Incommensurate Modulated Structure Research Articles

Aperiodic approximants bridging quasicrystals and modulated structures

Aperiodic crystals constitute a class of materials that includes incommensurate (IC) modulated structures and quasicrystals (QCs). Although these two categories share a common foundation in the concept of superspace, the relationship between them has remained enigmatic and largely unexplored. Here, we show “any metallic-mean” QCs, surpassing the confines of Penrose-like structures, and explore their connection with IC modulated structures. In contrast to periodic approximants of QCs, our work introduces the pivotal role of “aperiodic approximants”, articulated through a series of k-th metallic-mean tilings serving as aperiodic approximants for the honeycomb crystal, while simultaneously redefining this tiling as a metallic-mean IC modulated structure, highlighting the intricate interplay between these crystallographic phenomena. We extend our findings to real-world applications, discovering these tiles in a terpolymer/homopolymer blend and applying our QC theory to a colloidal simulation displaying planar IC structures. In these structures, domain walls are viewed as essential components of a quasicrystal, introducing additional dimensions in superspace. Our research provides a fresh perspective on the intricate world of aperiodic crystals, shedding light on their broader implications for domain wall structures across various fields.

Anharmonic incommensurate structure modulation in Ni-Mn-Ga martensite exhibiting highly mobile twin boundaries

We perform neutron diffraction on a bulk Ni50.0Mn27.5Ga22.5 single crystal to investigate the evolution of its five-layered modulated (10M) martensite structure, from 300 K down to 10 K. Close to martensite transformation, the modulation is nearly commensurate, with q = 0.402 in a modulation vector (q, q, 0). Upon cooling, the shift in diffraction satellites indicates a transition to an incommensurate modulation with increasing q. However, the observed fifth-order diffraction satellites below 260 K and even more complex satellite landscape at 10 K cannot be explained by incommensurability alone. Our analysis reveals that the modulation function is highly anharmonic, encompassing Fourier components up to the eighth order. We have developed a single-parameter description of the evolution of modulation across the entire temperature interval of the 10M phase existence. Surprisingly, the structure evolution from commensurate to incommensurate modulation has no significant effect on twin boundary mobility.

Enhanced energy-storage density in (Pb0.98La0.02)(Zr0.45-xSn0.55Hfx)0.995O3 antiferroelectric ceramics

Antiferroelectric ceramics with unique characteristics of electric-field-induced antiferroelectric to ferroelectric phase transitions and ultra-fast charging-discharging rate, which are demanded in high pulse power devices. In this work, (Pb0.98La0.02)(Zr0.45-xSn0.55Hfx)0.995O3 (PLZSH) antiferroelectric ceramics with x = 0∼0.20 were prepared via a conventional solid-state reaction approach. Incommensurate modulation structure with a 1/n<110> superlattice was observed in AFE at room temperature, which was associated with the unique domain boundaries, contributing the obvious declined AFE-FE phase transition electric field as well as the enhanced energy storage density. It was also observed that the maximum recoverable energy storage density (Wrec) increases from 2.93 J/cm3 (x = 0) to 5.72 J/cm3 (x = 0.15), and the corresponding energy-storage efficiency uprises from 82.4 % to 87.8 % with the increasing x at 280 kV/cm. The superior energy density and efficiency indicate that the obtained PLZSH antiferroelectric ceramics are promising for practical applications in pulse power devices.

Incommensurate local structural modulations in the uranium oxide system α-U3O8 studied on the diffuse X-ray scattering station ID28-II, ESRF

Incommensurate local structural modulations in the uranium oxide system α-U₃O₈ studied on the diffuse X-ray scattering station ID28-II, ESRF

Misfit-generated structural and optical anisotropies of the natural MoS2−PbS van der Waals heterostructure merelaniite

Significant efforts developing ${\mathrm{MoS}}_{2}\text{\ensuremath{-}}\mathrm{PbS}$-based heterostructure devices with several different architectures show promise for photosensor, solar cell, and chemical sensor applications. Merelaniite $({\mathrm{Mo}}_{4}{\mathrm{Pb}}_{4}{\mathrm{VSbS}}_{15})$ is a newly discovered natural van der Waals heterostructure of the cylindrite type, composed predominantly of heavily modulated pseudotetragonal PbS layers and pseudohexagonal ${\mathrm{MoS}}_{2}$ layers with large misfit-induced anisotropy. For an incommensurate modulated structure, the refined structural model from single-crystal x-ray diffraction analysis is in reasonable agreement with the results obtained by high-resolution scanning transmission electron microscopy, especially in light of the fact that that the two isolated single-crystal domains used for the x-ray and electron diffraction experiments were extracted from two different whiskers and subjected to different sample preparation methods. The effects of the misfit-induced structural anisotropy are studied using angle-resolved polarized Raman spectroscopy. The intensities of 12 Raman modes are studied as a function of incident polarization angle relative to merelaniite's whisker axis, and show maximal intensity with the polarization direction perpendicular to the whisker axis. Polarization-dependent anisotropic third-harmonic generation from ultrathin mechanically exfoliated flakes reveals the anisotropy of the third-order nonlinear susceptibility tensor. Merelaniite demonstrates an expanded structure-chemistry space for engineering stable layered materials for potential device applications.

