Incommensurate Crystals Research Articles

Weak coupling of pseudoacoustic phonons and magnon dynamics in the incommensurate spin-ladder compound Sr14Cu24O41

Intriguing lattice dynamics has been predicted for aperiodic crystals that contain incommensurate substructures. Here we report inelastic neutron scattering measurements of phonon and magnon dispersions in Sr14Cu24O41, which contains incommensurate one-dimensional (1D) chain and two-dimensional (2D) ladder substructures. Two distinct pseudoacoustic phonon modes, corresponding to the sliding motion of one sublattice against the other, are observed for atomic motions polarized along the incommensurate axis. In the long wavelength limit, it is found that the sliding mode shows a remarkably small energy gap of 1.7-1.9 meV, indicating very weak interactions between the two incommensurate sublattices. The measurements also reveal a gapped and steep linear magnon dispersion of the ladder sublattice. The high group velocity of this magnon branch and weak coupling with acoustic and pseudoacoustic phonons can explain the large magnon thermal conductivity in Sr14Cu24O41 crystals. In addition, the magnon specific heat is determined from the measured total specific heat and phonon density of states, and exhibits a Schottky anomaly due to gapped magnon modes of the spin chains. These findings offer new insights into the phonon and magnon dynamics and thermal transport properties of incommensurate magnetic crystals that contain low-dimensional substructures.

Aperiodic crystals and beyond.

Crystals are paradigms of ordered structures. While order was once seen as synonymous with lattice periodic arrangements, the discoveries of incommensurate crystals and quasicrystals led to a more general perception of crystalline order, encompassing both periodic and aperiodic crystals. The current definition of crystals rests on their essentially point-like diffraction. Considering a number of recently investigated toy systems, with particular emphasis on non-crystalline ordered structures, the limits of the current definition are explored.

Properties of Hartree-Fock solutions of the three-dimensional electron gas

In a previous letter, L. Baguet et al., (Phys. Rev. Lett. {\bf 111}, 166402 (2013)), we presented the ground state phase diagram of the homogeneous electron gas in three dimensions within the Hartree-Fock approximation yielding incommensurate crystal states at high density. Here, we analyze the properties of these solutions. In particular, at high density we find universal behavior of the incommensurate crystal strongly supporting the existence of a spin density wave ground state.

Moiré pattern as a magnifying glass for strain and dislocations in van der Waals heterostructures.

We consider the role of deformations in graphene heterostructures with hexagonal crystals (including strain, wrinkles and dislocations) on the geometrical properties of moiré patterns characteristic for a pair of two incommensurate misaligned isostructural crystals. By employing a phenomenological theory to describe generic moiré perturbations in van der Waals heterostructures of graphene and hexagonal crystals we investigate the electronic properties of such heterostructures.

Aperiodic Crystals

2014 is the International Year of Crystallography. During at least fifty years after the discovery of diffraction of X-rays by crystals, it was believed that crystals have lattice periodicity, and crystals were defined by this property. Now it has become clear that there is a large class of compounds with interesting properties that should be called crystals as well, but are not lattice periodic. A method has been developed to describe and analyze these aperiodic crystals, using a higher-dimensional space. In this lecture the discovery of aperiodic crystals and the development of the formalism of the so-called superspace will be described. There are several classes of such materials. After the incommensurate modulated phases, incommensurate magnetic crystals, incommensurate composites and quasicrystals were discovered. They could all be studied using the same technique. Their main properties of these classes and the ways to characterize them will be discussed. The new family of aperiodic crystals has led also to new physical properties, to new techniques in crystallography and to interesting mathematical questions. Much has been done in the last fifty years by hundreds of crystallographers, crystal growers, physicists, chemists, mineralogists and mathematicians. Many new insights have been obtained. But there are still many questions, also of fundamental nature, to be answered. We end with a discussion of these open questions.

