Hanna L. B.
Boström
,
Joshua A.
Hill
and
Andrew L.
Goodwin
*
Inorganic Chemistry Laboratory, Department of Chemistry, University of Oxford, South Parks Road, Oxford OX1 3QR, UK. E-mail: andrew.goodwin@chem.ox.ac.uk; Tel: +44 (0)1865 272137
First published on 20th October 2016
We introduce columnar shifts—collective rigid-body translations—as a structural degree of freedom relevant to the phase behaviour of molecular perovskites ABX_{3} (X = molecular anion). Like the well-known octahedral tilts of conventional perovskites, shifts also preserve the octahedral coordination geometry of the B-site cation in molecular perovskites, and so are predisposed to influencing the low-energy dynamics and displacive phase transitions of these topical systems. We present a qualitative overview of the interplay between shift activation and crystal symmetry breaking, and introduce a generalised terminology to allow characterisation of simple shift distortions, drawing analogy to the “Glazer notation” for octahedral tilts. We apply our approach to the interpretation of a representative selection of azide and formate perovskite structures, and discuss the implications for functional exploitation of shift degrees of freedom in negative thermal expansion materials and hybrid ferroelectrics.
In addition to conventional inorganic perovskites, there are several molecular perovskite analogues, including organic halide perovskites,^{9,10} dicyanamides,^{11–14} azides,^{15,16} Prussian blue analogues,^{17,18} dicyanometallates,^{19,20} thiocyanates,^{21–23} and formates.^{24–26} These are systems of strong scientific currency in which at least one component of the ABX_{3} perovskite structure is molecular: typically the A-site cation and/or the anionic linker X. An important consequence of the incorporation of molecular components is the emergence of new structural degrees of freedom for which there is no analogue in conventional perovskites. Examples include (i) the so-called “forbidden” tilts found in some azides, Prussian blue analogues, and dicyanometallates, in which neighbouring octahedra (no longer corner-sharing) rotate in the same sense as one another,^{20,27–29} and (ii) multipolar order associated with orientational degrees of freedom of molecular A-site cations.^{30,31} Coupling of these exotic degrees of freedom to the lattice then allows for entirely new symmetry-breaking mechanisms,^{32} and hence new crystal engineering strategies for targeting e.g. multiferroic or NTE responses.^{1,2,4,33}
It is natural then to ask: are there any other degrees of freedom of general relevance to the structural chemistry of molecular perovskites? Here, a key consideration is the energy scale associated with different deformations, since those with high energies (e.g. bond stretches or distortion of coordination geometries) are unlikely to behave as soft modes. Fortunately, simple geometric tools^{34} can be used to identify distortion modes that preserve bond lengths and coordination geometries and hence are predisposed to play a key role in the low-energy dynamics of materials; these are termed the rigid unit modes (RUMs) of a given topology. So, for example, it was shown in ref. 34 that the only RUMs supported by the conventional perovskite structure are the well-known octahedral tilts discussed above [Fig. 1(a)]. A similar analysis of the ABX_{3} lattice with molecular X components, however, revealed the existence of two types of rotational degrees of freedom (these are the conventional and forbidden tilts) together with a translational degree of freedom involving correlated displacements of columns of connected BX_{6} octahedra [Fig. 1(b) and (c)].^{35}
We refer to these columnar translations as shifts and argue here that they can indeed play an important role in the structural chemistry of certain families of molecular perovskites. Our paper begins by establishing the conceptual framework for interpreting and characterising columnar shifts. By focusing initially on a two-dimensional simplification of the perovskite framework, we explore the interplay between shift activation and symmetry breaking, the relationship to shear modes, and the potential for coupling with tilt degrees of freedom. We proceed to extrapolate this analysis to the interpretation of static symmetry-breaking distortions in three-dimensional molecular perovskites, drawing on topical case studies from the recent crystal engineering literature. The link to dynamical properties is then made via a simple lattice-dynamical model, which we then use to demonstrate that dynamic shift distortions have a distinctive NTE character. A simple density functional theory (DFT) phonon calculation for CdPd(CN)_{6} supports the relevance of shift modes in the low-energy phonon spectrum of this specific Prussian blue analogue. Our paper concludes with a discussion regarding the possibility of developing shift engineering approaches as an alternative mechanism for accessing polar states in molecular perovskites.
