Incommensurate Periods Research Articles

Mott Transition for a Lieb-Liniger Gas in a Shallow Quasiperiodic Potential: Delocalization Induced by Disorder.

Disorder or quasidisorder is known to favor localization in many-body Bose systems. Here, in contrast, we demonstrate an anomalous delocalization effect induced by incommensurability in quasiperiodic lattices. Loading ultracold atoms in two shallow periodic lattices with equal amplitude and either equal or incommensurate spatial periods, we show the onset of a Mott transition not only in the periodic case but also in the quasiperiodic case. Switching from periodic to quasiperiodic potential with the same amplitude, we find that the Mott insulator turns into a delocalized superfluid. Our experimental results agree with quantum MonteCarlo calculations, showing this anomalous delocalization induced by the interplay between the disorder and interaction.

RCWA4D: Electromagnetic solver for layered structures with incommensurate periodicities

We describe RCWA4D, an electromagnetic solver for layered structures with incommensurate periodicities. Our method is based on an extension of the rigorous coupled wave analysis. We illustrate our method on the example of twisted bilayer photonic crystal and show that various properties of such structures can be reliably simulated. The method can be generalized to multi-layer structures in general in which each layer is periodic or quasi-periodic.

Smectic pair-density-wave order in EuRbFe4As4.

The pair density wave (PDW) is a superconducting state in which Cooper pairs carry centre-of-mass momentum in equilibrium, leading to the breaking of translational symmetry1-4. Experimental evidence for such a state exists in high magnetic field5-8 and in some materials that feature density-wave orders that explicitly break translational symmetry9-13. However, evidence for a zero-field PDW state that exists independent of other spatially ordered states has so far been elusive. Here we show that such a state exists in the iron pnictide superconductor EuRbFe4As4, a material that features co-existing superconductivity (superconducting transition temperature (Tc) ≈ 37 kelvin) and magnetism (magnetic transition temperature (Tm) ≈ 15 kelvin)14,15. Using spectroscopic imaging scanning tunnelling microscopy (SI-STM) measurements, we show that the superconducting gap at low temperature has long-range, unidirectional spatial modulations with an incommensurate period of about eight unit cells. Upon increasing the temperature above Tm, the modulated superconductor disappears, but a uniform superconducting gap survives to Tc. When an external magnetic field is applied, gap modulations disappear inside the vortex halo. The SI-STM and bulk measurements show the absence of other density-wave orders, indicating that the PDW state is a primary, zero-field superconducting state in this compound. Both four-fold rotational symmetry and translation symmetry are recovered above Tm, indicating that the PDW is a smectic order.

Evidence of correlated incommensurate structural and magnetic order in highly oxygen-doped layered nickelate Nd2NiO4.23

$\mathrm{Nd}{}_{2}\mathrm{NiO}{}_{4+\ensuremath{\delta}}$ is a nonstoichiometric oxide, crystallizing in the Ruddlesden-Popper-type framework and exhibiting a wide range of oxygen nonstoichiometry $\ensuremath{\delta}$ with a complex structural phase diagram. Excess oxygen atoms insert holes, directly influencing the Ni valence state and associated electronic structure. We report here the existence of a correlated incommensurate structural and antiferromagnetic order below $T{}_{N}\phantom{\rule{4pt}{0ex}}\ensuremath{\approx}$ 150 K in highly oxygen-doped $\mathrm{Nd}{}_{2}\mathrm{NiO}{}_{4+\ensuremath{\delta}}$ with $\ensuremath{\delta} \ensuremath{\approx}$ 0.23. The crystal and magnetic structures are investigated by polarized and unpolarized single-crystal neutron diffraction studies together with synchrotron x-ray powder diffraction and macroscopic magnetization measurements. Due to the long-range ordering of excess oxygen atoms, the real structure of the compound is incommensurately modulated, represented by $\mathbf{Q}{}_{n}=\ifmmode\pm\else\textpm\fi{}0.813{\mathbf{a}}^{*}\ifmmode\pm\else\textpm\fi{}0.519{\mathbf{b}}^{*}$ wave vectors. The antiferromagnetic order is characterized by the presence of incommensurate Bragg peaks of type ($h\phantom{\rule{4pt}{0ex}}\ifmmode\pm\else\textpm\fi{}\phantom{\rule{4pt}{0ex}}\ensuremath{\varepsilon}$, 0, $l$/2), with $h$ and $l$ being odd-integer numbers and the magnetic incommensurability $\ensuremath{\varepsilon}\ensuremath{\approx}$ 0.36. The in-plane magnetic correlation length is found to be $\ensuremath{\xi}{}_{ab}$ = 184(1) \AA{} but is significantly reduced along the $c$ axis to $\ensuremath{\xi}{}_{c}$ = 39(2) \AA{}, indicating the quasi-two-dimensional nature of the antiferromagnetic correlations. Our study indicates that the excess oxygen atoms nevertheless strongly enhance the magnetic correlations between the ${\mathrm{NiO}}_{2}$ planes, resulting in a doubling of the magnetic unit cell along the $c$ axis. In contrast to $\mathrm{Nd}{}_{2}\mathrm{NiO}{}_{4.1}$, in which the incommensurate periodicity of oxygen and magnetic ordering was found to be identical, the structural and electronic orderings in heavily oxygen-doped ${\mathrm{Nd}}_{2}{\mathrm{NiO}}_{4.23}$ show two independent modulation vectors. Our study thus unravels a strong correlation between structural and electronic orderings in the highly oxygen-doped $\mathrm{Ln}{}_{2}\mathrm{NiO}{}_{4+\ensuremath{\delta}}$ system.

