Incommensurate Modulation Research Articles

Sublinear temperature dependence of thermal conductivity in the incommensurate phase of TlInTe2

Thermal conductivity $(\ensuremath{\kappa})$ of semiconducting and insulating solids generally show inverse linear temperature dependence above the Debye temperature $({T}_{\mathrm{D}})$ owing to dominant phonon-phonon scattering, i.e, $\ensuremath{\kappa}\ensuremath{\propto}{T}^{\ensuremath{-}1}$. Recently, in ultralow $\ensuremath{\kappa}$ materials, $\ensuremath{\kappa}$ is found to decrease sublinearly ($\ensuremath{\kappa}\ensuremath{\propto}{T}^{\ensuremath{-}\ensuremath{\alpha}}$ where $0<\ensuremath{\alpha}<1$) above ${T}_{\mathrm{D}}$ as interbranch wavelike tunneling contribution becomes significant. Here we show that the deviation from linearity can be unprecedently large in incommensurate (IC) phases as exemplified by archetypal Zintl-like semiconductor ${\mathrm{TlInTe}}_{2}$. This happens because two mutually incompatible translational symmetries allow spatially and temporally varying phase shifts, thus giving rise to two new heat-carrying modes: phasons and amplitudons. Using comprehensive transport and spectroscopy measurements combined with first-principles simulations of multichannel thermal transport, i.e., phonon-phonon scattering and tunneling contributions, we find that new modes contribute nearly 10-30% of the total $\ensuremath{\kappa}$ near the IC transition. The origin of this IC transition is rooted in the Tl lone pair and large polarizability of the electronic cloud, whose fingerprints are visible in the phonon linewidths. Our study paves the way for understanding ultralow $\ensuremath{\kappa}$ in IC phases, particularly for charge-density-wave materials where IC modulation is ubiquitous.

A Tour of Soft Atomic Motions: Chemical Pressure Quadrupoles Across Transition Metal–Main Group 1:2 Structure Types

The rational design of materials requires an understanding of the relationships between structure and properties. In some fields, advances in computational tools and modeling have mapped out many of these connections. In the realm of intermetallics, however, principles for such relationships are still needed. Recently, we illustrated how the DFT–Chemical Pressure (CP) analysis can link geometrical motifs within crystalline arrangements to the phonon band structures of intermetallics and thus the physical properties related to vibrations and atomic motion. The key indicators here are quadrupolar CP features─distributions in which positive and negative interatomic pressures are oriented perpendicular to one another. These quadrupoles then create low energy paths of motion for the atoms that can be linked to various structural phenomena, such as anisotropic vibrations and incommensurate modulations. In this article, we survey the presence of CP quadrupoles in a collection of TE2 structures (T = transition metal and E = main group element) and the roles they play in the behavior observed for these compounds. We begin by developing a metric for CP quadrupolar character, which can be calculated from the output of CP analysis. After compiling these metrics for 17 TE2 structure types, we delve deeper into four systems detected as exhibiting atomic sites with quadrupolar character: (1) the Co–Sn binary system, where the quadrupoles on the Co atoms in CuAl2-type CoSn2 direct the features of its superstructure in CoSn3; (2) arsenopyrite-type CoSb2, in which the highly quadrupolar CP distributions on the Sb atoms correlate with the motions followed during the phase transition to its high-temperature marcasite-type form; (3) the CrSi2 structure, in which the expansion of the lattice is predicted to open a helical path of potential soft motion analogous to that underlying the Nowotny Chimney Ladder series; and (4) the CdAs2 type, whose CP features anticipate the ability to incorporate Li atoms as guests. Together, these case studies highlight the potential for CP quadrupoles to link local atomic arrangements with a variety of emergent properties in intermetallic phases.