(1) |
In practice, the form of D(k) is simple for molecular perovskites. Even so, we consider first the (even simpler) 2D analogue of connected squares, for which
(2) |
These results translate directly to the three-dimensional case of molecular perovskites. The dynamical matrix now assumes the form
(3) |
What at face value might appear to be the simplest case—namely, activation of a single shift system with periodicity k = [0, 0]—turns out to give rise to a relatively complex situation. These shifts describe a shear of the perovskite lattice polarised along one of the lattice vectors a or b; the corresponding distortions are illustrated in Fig. 3(b) and (d). In both cases the vast majority of the symmetry elements present in the p4mm plane-group symmetry of the parent lattice are lost and the crystal symmetry is now reduced to p2. This symmetry lowering is so severe that activation of these shifts allows coupling to an entirely different type of rigid-body distortion—namely, the “forbidden” (in-phase) tilts also at k = [0, 0] [Fig. 4(a) and (b)]. In fact, these tilts provide a continuous pathway between k = [0, 0] shifts polarised along a, on the one hand, and those polarised along b, on the other hand, such that the former type of shift cannot be distinguished from a combination of the latter shift type together with an in-phase tilt (or vice versa). This confusing situation arises because k = [0, 0] shifts polarised along either a or b are characterised by the same irreducible representation; in other words, the two shift systems break the parent symmetry in identical ways.
Fig. 4 Some symmetry relationships in 2D shift systems. (a) Activation of zone-centre shifts leads to structures that are related to one another via in-phase tilts of the rigid units. (b) This transformation is continuous because the plane group symmetry elements of the shifted structures (2-fold rotation axes distributed as illustrated here) are compatible with the activation of in-phase tilts. (c) For some shift systems, such as the c2mm distortion shown here, the persistence of mirror symmetry elements (red lines) forbids mixing of shifts and tilts. (d) This particular shift system is related to the compliant structure of the MIL-53 family,^{37,38} shown here in polyhedral representation. |
Coupling to tilts is by no means a universal feature of shift distortions, and a counter-example is given by the case in which the two k = [0, 0] shifts are active to precisely the same extent. This situation corresponds to a shear polarised along the cell diagonal, which results in a much less severe symmetry-lowering process: the resulting plane group is now c2mm [Fig. 3(e)]. Importantly, the persistence of mirror symmetry elements bisecting the rigid units means that coupling to tilts can only occur by further symmetry lowering [Fig. 4(c)]. So in this case, the particular shift modes can be uniquely identified from the lattice symmetry. Of course, the transition from p4mm to c2mm structures—couched here in terms of activation of k = [0, 0] shifts—corresponds to a ferroelastic distortion of the lattice.^{39,40} The ferroelastic state is well known to be mechanically compliant,^{41,42} and as such is often associated with phenomena such as uniaxial NTE and negative linear compressibility (NLC).^{43–45} Indeed, the 2D model we consider here may be interpreted as a projection of the 3D “wine-rack” structure of well-known compliant framework materials such as the MIL-53 family, which is certainly known to exhibit both NTE and NLC [Fig. 4(d)].^{37,38}