Bose-Einstein condensates in quasiperiodic lattices: Bosonic Josephson junction, self-trapping, and multimode dynamics

A Bose-Einstein condensate loaded in a one-dimensional bichromatic optical lattice with constituent sublattices having incommensurate periods is considered. Using the rational approximations for the incommensurate periods, we show that below the mobility edge the localized states are distributed nearly homogeneously in the space and explore the versatility of such potentials. We show that superposition of symmetric and antisymmetric localized states can be used to simulate various physical dynamical regimes, known to occur in double-well and multiwell traps. As examples, we obtain an alternative realization of a bosonic Josephson junction, whose coherent oscillations display beatings or switching in the weakly nonlinear regime, and describe self-trapping and four-mode dynamics, mimicking coherent oscillations and self-trapping in four-well potentials. These phenomena can be observed for different pairs of modes, which are localized due to the interference rather than due to a confining trap. The results obtained using few-mode approximations are compared with the direct numerical simulations of the one-dimensional Gross-Pitaevskii equation. The localized states and the related dynamics are found to persist for long times even in the repulsive condensates. We also described bifurcations of the families of nonlinear modes, the symmetry breaking, and stable minigap solitons.

Localization of ultracold atoms in Zeeman lattices with incommensurate spin-orbit coupling

We consider a particle governed by a one-dimensional Hamiltonian in which artificial periodic spin-orbit coupling and Zeeman lattice have incommensurate periods. Using best rational approximations to such quasiperiodic Hamiltonian, the problem is reduced to description of spinor states in a superlattice. In the absence of a constant Zeeman splitting, the system acquires an additional symmetry, which hinders the localization. However, if the lattices are deep enough, then localized states can appear even for Zeeman field with zero or small mean value. Spatial distribution of localized modes is nearly uniform and is directly related to the topological properties of the effective superlattice: center-of-mass coordinates of modes are determined by Zak phases computed from the superlattice band structure. The best rational approximations feature the `memory' effect: each rational approximation holds the information about the energies and spatial distribution of the modes obtained under preceding, less accurate approximations. Dispersion of low-energy initial wavepackets is characterized by the law $\propto t^\beta$ with $\beta$ varying between $1/2$ at the initial stage and $1$ at longer, but still finite-time, evolution. The dynamics of initial wavepackets, exciting mainly localized modes, manifests quantum revivals.

Tunable Domain Switching Features of Incommensurate Antiferroelectric Ceramics Realizing Excellent Energy Storage Properties.

