Elastic and anelastic relaxations accompanying relaxor ferroelectric behaviour of Ba6GaNb9O30 tetragonal tungsten bronze from resonant ultrasound spectroscopy
Tetragonal tungsten bronze (TTB) structures offer some promise as lead-free ferroelectrics and have an advantage of great flexibility in terms of accessible composition ranges due to the number of crystallographic sites available for chemical substitution. The ferroic properties of interest are coupled with strain, which will be important in the context of stability, switching dynamics and thin film properties. Coupling of strain with the ferroelectric order parameter gives rise to changes in elastic properties, and these have been investigated for a ceramic sample of Ba6GaNb9O30 (BGNO) by resonant ultrasound spectroscopy. Room temperature values of the shear and bulk moduli for BGNO are rather higher than for TTBs with related composition which are orthorhombic at room temperature, consistent with suppression of the ferroelectric transition. Instead, a broad, rounded minimum in the shear modulus measured at ~1 MHz is accompanied by a broad rounded maximum in acoustic loss near 115 K and signifies relaxor freezing behaviour. Elastic softening with falling temperature from room temperature, ahead of the freezing interval, is attributed to the development of dynamical polar nanoregions (PNRs), whilst the nonlinear stiffening below ~115 K is consistent with a spectrum of relaxation times for freezing of the PNR microstructure.
KeywordsBulk modulus Ceramics Elastic properties Phase transitions Polar nanoregions (PNRs) Relaxor dielectrics Resonant ultrasound spectroscopy (RUS) Shear modulus Tetragonal tungsten bronzes (TTBs)
The high permittivity of polar dielectrics such as piezoelectrics or ferroelectrics has made these materials particularly useful in electronic devices, with broad application in devices such as capacitors and resonators . To date, much of the search for such new materials has largely centred on perovskite-based (ABO3) materials, but these systems have a number of commonly encountered drawbacks. Firstly, many of the best performing materials require the presence of lead (Pb) in the structure, with associated difficulties associated with its toxicity in manufacturing, use and disposal. Secondly, many of these materials exhibit poor dielectric properties (leakage between active ions) resulting in the inability to sustain an electric field sufficient for hysteretic switching. Thirdly, for multiferroic perovskite systems, exhibiting both electrical and magnetic order simultaneously, the temperature of magnetic ordering is often below ambient temperature, or even when this is not the case, the coupling between magnetic and electrical order at room temperature is often very weak . Nevertheless, some of these problems have been ameliorated either by finding new lead-free materials such as K1−xNaxNbO3 (piezoelectric) , doping by single site substitutions, or by solid solution formation between ferroelectric, antiferro- and ferromagnetic perovskites including BiFeO3–PbTiO3 and BiFeO3–BaTiO3 .
Recently, the tetragonal tungsten bronze (TTB) class of materials—a structure closely related to perovskites, has gathered the attention of the research community. The TTB structure: (A1)2(A2)4(C)4(B1)2(B2)8O30, due to the presence of crystallographically nonequivalent A- and B-sites and an extra C-site, provides supplementary degrees of freedom for manipulation of the structure, huge compositional flexibility allowing the insertion of various metals into the five different TTB sites , nevertheless offering the possibility of fine-tuning both electrical and magnetic behaviour [6, 7]. The TTB structure consists of a network of corner sharing BO6 octahedra formed around the perovskitic A1 site that creates further two types of channels: pentagonal A2 channels (which can be occupied by alkali, alkaline earth and rare earth cations) and smaller triangular C channels (mostly vacant, they can be filled/just partially filled by small low-charged cations like Li+—e.g., K6Li4Nb10O30). These materials, known to exhibit diverse properties as a result of compositional flexibility and by a higher probability for cation ordering, may offer better ways of attaining room-temperature ferroelectricity and (anti)ferromagnetism, multiferroic behaviour and eventually magnetoelectric coupling [2, 8]. Whilst ferroelectric TTBs (including Ba2NaNb5O15 [9, 10, 11] and (Ba,Sr)Nb2O6 [12, 13, 14]) were widely investigated during the 1960s and 1970s, our understanding of manipulating this structure type is still poor, with the research surprisingly limited compared to that in perovskites . Early attempts focused on tungsten bronzes of nominal composition A6B10O30 (mainly compositions where the C-sites are vacant). A particular interest was developed regarding the Nb-based TTBs [15, 16, 17] due to their enhanced ferroelectric properties over other analogues such as Ta [18, 19]. In the search for novel multiferroic and magnetoelectric materials, the effect of the A-site size in a family of unfilled ferroelectric TTBs Ba4RE0.67Nb10O30 (RE = La, Nd, Sm, Gd, Dy, Y)  and of the A-site strain on dipole stability in fully filled TTBs family A6GaNb9O30 (A = Ba, Sr, Ca) [7, 20] was studied. In addition to their ferroelectric and/or magnetic behaviour, the majority of TTBs reported in the literature exhibit relaxor properties [21, 22, 23, 24, 25, 26]. Most TTBs that have been investigated to date are also lead-free materials [21, 22, 27].
