structural and thermodynamic evolution of these systems. The most obvious effect is seen in the relationship between the preferred direction of incommensurate modulations and the orientation of the large strains due to Jahn–Teller distortions and octahedral tilting. The sinusoidal and cycloidal modulations have their propagation directions parallel to [010], the direction in which strain coupled to the Jahn Teller/tilt order parameters is positive (e₂ in table A1). In the other two orthogonal directions, the strains are negative, representing lattice contraction.

Linear strains accompanying the low temperature transitions are very much smaller. They do not have such an obvious orientation relationship with the magnetic structures, but they are clearly dependent on subtle differences in the evolution of the order parameters. On the other hand, the volume strain is essentially an integration of volume reductions associated with ordering of moments at each Mn³⁺ ion, irrespective of which long-range ordering scheme is adopted. The evolution of V_s is essentially the same for TbMnO₃ and GdMnO₃ and has the same form as in NdMnO₃ (figures A2(b) and A4(b)). In NdMnO₃ the main transition occurs at ∼85 K and is from a paramagnetic structure to an A-type antiferromagnetic structure (e.g. [1, 2, 68, 69]). It is notable, in particular, that the onset of volume strains in TbMnO₃ and GdMnO₃ can be detected from ∼90 K (figure A2(b)), indicating that short-range magnetic ordering develops well above T_N1 in a temperature range that is not dissimilar from that of the long-range ordering in NdMnO₃. There are small volume strains associated with ordering of the moments of Gd³⁺, Tb³⁺ and Nd³⁺, but these are an order of magnitude smaller.

The pattern of volume strains correlates with the pattern of phonon frequencies in the sense that selected modes of NdMnO₃, TbMnO₃ and GdMnO₃ all show an onset of softening through a similar temperature interval [2, 67, 80].

Precursor short-range ordering ahead of T_N1 is seen also in other properties. The dc magnetic susceptibility of both polycrystalline and single crystal samples of GdMnO₃ has been found to follow the Curie–Weiss law only down to ∼80 K [29, 41, 74]. In TbMnO₃ the onset of deviations from Curie–Weiss evolution of the inverse magnetic susceptibility is between ∼80 K, measured parallel to [100], and ∼200 K, measured parallel to [001] [52]. O'Flynn et al [52] already interpreted this, together with the magnetic field dependence of the heat capacity at 60 K and a significant precursor effect in the heat capacity (seen also in Sm_0.6Y_0.4MnO₃ [58]), in terms of short-range correlations of magnetic moments within the stability field of the paramagnetic structure ahead of T_N1. A similar conclusion was reached for GdMnO₃ by Ferreira et al [21] and Vilarinho et al [80] on the basis of their observation of changes in phonon frequencies below ∼100 K, again well ahead of the low temperature phase transitions. By way of contrast, the inverse magnetic susceptibility of GdAlO₃, which has magnetism arising only from Gd³⁺, remains linear right down to the Néel point of ∼4 K.

The overall implication is that freezing of local ferroelectric dipoles in these manganites is accompanied by the development of a well-developed precursor state involving coupling between local strain and static or dynamical short-range magnetic order. TbMnO₃ also has an anomaly in thermal conductivity which starts to develop below ∼150 K [66].

Acoustic attenuation near 150 K is most likely related to the increase in electrical conductivity with increasing temperature in all the samples, which accounts also for the steep increase in dielectric loss. In GdMnO₃, the activation energy for ac conductivity through the interval ∼125–175 K was reported by Pal and Murugavel [81] to be ∼0.13 eV and the conductivity mechanism was attributed to small polaron hopping. Analysis of a peak in pyrocurrent at ∼120 K by Zhang et al [82] yielded an activation energy of ∼0.11 eV and was also attributed to dipole reorientation or release of charge from localised states. This compares with E_a/r₂(β) values also close to ∼0.1 eV from fitting of the peak in Q⁻¹ (figure 11). The pattern of loss in this temperature range is also sensitive to the oxygen content of the sample, as seen by the difference between Q⁻¹ data above ∼120 K for TbMnO₃ crystal 1 (figure 5(b)), which was grown in air, and crystal 2 (figure 7(b)), which was grown under argon.

The loss peak near 80 K appears to be unique to orthorhombic manganite perovskites, as it was not observed in RUS measurements of Pbnm oxide perovskites that do not display magnetism, ferroelectricity or cooperative Jahn–Teller distortions, such CaTiO₃ [83] and BaCeO₃ [84]. It was also not observed in the multiferroic hexagonal manganite YMnO₃ [31] which has an antiferromagnetic ordering transition at ∼75 K. In view of the close relationship between the acoustic and dielectric properties shown in figure 13, there is little doubt that the loss mechanism involves freezing of electric dipoles with local strain coupling. The loss parameters from fitting the data in terms of a thermally activated process in figure 11(f), τ_o ∼ 10⁻¹⁰ s, E_a ∼ 0.04 eV, provide a measure of the dynamics of this mechanism.