An incommensurate modulated antiferroelectric phase is a key part of ideal candidate materials for the next generation of dielectric ceramics with excellent energy storage properties. However, there is less research carried out when considering its relatively low polarization response. Here, the incommensurate phase is modulated by stabilizing the antiferroelectric phase and the energy storage performance of the incommensurate phase under ultrahigh electric field is studied. The tape-casting method is applied to construct dense and thin ceramics. La3+ doping induces a room-temperature incommensurate antiferroelectric orthorhombic matrix. With little Cd2+ , the extremely superior energy storage performances arose as follows: when 0.03, the recoverable energy storage density reaches ≈19.3 J cm-3 , accompanying an ultrahigh energy storage efficiency of ≈91% (870kV cm-1 ); also, a giant discharge energy density of ≈15.4 J cm-3 emerges during actual operation. In situ observations demonstrate that these superior energy storage properties originate from the phase transition from the incommensurate antiferroelectric orthorhombic phase to the induced rhombohedral relaxor ferroelectric one. The adjustable incommensurate period affects the depolarization response. The revealed phase-transition mechanism enriches the existing antiferroelectric-ferroelectric transition. Attention to the incommensurate phase can provide a reference for the selection of the next generation of high-performance antiferroelectric materials.

Past and Present Trends in the Development of the Pattern-Formation Theory: Domain Walls and Quasicrystals

A condensed review is presented for two basic topics in the theory of pattern formation in nonlinear dissipative media: (i) domain walls (DWs, alias grain boundaries), which appear as transient layers between different states occupying semi-infinite regions, and (ii) two- and three-dimensional (2D and 3D) quasiperiodic (QP) patterns, which are built as a superposition of plane–wave modes with incommensurate spatial periodicities. These topics are selected for the present review, dedicated to the 70th birthday of Professor Michael I. Tribelsky, due to the impact made on them by papers of Prof. Tribelsky and his coauthors. Although some findings revealed in those works may now seem “old”, they keep their significance as fundamentally important results in the theory of nonlinear DW and QP patterns. Adding to the findings revealed in the original papers by M.I. Tribelsky et al., the present review also reports several new analytical results, obtained as exact solutions to systems of coupled real Ginzburg–Landau (GL) equations. These are a new solution for symmetric DWs in the bimodal system including linear mixing between its components; a solution for a strongly asymmetric DWs in the case when the diffusion (second-derivative) term is present only in one GL equation; a solution for a system of three real GL equations, for the symmetric DW with a trapped bright soliton in the third component; and an exact solution for DWs between counter-propagating waves governed by the GL equations with group-velocity terms. The significance of the “old” and new results, collected in this review, is enhanced by the fact that the systems of coupled equations for two- and multicomponent order parameters, addressed in this review, apply equally well to modeling thermal convection, multimode light propagation in nonlinear optics, and binary Bose–Einstein condensates.

Magnetoconductivity in quasiperiodic graphene superlattices

The magnetoconductivity in Fibonacci graphene superlattices is investigated in a perpendicular magnetic field B. It was shown that the B-dependence of the diffusive conductivity exhibits a complicated oscillatory behavior whose characteristics cannot be associated with Weiss oscillations, but rather with Shubnikov-de Haas ones. The absense of Weiss oscillations is attributed to the existence of two incommensurate periods in Fibonacci superlattices. It was also found that the quasiperiodicity of the structure leads to a renormalization of the Fermi velocity v_{F} of graphene. Our calculations revealed that, for weak B, the dc Hall conductivity sigma _{yx} exhibits well defined and robust plateaux, where it takes the unexpected values pm 4e^{2}/hslash left( 2N+1right) , indicating that the half-integer quantum Hall effect does not occur in the considered structure. It was finally shown that sigma _{yx} displays self-similarity for magnetic fields related by tau ^{2} and tau ^{4}, where tau is the golden mean.

A general fraction-of-time probability framework for chirp cyclostationary signals

The fraction-of-time (FOT) framework, a pseudo-probability framework, is proposed to promote the stochastic signal processing theory into practical applications. Technically, the ensemble averaging that requires infinite realizations is replaced by the time averaging of a single realization. This skill is further generalized for cyclostationary signals and accordingly the periodic FOT framework is obtained. However, neither the FOT nor the periodic FOT is proper for the chirp cyclostationary (CCS) signals which are utilized to model most of the nonstationary signals in mobile communications, radar/sonar systems. In this paper, a novel general FOT framework is proposed, which is obtained by constructing an isomorphic map associated with fractional Fourier transform (FrFT). Specifically, a CCS signal and its fractional shift form a Hilbert space, which is isomorphic to the Hilbert space formed by the CCS stochastic processes. The chirp-period FOT framework for CCS signals with a single chirp period is obtained directly, which is further extended for CCS signals with incommensurate chirp periods. Then the statistics of the CCS signals are defined and their properties are introduced in this framework. Finally, their estimators are investigated.