Whereas zone-centre shift modes describe ferroelastic distortions, those at the zone boundary give rise to antiferroelastic states. In the case of shifts polarised along a, the relevant zone-boundary periodicity is k = [0, ½]. Consequently, activation of this shift mode results in a doubling of the cell in the b direction with the corresponding plane-group symmetry now p2gm [Fig. 3(c)]. Once again, the persistence of mirror symmetry elements bisecting the rigid units forbids coupling to tilts. The equivalent shift mode polarised along b gives rise to an analogous distortion: the cell now doubles along a and the plane-group symmetry is p2mg [Fig. 3(g)]. In contrast to the situation for the corresponding zone-centre shift modes, in this case there is clearly no continuous pathway between the two states. Simultaneous activation of both zone-boundary shift modes to identical extents results in the appealing antiferroelastic distortion shown in Fig. 3(i). This distortion requires doubling along both cell axes and is described by the plane group p4gm. Once again, the point symmetry at the rigid unit site includes a mirror plane (at 45° to the cell axes) and so this particular shift system is symmetry forbidden from coupling with tilt modes. For completeness, we consider the final possibility in which a zone-centre shift mode polarised along one axis is combined with a zone-boundary shift polarised along the remaining axis. The corresponding distortions for the two possible axis choices are illustrated in Fig. 3(f) and (h). In both cases the cell doubles and in both cases the resulting plane-group symmetry is p2. Yet, while each shift distortion now has sufficiently low symmetry to couple with tilt modes (as above), there is no continuous path between the two: they are distinguishable by virtue of the particular axis along which the cell has doubled. Our key point in covering all these different possibilities is to demonstrate that activation of different shift modes results in different symmetry-breaking processes that can be fundamentally distinct from those accessible via tilt degrees of freedom—whether conventional^{46} or forbidden.^{20,27–29}
Like shifts, independent tilt systems can be associated with each of the three crystal axes. In conventional perovskites, rotations around the a axis (by way of example) can propagate with periodicity k = [k_{x}, ½, ½]. Hence, the particular tilt distortion associated with a single axis α is described by two terms: the tilt magnitude u_{α} and the relevant wave-vector component k_{α}, which—as discussed above—is usually either 0 (in-phase tilts) or ½ (out-of-phase tilts). Glazer condenses this information for each axis into a compound symbol λ^{μ}. The index μ ∈ {0, +, −} denotes whether a tilt is inactive (μ = 0; u_{α} = 0), in-phase (μ = +; k_{α} = 0) or out-of-phase (μ = −; k_{α} = ½); the primary symbol λ reflects the magnitude of an active tilt in order to show the existence or absence of symmetry relationships between tilts along different axes of the parent perovskite lattice. The untilted aristotype has Glazer symbol a^{0}a^{0}a^{0}; the term a^{−}a^{−}a^{−} denotes equal-magnitude out-of-phase tilts around each of the three crystal axes; and the term a^{+}b^{−}b^{−} denotes in-phase tilts around a with distinct equal-magnitude out-of-phase tilts around b and c. Howard and Stokes established a link between these labels and the corresponding space-group symmetries.^{6} We note that the index μ is equal to the value of u_{α}exp[2πik_{α}] if (i) the symbol + can be associated with 1 and − with −1, and (ii) u_{α} is taken to equal 1 for active tilt modes and 0 for inactive tilt modes.
The various shift distortions of the 2D molecular perovskite structure discussed above are also describable in terms of the magnitude and periodicity of collective translations along each crystal axis. This immediately suggests an analogous notation to that of Glazer's for tilts, with only one subtle conceptual modification: the periodicity implied by the index μ must now refer to the component of k perpendicular to the corresponding crystal axis. So, for example, the diagonal ferroelastic distortion discussed in terms of k = [0, 0] shifts along both a and b might be summarised by the Glazer symbol a^{+}a^{+}: here the + index would indicate k_{y} = 0 for shifts parallel to a and k_{x} = 0 for shifts parallel to b; likewise the use of the same primary symbol a would indicate that the shifts have identical magnitude along these two crystal axes. The corresponding symbols for each of the distortions originally presented in Fig. 3 are given in Table 1.
