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Edited by: Andrea Sanson, Università degli Studi di Padova, Italy

Reviewed by: Qiang Sun, Zhengzhou University, China; Lei Hu, University of Science and Technology Beijing, China

This article was submitted to Physical Chemistry and Chemical Physics, a section of the journal Frontiers in Chemistry

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Uniaxial negative thermal expansion (NTE) is known to occur in low _{n+1}B_{n}O_{3n+1} Ruddlesden–Popper (RP) layered perovskite series with a frozen rotation of BO_{6} octahedra about the layering axis. Previous work has shown that this NTE arises due to the combined effects of a close proximity to a transition to a competing phase, so called “symmetry trapping”, and highly anisotropic elastic compliance specific to the symmetry of the NTE phase. We extend this analysis to the broader RP family (

Ruddlesden–Popper (RP) oxides are an intriguing class of ceramic materials. They have the basic formula A_{n+1}B_{n}O_{3n+1} and consist of a perovskite block of _{6} octahedra separated by an AO rock salt layer. Blocks of octahedra are stacked perpendicular to the long crystallographic axis making this layering axis structurally distinct from the two in-plane axes. Neighboring blocks are de-phased from each other by a lattice translation of (0.5, 0.5, 0.5), and the aristotypical symmetry for any value of

In the A_{n+1}B_{n}O_{3n+1} Ruddlesden–Popper series, blocks consisting of _{3} perovskite structure are separated by a single layer of AO rock salt structure, with BO_{6} octahedra in the next block displaced by a (0.5, 0.5, 0.5) lattice translation. In the _{3} perovskite structure is recovered.

One of the most explored systems is the _{2}CuO_{4} on account of its high-temperature superconductivity, where doping of divalent A = Ba and Sr with trivalent rare earth cations has been extensively investigated (Dwivedi and Cormack, _{2}RuO_{4} (Mackenzie and Maeno, _{3}Ru_{2}O_{7} for its nematic orbitally-ordered phase (Borzi et al., _{3}Mn_{2}O_{7} and Ca_{3}Ti_{2}O_{7}, termed hybrid improper ferroelectricity (Benedek and Fennie, ^{0} criterion for ferroelectricity, as it does not require an off-centring of cations to drive the phase transition. Instead, this off-centring (_{1}) and rotation mode (_{2}) of the parent structure that are inherently unstable in some of these systems. This leads to a so-called trilinear term β_{1}_{2}_{1} and _{2} are inherently unstable, leads to a non-zero value of the polarization

Our contribution to this field of hybrid improper ferroelectricity was to provide experimental confirmation of this mechanism for the case of Ca_{3}Ti_{2}O_{7} (Senn et al., _{3}Mn_{2}O_{7} revealed an added complexity. What was believed to be a single phase at room temperature, having the polar symmetry _{1}

NTE is a rare property, that when it does occur is known to be caused by a diverse range of mechanisms in different materials. Even within inorganic perovskite-based systems, NTE has been found to originate due to coupling of the lattice parameters to: charge ordering (Azuma et al., _{3}Mn_{2}O_{7} that certain vibrational modes with RUM character would have negative Grüneisen parameters and be soft on account of the proximity of the system to the symmetry-forbidden phase transition to _{1}_{3}Mn_{2}O_{7} to systematically control and tune the uniaxial thermal expansion properties of the solid solution Ca_{3−x}Sr_{x}Mn_{2}O_{7} (Senn et al., _{1}

The presence of dynamic octahedral tilts in this Ca_{3−x}Sr_{x}Mn_{2}O_{7} system explained the thermodynamic driving force for NTE along the layering axis. However, the question remained open of why NTE was only observed in this _{3} perovskite phases, where dynamic octahedral tilts would still operate. We were able to answer this question in a recent computational study using density functional theory (DFT) and working within the quasi-harmonic approximation (QHA) to reproduce experimentally measured uniaxial NTE in the _{1}/_{2}MnO_{4} (Ablitt et al.,

Equation (1) (Grüneisen and Goens, _{η}(_{1} = α_{2} by symmetry). Equation (1) is explained in detail in Appendix

Equation (1) is illustrated on axes describing normal cell deformations using Voigt notation (e.g., where the vector [ϵ, ϵ, 0] corresponds to normal strains of ϵ of the

Until now, our research has focused on understanding uniaxial NTE in the low _{n+1}Ge_{n}O_{3n+1} system within the framework of DFT, we find that the magnitude of anisotropic elastic compliance is dependent upon the proportion of CaGeO_{3}:CaO interface in the structure—which may be conveniently expressed by the fraction 1/