In recent years, the research dedicated to novel TTB ferroelectric and ferroelectric-related (i.e., relaxors) materials has undergone a revival, with Ba6FeNb9O30 (BFNO) as a starting point; many related compositions [28, 29, 30, 31] or solid solutions, usually containing lanthanides [32, 33, 34, 35, 36], were studied. Arnold and Morrison , and subsequently Liu et al. , showed that these compounds display relaxor-type behaviour [38, 39, 40], with the peak maxima in the dielectric permittivity occurring in the temperature range 130–150 K. Earlier data indicated that BFNO is not electrically homogeneous , with oxygen vacancy gradients due to the variable oxidation state of Fe (Fe3+/Fe2+), as both low-temperature dielectric spectroscopy (DS) and high-temperature impedance spectroscopy (IS) data revealed a higher number of electroactive regions than expected . In order to avoid these additional complications whilst studying such materials, the replacement of Fe3+ with Ga3+ (similar in size) and other trivalent species like Sc3+ and In3+ was proposed . In previous research, temperature-dependent powder neutron diffraction (TDPND) and microstructural characterisation by scanning electron microscopy (SEM) confirmed the nature of the phases formed and contributed to their crystallographic identification . Moreover, the origin of the polar response and the nature of the relaxor behaviour were established by combining the results of the structural investigations with the dielectric properties inspected by immittance spectroscopy (IS) [7, 41], whilst the dynamics of dielectric relaxation of dipoles was understood by fitting the dielectric data (permittivity and loss) with the Vogel–Fulcher (VF) and the universal dielectric response (UDR) models .
The bulk ferroelectric and magnetic properties of interest in the context of potential device applications involve changes in structure or electronic state which are almost invariably accompanied by lattice distortions that can be characterised as strain. Anisotropic strains are likely to improve the capacity of a material to retain an imposed dipole orientation, but repeated switching is then likely to result in mechanical failure. In thin film applications, the influence of strain imposed by mismatch of lattice parameters with the substrate is also a major consideration in engineering desired properties, in terms of both the stability of induced dipoles and the dynamics of their switching. If there is a change in strain state, it inevitably follows that there is a change in elastic properties which might be fundamental to making a particular ferroelectric distortion stable (for large strains), or simply an associated consequence (small strains). Formally there is coupling between strain, e, and the ferroelectric (or magnetic) order parameter, Q, that can take the form λeQ, λeQ 2, λe2Q2, λe2Q, etc., depending on strict symmetry rules, where the coupling coefficient, λ, represents the strength of this coupling . Such coupling results in the interaction length of the order parameter being determined by the strain field, which is typically long ranging. In this case, the overall thermodynamic behaviour is likely to follow the precepts of mean field theory. Considered from this perspective, it is understandable why investigation of both the strains themselves and of the changes in elastic properties which accompany them can provide insights into the static and dynamic properties of ferroic and multiferroic materials .