a-shifts | b-shifts | Glazer symbol | Matrix symbol | Plane group |
---|---|---|---|---|
a As described in the main text, there is no unique assignment of shifts for this system since a^{+}b^{0}, a^{+}b^{+} and a^{0}b^{+} states are continuously interchangeable via activation of in-phase tilts. | ||||
Inactive | Inactive | a ^{0} a ^{0} | p4mm | |
In-phase | Inactive^{a} | a ^{+} b ^{0} (≡ a^{+}b^{+}) | p2 | |
In-phase | In-phase | a ^{+} a ^{+} | c2mm | |
Out-of-phase | Inactive | a ^{−} b ^{0} | p2gm | |
Out-of-phase | In-phase | a ^{−} b ^{+} | p2 | |
Out-of-phase | Out-of-phase | a ^{−} a ^{−} | p4gm |
We will come to show that an unambiguous extrapolation of this notation to 3D molecular perovskites is not straightforward, and so we present an alternative—albeit perhaps more cumbersome—approach similar to that developed in ref. 20 to describe “forbidden” tilts. Here the idea is to exploit the equivalence μ ≡ uexp[2πik] noted above. We assemble the matrix
(4) |
(5) |
By contrast, the more cumbersome matrix notation is straightforwardly extended to 3D shifts: we use the representation
(6) |
(7) |
Fig. 5 An antiferroelastic planar shift system characterised by displacements parallel to a, correlated with modulation wave-vector k = [0, 0, ½]. |
One possible approach to modifying the Glazer-type notation for these 3D shifts might be to exploit the Bradley–Cracknell abbreviations for high-symmetry points in the Brillouin zone.^{47} In some cases, the use of this abbreviation as the Glazer index μ would allow unambiguous identification of the two required wave-vector components. For example, the Pmma shift system discussed immediately above might be assigned the Glazer symbol a^{X}b^{0}c^{0}. Here, the index X of the first term signifies that shifts polarised along a are active and are modulated with a periodicity k ∈ 〈½, 0, 0〉. Since k_{x} must equal zero, we know that k = [0, ½, 0] or [0, 0, ½]; in the absence of active shifts along b or c these two periodicities give rise to symmetry-equivalent distortions. Despite the success of the nomenclature in this one example, it is straightforward to envisage scenarios in which unambiguous identification is not possible. Nevertheless, for each of the case studies below, we try to give both Glazer and matrix notations, with the understanding that future usage will likely determine limitations of the two approaches and identify of which of these is the more useful in practice.
(8) |
(9) |
Fig. 6 Static shift distortions in [NMe_{4}]Ca[N_{3}]_{3}. (a) A polyhedral/ball-and-stick representation of the crystal structure of [NMe_{4}]Ca[N_{3}]_{3}, as reported in ref. 48. Ca atoms are shown in yellow and N atoms in blue. The [NMe_{4}]^{+} cations have been omitted for clarity. Shifts are polarised along c (the vertical axis in this representation) and are related to those illustrated originally in Fig. 2(a). (b) 2D sections of the crystal structure using the same representations as in (a). These sections lie perpendicular to the a, b, and c axes (left–right) and relate the 2D shift systems enumerated in Fig. 3 with the matrix representation of the full 3D shift system active in this material. (c) A representation of the local environment of the [NMe_{4}]^{+} cation in this material; colours are as for (a) and (b), with C atoms shown in black and H atoms omitted for clarity. Thermal ellipsoids are given at 50% probability. There is a close match in A-site cation geometry and the framework distortion effected by shift activation. The relatively large thermal ellipsoids suggest substantial dynamic disorder in this system. |
As in a number of the simple 2D cases studied above, the particular shift distortion mode we observe in [NMe_{4}]Ca[N_{3}]_{3} retains a number of the mirror symmetry elements of the aristotype, which has the effect of preventing mixing between shifts and octahedral tilts. Indeed, there are no static active tilts in the reported structure. What is clear, however, is that there is likely a large degree of dynamic distortion, given the magnitude of the thermal ellipsoids. Consequently, it is possible that this system will exhibit displacive phase transitions on cooling; a re-examination using variable temperature methods may be rewarding in this case.
