Incommensurate chirality density wave transition in a hybrid molecular framework

TL;DR

This study uncovers a novel incommensurate chirality density wave phase in a hybrid molecular framework, revealing a new form of chiral symmetry breaking driven by coupling of lattice and molecular modes.

Contribution

It demonstrates the existence of an incommensurate chirality density wave phase in a hybrid framework, a previously unknown type of chiral order in solids.

Findings

Identification of the incommensurate phase transition at 235 K.

Observation of spontaneous chiral resolution in molecular conformations.

Proposal of coupling mechanisms between acoustic and molecular modes.

Abstract

Using single-crystal X-ray diffraction we characterise the 235\,K incommensurate phase transition in the hybrid molecular framework tetraethylammonium silver(I) dicyanoargentate, [NEt$_4$]Ag$_3$(CN)$_4$. We demonstrate the transition to involve spontaneous resolution of chiral [NEt$_4$]$^+$ conformations, giving rise to a state in which molecular chirality is incommensurately modulated throughout the crystal lattice. We refer to this state as an incommensurate chirality density wave (XDW) phase, which represents a fundamentally new type of chiral symmetry breaking in the solid state. Drawing on parallels to the incommensurate ferroelectric transition of NaNO$_2$ we suggest the XDW state arises through coupling between acoustic (shear) and molecular rotoinversion modes. Such coupling is symmetry-forbidden at the Brillouin zone centre but symmetry-allowed for small but finite modulation vectors $\mathbf q=[0,0,q_z]^\ast$. The importance of long-wavelength chirality modulations in the physics of this hybrid framework may have implications for the generation of mesoscale chiral textures, as required for advanced photonic materials.

Equations

$$(\nabla \psi )\varphi -\psi (\nabla \varphi )$$

Full text

Symmetry breaking processes that involve ordering of chiral degrees of freedom are implicated in the physics of a variety of exotic states of matter. Both the optical gyrotropy Orenstein_2013 and polar Kerr effect Xia_2006 ; Konig_2017 of underdoped cuprates arise from chiral ordering of loop-current anapoles Pershoguba_2013 . Likewise, chirality density wave (XDW note ) formation is implicated in the ‘hidden order’ state of URu2Si2 Kung_2015 . The possibility Kallin_2009 ; Kallin_2012 that chiral $p$-wave symmetry characterises the superconducting state of Sr2RuO4 has generated intense interest because the system should then support not only Majorana fermions Nobukane_2011 but also the non-Abelian winding statistics exploitable in quantum computing Nayak_2008 . Emergent chiral phases are observed in frustrated magnets Grohol_2005 ; Batista_2016 and in multi-$\mathbf{q}$ charge density wave (CDW) states, e.g. in TeSe2 Ishioka_2010 ; vanWezel_2010 ; Castellan_2013 . While in all these examples chirality emerges from peculiarities of the underlying electronic structure, there is intense fundamental interest in the phenomenology of chiral order Gopalan_2011 , irrespective of its particular physical origin.

In a chemical context, chirality arises from the spatial arrangement of molecules Pasteur_1848 , supramolecular assemblies Liu_2015 ; Roche_2016 , or extended lattices Kepert_2000 ; Wells_1977 . It is characterised formally by the absence of rotoinversion symmetry, which includes inversion and mirror symmetry. In the solid state, molecular (e.g. proteins) and lattice (e.g. $\beta$-quartz) chirality are usually fixed since interconversion between states of opposite handedness usually requires breaking of chemical bonds. By contrast, supramolecular assemblies—held together by weak non-covalent interactions—can in favourable cases reorganise dynamically to allow reversible chiral symmetry breaking; this is the mechanism responsible for the emergence of chiral cholesteric phases in liquid crystals, for example Dierking_2014 .

A conceptually related phenomenon is that of dynamic interconversion between locally-chiral states in the metal–organic framework (MOF) [Zn2(bdc)${}_{2}\cdot$dabco] (bdc2- = 1,4-benzenedicarboxylate anion; dabco = diazabicyclooctane) Gabuda_2008 ; Gabuda_2011 . Here chirality is associated with two equivalent conformations (left-handed twist; right-handed twist) of the dabco molecule. On cooling below 25 K, the system appears from 1H NMR measurements Gabuda_2011 to undergo a transition from dynamic enantiomer interconversion (‘parachiral’ state) to a state in which molecule chiralities are fixed on NMR timescales, reflecting at least a local—if not necessarily long-range—chiral symmetry breaking process. A key success of Refs. 24; 25 was to show that framework structures incorporating molecules whose chirality arises from conformational states (i.e. rather than chemical connectivity) may in principle exhibit thermally-accessible chiral ordering transitions.