Activation energies of ∼0.07 eV appear to be characteristic of polaronic relaxation in perovskites more generally [85]. This view is reinforced by comparison with the dielectric loss patterns in ferroelectric or incipient ferroelectrics which do not have magnetic transitions. For example, the incipient ferroelectric KTaO₃ remains cubic down to the lowest measuring temperatures but has an acoustic loss peak at ∼60 K with E_a ⩾ ∼0.09 eV, from a resonance with frequency near 740 kHz [86]. There is also a frequency dependent dielectric loss peak in the temperature interval ∼40–60 K with E_a ∼ 0.04 eV [87], which presumably has the same origin.

Precursor strains which occur in the same temperature interval as these Debye-like freezing processes have the same sign as the strains which continue to develop below T_N1 in GdMnO₃ and TbMnO₃ and below T_N2 in TbMnO₃ (figures A3(b) and (d)). On this basis, it appears that the local dynamical structural state is likely to be related to the configuration of long-range magnetic ordering in the colinear-sinusoidal and cycloidal structures. It should be noted, however, that Pal et al [27] reported activation energies of ∼0.03–0.04 eV obtained by analysis of a broad pyrocurrent signal at ∼50 K and a dielectric loss peak in GdMnO₃, which they suggested could be extrinsic due to defect dipoles associated with oxygen nonstoichiometry and mixed valence of Mn.

The importance of frustration effects, arising from ferromagnetic nearest-neighbour and antiferromagnetic next-nearest-neighbour interactions, in promoting the stability of magnetoelectric structures in RMnO₃ perovskites was emphasised from the start by Kimura et al [1]. Competition between different static magnetic ordering schemes results in progressive suppression of the A-type antiferromagnetic structure in the sequence La-, Pr-, Nd-, Sm-, Eu-, Gd-, TbMnO₃ associated with increasing lattice distortions that arise from the reductions in ionic radius. It now appears that suppression of long-range ordering in GdMnO₃ and TbMnO₃ is mitigated by local dynamical ordering which enhances the stability of the paramagnetic structure below ∼80–100 K. The more complex incommensurate and cycloidal structures only develop once the thermally activated fluctuations revealed by the patterns of acoustic and dielectric loss have frozen out. These dynamical effects involve electric dipoles which have a relative strong coupling with strain. In addition, however, there must be contributions from weaker coupling of magnetic moments with strain which freeze out within the stability field of the cycloidal structure. Evidence of these is provided by the acoustic loss peaks near 10 K in the RUS data from TbMnO₃ (figure 12).

Frustration effects could be modified by competing strain fields in doped crystals. Replacing 2% of Mn³⁺ by Fe³⁺ is not likely to affect the magnetic ordering but, by comparison with La substitution in PrAlO₃ [88], corresponds almost exactly with the substitution limit at which strain fields around one dopant cation would start to overlap with the strain fields around nearby dopant cations and thereby modify the properties of the whole crystal. f² for several resonance modes from TbMn_0.98Fe_0.02O₃ (figure 9) show slight softening with falling temperature but there is no evidence of any of the steep softening which would be expected if there was any marked tendency to stabilise a more typical ferroelectric structure, such as in BaTiO₃ for example. The origin of the softening therefore is more likely to be a slight adjustment to the coupling of small strains with the magnetic order parameter(s).

Coupling between strain and magnetic or electric dipoles is extensively exploited in thin film applications since the substrate allows the functional properties to be manipulated through the choice of imposed strain fields. Of particular interest in the present context is the fact that thin films of TbMnO₃ and GdMnO₃ contain abundant self-organised twin domains (e.g. [9, 10, 15, 89, 90]). While it might not be possible to switch these domains by any external field because of their attachment to the substrate, they are in effect ferroelastic twins due to lowering of symmetry from cubic or tetragonal to orthorhombic. Because the tilt, Jahn–Teller and magnetic order parameters all couple with strain, it is inevitable that steep strain gradients through ferroelastic twin walls will cause the magnetic and ferroelectric properties of the twin walls to differ from those of the twin domains. For example, Daumont et al [90] have proposed that a high density of twin walls in a thin film of TbMnO₃ on SrTiO₃ might contribute to ferromagnetism which is not seen in bulk samples.

There do not appear to be reports of ferroelastic twinning in single crystals but if it was possible to grow crystals containing multiple ferroelastic domains walls, such as have been observed in LaMnO₃ for example (see figure 1(a) of Dechamps et al [91]), they would be expected to display quite different magnetoelectric properties and should lead to new possibilities for engineering of functional nanostructures. The mobility of ferroelastic and multiferroic domain walls in such crystals will be constrained significantly by the polaronic freezing effects identified here but, if they are sufficiently broadened by the influence of their magnetic component, they might become able to ride over the strain fields of local pinning points. In DyMnO₃, for example, domain walls between ab- and bc-cycloid domains appear to be mobile on a timescale of less than ∼10⁻⁶ s down to at least 5 K [75].