But what drives the presence of static shifts in this system? We offer two suggestions. The first concerns the coordination preference of the azide anion as a bridging linker. It has long been known that the preferred end-to-end bridging geometry involves substantially bent M–N–N angles; together with the trans-EE coordination of the N^{−}_{3} ion this is presumably what allows such large (≃1.3 Å) displacements between neighbouring Ca^{2+} ions [Fig. 6(a)]. Indeed this propensity of azide to allow activation of shifts is likely a general phenomenon, even if this point does not explain why it is this particular a^{0}a^{0}c^{M} shift system that is adopted here. So our second observation concerns the relationship between the geometry of the [NMe_{4}]^{+} cation and the structural distortions to the A-site cavity that occur as a result of columnar shifts. In the aristotype structure, the point symmetry at the A site is mm (O_{h}), which is a supergroup of the 3m (T_{d}) symmetry of tetramethylammonium; consequently the cation must exhibit orientational disorder in this parent structure. On activation of the a^{0}a^{0}c^{M} shifts, the A-site point symmetry is reduced to 2m (D_{2d}), a subgroup of 3m. This allows orientational order of the cation. Indeed, there is a close match between the geometry of the (ordered) cation and the shape of the A-site cavity that suggests the distortions is driven largely by packing and cation-framework interactions [Fig. 6(c)].
(10) |
Fig. 7 Static shift distortions in [NMe_{2}H_{2}]Mn[N_{3}]_{3}. (a) A representation of the crystal structure as reported in ref. 50. Mn atoms are shown as pink polyhedra; N atoms as blue spheres; [NMe_{2}H_{2}]^{+} cations have been omitted for clarity. There are two orthogonal shift distortions active in this system. One is precisely the same as that shown in Fig. 6(a) and gives rise to alternating columnar displacements polarised along c (the vertical axis in our representation here). At the same time there is an antiferroelastic planar shift system polarised along the b direction (the horizontal axis in this representation) that by itself gives rise to the type of distortion shown in Fig. 5 (albeit with axes relabelled). (b) Thermal ellipsoid representation of the local environment of [NMe_{2}H_{2}]^{+} cations within the distorted perovskite framework. C atoms are shown in black and H atoms have been omitted for clarity. |
The arguments presented to explain the activation of a^{0}a^{0}c^{M} shifts in [NMe_{4}]Ca[N_{3}]_{3} appear to hold again for the a^{0}b^{X}c^{M} shifts we find in [NMe_{2}H_{2}]Mn[N_{3}]_{3}. Clearly the azide linker is common to both, but we find also that the point symmetry at the A site is reduced in order to allow orientational order of the [NMe_{2}H_{2}]^{+} cation [Fig. 7(b)]. The crystallographic point symmetry of this site is 2 (C_{2}) in the Cmce structure, which is a subgroup of the idealised 222 (C_{2v}) molecular point symmetry.
One effect of the activation of multiple shift systems is that the crystal symmetry is now sufficiently low that a set of octahedral tilts couples to the shift-induced distortions. This tilt system is characterised by the (conventional) Glazer label a^{−}b^{0}b^{0} and cannot by itself account for the Cmce symmetry. In other words, octahedral tilts do not act as the primary order parameter in this system.
(11) |
Fig. 8 Representation of the crystal structure of [NMe_{2}H_{2}]Mn[HCOO]_{3}, as reported in ref. 25. Mn coordination polyhedra are shown in pink, O atoms as red spheres, and C atoms as black spheres. [NMe_{2}H_{2}]^{+} cations and H atoms have been omitted for clarity. Here the shift distortions are associated with macroscopic shear of the lattice. |
This distortion reduces the Pmm aristotype symmetry to Rm ( and ), which is a minimal supergroup of the observed space group Rc () and so cannot act as a primary order parameter. Instead, it is the conventional octahedral tilt distortion (Glazer notation a^{−}a^{−}a^{−}) that is responsible for breaking the aristotypic symmetry; so, in this case, the shifts couple to the tilts.