It was in this spirit that we chose to study the phase behaviour of the hybrid molecular framework tetraethylammonium silver(I) dicyanoargentate, [NEt4]Ag3(CN)4 (Et = ethyl) Hill_2016 . Though achiral in both solution and the gas phase, the [NEt4]+ cation can nevertheless adopt a geometry in condensed phases with non-superimposable left- and right-handed conformations [Fig. 1(a)]. In the room-temperature crystal structure of [NEt4]Ag3(CN)4, both conformations co-exist in a disordered fashion: the electron density at the molecular cation site reflects an equal population of left- and right-handed conformations, with the two related by a crystallographic mirror plane [Fig. 1(b)]. We reported in Ref. 26 the existence of an incommensurate phase transition at $T_{\rm c}=235$ K in this system. In this Letter, we characterise this transition and show it to involve spontaneous resolution of [NEt4]+ conformations, giving rise to a state in which molecular chirality is incommensurately modulated throughout the crystal lattice: we describe this as an incommensurate XDW phase (by analogy to Kung_2015 ) and suggest that it represents a fundamentally new type of chiral symmetry breaking in the solid state.

Following the approach of Ref. 26 we prepared single crystals of [NEt4]Ag3(CN)4. Two separate sets of single-crystal X-ray diffraction measurements were performed: the first involved data collection at 100–300 K using an Oxford Diffraction Supernova diffractometer fitted with an Oxford Cryosystems 700 Plus open-flow nitrogen cooling device; the second involved a single data collection at 30 K on the I19 Beamline at the Diamond Light Source, making use of an Oxford Cryosystems HeliX open-flow helium cryostat Goeta_1999 . In both cases, crystals were mounted in perfluoropolyether oil on a MiTeGen loop. The CrysAlisPro and CrystalClear software were used for data collection and reduction as appropriate Crysalis_2010 ; CrystalClear_2009 ; structure refinement was carried out in JANA2006 Petricek_2014 .

At ambient temperature, the crystal structure of [NEt4]Ag3(CN)4 has orthorhombic $Pnma$ symmetry Hill_2016 : three crystallographically-related diamondoid [Ag3(CN)4]- lattices interpenetrate, with the conformationally-disordered [NEt4]+ counterions situated in extra-framework cavities [Fig. 1(c)]. Argentophilic interactions Jansen_1987 ; Schmidbaur_2015 connect adjacent frameworks, presumably stabilising the particular mode of interpenetration adopted. There is no appreciable void space in this structure; hence one anticipates strong coupling between cation orientations and framework displacements, as in e.g. the organic lead halide perovskites Frost_2016 . The single crystallographically-distinct [NEt4]+ cation is located on the $4c$ Wyckoff position, the site symmetry of which is $.m.$ (i.e., with the mirror plane lying perpendicular to the $\mathbf{b}$ crystal axis). The important point is that this site symmetry is incompatible with the molecular point symmetry of any chiral species.

On cooling below $T_{\rm c}$ we observe the emergence of a set of additional reflections associated with wave-vectors $\mathbf{Q}=\mathbf{G}+m\mathbf{q}$, where $\mathbf{G}$ is a reciprocal lattice vector, $\mathbf{q}$ a modulation vector $\mathbf{q}=[0,0,q_{z}]$, and $m\in\mathbb{Z}$ the superspace index [Fig. 2(a)]. The first-order ($|m|=1$) superlattice reflection intensities vary as the square of the order parameter, the temperature dependence of which is shown in Fig. 2(a). The transition is probably first-order in nature, given that there is a small discontinuity in the $c$ unit cell parameter at $T_{\rm c}$ (see SI). The modulation wave-vector component $q_{z}$ is evidently independent of temperature [Fig. 2(b)]: we find $q_{z}=0.12867(16)$ for our 100–300 K data set and $q_{z}=0.1278(3)$ for the 30 K data set (the variation in $q_{z}$ here is within the uncertainty expected for data collected using different diffractometers and different crystals Cummins_1990 ). Temperature-independent modulation wave-vectors are not uncommon amongst incommensurate phase transitions: notable examples are the transitions in quartz, NaNO2, and biphenyl McConnell_1991 ; Cummins_1990 . So our experimental measurements indicate a low-temperature modulation of the [NEt4]Ag3(CN)4 crystal structure by a set of distortions that repeat along the $\mathbf{c}$ crystal axis with wavelength $c/q_{z}\simeq 7$ nm.

