Rigid Unit Modes Research Articles

Relaxor unveils geometrical frustration

For over a century, materials science has been focused on comparing magnetism and ferroelectricity, revealing similarities in order types, domains, hysteresis, and responses to external stimuli. The exceptional physical properties of magnets often originate from frustration, which can result from site disorder and lattice geometry. While disordered relaxor ferroelectrics exhibit ultra-high dielectric and piezoelectric properties and fingerprinted behavior similar to spin glasses, ferroelectric-like materials with clear geometrically frustrated states have not been previously identified. Here we introduce a new type of relaxor material, a geometrically frustrated relaxor, exemplified by a Bi2Ti2O7 single crystal with a compositionally ordered pyrochlore structure. Our findings demonstrate canonical relaxor behavior, including dielectric anomalies, dipole freezing, non-ergodicity and a lack of spontaneous phase transitions. Furthermore, we show the emergence of a unique dipole ice state upon cooling, arising from conflicting order parameters related to two mutually exclusive rigid unit modes of Bi4Oʹ tetrahedral rotations. The coexistence of these rotations does not satisfy symmetry constraints, causing geometrical frustration and suppressing the long-range order. The results provide the structural mechanism of geometrical frustration manifestation in Bi-containing pyrochlores, opening up avenues for exploring exotic ground states and advanced functionalities in dielectric materials.

Unconventional floppy network structures in titanate glasses

Annealed titanate glasses are important candidate materials for high-performance electronic components, but their structural response to temperature-induced properties upon thermal annealing remains elusive. Here, using high-energy synchrotron X-ray scattering, high-temperature Raman spectroscopy, and empirical potential structure refinement simulation, we track the structural evolution in situ of a barium dititanate (BaTi2O5) glass upon annealing around and below the glass transition temperature. We find that the network structure is intrinsically floppy, as the network units ([TiOm] polyhedra) themselves and their topological packing can be easily modulated by annealing temperature. The floppy nature of the glassy titanate network challenges the currently well-cognized Rigid-Unit Mode model explaining temperature-driven structural reorganization of prototypical network glasses (e.g., silicate/borate glasses). As temperature increases, the floppy network of BaTi2O5 glass undergoes a non-monotonic, two-stage network rearrangement. After cooling back to room temperature, the distortion of network units is found recoverable whereas the change of network connectivity in the intermedium-range order is irreversible. Simulation ensembles further show the irreversible connectivity change can be ascribed to the local development of crystal-like cationic motifs, which in turn induce excess enthalpy relaxation. Our findings provide an atomic-scale perspective on temperature-dependent structural and enthalpy relaxation of a model titanate glass upon annealing, which are crucial for understanding thermal-related changes in the properties of titanate glasses.

Which phonons contribute most to negative thermal expansion in ScF3?

Using calculations of the phonon distribution in ScF3 across the whole of reciprocal space, we show that the important phonons for the negative thermal expansion in this material are those associated with the rigid unit modes (RUMs) and associated quasi-RUMs. We discuss the role of the bond-bending flexibility within the ScF6 octahedra, and how this enables other phonons to make an additional but ultimately much weaker contribution to negative thermal expansion. These results inform recent discussions on the role of correlated atomic motions in giving rise to negative thermal expansion in ScF3.

Revealing the effect of medium-range structure on silicate glass hardness

Atomic structure determines physical properties, but for glassy materials, the nature of structure-property relationships remains ambiguous. Since glass properties are governed by both chemistry and structure, it is difficult to dissociate these two effects. Here, the sole effect of the structure on property is isolated by treating an industrial aluminosilicate glass with either thermal-annealing or pressure-quenching processes to produce glasses with varying densities and hardnesses (at constant composition). To explore the underlying structural origin of property changes, neutron total-scattering patterns of these glasses were measured. These results confirm the applicability of rigid-unit mode theory since the short-range tetrahedra were found to remain unaffected. In contrast, close correlations are derived between properties and medium-range structure (as encoded in various features of the first sharp diffraction peak). Overall, it reveals that the increase in the medium-range order is the structural origin of the extra extent of hardness increase beyond the densification effects.

Biaxial negative thermal expansion in carbon nets - graphyne and derivatives.