Magneto-Elastic Coupling in a Sinusoidal Modulated Magnet Cr2GaN

We use neutron powder diffraction to investigate the magnetic and crystalline structure of Cr2GaN. A magnetic phase transition is identified at T ≈ 170 K, whereas no trace of structural transition is observed down to 6 K. Combining Rietveld refinement with irreducible representations, the spin configuration of Cr ions in Cr2GaN is depicted as an incommensurate sinusoidal modulated structure characterized by a propagating vector k = (0.365, 0.365, 0). Upon warming up to the paramagnetic state, the magnetic order parameter closely resembles to the temperature dependence of c-axis lattice parameter, suggesting strong magneto-elastic coupling in this compound. Therefore, Cr2GaN provides a potential platform for exploration of magnetically tuned properties such as magnetoelectric, magnetostrictive and magnetocaloric effects, as well as their applications.

Pressurizing the van der Waals magnet FeOCl at low temperatures: Phase transitions and structural evolution

Magnetic order is frustrated on the orthorhombic lattice of van der Waals layered FeOCl. Antiferromagnetic (AFM) order is attained at ambient pressure upon cooling through ${T}_{N}=81$ K, due to an accompanying monoclinic lattice distortion lifting the magnetic frustration. Within the paramagnetic state at 293 K, an incommensurate structural modulation appears above a critical pressure of ${p}_{c}\ensuremath{\approx}15$ GPa, while orthorhombic symmetry is retained. This modulation is related to an optimization of the packing of chlorine atoms within the van der Waals gap. Here, we report four new phases in the pressure-temperature $(p,T)$ phase diagram of FeOCl below room temperature. High-pressure--low-temperature single-crystal x-ray diffraction (SXRD) up to 37.8 GPa reveals that, at 100 K, the AFM transition occurs at $p=7.3\ifmmode\pm\else\textpm\fi{}1.3$ GPa. The pressure coefficient of $\mathrm{\ensuremath{\Delta}}{T}_{N}/\mathrm{\ensuremath{\Delta}}p=2.13$ K/GPa explains that FeOCl remains paramagnetic up to the highest measured pressure of 33.3 GPa at 293 K. At 6 and 100 K, the structural modulation appears around ${p}_{c}\ensuremath{\approx}15$ GPa within the AFM ordered phase with monoclinic symmetry. The monoclinic and triclinic lattice distortions increase with pressure up to $\ensuremath{\gamma}=90.64{(1)}^{\ensuremath{\circ}}$, much larger than the maximum value of $90.{1}^{\ensuremath{\circ}}$, that can be reached upon cooling at ambient pressure. The structural evolution provides the geometrical basis for the increase with the pressure of direct $3d\ensuremath{-}3d$ exchange and superexchange interactions. It is proposed, that a strong monoclinic lattice distortion may be of importance for understanding the properties of single-layer FeOCl materials.

Compressibility of structural modulation waves in the chain compounds BaCoX2O7 (X = As, P): a powder study.

BaCoX2O7 (X = As, P) are built on magnetic 1D units in which strong aperiodic undulations originate from incommensurate structural modulations with large atomic displacive amplitudes perpendicular to the chain directions, resulting in very unique multiferroic properties. High-pressure structural and vibrational properties of both compounds have been investigated by synchrotron X-ray powder diffraction and Raman spectroscopy at room temperature and combined with density functional calculations. A structural phase transition is observed at 1.8 GPa and 6.8 GPa in BaCoAs2O7 and BaCoP2O7, respectively. Sharp jumps are observed in their unit-cell volumes and in Raman mode frequencies, thus confirming the first-order nature of their phase transition. These transitions involve the disappearance of the modulation from the ambient-pressure polymorph with clear spectroscopic fingerprints, such as reduction of the number of Raman modes and change of shape on some peaks. The relation between the evolution of the Raman modes along with the structure are presented and supported by density functional theory structural relaxations.

