Control of Uniaxial Negative Thermal Expansion in Layered Perovskites by Tuning Layer Thickness

Uniaxial negative thermal expansion (NTE) is known to occur in low n members of the An+1BnO3n+1 Ruddlesden–Popper (RP) layered perovskite series with a frozen rotation of BO6 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 (n = 1, 2, 3, 4, …, ∞), demonstrating that by changing the fraction of layer interface in the structure (i.e., the value of 1/n) one may control the anisotropic compliance that is necessary for the pronounced uniaxial NTE observed in these systems. More detailed analysis of how the components of the compliance matrix develop with 1/n allows us to identify different regimes, linking enhancements in compliance between these regimes to the crystallographic degrees of freedom in the structure. We further discuss how the perovskite layer thickness affects the frequencies of soft zone boundary modes with large negative Grüneisen parameters, associated with the aforementioned phase transition, that constitute the thermodynamic driving force for NTE. This new insight complements our previous work—showing that chemical control may be used to switch from positive to negative thermal expansion in these systems—since it makes the layer thickness, n, an additional design parameter that may be used to engineer layered perovskites with tuneable thermal expansion. In these respects, we predict that, with appropriate chemical substitution, the n = 1 phase will be the system in which the most pronounced NTE could be achieved.

where ε i (σ i ) is component i of ε (σ) and ε (σ) is a 6-dimensional vector expressed in Voigt notation so that the first three components (i = 1, 2, 3) describe normal strains (stresses) of the crystal and the latter three components (i = 4, 5, 6) describe shear strains (stresses). s is therefore a 6×6 matrix.
Using this definition of s, the general anisotropic thermal expansion, α, of a material is given by the equation where α is the anisotropic thermal expansion vector expressed in Voigt notation, in response to an anisotropic driving force for thermal expansion, that we express by the vector Φ.
In a tetragonal material, the a and b axes are equivalent, and thus in the compliance matrix s 11 = s 22 and s 13 = s 12 . Furthermore, all normal-shear coupling terms, s ij (i = 1, 2, 3; j = 4, 5, 6), are 0 by symmetry and thus shear components of Φ may not contribute to normal components of α. If we assume that we have a tetragonal material that remains tetragonal, and thus undergoes no shear deformations, Equation A2 simplifies to: s 11 s 12 s 13 s 12 s 22 s 13 s 13 s 13 s 33 In Equation A3 we assume that s is temperature independent to a first approximation and therefore the temperature dependence of α is given by the driving force for thermal expansion Φ (T ). We may then express Φ (T ) in terms of mode specific heat capacities, C i v (T ), and anisotropic mode Grüneisen parameters, γ i , by the equation In Equation A4 the summation over indices i is really of every discrete phonon mode at every phonon wavevector on a sufficiently dense grid to approximate an integral over the Brillouin zone. The specific heat capacity of mode i is a function of the frequency of that mode, ω i , and temperature T , where the derivative describes how the population of that mode increases with increasing T . The component γ i η of the vector γ i then describes the contribution of mode i to thermal expansion of lattice parameter η, and is defined as such that if γ i η > 0 mode i contributes to PTE of η and likewise if γ i η < 0 mode i contributes to NTE. Equation A3 implies that even a Φ vector with all positive components could be transformed into a uniaxial or biaxial NTE regime (α with one or two negative components respectively) by a sufficiently anisotropic compliance matrix. This scenario is illustrated for a tetragonal material in Figure 2 in the main manuscript where a Φ driving bulk PTE is transformed by a highly anisotropic s into the quadrant corresponding to uniaxial NTE of the c axis.
The degree of anisotropy can be quantified by the ratio, κ, of the highest and lowest eigenvalues of s, s H and s L respectively, given as If κ = 1, the quadratic form of s in Figure 2 would be a sphere and s would not alter the direction of the vector Φ in Equation A2. However, as κ becomes greater, the quadratic form of s becomes more ellipsoidal and thus s has the potential to rotate the direction of Φ. κ is thus a good metric to consider the potential for s to transform Φ driving bulk PTE into α corresponding to anisotropic NTE.