Materials that shrink during heating are important for advanced applications. Graphene has shown negative thermal expansion (NTE) up to 1000 K, which drives further research towards new two-dimensional allotropes of carbon to achieve higher performance. In this article, we have shown high NTE for Graphynes having sp-sp connectivity and that are stable at high temperature. The effect of heteroatom substitution in NTE for some graphynes and their periodic trends are also studied. Quasi-harmonic approximation (QHA) calculations show that thermal expansion remains negative at least until 1000 K for some graphynes. The results are in good agreement with ab-initio molecular dynamics (AIMD) simulations. The high NTE in the graphynes is understood based on their rigid unit modes (RUMs).

Anisotropic thermal expansion of silicon monolayer in biphenylene network.

Materials with a negative thermal expansion property are of great importance in the emerging family of two-dimensional materials. For example, mixing two materials with negative and positive coefficients of thermal expansion avoids volume changing with temperature. In this work, based on first-principles calculations and Grüneisen's theory, we investigated the thermal expansion properties of a silicon monolayer in biphenylene networks. Our results show that the thermal expansion is greatly negative and anisotropic, as the linear thermal expansion coefficient along the a-direction is significantly smaller than the one along the b-direction, even at high temperatures. At 300 K, the thermal expansion coefficients along the two lattice directions are -17.010 × 10-6 K-1 and -2.907 × 10-6 K-1, respectively. By analyzing the Grüneisen parameters and the elastic compliance, we obtained an understanding of the giant negative thermal expansion of the material. Rigid unit modes are also responsible for the negative thermal expansion behavior. Our work provides fundamental insights into the thermal expansion of silicon monolayer in biphenylene networks and should stimulate the further exploration of the possible thermoelectric and thermal management applications of the material.

The contribution of phonons to the thermal expansion of some simple cubic hexaboride structures: SmB6, CaB6, SrB6 and BaB6.

We have performed first-principles calculations of the structure and lattice dynamics in the metal hexaborides SmB6, CaB6, SrB6 and BaB6 using Density Functional Theory in an attempt to understand the negative thermal expansion in the first of these materials. The focus is on the role of Rigid Unit Modes involving rotations of the B6 octahedra similar to the rotations of structural polyhedra connected by bonds in Zn(CN)2, Prussian Blue and Si(NCN)2. However, it was found that there is very low flexibility of the network of connected B6 octahedra, and the lattice dynamics do not support negative thermal expansion except possibly at very low temperature. Thus the negative thermal expansion observed in SmB6 probably has an electronic origin.

Anharmonic Interaction in Negative Thermal Expansion Material CaTiF6.

Although the quasi-harmonic approximation (QHA) method applies to many materials, it is necessary to study the anharmonic interaction for extremely anharmonic materials. In this work, the unusual negative thermal expansion (NTE) property of CaTiF6 is studied by combing QHA and anharmonic interaction. The improved self-consistent phonon approximation (ISCPA), which treats anharmonic effects in solids nonperturbatively, is employed. The agreement of NTE behavior between the calculation and the experiment can be further promoted from qualitative consistency by QHA to quantitative consistency by the ISCPA. From mode Grüneisen parameters, it is found that the low-frequency phonons, especially acoustic phonons, contribute greatly to the NTE behavior of CaTiF6. The rigid unit modes (RUMs) of low-frequency optical phonons can be identified. The phonon lifetime of CaTiF6 is calculated from three-phonon interactions; thereby, the NTE mechanism can be further explored by phonon lifetimes of phonons with different frequencies on heating. The anomalous lattice thermal conductivity (LTC) is predicted using the Boltzmann transport equation within the relaxation time approximation. The glasslike LTC can occur in crystal CaTiF6.

Negative Thermal Expansion Induced in Tri-graphene and T-graphene by the Rigid-Unit Modes.

Materials which contract on heating (negative thermal expansion, NTE) are of significant interest for advanced applications. Graphene has shown NTE up to 1000 K, which motivates further improvements in two-dimensional carbon to attain superior performance. In this Communication, very large negative thermal expansion coefficients (αT) are reported for tri-graphene (TrG) and T-graphene (TG). Quasi-harmonic approximation calculations show that αT remains negative until 4200 K and 2900 K for TrG and TG, respectively. The high NTE for these systems is understood on the basis of the soft phonon modes, which induce rotation of the 3-membered and the 4-membered rings in TrG and TG, respectively, and ab initio molecular dynamics simulations. The local distortions for the 3-12 rings (in TrG) and 4-8 rings (in TG) have structural resemblance with the rigid-unit modes that are usually envisioned for bulk systems.