High-Temperature Evolution of the Incommensurate Composite Crystal Ca0.83CuO2

The crystal structure of the composite crystal Ca0.83CuO2 was investigated by synchrotron powder diffraction at high temperature. The incommensurate modulated structure was firstly analyzed at room temperature (RT) and successfully solved by adopting the (3D + 1)-dimensional symmetry P21/m(α0γ)0s. The composite crystal is featured by a non-uniform distribution of Ca ions occupying octahedral sites formed by the spatial arrangement by the infinite 1D CuO2 chains. By approaching 500 K, Ca0.83CuO2 undergoes a structural rearrangement ruled by the shrinking of the Ca interatomic distances. The high-temperature crystalline phase is characterized by a different incommensurate periodicity requiring the recombination of the Ca/CuO2 balance featuring the composite intergrowth of the two almost independent sub-structures. We ascertain that the new crystalline form is stable up to 950 K near to the limit of the thermal decomposition.

The ideal commensurate value of Sc and the superconducting phase under high pressure

The host–guest structure of scandium is described as being built of two penetrating substructures with the incommensurate periods of the channels along the c axis. We present the ideal commensurate value of 4/3 in Sc-II using ab initio calculations. We reveal that the 3cH and 4cG structures do interpenetrate and combine to the commensurate value of 4/3 of Sc-II at a pressure of 70 GPa. Ab initio molecular dynamics confirms the stability of the commensurate value 4/3 of the host–guest structure at 300 K and 72 GPa. The pressure-induced structural phase transitions in scandium under high pressure up to 200 GPa are investigated. We use ab initio random structure searching to predict the crystal structure of Sc-III: it is the tetragonal structure with space group P41212. Our calculations show that superconductivity arises in the P41212 structure. This high pressure structure is not only a superconducting phase but also has been reported for the first time in this group of elements.

Disentangled Cooperative Orderings in Artificial Rare-Earth Nickelates.

Coupled transitions between distinct ordered phases are important aspects behind the rich phase complexity of correlated oxides that hinder our understanding of the underlying phenomena. For this reason, fundamental control over complex transitions has become a leading motivation of the designer approach to materials. We have devised a series of new superlattices by combining a Mott insulator and a correlated metal to form ultrashort period superlattices, which allow one to disentangle the simultaneous orderings in RENiO_{3}. Tailoring an incommensurate heterostructure period relative to the bulk charge ordering pattern suppresses the charge order transition while preserving metal-insulator and antiferromagnetic transitions. Such selective decoupling of the entangled phases resolves the long-standing puzzle about the driving force behind the metal-insulator transition and points to the site-selective Mott transition as the operative mechanism. This designer approach emphasizes the potential of heterointerfaces for selective control of simultaneous transitions in complex materials with entwined broken symmetries.

Ground-state properties of Na2IrO3 determined from an ab initio Hamiltonian and its extensions containing Kitaev and extended Heisenberg interactions

We investigate the ground state properties of $\mathrm{Na_2IrO_3}$ based on numerical calculations of the recently proposed {\it ab initio} Hamiltonian represented by Kitaev and extended Heisenberg interactions. To overcome the limitation posed by small tractable system sizes in the exact diagonalization study employed in a previous study (Yamaji $\textit{et al.}$, Phys. Rev. Lett. $\textbf{113}$, 107201 (2014)), we apply two-dimensional density matrix renomalization group, and infinite-size tensor-network method. By calculating at much larger system sizes, we critically test the validity of the exact diagonalization results. The results consistently indicate that the ground state of $\mathrm{Na_2IrO_3}$ is a magnetically ordered state with zigzag configuration in agreement with experimental observations and the previous diagonalization study. Applications of the two independent methods in addition to the exact diagonalization study further uncover a consistent and rich phase diagram near the zigzag phase beyond the accessibility of the exact diagonalization. For example, in the parameter space away from the $\textit{ab initio}$ value of $\mathrm{Na_2IrO_3}$ controlled by the trigonal distortion, we find three phases: (i) an ordered phase with the magnetic moment aligned mutually in $120$ degrees orientation on every third hexagon, (ii) a magnetically ordered phase with a $16$-site unit-cell, and (iii) an ordered phase with presumably incommensurate periodicity of the moment. It suggests that potentially rich magnetic structures may appear in $A_\mathrm{2}\mathrm{IrO_3}$ compounds for $A$ other than Na. The present results also serve to establish the accuracy of the first-principles approach in reproducing the available experimental results thereby further contribute to find a route to realize the Kitaev spin liquid.