The layout of the paper is as follows: section 2 gives details of simulation parameters used in this work; section 3 is then split into four subsections: section 3.1 presents the key result of the paper, showing how the magnitude of anisotropic compliance, linked with uniaxial NTE, varies with 1/_{6} octahedra in Ca_{n+1}Ge_{n}O_{3n+1}, and compare these results against experimental phase diagrams constructed with data taken from the literature of the analogous Ca_{n+1}Mn_{n}O_{3n+1} system. Additionally, in the associated Appendices file, Appendix

Calculations were performed using CASTEP, a plane-wave DFT code, version 7.0.3 (Clark et al., _{2}GeO_{4}, with grids of equivalent reciprocal space density used for all other structures (high-symmetry and rotation phases for ^{−4} eV/Å and a stress tolerance of 10 MPa.

We expect our Ca_{n+1}Ge_{n}O_{3n+1} system to be well-described by conventional GGA density functionals. There are other members of the chemical space that might require more careful consideration in terms of the appropriate methodology, such as hybrid functionals, DFT+U, or potentially even DMFT in order to accurately describe the physics associated with localized d and f-electrons.

Elastic constants were computed by fitting second order polynomials to the energies of cells with applied strains of ±0.2, 0.4% from the fully relaxed cell, where the internal degrees of freedom (the atomic positions) were free to relax. The quadratic terms to these fits were used to construct terms within the elastic constant matrix,

Bulk moduli, ^{−1}.

Density functional perturbation theory (DFPT) was used within CASTEP (Refson et al., _{3}) and for

It was previously shown in first-principles calculations performed on the NTE phase of Ca_{2}GeO_{4} (i.e., the _{1}/_{H}) and lowest (_{L}) eigenvalues of _{H}, _{L}, and κ evolve with varying _{n+1}Ge_{n}O_{3n+1} series.

_{L} and highest, _{H}, eigenvalues to the compliance matrix, _{n+1}Ge_{n}O_{3n+1} series against the mole fraction of CaGeO_{3}:CaO interface (1/_{3} (1/_{2}GeO_{4} (1/

In the composite mechanics community, the elastic properties of laminates are typically described by the properties of the constituent phases, weighted by the relative fraction of that phase (Sarlosi and Bocko, _{3} and CaO phases or (ii) as being comprised of CaGeO_{3} and the CaGeO_{3}:CaO interface.

From Equation (3), it is clear that (i) the mole fraction of CaO in Ca_{n+1}Ge_{n}O_{3n+1} is given by the ratio 1/(_{2}GeO_{4}, which represents the fraction of CaGeO_{3}:CaO interface in the structure, is given by the ratio 1/_{H}, _{L}, and κ are plotted as a function of 1/_{3} (1/_{2}GeO_{4} (1/

The least compliant eigenvector, _{L}, corresponds to isotropic expansion/contraction for all structures investigated (see Table _{L} increases linearly with higher Ca_{2}GeO_{4} mole fraction but is invariant to changes in symmetry for a given _{L} lie on the line interpolating between CaGeO_{3} and Ca_{2}GeO_{4} high-symmetry end members regardless of phase symmetry implying that _{L} is determined mainly by the composition.

_{H} also increases in magnitude with Ca_{2}GeO_{4} content for all RP phases. However, unlike _{L}, _{H} is greatly enhanced in the phase with a frozen rotation compared to the high-symmetry phase, and the rate of increase in _{H} for rotation phases with 1/_{H} lies in a strain direction corresponding to a cooperative increase in in-plane lattice parameters, _{3}:CaO layer interface to closely couple the _{3}:CaO interface, it is interesting to note that _{H} in the rotation phase is linearly dependent upon the mole fraction of this interface in the structure, increasing as this interface fraction becomes greater, and thus _{H} for intermediate values of 1/_{H} values for CaGeO_{3} (with no interface) and Ca_{2}GeO_{4} (maximum interface) rotation phases.

This steeper increase in _{H} for rotation phases than high-symmetry phases with interface mole fraction (1/

Figure _{3}:CaO interface fraction (1/_{ij}, vary with 1/_{3} and CaO constituents and (ii) between CaGeO_{3} and CaGeO_{3}:CaO interface constituents, to predict the compliance components of intermediate values of 1/

Figure _{n+1}Ge_{n}O_{3n+1} series. Since all phases are (pseudo-)tetragonal, only the four symmetrically distinct _{ij} components identified in Equation (1) are plotted. The full