Diffuse scattering and phase transitions in aperiodic inclusion compounds

Small molecules, such as urea, thiourea, perhydrotriphenylene can be co-crystallised with long-chain hydrocarbon molecules to form inclusion compounds. The guest chains are confined to narrow, approximately cylindrical, channels created by the host small-molecule lattice. The stoichiometry and the conformations of the chains included inside the channels are function of internal interactions such as intra-chain interaction, but also of overall co-operative properties of the resulting three dimensionally ordered single crystal. These intergrowth compounds may form incommensurate composite crystals. A prototype example of such uniaxial intergrowth aperiodic crystals is n-alkane (CnH2n+2)/urea (CO(NH2)2). In these supra-molecular systems, urea molecules are connected by H-bonds and form helical ribbons, which repeat every six urea molecules to form a series of linear, hexagonal tunnels that can accommodate linear alkanes. Because the channels (~0.53 nm) are larger than the hydrocarbon chains, guests are held loosely and can undergo substantial motions. A significant amount of diffuse scattering of the first and second kinds can be depicted in scattering experiments, static or dynamic. These materials undergo a large variety of continuous or weakly first order structural phase transitions when changing the alkane molecule length and giving place to large pre-transitional effects. The talk will give an overview of the diffuse scattering in these compounds and will focus on connection with aperiodicity.

Transient spectral hole burning observed on the single-molecule level in terrylene-doped biphenyl

We use the method of fluorescence correlation spectroscopy to analyze the single-molecule (SM) spectroscopy data earlier recorded for a special type of terrylene SM impurity center (referred as “spectrally confined unstable molecule”, SCM) in an incommensurate single crystal of biphenyl. The SCM's SM line seems to be chaotically jumping around within a broad “spectral envelope” and was first considered being subject to a peculiar spectral diffusion behavior. However, our correlation analysis reveals that all the features observed for SCM at 1.8K are consistent with an assumption that this SM center participates in a process of reversible (transient) spectral hole burning (THB) earlier observed for terrylene-doped polycrystalline biphenyl. No observations of THB processes on SM level have been so far reported for this impurity system, partially due to a low concentration of relevant impurity centers. Another reason making searching for such centers experimentally challenging is an unusual SM line behavior: the photoinduced transition to a metastable “dark state” leads to the SM line saturational broadening, which is much stronger than the triplet broadening. Hence required prolonged observation is often prevented by an SM act of persistent spectral hole burning.

Hartree-Fock Ground State Phase Diagram of Jellium

We calculate the ground state phase diagram of the homogeneous electron gas in three dimensions within the Hartree-Fock approximation and show that broken symmetry states are energetically favored at any density against the homogeneous Fermi gas state with isotropic Fermi surface. At high density, we find metallic spin-unpolarized solutions where electronic charge and spin density form an incommensurate crystal having more crystal sites than electrons. For r(s)→0, our solutions approach pure spin-density waves, whereas the commensurate Wigner crystal is favored at lower densities, r(s)≳3.4. Decreasing the density, the system undergoes several structural phase transitions with different lattice symmetries. The polarization transition occurs around r(s)≈8.5.

Flexoelectric spectroscopy

Flexoelectricity is an increasingly popular subject because it can be extremely large in thin films and permits switching of devices in nonpolar (non-piezoelectric) crystals via application of inhomogeneous stresses. However, recent work has been limited to macroscopic measurement of voltage or strain. Here, we discuss the vibrational spectroscopy of flexoelectricity as a recommended new tool for thin-film characterization, with special emphasis upon incommensurate crystals.

Phase diagram of4He on graphene

The low temperature phase diagram of $^4$He adsorbed on a single graphene sheet is studied by computer simulation of a system comprising nearly thousand helium atoms. In the first layer, two commensurate solid phases are observed, with fillings 1/3 and 7/16 respectively, separated by a domain wall phase, as well as an incommensurate crystal at higher coverage. No evidence of a thermodynamically stable superfliuid phase is found for the first adlayer. Second layer promotion occurs at a coverage of 0.111(4) $\AA^{-2}$. In the second layer two phases are observed, namely a superfluid and an incommensurate solid, with no commensurate solid intervening between these two phases. The computed phase diagram closely resembles that predicted for helium on graphite.