Compound | Glazer symbol | ε _{ a } (%) | ε _{ b } (%) | ε _{ c } (%) | Ref. |
---|---|---|---|---|---|
[NMe_{4}]Ca[N_{3}]_{3} | a ^{0} a ^{0} c ^{M} | 0 | 0 | 18.6 | 48 |
[NMe_{2}H_{2}]Mn[N_{3}]_{3} | a ^{0} b ^{X} c ^{M} | 0 | 23.2 | 22.2 | 50 |
[NMe_{2}H_{2}]Mn[HCOO]_{3} | a ^{Γ} a ^{Γ} a ^{Γ} | 7.9 | 7.9 | 7.9 | 25 |
[NPr_{4}]Mn[N(CN)_{2}]_{3}-LT | a ^{M} a ^{M} c ^{M} | 6.9 | 6.9 | 12.4 | 53 |
CsCd[SCN]_{3} | a ^{Γ} a ^{Γ} c ^{Γ} | 19.5 | 19.5 | 0.6 | 22 |
In selecting the various case studies included here we have intentionally focussed on systems for which the active shift distortions are relatively straightforward. A more general approach for identifying shifts would be to employ group-theoretical analysis of the corresponding distortion modes, as implemented in e.g. ISODISTORT.^{49,54} Irrespective of how shifts are identified, there is absolutely no difficulty in anticipating complicating factors in other systems that would make the kind of analysis we present much trickier. We briefly highlight some of these factors here, noting that many of these are complications also in the characterisation of octahedral tilts in conventional perovskites.
First, there will be systems for which the difference in magnitude of shifts for different directions will meaningfully affect the symmetry of the distorted state. Glazer notation allows this distinction to be made through the use of different primary symbols λ; however, the matrix notation as presented would need to be modified to reflect this variation—perhaps through the use of variables or constants ∉ {0, ±1} in the matrix itself. Second, we have focused on shifts characterised by periodicities at the zone centre or zone boundary. More complex periodicities are allowed: an example occurs in the material [NPr_{4}]Ni[N(CN)_{2}]_{3} (Pr = C_{3}H_{7}), for which a-shifts are active and modulated by the wave-vector k = [0, ¼, ¼].^{13} One might anticipate the use of the Bradley–Cracknell symbol Σ in the corresponding Glazer notation; likewise there is in principle no reason why complex (or in this case, imaginary) values of exp[2πik_{α}] might not be used in the matrix notation. Nevertheless, in both cases there are issues of distortion phase that are probably too difficult to be unambiguously resolved by a terse symbolic representation. And, third, it is perfectly feasible for a system to support more than one shift distortion along a given axis. Indeed, this may not be particularly rare, given that zone-centre shifts correspond to shear modes.^{50} This situation is akin to the well-known case of compound tilts found in the study of some inorganic perovskites.^{55}
(12) |
Parameter | Value |
---|---|
Space group | Pmm |
a (Å) | 5.0 |
m(B) (a.m.u.) | 54.94 |
m(X) (a.m.u.) | 16.00 |
k _{harm}(B–X) (eV Å^{−2}) | 1.0 |
r _{0}(B–X) (Å) | 2.0 |
k _{harm}(X–X) (eV Å^{−2}) | 1.0 |
r _{0}(X–X) (Å) | 1.0 |
k _{harm}(X⋯X) (eV Å^{−2}) | 1.0 |
r _{0}(X⋯X) (Å) | 2.828 |
k _{angle}(X–B–X) (eV rad^{−2}) | 1.0 |
θ _{0}(X–B–X) (°) | 90.0 |
k _{angle}(X–X–X) (eV rad^{−2}) | 0.01 |
θ _{0}(X–B–X) (°) | 135.0 |
We proceeded to calculate the harmonic phonon dispersion relation for this simple lattice-dynamical model, making use of a k-grid of roughly 0.025 reciprocal lattice units. The corresponding phonon dispersion curves along specific high-symmetry directions are shown in Fig. 10(a). We do not attach any significance to the absolute energy scale of these excitations, since we have not aimed to replicate experimental values in our choice of harmonic spring constants. What is significant is the partitioning of the spectrum into a low-energy regime (which we will come to show dominates NTE behaviour) and a higher-energy regime. With respect to the low-energy component, we note the anomalous slope of the transverse acoustic branch along the Γ–X direction that is diagnostic of a shear instability, the existence of multiple dispersionless bands (evidence of localised degrees of freedom), and also the presence of zone-boundary soft modes.