_{ij} of the elastic compliance matrix for a tetragonal material plotted for high-symmetry and rotation phases in the Ca_{n+1}Ge_{n}O_{3n+1} series against the mole fraction of CaGeO_{3}:CaO interface (1/_{3} (1/_{2}GeO_{4} (1/_{3} phase and CaO rock salt structure computed as a function of CaO mole fraction 1/(

The bulk elastic compressibility, β, increases linearly with 1/

The normal compliance components, _{11} and _{33}, also increase with 1/

The sign of the off-diagonal compliance components, _{12} and _{13}, that couple normal stresses to normal strains between axes, are negative for all compounds. This indicates that all materials have all positive Poisson ratios, ν_{ij}, where ν_{ij} describes the normal strain of axis _{ij} > 0, so these NTE RP phases are not auxetic (ν_{ij} < 0), even though auxetic materials have been linked with materials that exhibit anisotropic NTE (Wang et al.,

Despite the negative sign, the behavior of _{13} is similar to that of _{11} and _{33}: compliance increases with 1/_{12}, on the other hand, displays the opposite trend since the magnitude of coupling decreases with 1/

As in Figure _{3} (1/_{2}GeO_{4} (1/_{3} and CaO rock salt constituent phases. Because the mole fraction of CaO is actually expressed as 1/(

The trend in β follows that that would be predicted by modeling the RP series as a laminate of CaGeO_{3} and CaO, suggesting that β is determined predominantly by the composition. Since bulk volume thermal expansion, α, is proportional to the bulk compressibility, β, this implies that the magnitude of α is heavily dependent on chemistry. This result echoes recent work showing that experimental measurements of many thermodynamic properties of RP structures may be predicted by interpolating between values of their chemical constituents (Glasser,

Whereas, β could be approximated well as a function of CaO content for RP phases, _{11} of high-symmetry phases increases above that predicted by the cyan curve. This indicates that even in the high-symmetry phase, the CaGeO_{3} and CaO layers do not behave independently and are affected by the interface between them. The prediction for _{33} based on the CaO content is quite good, which may be because _{33} corresponds to deformations along the layering axis (with the ^{1}_{11} and _{33}, the rotation phases follow a linear relationship with 1/_{3}:CaO interface in the structure (red dotted line). However, in the high-symmetry phase the _{3} and Ca_{2}GeO_{4} values (blue dot-dashed line). This is surprising since it is not immediately obvious how the structure of higher _{3} and Ca_{2}GeO_{4} and therefore what additional compliance mechanisms could operate.

For both _{12} and _{13}, modeling the compliance according to the mole fraction of CaO is a poor approximation, so much so that this prediction actually gives the wrong sign of the change in _{12} with 1/

In section 3.2 we showed that certain elastic properties, such as the bulk compressibility, β, are insensitive to small changes in crystal symmetry and may be accurately predicted by interpolating between the value of β in CaGeO_{3} and CaO end member structures based on the mole fraction of CaO. However, components of the anisotropic compliance matrix, _{ij}, typically differ in magnitude between high and low symmetry phases and are generally more compliant than a CaGeO_{3}:CaO interpolation predicts. Figure _{13} vs. 1/_{13} behavior of different structures into regimes of increasingly enhanced compliance. By separating the compliance regimes in this way, in this section we discuss the atomic displacements allowed in each regime by the phase symmetry and thus propose atomic mechanisms that may explain these enhancements in the _{13} axis coupling parameter. In many cases (although not discussed here) this analysis may be used to explain the different regimes of the _{11}, _{33}, and _{12} components in Figure

The _{13} term of the elastic compliance matrix (corresponding to coupling between an in-plane axis, _{n+1}Ge_{n}O_{3n+1} series against the mole fraction of CaGeO_{3}:CaO interface (1/

Taking the value of _{13} that would be predicted by interpolating between values in the CaGeO_{3} and CaO constituent structures as a base (the dashed cyan curve in Figure _{2}GeO_{4} _{3} perovskite, the A cations by symmetry have the same _{3}:AO interface such that the apical O and interfacial A ions are no longer restricted to the same _{13} coupling illustrated in Figure _{6} octahedra below the interfacial A cation changes in size, but the rumpling adds a degree of freedom to the

Strain coupling mechanisms that make structures particularly compliant to cooperative strains. Since these mechanisms couple the _{13} off-diagonal components of the compliance matrix. _{1}–_{4}) by changing the angle of in-plane and out-of-plane hinges (described by the angles θ and α as shown);