Octahedral Tilting and Modulation Structure in Perovskite‐Related Compound La1/3NbO3

The crystal structure of the A‐site‐deficient perovskite La1/3NbO3 is investigated using X‐ray powder diffraction, transmission electron microscopy (TEM), and computational methods. The NbO6 octahedral tilting is observed and confirmed to be in an a − b 0 c 0 manner in Glazer's notation. An incommensurate modulation structure in La1/3NbO3 is observed. Using scanning TEM (STEM) analyses and Monte Carlo simulations, the modulated structure is shown to originate from a striped arrangement of La and vacancies. The satellite reflections arising from the modulation structure are observed as a diffuse scatter in the electron diffraction patterns. By analyses with STEM and molecular dynamics calculations, it is concluded that the origin of diffuse scattering is the existence of nanoscale twin variants with different crystal orientations. The Nb ions in NbO6 octahedra represent an off‐center shift like a pseudo‐Jahn–Teller distortion. The shift of Nb ions is found to be cooperative with the concentration modulation of La ions.

Theory and application of the vector pair correlation function for real-space crystallographic analysis of order/disorder correlations from STEM images

Deviations of local structure and chemistry from the average crystalline unit cell are increasingly recognized to have a significant influence on the properties of many technologically important materials. Here, we present the vector pair correlation function (vPCF) as a new real-space crystallographic analysis method, which can be applied to atomic-resolution scanning transmission electron microscopy (STEM) images to quantify and analyze structural order/disorder correlations. Our STEM-based vPCFs have several advantages over radial PCFs and/or 3D pair distribution functions from x-ray total scattering: vPCFs explicitly retain crystallographic orientation information, are spatially resolved, can be applied directly on a sublattice basis, and are suitable for any material that can be imaged with STEM. To show the utility of our approach, we measure partial vPCFs in Ba5SmSn3Nb7O30 (BSSN), a tetragonal tungsten bronze (TTB) structured complex oxide. Many TTBs are known to be classical or relaxor ferroelectrics, and these properties have been correlated with the presence of superlattice ordering. BSSN, specifically, exhibits relaxor behavior and an incommensurate structural modulation. From the vPCF data, we observe that, of the cation sites, only the Ba (A2) sublattice is structurally modulated. We then infer the local modulation vector and reveal a marked anisotropy in its correlation length. Finally, short-range correlated polar displacements on the B2 cation sites are observed. This work introduces the vPCF as a powerful real-space crystallography technique, which enables direct, robust quantification of short-to-long range order on a sublattice-specific basis and is applicable to a wide range of complex material types.

Incommensurately modulated structure zone in α-Zr instigated by the drag force on a grain boundary

Among several microstructural features found in deformed Zircaloy-4 alloy, the formation of incommensurately modulated structure (IMS) has not been studied extensively. In this study, an IMS zone in the α-Zr matrix of forged Zircaloy-4 alloy was investigated by transmission electron microscopy and related techniques. The results revealed a new intensity sequence along the (011 2) and (0113) planes, implying that an IMS was formed adjacent to a second phase particle (SPP) located at a grain boundary (GB) in the α-Zr matrix. The GB experienced maximum dragging force parallel to the [0110] direction of the α-Zr matrix, resulting in the formation of the IMS zone in the α-Zr matrix adjacent to the SPP.

Competing electronic instabilities in the quadruple perovskite manganite PbMn7O12

Structural behaviour of PbMn$_{7}$O$_{12}$ has been studied by high resolution synchrotron X-ray powder diffraction. This material belongs to a family of quadruple perovskite manganites that exhibit an incommensurate structural modulation associated with an orbital density wave. It has been found that the structural modulation in PbMn$_{7}$O$_{12}$ onsets at 294 K with the incommensurate propagation vector $\mathbf{k}_s=(0,0,\sim2.08)$. At 110 K another structural transition takes place where the propagation vector suddenly drops down to a \emph{quasi}-commensurate value $\mathbf{k}_s=(0,0,2.0060(6))$. The \emph{quasi}-commensurate phase is stable in the temperature range of 40K - 110 K, and below 40 K the propagation vector jumps back to the incommensurate value $\mathbf{k}_s=(0,0,\sim2.06)$. Both low temperature structural transitions are strongly first order with large thermal hysteresis. The orbital density wave in the \emph{quasi}-commensurate phase has been found to be substantially suppressed in comparison with the incommensurate phases, which naturally explains unusual magnetic behaviour recently reported for this perovskite. Analysis of the refined structural parameters revealed that that the presence of the \emph{quasi}-commensurate phase is likely to be associated with a competition between the Pb$^{2+}$ lone electron pair and Mn$^{3+}$ Jahn-Teller instabilities.