Incommensurate Modulation and Competing Ferroelectric/Antiferroelectric Modes in Tetragonal Tungsten Bronzes

The nanoscale structure of Sr0.61Ba0.39Nb2O6, a classic uniaxial relaxor ferroelectric crystallizing with the tetragonal tungsten bronze (TTB) structure and exhibiting an incommensurate modulation, has been determined by atomistic refinements using combined data from variable-temperature neutron total scattering, extended X-ray absorption fine structure, and three-dimensional single-crystal diffuse scattering. We found the modulation to arise from the intergrowth of structural slabs featuring distinct types of octahedral rotations directed to minimize octahedral deformations. This modulation involves displacements of the A cations (Sr and Ba) but exerts no significant effect on the polar displacements of Nb. Our results demonstrated the coexistence of competing polar Γ3– and antipolar Γ2– modes for off-centering of Nb above the nominal ferroelectric transition temperature of 350 K. The nanoscale-correlated antipolar distortions, which exhibit their largest amplitudes at 425 K, are suppressed below the transition. We identified coupling between the polar off-centering of Nb on its two symmetrically distinct sites as a fundamental characteristic controlling the ferroelectric-to-relaxor crossover in TTBs. We then used it to propose a simple microscopic interpretation for the previously established empirical trend that links crystal–chemical parameters in these systems to their polar response.

Breakdown of the correspondence between the real-complex and delocalization-localization transitions in non-Hermitian quasicrystals

The correspondence between the real-complex transition in energy and delocalization-localization transition is well-established in a class of Aubry-Andr'e-Harper model with exponential non-Hermitian on-site potentials. In this paper, we study a generalized Aubry-Andr'e model with off-diagonal modulation and non-Hermitian on-site potential. We find that, when there exists an incommensurate off-diagonal modulation, the correspondence breaks down, although the extended phase is maintained in a wide parameter range of the strengths of the on-site potential and the off-diagonal hoppings. An additional intermediate phase with a non-Hermitian mobility edge emerges when the off-diagonal hoppings become commensurate. This phase is characterized by the real and complex sections of the energy spectrum corresponding to the extended and localized states. In this case, the aforementioned correspondence reappears due to the recovery of the PT-symmetry.

Incommensurate Magnetic Order in Sm3BWO9 with Distorted Kagome Lattice

We investigate the magnetic ground state of Sm3BWO9 with a distorted kagome lattice. A magnetic phase transition is identified at T N = 0.75 K from the temperature dependence of specific heat. From 11B nuclear magnetic resonance measurements, an incommensurate magnetic order is shown by the double-horn type spectra under a c-axis magnetic field, and absence of line splitting is observed for field oriented within the ab-plane, indicating the incommensurate modulation of the internal field strictly along c-axis. From the spin dynamics, the critical slowing-down behavior is observed in the temperature dependence of 1/T 1 with μ 0 H⊥c-axis, which is completely absent in the case with μ 0 H||c-axis. Based on the local symmetry of 11B sites, we analyze the hyperfine coupling tensors and propose two constraints on the possible magnetic structure. The single ion anisotropy should play an important role in determination of contrasting ground states of Sm3BWO9 and Pr3BWO9.

High-quality resonances in quasi-periodic clusters of scatterers for flexural waves

Multiple scattering theory is applied to the study of clusters of point-like scatterers attached to a thin elastic plate and arranged in quasi-periodic distributions. Two types of structures are specifically considered: the twisted bilayer and the quasi-periodic line. The former consists in a couple of two-dimensional lattices rotated a relative angle, so that the cluster forms a moiré pattern. The latter can be seen as a periodic one-dimensional lattice where an incommensurate modulation is superimposed. Multiple scattering theory allows for the fast and efficient calculation of the resonant modes of these structures as well as for their quality factor, which is thoroughly analyzed in this work. The results show that quasi-periodic structures present a large density of states with high quality factors, being therefore a promising way for the design of high quality wave-localization devices.