In this paper, we present results for the elastic properties of the relaxor dielectric Ba6GaNb9O30 (BGNO) ceramic material, with the TTB structure, obtained using resonant ultrasound spectroscopy (RUS), and relate the temperature-dependent elastic behaviour to the structural phase transitions previously observed by low-temperature dielectric spectroscopy [5, 6]. Resonant ultrasound spectroscopy is a powerful thermal analysis technique—belonging to the mechanical subgroup together with the thermomechanical analysis (TMA) and dynamical mechanical analysis (DMA)—dedicated to the study of fundamental properties involving elasticity [44, 45, 46, 47, 48, 49]. When mechanically excited, solid bodies exhibit natural frequencies (depending on the elastic moduli, size, mass and shape) at which they vibrate [47, 48]. By exploiting this property of solids, the RUS technique determines the elastic tensor (elastic constants) of the material as a function of temperature from the resonance frequencies of the freely vibrating sample  and is an effective tool to monitor variations associated with structural transitions mainly due to its direct nature of sensing the structure . Recently, the elastic properties of several relaxor systems were investigated by the RUS technique [50, 51, 52, 53], and it has been applied by Pandey et al. [54, 55, 56, 57] to investigate relaxor behaviour of Ca- and Sr-doped Ba5Nb10O30 TTBs. We report new experimental determinations of the bulk and shear moduli of polycrystalline ceramic BGNO at room temperature, including corrections for the effects of for porosity , and compare them with values obtained for single-crystal TTB compounds with similar structures . Variations of the shear modulus and of acoustic loss as a function of temperature through interval 10–300 K are interpreted in terms of the dynamics and freezing behaviour of PNRs.
Ba6GaNb9O30 (BGNO) powder was synthesised by standard solid-state techniques. Stoichiometric ratios of dried BaCO3, Nb2O5 and Ga2O3 (all Aldrich, 99+ %) were ball-milled under ethanol until homogenised, 5 min at 400 rpm, using a Fritsch Pulverisette 7 system with agate mortar and balls. The produced powder was placed on platinum foil, in alumina boats and fired in a muffle furnace (static air atmosphere) straight to 600 °C. It was heated from 600 to 1000 °C, left to decarbonate for 1 h and then fired for 12 h at 1250 °C. Afterwards, it was quenched to room temperature, reground and heated back at 1250 °C for 12 h more in alumina boats (placed on platinum foil) inside the same muffle furnace. Finally, a third thermal process was carried out for 12 h at 1300 °C; the boats containing the previously ball-milled powders were placed in the middle of a tube furnace, and radiation shields were used at both ends of the alumina tube to maintain a more stable temperature (the conditions inside the tube were still at atmospheric pressure). For obtaining Ba6GaNb9O30 pellets, the powders were pressed in a 10-mm-diameter uniaxial stainless steel die to 1 tonne, producing a pressure of around 200 MPa followed by sintering in a tube furnace for 12 h at 1250 °C (heating and cooling rates were of 10 K min−1). Phase formation was confirmed by powder X-ray diffraction and temperature-dependent powder neutron diffraction; at room temperature, diffraction data were refined with the centrosymmetric tetragonal space group P4/mbm [5, 41, 42].
Rectangular parallelepipeds within millimetre range of the edge lengths were cut from the pellets described above using a fine annular diamond saw lubricated with paraffin. The pellets were initially glued onto a flat glass surface with Crystalbond glue (with the softening temperature: 120–130 °C), and the cuts were made perpendicular to the glued surface whilst rotating by 90 several times until obtaining complete rectangular parallelepipeds. The glue was removed by heating and washing with acetone. The parallelepipeds were subsequently polished using a stable vertical steel lapping jig, fine polishing paper and a bed of acetone to remove cracks and until achieving optical quality. A parallelepiped-shaped sample of m = 0.1991 mg and ρr = 94.43 %, with dimensions 2.346 × 3.241 × 4.757 mm3, was employed for the resonant ultrasound spectroscopy measurements.