Fig. 10 Phonon dispersion curves for our lattice-dynamical model and their interpretation in terms of shift degrees of freedom. (a) The entire phonon dispersion across selected high-symmetry directions in reciprocal space, as determined using GULP.^{61} The shaded region at frequencies below 100 cm^{−1} contains the modes responsible for NTE behaviour. (b) The low-frequency region of the phonon dispersion (as shown in (a)) where the branches have been broadened according to the corresponding value of ρ(k, ν). Consequently, those branches that appear bold in this representation correspond to modes with significant translational components. (c) The same low-frequency region of the phonon spectrum now coloured and broadened according to the value of the mode Grüneisen parameter: blue values correspond to γ > 0 and red to γ < 0. The branches that appear bold and red are the most important for NTE; our key result is that these include the shift modes as discussed in the text. |
In order to better understand the distribution of shift modes throughout this phonon dispersion, we exploited the observation that shifts are associated with eigenvectors e(k, ν) uniformly polarised along a single Cartesian axis. Consequently, the projections
(13) |
What is immediately clear is that shifts play an active role in the low-energy dynamics for those branches along which they are allowed. For the Γ–X direction, by way of example, the soft acoustic branch is almost entirely accounted for in terms of shift distortions. This branch is doubly degenerate; its low energy reflects the ease with which planar shifts can be accommodated in this simple model. As k → X, this branch anti-crosses with a rotational RUM branch, such that at the X point itself the shifts correspond to the set of modes with the second lowest phonon frequencies. Note that the longitudinal acoustic branch has increased significantly in energy at this point, such that translations polarised along the same direction as k have very much higher energies. Across the X–M direction, one of the two shift degrees of freedom accessible at X becomes increasingly stiff. As k → M there is only one shift degree of freedom remaining at the lowest energies; this degree of freedom couples strongly with the rotational RUMs such that it contributes to all three lowest-energy phonon branches. These observations are entirely consistent with the RUM analysis of Section 2.
(14) |
The combination of large negative Grüneisen parameters and low phonon frequencies also suggests shift-type vibrational modes are likely to show strongly anharmonic behaviour. Hence, the soft mode instabilities normally associated with octahedral tilts and/or ferroelectric displacements may also involve correlated shifts in perovskite analogues. The equilibrium geometry of the B–X–X angle and the presence and charge distribution of A-site cations will help shape the phonon dispersion and—by virtue of the close match in A-site geometry and perovskite deformation noted in the various case studies above—might also be expected to drive phonon softening in suitable cases. Molecular dynamics studies, such as those used to interrogate negative thermal expansion in Zn(CN)_{2},^{68,69} would provide valuable insight into the possible existence and phenomenology of displacive transitions involving shift degrees of freedom.
Using the CASTEP code,^{70} we optimised the Fmm geometry of the CdPd(CN)_{6} crystal structure^{60} and calculated its phonon spectrum. For these calculations, we used plane-wave basis sets, with the GGA-PBE functional and CASTEP's on-the-fly norm-conserving pseudopotentials for Cd, Pd, C, and N.^{71,72} We did not include any empirical dispersion corrections, and treated the system explicitly as non-metallic. We used a plane wave cutoff of 750eV, a single wave-vector for integration of electronic states across the Brillouin zone, and standard convergence tests for convergence (energy shift per atom of 2μeV). The CASTEP phonon calculations makes use of density-functional perturbation theory;^{73} we used a 2 × 2 × 2 Monkhorst–Pack grid^{74} with a phonon convergence tolerance of 10^{−5}eV Å^{−2} and explicitly accounted for LO/TO splitting and enforcement of the acoustic sum rule. Phonons were calculated at evenly-spaced intervals across the Brillouin zone path Γ–X–M–Γ–R–M–X–R.