Peculiarities of incommensurate modulation behavior during superstructure creation in case of potential symmetry n = 4

Type or paste your abstract here as prescribed by the journal’s instructions for authors. Type or paste your abstract here as prescribed by the journal’s instructions for authors. Type or paste your abstract here as prescribed by the journal’s instructions for authors. Type or paste your abstract here. This paper deals with dynamics of incommensurate superstructure at the moment of its formation. Lyapunov exponent value spectra, phase portraits, incommensurate modulation wave vector behavior, Fourier spectra and maps of incommensurate phase dynamic modes under changed parameters of long-range interaction (Т) and anisotropic interaction (K) in interval 0.0 ÷ 1.0 were analyzed. The calculations were made in Python environment applying Skipy, JiTCODE libraries. Under low values of Т and K parameters (0.0015 ÷ 0.03) a chaotic state occurs in this system. It is determined by the interaction between different spatial areas of the correlated motion of tetrahedral groups. The interaction of spatial areas of the correlated motion of tetrahedral groups violates the modulation pattern of tetrahedral group motion causing the superstructure chaotic state. The study of Fourier spectra has shown that interaction of spatial areas of tetrahedral group correlated motion of causes the chaotic state that is non-stable and is destroyed by long-range interaction. The impact of surface energy on the interaction of spatial areas of tetrahedral group correlated motion is determined by decreased amplitude of vibrations of tetrahedral groups causing the disappearance of the chaotic state.

Fully Two-Dimensional Incommensurate Charge Modulation on the Pd-Terminated Polar Surface of PdCoO2.

Here, we use low-temperature scanning tunneling microscopy and spectroscopy to study the polar surfaces of PdCoO2. On the CoO2-terminated polar surface, we detect the quasiparticle interference pattern originating from the Rashba-like spin-split surface states. On the well-ordered Pd-terminated polar surface, we observe a regular lattice that has a larger lattice constant than the atomic lattice of PdCoO2. In comparison with the shape of the hexagonal Fermi surface on the Pd-terminated surface, we identify this regular lattice as a fully two-dimensional incommensurate charge modulation that is driven by the Fermi surface nesting. More interestingly, we also find the moiré pattern induced by the interference between the two-dimensional incommensurate charge modulation in the Pd layer and its atomic lattice. Our results not only show a new charge modulation on the Pd surface of PdCoO2 but also pave the way for fully understanding the novel electronic properties of this material.

Phase transitions and ferroelectricity of perovskite layered Sr2Nb2O7 ceramics

Spontaneous polarization (Ps) is the fundamental for ferroelectrics. Compared with the relative explicit origin of Ps of perovskite ferroelectrics and bismuth layer structured ferroelectrics, that of the perovskite layer structured ferroelectrics (PLSFs) is still unclear and controversial due to their unique layer structures. Sr2Nb2O7 compound with a high Curie temperature of 1342 °C is a typical representative in the family of PLSFs. Here, we report our investigations about the phase transitions and the mechanism of ferroelectricity of Sr2Nb2O7 ceramics from the viewpoint of layered crystal structure experimentally and theoretically. Sr2Nb2O7 has two phase transitions at about 215 °C and 1340 °C. Selected area electron diffraction (SAED) images show superlattice along the [001] zone axis related to an incommensurate modulation and confirm it's incommensurate-commensurate phase transition at around 215 °C. The latter on at 1340 °C corresponds to the paraelectric-ferroelectric phase transition which is originated from oxygen octahedral rotation along c-axis, named a0b0c+ in Glazer's notation. Phonon spectra shows the origin of Ps for Sr2Nb2O7 is from mainly the oxygen octahedral rotations and partially the displacement of Sr2+ ions. This study not only demonstrates the rotation of oxygen octahedron is the domination that affects ferroelectric polarization but also opens up a new direction to design high ferroelectric and piezoelectric PLSF ceramics.

Enumeration and tabulation of magnetic (3+d)-dimensional superspace groups.

A magnetic superspace group (MSSG) simultaneously constrains both the magnetic and non-magnetic (e.g. displacive, occupational, rotation and strain) degrees of freedom of an incommensurately modulated magnetic crystal. We present the first enumeration and tabulation of all non-equivalent (3+d)-dimensional magnetic superspace groups for d = 1, 2 and 3 independent incommensurate modulations, along with a number, symbol and reference setting for each group. We explain the process for generating an exhaustive set of inequivalent magnetic superspace groups, describe several examples, and show how the tables can be accessed via the ISO(3+d)D interface within the ISOTROPY Software Suite. We recommend that published incommensurate magnetic structures indicate a magnetic superspace-group number and symbol from these tables, as well as the transformation matrix from the published group setting to the reference setting used in these tables.