The RUS spectra of the BGNO sample were first collected at room temperature (RT) by exciting the sample at frequencies between 0.1 and 1.2 MHz (65,000 data points per spectrum). The parallelepiped was remounted in different orientations between the two piezoelectric transducers (across corners, edges, faces) for successive spectra so as to ensure that all resonances are observed at least two times. The spectra were analysed offline using the software package Igor Pro (WaveMetrics), whilst the stack of spectra were plotted here as amplitude versus frequency in Origin7.5 software. Bulk (K) and shear (G) moduli were determined by matching—nonlinear least-squares refinement—the observed resonance frequencies with the calculated resonance frequencies using the RPR fitting procedure . The obtained values of the elastic moduli were corrected for porosity following the correction equations proposed by Ledbetter et al.  and compared with those obtained for two single-crystal TTB compounds with similar structure ; the two single-crystal TTBs Ba2.8Sr1.2Na2Nb10O30 and Ba1.2Sr2.8Na2Nb10O30 used for this comparison belong to the (Ba1−xSrx)4Na2Nb10O30 class and were investigated by means of ultrasonic measurements, which provided the entire set of characteristic elastic constants , whilst we have calculated further their isotropic equivalent elastic moduli by employing these previously reported data and the relevant equations facilitating the conversion to a polycrystalline ensemble from the moduli in the original orthorhombic symmetry.
Low-temperature RUS measurements were carried out with the sample resting lightly between the transducers across a pair of faces. Again the sample was excited at frequencies between 0.1 and 1.2 MHz with 65,000 data points per spectrum. The head assembly was held vertically in a helium-flow cryostat in an atmosphere of few millibars of He to facilitate heat transfer, as described previously by McKnight et al. [60, 61]. Sequences of spectra were collected during cooling and heating between ~300 and ~10 K, and a period of 20 min was allowed for thermal equilibration at each temperature before data collection. The frequency, fp, and width, Δf, for selected resonance peaks were obtained by fitting with an asymmetric Lorentzian function. The elastic constants which determine the resonant behaviour of an individual mode scale with fp2 for that mode, and the acoustic loss is given in terms of the inverse mechanical quality factor, Q−1, which can be expressed as Δf/fp. Most mechanical resonances of a small object are determined predominantly by shearing motions so that the temperature dependence of fp2 provides a good representation of the temperature dependence of the shear modulus.
Results and discussion
Usually, the most common technique employed to investigate ferroelectric and relaxor properties of bulk ceramics is dielectric spectroscopy. However, as stated in the introductory part of this paper, mechanical spectroscopy is a useful tool which completes the general understanding of materials’ structure, the structural evolution and structure–property relationships, by revealing the elastic properties of materials. Generally, elastic constants change due to the high sensitivity of the material to induced strain, regardless of the excitation applied: mechanical, thermal, electric, magnetic, etc. In our particular case, differences between the results obtained from dielectric and mechanical spectroscopies might be due to the relatively strong and short-ranging interactions between electric dipoles on one side, and due to the weaker but long-ranging elastic strain fields on the other . In a tetragonal structure, for example, there is no strain across twin walls between 180° domains which are nonetheless ferroelectric and thus respond only to an applied electric field, whilst the twin walls between 90° domains are both ferroelectric and ferroelastic, yielding an extrinsic response to a stress field; as a result, the dielectric loss and the acoustic loss will in general be different . It is clear that the advantages of using a mechanical spectroscopy technique like RUS lie in the interesting attribute strain coupling phenomena to all structural changes that may be otherwise found from dielectric spectroscopy, magnetic studies, calorimetry, X-ray and neutron diffraction, etc. In the case of RUS, mechanical energy losses are not affected by bonding agents, transducers and issues such as the scattering of laser beams is not an issue and true sample dissipation may be directly observed (RUS resonances are standing waves) .