Ultrafast filling of an electronic pseudogap in an incommensurate crystal

We investigate the quasiperiodic crystal (LaS)1.196(VS2) by angle and time resolved photoemission spectroscopy. The dispersion of electronic states is in qualitative agreement with band structure calculated for the VS2 slab without the incommensurate distortion. Nonetheless, the spectra display a temperature dependent pseudogap instead of quasiparticles crossing. The sudden photoexcitation at 50 K induces a partial filling of the electronic pseudogap within less than 80 fs. The electronic energy flows into the lattice modes on a comparable timescale. We attribute this surprisingly short timescale to a very strong electron-phonon coupling to the incommensurate distortion. This result sheds light on the electronic localization arising in aperiodic structures and quasicrystals.

Equivalence of superspace groups.

An algorithm is presented which determines the equivalence of two settings of a (3 + d)-dimensional superspace group (d = 1, 2, 3). The algorithm has been implemented as a web tool findssg on SSG(3+d)D, providing the transformation of any user-given superspace group to the standard setting of this superspace group in SSG(3+d)D. It is shown how the standard setting of a superspace group can be directly obtained by an appropriate transformation of the external-space lattice vectors (the basic structure unit cell) and a transformation of the internal-space lattice vectors (new modulation wavevectors are linear combinations of old modulation wavevectors plus a three-dimensional reciprocal-lattice vector). The need for non-standard settings in some cases and the desirability of employing standard settings of superspace groups in other cases are illustrated by an analysis of the symmetries of a series of compounds, comparing published and standard settings and the transformations between them. A compilation is provided of standard settings of compounds with two- and three-dimensional modulations. The problem of settings of superspace groups is discussed for incommensurate composite crystals and for chiral superspace groups.

Static and dynamic friction in sliding colloidal monolayers

In a pioneer experiment, Bohlein et al. realized the controlled sliding of two-dimensional colloidal crystals over laser-generated periodic or quasi-periodic potentials. Here we present realistic simulations and arguments that besides reproducing the main experimentally observed features give a first theoretical demonstration of the potential impact of colloid sliding in nanotribology. The free motion of solitons and antisolitons in the sliding of hard incommensurate crystals is contrasted with the soliton–antisoliton pair nucleation at the large static friction threshold F s when the two lattices are commensurate and pinned. The frictional work directly extracted from particles’ velocities can be analyzed as a function of classic tribological parameters, including speed, spacing, and amplitude of the periodic potential (representing, respectively, the mismatch of the sliding interface and the corrugation, or “load”). These and other features suggestive of further experiments and insights promote colloid sliding to a unique friction study instrument.

Pattern-Stabilized Decorated Polar Liquid-Crystal Fibers

Geometric frustration gives rise to new fundamental phenomena and is known to yield the formation of exotic states of matter, such as incommensurate crystals, modulated liquid-crystalline phases, and phases stabilized by defects. In this Letter, we present a detailed study of polar structure of freely suspended fluid filaments in a polarization modulated liquid-crystal phase. We show that a periodic pattern of polarization-splay stripes separated by defect boundaries and decorating smectic layers can stabilize the structure of fluid fibers against the Rayleigh-Plateau instability. The instability is suppressed by the resistance of the defect structure to a radial compression of the cylindrical fibers. Our results provide direct experimental observation of a link between the stability of the liquid fibers, internal polar order, and geometrical constraints. They open a new perspective on a wide range of fluid polar fiber materials.

Hartree-Fock phase diagram of the two-dimensional electron gas

We calculate the ground-state phase diagram of the homogeneous electron gas in two dimensions within the Hartree-Fock approximation. At high density, we find stable solutions, where the electronic charge and spin density form an incommensurate crystal having more crystal sites than electrons, whereas the commensurate Wigner crystal is favored at lower densities, ${r}_{s}\ensuremath{\gtrsim}1.22$. Our explicit calculations demonstrate that the homogeneous Fermi-liquid state, though being an exact stationary solution of the Hartree-Fock equations, is never the Hartree-Fock ground state of the electron gas.

Convergent-beam electron diffraction

This paper reviews the convergent-beam electron diffraction (CBED) technique. Point- and space-group determination methods of ordinary crystals are described, along with an example of the determination method for Sr₃Ru₂O₇. The symmetry determination of one-dimensionally incommensurate crystals and quasicrystals is explained. The large-angle CBED technique, which is indispensable for lattice defect and lattice strain analysis, is also described. A real procedure for lattice strain analysis is provided, using an example of a multilayer Si₁-xGe(x)/Si material. A nanometer-scale crystal structure refinement method and charge density and crystal potential determination method by CBED are briefly described.