We found the system to be unstable with respect to a number of distortions characterised by phonon wave-vectors k distributed throughout the Brillouin zone. The most significant of these distortion modes actually occurs at the zone centre; these modes—which may be interpreted as low-energy modes of the finite-temperature parent structure—are illustrated in Fig. 11. An important point is that the double-perovskite structure of CdPd(CN)_{6} means that k = 0 distortions for this system include all those characterised by periodicities Γ, X, M, and R of the parent perovskite lattice. So we should expect to find zone-boundary (X-type and M-type) shifts amongst these distortions if they are indeed relevant to the dynamical behaviour of CdPd(CN)_{6}. There are in fact 21 unstable modes for CdPd(CN)_{6} at k = 0 (we show the 15 most significant), which fall into three categories. Those corresponding to the largest imaginary frequencies (98i cm^{−1}) are conventional tilt distortions: the six such modes correspond to in-phase and out-of-phase tilt combinations for rotations around each of the three Cartesian axes. The second category corresponds to columnar shift distortions, with nine modes in total: two X-type (95i cm^{−1}) and one M-type (80i cm^{−1}) shift combination for displacements parallel to each of three Cartesian axes. The third and final set of unstable modes are distortive tilts, which are much less significant (50i cm^{−1}). In Fig. 11(b) we highlight the atomic displacement pattern associated with one of the M-type instabilities, which we note is of precisely the same form as the static distortion observed in [NMe_{4}]Ca[N_{3}]_{3} [Fig. 6].
Fig. 11 Key zone-centre displacive instabilities in CdPd(CN)_{6} as determined using DFT calculations. (a) Conventional tilts (highlighted in green) are the least stable modes, followed by X-type and M-type shifts. (b) A representation of the phonon eigenvector associated with one of the M-type shift instabilities (highlighted in panel (a)), together with a schematic relating the collective displacements to the cartoon shown in Fig. 2(a). |
Our key result in this simple DFT study is that all possible octahedral shifts are amongst the set of phonon instabilities for this molecular perovskite analogue. This reinforces our point that shifts play a role in the low-energy dynamics of these systems and, in suitable cases, may be responsible for displacive phase transitions and/or NTE behaviour (including e.g. in the MPt(CN)_{6} family^{65}). There is clearly scope for a more detailed DFT investigation of this—and related—systems, including determination of mode Grüneisen parameters as carried out in ref. 67.
A crucial result of our study has been to show that shift distortions can give rise to symmetry-lowering processes inaccessible through e.g. octahedral tilt mechanisms. The importance of this result lies in the emerging interest in exploiting compound distortions as indirect mechanisms of driving polarisation:^{33,75,76} this is the strategy of so-called “tilt engineering”, which is allowing access to entirely new families of multiferroic materials.^{2} The new symmetry-breaking mechanisms we identify here allow, in principle, for analogous “shift engineering” approaches, where combinations of various correlated shifts—perhaps coupled with tilt or cation order—might be used to break inversion symmetry. For instance, we find that the combination of a^{X}b^{0}c^{0} shifts characterised by the matrix
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Finally, we emphasise the point made in ref. 35 that the incorporation of defects—and in particular B-site vacancies—can influence the number of shift-type instabilities in perovskite systems. The key geometric effect of vacancies is to relax the constraint k_{α} = 0 for shift displacements along the direction α ∈ {a, b, c}: chains on opposite sides of a B-site vacancy are in principle free to displace in different directions via compression of the vacancy. Since defects increasingly appear to be ubiquitous amongst perovskite analogues—and not least so for the Prussian blue analogues, which can support extremely high defect concentrations^{17}—the use of defect structures to influence the importance and nature of shift-type distortions may indeed prove a fruitful avenue for future research.^{78}
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