Elastic properties at room temperature
In order to refine K and G values for polycrystalline Ba6GaNb9O30 (BGNO) from the measured resonance frequencies, it was necessary to start with trial values taken from the literature. Data from ferroelectric single crystals of (Ba1−xSrx)4Na2Nb10O30 (x = 0.3 and 0.7) solid solutions which were previously investigated by means of ultrasonic measurements to provide the entire set of characteristic elastic constants  were utilised for this purpose here. Although the end members of this family are known to have orthorhombic (in the Sr-rich compound) and tetragonal (in the Ba-rich compound) symmetry at room temperature, for the two single-crystal compounds investigated in Ref.  it is practically impossible to establish the symmetry for certain, and a morphotropic phase boundary has been proposed for x = 0.6 [59, 64]. However, the discussion regarding the elastic properties was carried out, further assuming orthorhombic symmetry and 17 independent constants: 9 elastic, 5 piezoelectric and 3 dielectric .
Bulk and shear elastic moduli of (Ba1−xSrx)4Na2Nb10O30 (x = 0.3 and 0.7) solid solutions at room temperature, according to Voigt limit, Reuss limit and Voigt–Reuss–Hill average
By supporting the BGNO sample in opposite corners between the piezoelectric transducers, the loading is reduced on the sample, elasticity is maximal compared to the case of edges and faces, and therefore the coupling to all normal vibrational modes led to the impossibility of losing resonance . Furthermore, other mounting orientations of the BGNO parallelepiped—edges and faces—were utilised several times on different parts of the sample in order to be sure of a sufficiently complete data collection, giving the presence of all resonance frequency peaks, preferably with optimal shapes and amplitudes, which are mainly dependent on the mechanical coupling between the samples than the transducers (Fig. 1b). All in all, the resonant ultrasound spectra of BGNO dense ceramic contain uniformly nondegenerate resonances, since rectangular parallelepipeds were used, and all modes were observed individually.
Results of the RPR combined problem for BGNO: fexperimental versus fcalculated and contribution of K versus G
Observation of the first few peaks at low frequencies, and in particular the first one, is of key importance for the fitting procedure and the calculation of the elastic constants and elastic moduli; specifically, the first resonance peak has a 100 % shear contribution (Table 2), and thus an error in this peak can lead to detrimental shifts throughout the pattern. The elastic constants (c11 and c44) of the sample and thus the elastic moduli (K and G) of polycrystalline BGNO ceramic were determined by the solution of a combined problem, using the method known as rectangular parallelepiped resonance (RPR) [70, 71] implemented in the RPR programme. This combined problem involves on one side a “forward” problem consisting of the calculation of the expected resonance frequencies from inputting data such as the dimensions, mass, density and guesses of elastic constants  and on the other side a “inverse” problem consisting in calculating the elastic constants from the experimental resonance frequencies of the eigenmodes . The Lagrangian minimisation of the free energy function was used by applying the Rayleigh–Ritz approximation method for solving the “forward” problem , and further the Levenberg–Marquardt algorithm was used to carry out the least-squares fitting procedure between the guess elastic constants and the calculated elastic constant for solving the “inverse” problem and convergence . Moreover, if a peak is not observed, the RPR programme gives certain signals that it is missing and thus its presence may be included in the input file either through direct observation on the room-temperature measurements, or if, for example, one peak out of more than 20 is not observed in the measurements, it can be omitted and left as an unrefined dummy value. This was not required for this sample, but is standard practice when the specific coupling has not yielded every peak, or where accidental degeneracy obscures one of a pair of peaks, for example.
Bulk and shear elastic moduli of Ba6GaNb9O30 polycrystalline ceramic before and after correcting for porosity, in comparison with single-crystal (Ba1−xSrx)4Na2Nb10O30 (x = 0.3 and 0.7) solid solutions
Both the bulk and shear intrinsic moduli values of BGNO are higher than those of Ba2.8Sr1.2Na2Nb10O30 and Ba1.2Sr2.8Na2Nb10O30 (Km = 108.39 GPa and Gm = 66.12 GPa, respectively, Km = 97.56 GPa and Gm = 37.87 GPa), indicating a tendency of increasing material resistance to both uniform compression and shear stress with increasing Ba2+ content. This is likely due to the bigger cation size on the A-site within the TTB structure which will lower the temperature for a tetragonal—orthorhombic transition. In other words, BGNO at room temperature has the relatively stiff elastic properties of the high-symmetry TTB structure, whilst the Sr phases probably show the effects of elastic softening associated with transformation to the orthorhombic structure.