A Deep-UV Nonlinear Optical Borosulfate with Incommensurate Modulations.

Despite borate-sulfates composed of isolated borate and sulfate groups have been discovered for more than one century, borosulfates of B-O-S bonding have not been synthesized until 2012 and their properties remain rarely explored. Here, we report a new borosulfate, namely LiRb4 [B(SO4 )4 ], which shows potential as a bran-new class of deep-UV nonlinear optical (NLO) material. Remarkably, according to the in situ single-crystal X-ray diffraction, LiRb4 [B(SO4 )4 ] undergoes successive first-order phase transitions with incommensurate structural modulations under room temperature and low temperature with (3 + 1)D superspace groups of X2(α0γ)0 and P21 (α0γ)0, respectively. The first-principle calculations and structural analyses reveal that NLO properties, phase transitions, and incommensurate modulations are closely related to the [B(SO4 )4 ]5- super-tetrahedral units with B-O-S bonding. We anticipate that the unprecedented coexistence of deep-UV transparency, NLO responses, and incommensurate structural modulations in one borosulfate will bring new possibilities to understand the underlying concepts of physics and chemistry in aperiodic materials.

A Deep‐UV Nonlinear Optical Borosulfate with Incommensurate Modulations

AbstractDespite borate‐sulfates composed of isolated borate and sulfate groups have been discovered for more than one century, borosulfates of B‐O‐S bonding have not been synthesized until 2012 and their properties remain rarely explored. Here, we report a new borosulfate, namely LiRb4[B(SO4)4], which shows potential as a bran‐new class of deep‐UV nonlinear optical (NLO) material. Remarkably, according to the in situ single‐crystal X‐ray diffraction, LiRb4[B(SO4)4] undergoes successive first‐order phase transitions with incommensurate structural modulations under room temperature and low temperature with (3 + 1)D superspace groups of X2(α0γ)0 and P21(α0γ)0, respectively. The first‐principle calculations and structural analyses reveal that NLO properties, phase transitions, and incommensurate modulations are closely related to the [B(SO4)4]5− super‐tetrahedral units with B‐O‐S bonding. We anticipate that the unprecedented coexistence of deep‐UV transparency, NLO responses, and incommensurate structural modulations in one borosulfate will bring new possibilities to understand the underlying concepts of physics and chemistry in aperiodic materials.

Incommensurate modulated structure in zirconium under three-point bending deformation

Three-point bending tests (bending angle ~46°) were carried out on pure zirconium. The characteristic of microstructure after deformation was studied using transmission electron microscopy (TEM). Results showed that two types of face-centered cubic (FCC) phases were induced and a unique incommensurate modulated structure (IMS) was observed between these two FCC lamellae. In addition, the modulation wave vector q = −0.129b* + 0.318c*. Most importantly, the IMS was induced by periodic deformation fluctuations during hexagonal close-packed (HCP) to FCC phase transformation by the bending deformation.

Multiferroic BaCoX2O7 (X = P, As) Compounds with Incommensurate Structural Waves but Collinear Spin Ingredients

AbstractA new paradigm in multiferroics is observed in BaCoX2O7 (X = As, P) compounds. They consist of one dimensional (1D) antiferromagnetic chains undulated by incommensurate structural modulations with unusually large atomic displacive waves, giving a mixed 1D/2D “real” magnetic topology. The magnetic ground state is antiferromagnetic (AFM) with k = [½ 0 0], leading to a nonmodulated collinear spin lattice despite the aperiodic atomic framework, and allows developing spin‐induced multiferroicity below TN. Severe arguments against the identified mechanisms for type‐II multiferroics, i.e., by inverse Dzyaloshinskii–Moriya, exchange striction and spin‐dependent p–d hybridizations, suggest an original scenario in which the atomic waves, the collinear magnetic structure, and magnetic dipole–dipole interactions may interact as crucial ingredients of the spin‐induced ferroelectric phase. Here, the specific role of the Co2+ spin–orbit coupling in the magnetoelectric (ME) phase diagram is demonstrated by comparison with the novel Heisenberg BaFeP2O7 isomorph, similarly structurally modulated. This compound shows a noncollinear modulated AFM ordering, while no ME coupling is detected in its case. Accordingly, both BaCoX2O7 and BaFeP2O7 also undergo metamagnetic transitions above 5–6 T promoted by the modulated distribution of spin exchanges, but the spin‐flop progressive alignment of the spins in the noncollinear spin structure (Fe2+ case) turns into an abrupt flip‐like transition in the uniaxial spin structure (Co2+ case).