APPENDIX 2 PHASE DIAGRAM AND SYMMETRY OF RUDDLESDEN-POPPER PHASES
In Figure 1 in the main manuscript, members of the A n+1 B n O 3n+1 Ruddlesden-Popper series were displayed in the high-symmetry I4/mmm parent structure. Figure A1 shows the phase diagrams relevant for NTE in the n = 1 and n = 2 systems. In Ca 2 MnO 4 and Ca 3 Mn 2 O 7 , the uniaxial NTE phase has an anti-phase frozen octahedral rotation about the c axis, corresponding to the I4 1 /acd or Acaa space groups respectively. In n = 1 I4 1 /acd, this rotation is anti-phase between adjacent equivalent BO 6 perovskite layers in different unit cells -the corresponding distortion is at P = ( 1 /2, 1 /2, 1 /2) -whereas in n = 2 Acaa the rotations are anti-phase within each BO 6 block but with no doubling of the I4/mmm unit cell along c -corresponding to a distortion at X = ( 1 /2, 1 /2, 0). In both systems, this NTE phase with anti-phase rotations competes with a ground-state phase with both frozen rotations (about c) and tilts (with rotation axes in the layering plane) of BO 6 octahedra, that is a child of an alternative rotation phase with in-phase rotations. The ground-state phase shown is found to be the lowest energy structure computed using DFT in Ca n+1 Ge n O 3n+1 . The analogous phase diagram for an ABO 3 perovskite (the n = ∞ RP end-member) is also shown for comparison, even though ABO 3 perovskites typically do not exhibit uniaxial NTE in their I4/mcm phase with anti-phase rotations.
We previously used the concept of symmetry trapping to explain the presence of soft (low frequency), yet stable (real phonon frequencies), octahedral tilts driving uniaxial NTE in n = 2 Ca 3 Mn 2 O 7 (Senn et al. (2015)). This idea stems from the fact in the n = 1, 2 phase diagrams, that the crystal cannot transform from the NTE metastable phase to the ground-state phase without the frozen octahedral rotations changing sense -anti-phase rotations about c need to "unwind" to form the in-phase rotations in the ground-state phase. Thus soft phonons are able to persist without the structure undergoing a soft-mode phase transition to the ground state. In the ABO 3 perovskite, the c axis is not as strongly defined as in layered RP compounds (since there is no inherent layering topologically distinguishing a particular axis in the ABO 3 structure). Therefore, although there is no group-subgroup relationship between I4/mcm and the ground-state P nma phases, there is no clear distinction between rotations and tilts and a phase transition with a relatively low activation barrier corresponding to a rotation of the direction of the out-of-phase octahedral tilting can be envisaged. Figure A1. Space-group diagrams showing the relevant phases for uniaxial NTE in low n A n+1 B n O 3n+1 Ruddlesden-Popper systems. The NTE phase has a frozen rotation of BO 6 octahedra about the layering axis, which is out-of-phase between adjacent unit cells (P 4 irrep) if n = 1 and out-of-phase with each perovskite block but in-phase between adjacent unit cells (X − 1 irrep) in the n = 2 system. This NTE phase competes with a ground state phase that has an in-phase frozen octahedral rotation about the layering axis and frozen octahedral tilt in the plane of the layering axis. For higher n, the n = 1 picture extends to odd values of n and the n = 2 to even values of n. An analogous phase diagram for the n = ∞ extreme of an ABO 3 perovskite is also shown even though the I4/mcm phase seldom exhibits NTE.
The n = 1 phase diagram may be extended to all odd n in the Ruddlesden-Popper series and the n = 2 phase diagram to all even n. In this work, we distinguish between high-symmetry phases -the I4/mmm RP or P m3m perovskite parents -and rotation phases -by which we mean the NTE (or equivalent) phase Frontiers