Temperature-dependent elastic properties
The great interest in following changes in the elastic properties of materials with temperature variation derives from a need to understand the structural and thermodynamic behaviour of those materials, across phase transitions. During phase transitions, the elastic properties exhibit important anomalies, often in the form of elastic softening, with specific elastic moduli tending to zero in some classes of phase transition. Structural phase transitions can be related to elastic behaviour through the use of Landau theory , briefly defined as a “phenomenological description of the free energy across the transition as a function of temperature, in terms of the driving order parameter, the strains and the coupling between the order parameter and the strains” .
The softening of fp2 to a broad, rounded minimum without obvious hysteresis and subsequent stiffening are accompanied by a broad, asymmetric peak in Q−1 in both cases. The temperatures of the minima are the same within experimental uncertainty, and small differences between the fp2 values for the two peaks are due to slightly different contributions of K (2 vs. 9 %) and G (98 vs. 92 %) to each resonance mode (see Table 2 for peaks number 21 and 23). Below 90 K, a high level of acoustic loss, combined with decreased amplitude from the piezoelectric transducers, makes fitting of the peaks difficult, hence the larger scatter and occasional missing data, particularly for Q−1. The imaginary part of dielectric spectra (dielectric loss) collected at 1 MHz is shown in Fig. 4c for comparison (see Ref. ).
This overall pattern is closely similar to that seen for the classic relaxor ferroelectric Pb(Mg1/3Nb2/3)O3 (PMN) described by Carpenter et al.  and is consistent with freezing of PNRs in a temperature interval below ~150 K. The onset of elastic softening for a relaxor ferroelectric appears to coincide with the Burns temperature (i.e., see Ref. ) which must be somewhat above room temperature for BGNO. The softening mechanism is likely to be due to coupling of acoustic modes with a central mode that is due to the dynamic PNRs. Increasing acoustic loss below ~150 K signifies slowing down of the PNR dynamics, and the peak at ~60 K signifies their freezing point (ωτ = 1, where ω is the measuring frequency and τ the relaxation time) at a measuring frequency of ~1 MHz. The evolution of the modulus at low temperature is required to undergo a flattening of the sort described by Varshni , and, as such, the continued stiffening at low temperatures is also anomalous. It can be explained if slowing down and/or freezing of defects  coupled with strains continues down to the lowest temperatures, indicating a widespread spectrum of relaxation times, which is a defining property of relaxor behaviour. A multitude of relaxational processes occurring and overlapping in the temperature range 50–100 K may explain the poor results obtained previously for extrapolating the freezing temperature of dipoles of BGNO from employing both universal dielectric response model (UDR) [78, 79] and the maximum of unit-cell tetragonality as a function of temperature (Figures 4d & 6a from Ref. ).
The elastic and anelastic properties of Ba6GaNb9O30 (BGNO) reveal a material which is significantly stiffer than related ferroelectric structures with nearby compositions, probably due to suppression of the ferroelectric transition. Instead, BGNO undergoes relaxor freezing in a temperature interval below ~150 K with anomalies in elastic properties which show that development of local ferroelectric dipoles also involves local strain variations; before the relaxor freezing, there is the formation of polar nanoregions (PNRs), which implies an elastic softening. A broad minimum in the shear modulus and a broad maximum in Q−1 reflect the wide spectrum of relaxation times which is believed to be at the heart of relaxor behaviour, and freezing of some parts of the PNR microstructure appears to extend down to the lowest temperatures. The sensitivity to composition could be exploited in the engineering of relaxor properties for device applications, and the significance of strain coupling is that these properties could also be engineered by the choice of strain imposed from a substrate.
This work was supported by the strategic Grant POSDRU/159/1.5/S/133255, Project ID 133255 (2014), co-financed by the European Social Fund within the Sectorial Operational Program Human Resources Development 2007–2013. RUS facilities in Cambridge were established with funding from the Natural Environment Research Council (Grants NE/B505738/1, NE/F017081/1) and from the Engineering and Physical Sciences Research Council (EP/I036079/1).
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