The particular superlattice reflection conditions of the low-temperature diffraction pattern of [NEt4]Ag3(CN)4 unambiguously identify the low-temperature superspace group as $Pnma(00\gamma)0s0$. Crucially, the symmetry-lowering process described by a transition from $Pnma$ to this superspace group removes precisely those mirror symmetry elements incompatible with chiral order at the [NEt4]+ site and replaces them with superspace ($s$-)glides in the same orientation. We will come to show that key consequence is that the two enantiomeric [NEt4]+ conformations are no longer disordered throughout the crystal, but are instead modulated by the incommensurate wave-vector $\mathbf{q}$: the $s$-glides map right- and left-handed conformations onto one another.

Superstructure refinement using the data collected at 100 K reveals two dominant contributions to the structural modulation activated at $T_{\rm c}$: (i) a continuous displacement modulation of both framework and [NEt4]+ counterions that is polarised predominantly along the $\mathbf{b}$ crystal axis, and (ii) a crenel-type occupancy modulation of the two enantiomorphic conformations of the [NEt4]+ ion. The amplitudes of these two components as a function of modulation coordinate are shown in Fig. 3(a). We note that the displacement magnitude is relatively large—a shift of ca 1 Å between maximum and minimum $b\Delta y$ values—and is best modelled using sinusoidal and saw-tooth modulation functions for the framework and cation, respectively (see SI for further discussion and full details of our refinements). But our key result is the incommensurate modulation of chirality throughout the [NEt4]Ag3(CN)4 structure; this is the XDW state, which to the best of our knowledge has not previously been reported. Because $q_{z}\simeq\frac{1}{8}$ we can generate an approximant of the modulated structure with $\mathbf{c}_{\textrm{XDW}}\simeq 8\mathbf{c}_{Pnma}$. Graphical representations of the 100 K framework displacements, [NEt4]+ molecular ion displacements, and [NEt4]+ enantiomeric states present in this approximant model are shown in Fig. 3(b,c); the model itself is included as Supporting Information.

A transition with close conceptual similarity occurs in the inorganic salt sodium nitrite, NaNO2 Yamada_1963 ; Heine_1984 . The NO${}_{2}^{-}$ ion has a bent geometry and so supports a local dipole moment. At temperatures above $T_{\textrm{IC}}=436$ K, NaNO2 is in a paraelectric state, with NO${}_{2}^{-}$ ion orientations disordered over two positions corresponding to inverse polarisation states. The system begins to order on cooling below $T_{\textrm{IC}}$, but enters an incommensurate regime before settling into a ferroelectric state at $T_{\textrm{C}}=434$ K. The crystal symmetry of NaNO2 is such that chiral degrees of freedom do not play a role in its phase transition behaviour. Nevertheless the two polarisation states of the NO${}_{2}^{-}$ ion act as an Ising variable (or pseudospin) in precisely the same way as do the two chirality states of [NEt4]+; both local degrees of freedom are odd under inversion, and hence the elegant body of symmetry analysis developed to understand NaNO2 Heine_1981 ; LyndenBell_1994 is directly applicable to [NEt4]Ag3(CN)4.

The key rationale for an incommensurate phase transition in systems such as NaNO2 is the interplay of two structural distortion modes $\psi,\phi$ of different symmetries. This is the ‘structrual resonance’ mechanism of Heine and McConnell Heine_1984b , which is applicable not only to NaNO2 but also to a diverse range of systems including minerals Levanyuk_1976 , molecular solids Bak_1976 , and alloys Friedel_1977 . The central concept is that modulation by some wave-vector $\mathbf{q}$ allows the two distortion modes to interact; moreover for degrees of freedom with odd parity (such as dipoles or chiral states) the strength of this interaction scales as $|\mathbf{q}|^{2}$ and so becomes large for some small but finite $\mathbf{q}\neq[0,0,0]$. A characteristic signature of the mechanism is a phase shift of ca $\pi/2$ in the modulation of the two structural distortions, which is a consequence of the role played by the Lifshitz invariant

[TABLE]

in the free energy expansion McConnell_1991 ; Lifshitz_1942 . In the case of NaNO2 the two distortions are (i) shear and (ii) ferroelectric order of NO${}_{2}^{-}$ ions.