Hierarchical domain structures in (Pb,La)(Zr, Sn, Ti)O3 antiferroelectric ceramics

(Pb, La)(Zr, Sn, Ti)O3 (PLZST) ceramic is one of the most prospective antiferroelectric (AFE) materials for variety of functional applications including energy storage and converter. Systematic structural investigation of domain structures should be of fundamental importance for understanding the structure-property relationship in AFE ceramics. In this study, the hierarchical domain structures and modulated structures correlated to the compositional variation in (Pb0.97La0.02) (Zr0.50SnxTi0.50-x)O3 (x = 0.375, 0.45 and 0.50) were observed and investigated in details by transmission electron microscopy. The PLZST ceramics show exclusively incommensurate modulated structures (IMS) whose modulation period changed from 9.37 to 6.15 and to 4.04 with increasing of the x value. The hierarchical domain structures include, in decreasing scales, AFE domains, incommensurate domains and nanodomains. The elementary domains in PLZST ceramics are pinstriped nanodomains which were formed based on IMS configuration but by frequent modulation of IMS periodicity and formation of faults. Nanodomains accumulated and then dissociated into incommensurate domains and AFE domains successively. The presently revealed structural characteristics in antiferroelectric PLZST may stimulate future researches on the evolution of IMS-based hierarchical domains under external physical fields, e.g. thermal or electrical, and their correlation to the physical performance.

New modulated structures of solid bromine at high pressure

Crystal structures of bromine under pressure have been studied through the advanced structure search method in combination with first-principles calculations. At 10 and 90 GPa, we successfully reproduced the experimentally observed molecular phase I and monatomic body-centred orthorhombic phase II, respectively. At 120 and 200 GPa, the lowest-enthalpy structures predicted are body centered tetragonal and face-centered cubic phase, respectively, which are also consistent with previously theoretical study. Interestingly, in the pressure range of 50–80 GPa, our structural searches predicted four structurally similar and modulated structures, oF20, oC24, oF28 and oC12, whose magnitude of the modulation vectors k are 2/5, 1/6, 2/7 and 1/3, respectively. Enthalpy calculations show that oF20, oC24, oF28 and oC12 are thermaldynamically stable in the pressure range of 52–53, 53–58, 58–65 and 65–79 GPa, respectively. Through comparison of the structural characterization and X-ray diffraction patterns of the predicted oF28, oC12 in bromine and the known incommensurate modulated phase V in iodine, we suggest that bromine might also adopt an incommensurate modulated structure. In addition, the prediction of two new phases (oF20 and oC24) between molecular phase I and incommensurate phases are consistent with Raman scattering and X-ray absorption spectroscopy (XAS) experimental results. For the first time, we presented the structure information between molecular phase I and monatomic phase II in bromine. The calculated elastic stiffness coefficients indicate that the four modulated phases are mechanically stable. The calculated electronic density of states for these four predicted phases show that they are all metallic.

New tools in JANA2006/JANA2020 to study and characterize π–π stacking of incommensurate modulated structures: α- and β-Mn(dmp)Cl2

New tools in <i>JANA2006</i>/<i>JANA2020</i> to study and characterize π–π stacking of incommensurate modulated structures: α- and β-Mn(dmp)Cl₂

Property-structure relationship in lead-free relaxors Ba5RSn3Nb7O30 with tungsten bronze structure

Lead-free relaxors Ba5RSn3Nb7O30 (R = La, Nd, Sm) with a tetragonal tungsten bronze structure were prepared by a standard solid-state reaction process, and the dielectric and ferroelectric properties were systematically determined together with the structures and microstructures. Rietveld refinement results suggested that a single-phase tungsten bronze structure in space group P4/mbm was obtained for all the compositions. Both the dielectric constant (ε′) and the dielectric loss (tan δ) exhibited a broad maximum with evident frequency dispersion far below room temperature, which indicated the typical relaxor nature. The selected area electron diffraction patterns revealed that the relaxor behavior in the present system originated from the incommensurate structure modulation. The dark-field transmission electron microscopy images confirmed the presence of polar nanoregions in the areas generating the satellite reflections of incommensurate structure modulation, which implied a possible correlation between the incommensurate structure modulation and polar nanoregions in tungsten bronze relaxors.