Vibrational and thermodynamic properties and molecular motions in the incommensurate crystal of morpholinium tetrafluoroborate studied by 1H NMR

Dynamics of molecules in morpholinium tetrafluoroborate, [NH 2(C 2H 4) 2O][BF 4], has been studied in a wide temperature range by means of AC calorimetry, 1H NMR and infrared spectroscopy. Simultaneously, DFT (density functional theory) calculations of the molecular structure and normal vibration frequencies have been performed by using various functionals for initial structural data of the low temperature phase. Infrared spectra of polycrystalline [NH 2(C 2H 4) 2O][BF 4] have been analyzed in a frequency range 4000–400 cm −1. Substantial changes in a temperature evolution of internal modes of both morpholinium cations and [BF 4] − anions are due to the freezing of these moieties motions below 117 K. A dynamic nonequivalence of protons involved in N–H⋯F and N–H⋯O hydrogen bonds has been detected. Heat capacity anomalies around structural phase transitions detected by AC calorimetry (117 and 153 K) have been evaluated and described. A molecular mechanism of the phase transitions is discussed on the basis of presented results.

Piezo-optic properties of incommensurately modulated K2ZnCl4 crystals

The effect that uniaxial pressures along the principal crystallophysical directions have on the dispersion and temperature dependences of the birefringence in incommensurate K2ZnCl4 crystals has been investigated in a wide temperature range. This parameter has been established to be fairly sensitive to uniaxial pressures. The spectral and temperature dependences of combined piezo-optic coefficients are analyzed. Significant anomalies of these coefficients are found at the incommensurate-commensurate phase transition. They are due to the change in the induced birefringence that results from spontaneous polarization and to the significant effect of uniaxial stress on the soliton structure of the crystal. The contributions that the electrooptic effect, the elasto-optic effect, and the order parameter make to the spontaneous changes in the piezooptic coefficients are separated.

Defining Contact at the Atomic Scale

Molecular dynamics simulations are used to study different definitions of contact at the atomic scale. The roles of temperature, adhesive interactions and atomic structure are studied for simple geometries. An elastic, crystalline substrate contacts a rigid, atomically flat surface or a spherical tip. The rigid surface is formed from a commensurate or incommensurate crystal or an amorphous solid. Spherical tips are made by bending crystalline planes or removing material outside a sphere. In continuum theory the fraction of atomically flat surfaces that is in contact rises sharply from zero to unity when a load is applied. This simple behavior is surprisingly difficult to reproduce with atomic scale definitions of contact. Due to thermal fluctuations, the number of atoms making contact at any instant rises linearly with load over a wide range of loads. Pressures comparable to the ideal hardness are needed to achieve full contact at typical temperatures. A simple harmonic mean-field theory provides a quantitative description of this behavior and explains why the instantaneous forces on atoms have a universal exponential form. Contact areas are also obtained by counting the number of atoms with a time-averaged repulsive force. For adhesive interactions, the resulting area is nearly independent of temperature and averaging interval, but usually rises from zero to unity over a range of pressures that is comparable to the ideal hardness. The only exception is the case of two identical commensurate surfaces. For nonadhesive surfaces, the mean pressure is repulsive if there is any contact during the averaging interval $\Delta t$. The associated area is very sensitive to $\Delta t$ and grows monotonically. Similar complications are encountered in defining contact areas for spherical tips.

Single-molecule probing of incommensurate biphenyl

Our data on the distribution of purely electronic linewidths of single molecules of terrylene in incommensurate biphenyl crystals are compared with the data obtained by other groups for different low-temperature organic solid hosts and with results of numerical simulations. The first two moments of the distributions measured within a narrow temperature interval have been used to calculate a single dimensionless parameter characterizing each of the respective hosts—the variation coefficient. It appears that different amorphous hosts have similar values of this coefficient, but the value obtained for the incommensurate crystal of biphenyl is significantly different. One can conclude that the remarkable single-molecule line broadening in biphenyl at 1.8K cannot be solely explained by the interaction with two-level systems, which is considered to cause the broadening in amorphous hosts.