Structural Features and Defect Equilibrium in Cubic PrBa1-xSrxFe2O6-δ.

The structure, oxygen non-stoichiometry, and defect equilibrium in perovskite-type PrBa1−xSrxFe2O6−δ (x = 0, 0.25, 0.50) synthesized at 1350 °C were studied. For all compositions, X-ray diffraction testifies to the formation of a cubic structure (S.G. ), but an electron diffraction study reveals additional diffuse satellites around each Bragg spot, indicating the primary incommensurate modulation with wave vectors about ±0.43. The results were interpreted as a sign of the short-order in both A-cation and anion sublattices in the areas of a few nanometers in size, and of an intermediate state before the formation of an ordered superstructure. An increase in oxygen deficiency was found to promote the ordering, whereas partial substitution of barium by strontium caused the opposite effect. The oxygen content in oxides as a function of oxygen partial pressure and temperature was measured by coulometric titration, and the data were used for the modeling of defect equilibrium in oxides. The simulation results implied oxygen vacancy ordering in PrBa1−xSrxFe2O6−δ that is in agreement with the electron diffraction study. Besides oxidation and charge disproportionation reactions, the reactions of oxygen vacancy distribution between non-equivalent anion positions, and their trapping in clusters with Pr3+ ions were taken into account by the model. It was demonstrated that an increase in the strontium content in Pr0.5Ba0.5−xSrxFeO3−δ suppressed ordering of oxygen vacancies, increased the binding energy of oxygen ions in the oxides, and resulted in an increase in the concentration of p-type carriers.

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.

Order and disorder in the modulated phase hp‐Mg9Ge5.1

AbstractSingle crystals of Mg9Ge5.1(2) were obtained at a pressure of 5 GPa in the temperature range 723 to 1273 K. The metastable high‐pressure phase exhibits a crystal structure, which may be described as a host‐guest variety of an UCl3‐type framework. The atomic arrangement is characterized by a host crystal structure Mg9Ge3 with disordered germanium guests (≈2 Ge per formula unit) in the channels. The composite arrangement undergoes incommensurate modulation. Upon heating at ambient pressure, the phase transforms into the thermodynamically stable compounds Mg2Ge and Ge at 496(5) K. Measurements of the magnetic susceptibility indicate diamagnetic behavior between 75 and 350 K.

Hybrid improper dipolar density wave in NaLaCoWO6

Hybrid improper ferroelectricity (HIF) allows the generation of an electrical polarization in the AA${}^{\ensuremath{'}}$BB${}^{\ensuremath{'}}{\mathrm{O}}_{6}$ double perovskite materials thanks to the combination of two nonpolar octahedral distortions. Nevertheless, for selected combination of the A/A${}^{\ensuremath{'}}$ cations a nonpolar incommensurate phase is observed with average symmetry $C2/m$. Thanks to a detailed crystallographic description of the incommensurate phase, based on electron, neutron, and x-ray diffraction data, we show that the incommensurate modulation is related to an abrupt change of the out-of-phase tilting along the $a$ and $c$ axis, whereas the tilting along the $b$ axis remains constant across the structure. By using group theory and symmetry analysis we show that we observe an incommensurate analog of HIF, which induces a hybrid improper dipolar density wave in ${\mathrm{NaLaCoWO}}_{6}$. The dipolar ordering is due also in this case to a trilinear invariant involving the commensurate and incommensurate octahedra tilting.

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.