It was commented in the preceding section that the enhancement in compliance from the interpolation between CaGeO_{3} and Ca_{2}GeO_{4} values to _{3}:CaO interface (with rumpling of the CaO _{3} (that one might expect to behave as bulk cubic CaGeO_{3}). Close inspection of the _{2}BO_{4} structures, the length of all apical B–O bonds are equal due to the mirror symmetry plane lying in each BO_{6} layer. Similarly the angle between epitaxial B–O and apical B–O bonds must be 90° by the same reasoning. However, in A_{3}B_{2}O_{7} _{6} internal angles must be equal and the apical B–O bond lengths between the two inner BO_{6} layers must be equal. There is no restriction by symmetry that all apical B–O bond lengths must be equal or in fact that all BO_{6} internal angles must be 90°. These weaker restrictions create internal degrees of structural freedom that may facilitate greater compliance, since there is greater freedom for the atoms to move in response to external strains. In structures relaxed using DFT, we find that there are slight differences in these two bondlengths: 1.87 and 1.90 Å for the outer and inner apical B–O bonds, respectively, and the angle between outer apical B–O and epitaxial B–O bonds is 91.2°. This same argument may be applied to all _{13} of _{3}:Ca_{2}GeO_{4} interpolation is not also seen in the significantly more compliant rotation phases may be because in these rotation phases, additional compliance mechanisms operate that dwarf the effect described by arrow ②.

Arrow ③ in Figure _{3} phase to an _{6} octahedra rotating, but not also large changes in _{3} (Hatt and Spaldin, _{3} (Weber et al., _{6} units with compression of the epitaxial B–O bonds and extension of the apical B–O bonds.

Finally, arrow ④ in Figure _{13} coupling between rotation phases of CaGeO_{3} perovskite and RP phases with a frozen octahedral rotation. In these phases, the rumpling of the interfacial AO layer identified in the high-symmetry structure is still present. However, whereas in the high-symmetry structure, in-plane strains necessarily involved deformations of stiff epitaxial B–O bonds, in this lower-symmetry phase, the frozen in-plane octahedral rotation adds an internal degree of freedom in the in-plane epitaxial O positions. There are thus internal degrees of freedom in both in-plane and layering axes in RP rotation phases. In our previous paper (Ablitt et al., _{13} coupling in RP1 rotation phases but not in _{3} phases. This coupling mechanism in theory allows _{1}/_{2}GeO_{4} structure, by only changing two bond angles, labeled θ and α in Figure

The net result of these interfacial strain coupling mechanisms in RP phases, yielding an enhanced off-diagonal _{13} compliance term, is rather reminiscent of the “wine-rack” mechanism such as that which operates in methanol monohydrate (Fortes et al., _{13} compliance parameter as a function of RP layer thickness,

where _{θ} and _{α} represent the harmonic stiffness of the θ and α-hinges, respectively, and f(θ, α) is an expression of trigonometric functions of θ and α

Under the constraints of this model, α is explicitly dependent on θ by the equation
_{1}-_{4} (as defined in Figure

In the limit that _{α} >> _{θ}, the AO interface is stiffer than the ABO_{3} perovskite blocks and _{13} loses dependence on _{θ} >> _{α} and changing the in-plane rotation angle is the main obstacle to strain, _{13}∝1/_{θ} >> _{α} limit as we observe linear behavior of _{13} with 1/_{1}) and all apical B–O bond lengths (_{2}) are equal, the bond lengths _{2} and _{3} do not feature in Equation (4) and _{1} and θ only need refer to the in-plane bond lengths and rotation angle in the outer layer of the perovskite block. Therefore, this mechanism is compatible with a distribution of possible bond lengths and rotation angles in different layers of a perovskite block if

We accept the limitations of such simple models but developing the study of how symmetry-allowed local distortions can give rise to new compliance mechanisms in different crystallographic phases, such as those identified using arrows ①–④ in Figure

Therefore, looking for other materials with such high cross compliances, using symmetry analysis as a guide to narrow the pool of structures, may prove a more general method for searching for novel NTE materials. Indeed, by considering this analysis and our thermodynamic criteria requiring a proximity to a competing phase transition to provide Φ, we have already been able to identify (Ablitt et al., _{2}MgWO_{6} (Achary et al., _{4} (Cordrey et al.,

So far we have not addressed the thermodynamic driving force for thermal expansion Φ(_{n+1}Ge_{n}O_{3n+1} series, this latter condition is not met, and therefore full computation of Φ(_{1}_{3}Ti_{2}O_{7} phase after 30 GPa hydrostatic pressure has been applied and in which these octahedral tilts are already frozen (Huang et al., _{n+1}Ge_{n}O_{3n+1} series is being treated as an analog to Ca_{n+1}Mn_{n}O_{3n+1} for comparison against experimental data to avoid expensive magnetic calculations since Ge^{4+} and Mn^{4+} are known to have equal ionic radii (Shannon,