Unfolding the terahertz spectrum of soft porous crystals: rigid unit modes and their impact on phase transitions

The flexibility of soft porous crystals is influenced by RUMs. Via static DFT calculations, the effect of building block substitutions on the RUMs in winerack MOFs is investigated, giving insight in the flexible nature of these frameworks.

Soft-mode anisotropy in the negative thermal expansion material ReO3

We use a symmetry-motivated approach to analyse neutron pair distribution function data to investigate the mechanism of negative thermal expansion (NTE) in ReO$_3$. This analysis shows that the local structure of ReO$_3$ is dominated by an in-phase octahedral tilting mode and that the octahedral units are far less flexible to scissoring type deformations than the octahedra in the related compound ScF$_3$. These results support the idea that structural flexibility is an important factor in NTE materials, allowing the phonon modes that drive a volume contraction of the lattice to occupy a greater volume in reciprocal space. The lack of flexibility in ReO$_3$ restricts the NTE-driving phonons to a smaller region of reciprocal space, limiting the magnitude and temperature range of NTE. In addition, we investigate the thermal expansion properties of the material at high temperature and do not find the reported second NTE region. Finally, we show that the local fluctuations, even at elevated temperatures, respect the symmetry and order parameter direction of the observed $P4/mbm$ high pressure phase of ReO$_3$. The result indicates that the motions associated with rigid unit modes are highly anisotropic in these systems.

Extraordinary negative thermal expansion of monolayer biphenylene

Recently, the two-dimensional biphenylene with sp2-hybridized carbon atoms has been successfully fabricated by experiment [Science,372, 852 (2021)]. In this paper, the thermal expansion properties of monolayer biphenylene (MBP) are investigated by using the density functional theory combined with Grüneisen's theory. It is found that MBP exhibits anisotropic and negative in-plane thermal expansion and the negative thermal expansion is up to 1000 K. Although MBP and graphene have similar sp2 hybridized planar structure, the MBP exhibits significantly larger thermal contraction than graphene over a wide temperature range (0–1000K). At 300 K, the linear thermal expansion coefficient of MBP along the a-direction (1.10 × 10−5 K−1) is about three times that of graphene (3.7 × 10−6 K−1). The extraordinary negative thermal expansion not only originates from the ripple effect as in graphene only, but also the rigid unit modes are responsible for the thermal expansion behavior.

Theoretical and computational improvements to the algebraic method for discovering cooperative rigid-unit modes

A linear-algebraic algorithm for identifying rigid-unit modes in networks of interconnected rigid units has recently been demonstrated. This article presents a series of enhancements to the original algorithm, which greatly improve its conceptual simplicity, numerical robustness, computational efficiency and interpretability. The improvements include the efficient isolation of constraints, the observation of variable-block separability, the use of singular value decomposition and a quantitative measure of solution inexactness.

The ISOTILT software for discovering cooperative rigid-unit rotations in networks of interconnected rigid units.

A user-friendly web-based software tool called 'ISOTILT' is introduced for detecting cooperative rigid-unit modes (RUMs) in networks of interconnected rigid units (e.g. molecules, clusters or polyhedral units). This tool implements a recently described algorithm in which symmetry-mode patterns of pivot-atom rotation and displacement vectors are used to construct a linear system of equations whose null space consists entirely of RUMs. The symmetry modes are first separated into independent symmetry-mode blocks and the set of equations for each block is solved separately by singular value decomposition. ISOTILT is the newest member of the ISOTROPY Software Suite. Here, it is shown how to prepare structural and symmetry-mode information for use in ISOTILT, how to use each of ISOTILT's input fields and options, and how to use and interpret ISOTILT output.

Crystal flex bases and the RUM spectrum

A theory of infinite spanning sets and bases is developed for the first-order flex space of an infinite bar-joint framework, together with space group symmetric versions for a crystallographic bar-joint framework ${{\mathcal {C}}}$. The existence of a crystal flex basis for ${{\mathcal {C}}}$ is shown to be closely related to the spectral analysis of the rigid unit mode (RUM) spectrum of ${{\mathcal {C}}}$ and an associated geometric flex spectrum. Additionally, infinite spanning sets and bases are computed for a range of fundamental crystallographic bar-joint frameworks, including the honeycomb (graphene) framework, the octahedron (perovskite) framework and the 2D and 3D kagome frameworks.