Unusual structure and properties of germanium under pressure

Structural phase transitions in group 14 element germanium (Ge) under high pressure were investigated through first-principle calculations along with unbiased CALYPSO structure prediction method. We successfully reproduced experimental structures Fd-3m, I41/amd, Imma, P6/mmm, Cmca and P63/mmc were in our study on crystal structures. In addition, we found four new high-pressure metastable phases (Fmmm(I) and Cmcm at 75 GPa, Fmmm(II) and Fmm2 at 130 GPa). It was found in electronic density of states (DOS) and electronic band structure diagrams that all predicted new phases were metallic phases. The results of crystal structures, X-ray diffraction and phonon dispersion curve showed that our predicted phases (Cmcm and Fmmm(II)) could be incommensurate modulation structures. The findings of this study enriched the phases of Ge under high pressures and offered fundamental insight for further investigation of structural phase transitions in other group 14 elements under high pressures.

High-field magnetic structure of the triangular antiferromagnet RbFe(MoO4)2

The magnetic H - T phase diagram of a quasi-two-dimensional antiferromagnet RbFe(MoO4)2 with an equilateral triangular lattice structure is studied with 87Rb NMR and neutron diffraction techniques. This combination of experimental techniques allows us to determine the ordered components of the magnetic moments on the Fe3+ ions within various high-field phases - the Y, UUD, V, and fan structures, stabilized in the compound by the in-plane magnetic field. It is also established that the transition from the V to the fan phase is of first-order, whereas the transition from the fan phase to the polarized paramagnetic phase is continuous. An analysis of the NMR spectra shows that the high-field fan phase of RbFe(MoO4)2 can be successfully described by a periodic commensurate oscillation of the magnetic moments around the field direction in each Fe layer combined with an incommensurate modulation of the magnetic structure perpendicular to the layers.

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.

Charge Order at High Temperature in Cuprate Superconductors

The presence of different electronic orders other than superconductivity populating the phase diagram of cuprates suggests that they might be the key to disclose the mysteries of this class of materials. In particular charge order in the form of charge density waves (CDW), i.e., the incommensurate modulation of electron density in the CuO$_2$ planes, is ubiquitous across different families and presents a clear interplay with superconductivity. Until recently, CDW had been found to be confined inside a rather small region of the phase diagram, below the pseudogap temperature and the optimal doping. This occurrence might shed doubts on the possibility that such "low temperature phenomenon" actually rules the properties of cuprates either in the normal or in the superconducting states. However, recent resonant X-ray scattering (RXS) experiments are overturning this paradigm. It results that very short-ranged charge modulations permeate a much wider region of the phase diagram, coexisting with CDW at lower temperatures and persisting up to temperatures well above the pseudogap opening. Here we review the characteristics of these high temperature charge modulations, which are present in several cuprate families, with similarities and differences. A particular emphasis is put on their dynamical character and on their coupling to lattice and magnetic excitations, properties that can be determined with high resolution resonant inelastic x-ray scattering (RIXS).

Enhanced energy-storage performance in a flexible film capacitor with coexistence of ferroelectric and polymorphic antiferroelectric domains

Advances in flexible electronics are driving dielectric capacitors with high energy storage density toward flexibility and miniaturization. In the present work, an all-inorganic thin film dielectric capacitor with the coexistence of ferroelectric (FE) and antiferroelectric (AFE) phases based on Pb0.96La0.04(Zr0.95Ti0.05)O3 (PLZT) was prepared on a 2D fluorophlogopite mica substrate via a simple one-step process. The flexible capacitor exhibits a high recoverable energy density (Urec) of ≈ 44.2 J/cm3, a large electric breakdown strength (EBDS) of 3011 kV/cm, excellent frequency stability (500 Hz-20 kHz) and high thermal stability over 30–190 °C. It also demonstrates an outstanding bending endurance, which can maintain a high energy storage performance under various bending radii (R = 2–10 mm) or 103 repeated bends at 4 mm. The FE phase is stable near the film surface and the interface with the bottom electrode. The AFE phase with multi-domains has incommensurate modulation structures with super-periodicity of 6.5, 6.9 and 5.2. It indicates that the PLZT/LNO/F-Mica capacitor has high potential for energy storage application and may provide great opportunities for exploring new energy storage materials.