We suggest that the incommensurate XDW state in [NEt4]Ag3(CN)4 can be rationalised using entirely analogous arguments. In the $\mathbf{k}\rightarrow\mathbf{0}$ limit, the framework modulation shown in Fig. 3(b) maps onto a $yz$ shear distortion. This distortion mode is characterised by the $\Gamma_{3}^{+}$ irreducible representation (irrep.) and, if active, would reduce the space-group symmetry of the parent structure from $Pnma$ to $P\frac{2_{1}}{n}11$ ($\equiv P2_{1}/c$) Stokes_1988 . Group-theoretical analysis using the ISOTROPY software Campbell_2006 shows the $\mathbf{k}\rightarrow\mathbf{0}$ limit of the chiral ordering pattern illustrated in Fig. 3(c) to be characterised by the $\Gamma_{4}^{-}$ irrep. If activated in isolation, this second distortion would reduce the parent space-group symmetry to $Pn2_{1}a$ ($\equiv Pna2_{1}$). Hence the two distortions are symmetry-forbidden from mixing in a commensurate state. For modulation wave-vectors of the type $\mathbf{q}=[0,0,\xi]^{\ast}$, both distortions transform as the $\Lambda_{3}$ irrep., and so can couple in accordance with the theory outlined in Refs. 36; 40. The $\Lambda_{3}$ irrep. can be shown to act as the primary order parameter for descent of symmetry from $Pnma$ to $Pnma(00\gamma)0s0$ Stokes_2013 . Furthermore, the empirical phase shift between the modulation of [NEt4]+ chirality and framework displacement is close to $\pi/2$ [i.e., $\Delta t\simeq\frac{1}{4}$ in Fig. 3(a)].

So the incipient chiral order in ambient-phase [NEt4]Ag3(CN)4 is relatively straightforward and involves a tendency to form alternate chains of right- and left-handed [NEt4]+ conformations. But the interplay of this ordering motif with a shear instability leads to an incommensurate XDW state, essentially as a consequence of crystal symmetry.

One might then anticipate that [NEt4]Ag3(CN)4 may not be unique amongst molecular frameworks in exhibiting such a state: shear instabilities are relatively common for this broader family, primarily as a result of the low energy penalty associated with framework deformation (see e.g. Cairns_2013 ). In designing XDW systems one obvious requirement is the presence of an optically-active molecular component, which imposes its own size constraints. Hence XDW states are more likely to be found in open framework structures, such as MOFs Hoskins_1990 ; Rowsell_2004 or their inorganic counterparts Hill_2016 ; Li_2017 , capable of containing chiral molecules within their framework cavities. In such cases, the onset temperature of the XDW state will be a function of the barrier height for enantiomeric interconversion—and hence molecules with chiral states that arise from conformation rather than topology will favour XDW transitions at accessible temperatures since interconversion does not require bond breaking.

From a conceptual viewpoint the physics at play in [NEt4]Ag3(CN)4 shares much in common with the established phenomenology of translation–rotation coupling in molecular solids LyndenBell_1994 . The key difference is the thermal accessibility of internal degrees of freedom beyond molecular translations and rotations alone. While a variety of different types of conformational degrees of freedom are very possibly relevant in many hybrid frameworks (e.g. staggered–eclipsed NH3CH${}_{3}^{+}$ conformations in methylammonium lead iodide Whitfield_2016 ), here we have focused on chirality because rotoinversion is the obvious point symmetry operation absent from the ‘classical’ systems of Ref. 38. The ability to modulate chirality over nanometre lengthscales suggests an attractive bottom-up approach for developing next-generation photonic materials that exploit the interaction between chiral matter and circularly polarised light Lodahl_2017 . Moreover, there is genuine scope for control: taking [NEt4]Ag3(CN)4 as an example, the modulation wavelength $\lambda=c/q_{z}$ will be affected by the shear modulus of the $[$M3(CN)${}_{4}]^{-}$ framework, which in turn could be tuned by chemical substitution e.g. of Ag for Cu or Au. Likewise there is enormous scope for substitution on the [NEt4]+ site, and indeed variation in network topology Hill_2016 ; Schlueter_2005 . Whether or not it eventually proves possible to use XDW states to generate mesoscale chiral textures, our study has firmly established the ability of hybrid frameworks to support unconventional chiral order.