Uniaxial negative thermal expansion behavior of β -CuSCN

Negative thermal expansion (NTE) as an interesting physical behavior is promising for thermal expansion control engineering. β-CuSCN consists of linear chain units with NTE along the c-axis. The NTE mechanism of β-CuSCN is investigated by variable temperature x-ray diffraction, temperature- and pressure-dependent Raman spectra, and first-principles calculations. It is found that the quasi rigid unit modes associated with the rotations of S–C≡N–Cu chains driven by Cu and S antiphase transverse vibrations and longitudinal acoustic and transverse acoustic modes involving the collective motions of atoms have large negative Grüneisen parameters, contributing significantly to the NTE of c-axis. Translational and librational motions of C≡N units, in which C and N atoms vibrate in the same and opposite directions have much smaller negative Grüneisen parameters, contribute only a minor part to the NTE, which is different from the known NTE mechanism of cyanides and Prussian blue analogous.

Framework structure crystalline materials and Rigid Unit Modes (RUMs). Introducing the new concept of MLRUMs and skeletions Authors

This article reviews Framework Structures (FWSs), defined as crystalline materials built of rigid AXn polyhedra sharing vertices (like perovskites, tungsten bronzes, Dion-Jacobson, Ruddlesden-Popper, and Aurivillius phases, quartz, silicates, and others), and their pecularities resulting from this linkage. The situation of rigid units linked by common vertices may allow the units to accomplish concordant rotations without deformation, which gives rise to soft phonon modes called “Rigid Unit Modes” (RUMs). The condensation of a RUM can trigger structural phase transitions to a structure of lower symmetry, with tilted polyhedra, at the origin of spontaneous ferroic or multiferroic properties. We overview results precedently obtained on RUMs in perovskites, tetragonal tungsten bronzes, and quartz, and detail new results on “maximally localized RUMs” (MLRUMs), a fundamental new concept in the physics of RUMs. We introduce also the related new concept of “skeletions” that allows to generate all ferroelastic phases found in these systems, and generalizes the Glazer's tilt-system approach.

Symbol functions for symmetric frameworks

We prove a variant of the well-known result that intertwiners for the bilateral shift on ℓ2(Z) are unitarily equivalent to multiplication operators on L2(T). This enables us to unify and extend fundamental aspects of rigidity theory for bar-joint frameworks with an abelian symmetry group. In particular, we formulate the symbol function for a wide class of frameworks and show how to construct generalised rigid unit modes in a variety of new contexts.

Quantitative understanding of negative thermal expansion in scandium trifluoride from neutron total scattering measurements

Negative thermal expansion (NTE) - the phenomenon where some materials shrink rather than expand when heated - is both intriguing and useful, but remains poorly understood. Current understanding hinges on the role of specific vibrational modes, but in fact thermal expansion is a weighted sum of contributions from every possible mode. Here we overcome this difficulty by deriving a real-space model of atomic motion in the prototypical NTE material scandium trifluoride, ScF3, from total neutron scattering data. We show that NTE in this material depends not only on rigid unit modes - the vibrations in which the scandium coordination octahedra remain undistorted - but also on modes that distort these octahedra. Furthermore, in contrast with previous predictions, we show that the quasiharmonic approximation coupled with renormalisation through anharmonic interactions describes this behaviour well. Our results point the way towards a new understanding of how NTE is manifested in real materials.

Negative thermal expansion of cubic silicon dicarbodiimide, Si(NCN)2, studied by ab initio lattice dynamics

We report an ab initio calculation of crystal structure and lattice dynamics of cubic silicon dicarbodiimide, Si(NCN)2, using density functional theory methods. The calculations reveal a low-energy spectrum of rigid unit modes that are shown to be associated with negative thermal expansion. Comparisons are drawn with the closely-related materials Zn(CN)2 and the cubic-cristobalite phase of SiO2. Instabilities in the spectrum of rigid unit modes point to the existence of disorder of the positions and orientations of the dicarbodiimide molecular anions.