The lowest frequency harmonic phonon mode with octahedral tilt character found in DFT simulations at high-symmetry q-points in the Ca_{n+1}Ge_{n}O_{3n+1} series against interface fraction, 1/_{n+1}Ge_{n}O_{3n+1} structure to Ca_{n+1}Mn_{n}O_{3n+1}. DFT only simulates the system at 0 K whereas phonon frequencies may harden with increased temperature. This means that many octahedral tilts are predicted with imaginary frequencies (shown as negative) even if the mode has a real frequency and the structure is stable at some higher temperature. To give an idea of how the equilibrium phase changes with temperature, the inset graphs plot the experimentally observed phase diagram for each _{n+1}Mn_{n}O_{3n+1} system. The data to make these illustrations was taken from: Ca_{2}MnO_{4} (Takahashi and Kamegashira, _{3}Mn_{2}O_{7} (Senn et al., _{4}Mn_{3}O_{10} (Battle et al., _{3} (Taguchi et al.,

It can be seen from Figure _{2}GeO_{4} content, indicating that the inclusion of the CaGeO_{3}:CaO interface reduces the propensity of octahedra to tilt. The tilt mode is unstable (has an imaginary frequency) in all high-symmetry parent phases, which is unsurprising since these phases are not observed experimentally at low temperatures at any _{n+1}Mn_{n}O_{3n+1} series. The tilt stiffens between the parent and child phases for all _{2}GeO_{4} mole fraction, although the stiffening effect going between the high- and low- symmetry phases becomes greater at higher _{2}GeO_{4} fraction). The CaO rock salt layer also stiffens octahedral rotations and thus the angle of the frozen rotation increases with _{1}/_{2}GeO_{4} (_{3} (_{3} means that the competitive coupling between the frozen rotation and dynamic tilt is greatest for this largest

In the 0 GPa relaxed Ca_{n+1}Ge_{n}O_{3n+1} structures, the tilt is still unstable for all _{2}GeO_{4} and Ca_{2}MnO_{4} in-plane lattice parameters in the _{1}/_{1}/_{2}GeO_{4} rotation phase has all real mode frequencies, but the softest tilt in all structures with

Ca_{3}Mn_{2}O_{7} is found at low temperature in the improper ferroelectric _{1}_{1}_{3}, transforms around 1166 K from a _{4}Mn_{3}O_{10} from 5 K up until room temperature is ^{2}

As well as having optimal elastic anisotropy to facilitate uniaxial NTE, the _{2}MnO_{4} _{1}/_{n+1}Mn_{n}O_{3n+1} compounds should transform to a phase in which the tilt frequencies are real and soft, at least over some temperature range. Furthermore, we have demonstrated previously in the Ca_{3−x}Sr_{x}Mn_{2}O_{7} system, that for a given layer thickness

We have shown that the elastic anisotropy ratio, κ, found previously to be an essential ingredient for uniaxial NTE, increases linearly in the RP Ca_{n+1}Ge_{n}O_{3n+1} series (_{3}:CaO content (expressed by the ratio 1/_{3}:CaO interface in phases with a frozen octahedral rotation about the layering axis and therefore explains the trend that anisotropic compliance correlates with the fraction of interface in these phases. This local atomic compliance mechanism is analogous in certain ways to the wine-rack mechanism that operates in many much softer framework materials. The compliance matrices can be rapidly calculated by DFT methods and diagonalized to assess them for cross coupling terms that promote pronounced uni or biaxial NTE. This makes them suitable descriptors for high throughput computational searching for novel NTE materials, especially when symmetry constraints may be employed to narrow the space of candidate phases.

We further investigated the trend in frequency of octahedral tilts with RP layer thickness and found that the 0 K tilt frequencies in NTE or analogous structures become softer with increasing _{3}:CaO interface layers in the structure. On the basis of this analysis, we thus predict that the _{2}MnO_{4}, will be the RP systems in which the maximum NTE can be achieved via chemical substitution.

Data underlying this article can be accessed on figshare at DOI:10.6084/m9.figshare.6729287, and used under the Creative Commons Attribution licence.

CA performed the calculations and data analysis. All authors contributed to the design of the study and the analysis and interpretation of the results. The paper was drafted by CA and MS, and all authors contributed to its development into final form.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The Supplementary Material for this article can be found online at:

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^{1}However, we note that under these conditions one should use a Voigt average to interpolate _{33}, whereas by drawing a straight line of compliance vs mole fraction we have actually performed a Reuss interpolation. In Figure

^{2}Although we do note that the high-symmetry