Features of Highly Correlated Electronic States in the Simple Perovskite Manganite Sr1−xSmxMnO3 with 0.15 ≤ x ≤ 0.50

The C- and A-type antiferromagnetic orderings have been reported in the highly correlated electronic manganite Sr1−xSmxMnO3 (SSMO) with the simple perovskite structure in the composition range of 0.34 ≤ x ≤ 0.46. Because these antiferromagnetic orderings should be coupled to orbital orderings, the crystallographic features of SSMO samples prepared with 0.15 ≤ x ≤ 0.50 have been investigated, mainly by X-ray powder diffraction and transmission electron microscopy, to understand the detailed features of the corresponding orbital-ordered states. It was found that around 100 K, for instance, the C-type orbital-ordered (COO) state was observed for 0.15 ≤ x ≤ 0.42, the orbital-modulated (OM) state for 0.44 ≤ x < 0.50, and the coexistence state of both ferromagnetic Pnma (I) and paramagnetic Pnma (II) regions for x = 0.50, where the OM state was characterized by a transverse lattice modulation with an incommensurate periodicity. Neither the A-type orbital-ordered state nor the CE-type state involving both orbita...

Signatures of the single-particle mobility edge in the ground-state properties of Tonks-Girardeau and noninteracting Fermi gases in a bichromatic potential

We explore the ground state properties of cold atomic gases, loaded into a bichromatic lattice, focusing on the cases of non-interacting fermions and hard-core (Tonks-Girardeau) bosons, trapped by the combination of two potentials with incommensurate periods. For such systems, two limiting cases have been thoroughly established. In the tight-binding limit, the single-particle states in the lowest occupied band show a localization transition, as the strength of the second potential is increased above a certain threshold. In the continuous limit, when the tight-binding approximation does not hold anymore, a mobility edge is found, whose position in energy depends upon the strength of the second potential. Here, we study how the crossover from the discrete to the continuum behavior occurs, and prove that signatures of the localization transition and mobility edge clearly appear in the generic many-body properties of the systems. Specifically, we evaluate the momentum distribution, which is a routinely measured quantity in experiments with cold atoms, and demonstrate that, even in the presence of strong boson-boson interactions, the single particle mobility edge can be observed in the ground state properties.

Localization of interacting Fermi gases in quasiperiodic potentials

We investigate the zero-temperature metal-insulator transition in a one-dimensional two-component Fermi gas in the presence of a quasi-periodic potential resulting from the superposition of two optical lattices of equal intensity but incommensurate periods. A mobility edge separating (low energy) Anderson localized and (high energy) extended single-particle states appears in this continuous-space model beyond a critical intensity of the quasi-periodic potential. In order to discern the metallic phase from the insulating phase in the interacting many-fermion system, we employ unbiased quantum Monte Carlo (QMC) simulations combined with the many-particle localization length familiar from the modern theory of the insulating state. In the noninteracting limit, the critical optical-lattice intensity for the metal-insulator transition predicted by the QMC simulations coincides with the Anderson localization transition of the single-particle eigenstates. We show that weak repulsive interactions induce a shift of this critical point towards larger intensities, meaning that repulsion favors metallic behavior. This shift appears to be linear in the interaction parameter, suggesting that even infinitesimal interactions can affect the position of the critical point.

Features of Electronic States in the Highly-Correlated Electronic System Sr&lt;sub&gt;1-&lt;i&gt;x&lt;/i&gt;&lt;/sub&gt;Nd&lt;i&gt;&lt;sub&gt;x&lt;/sub&gt;&lt;/i&gt;MnO&lt;sub&gt;3&lt;/sub&gt; around &lt;i&gt;x &lt;/i&gt;= 0.50

In the highly-correlated electron system Sr1-xNdxMnO3 (SNMO) having the simple perovskite structure, there are interesting electronic states, which are related to degrees of charge, orbital, and spin freedoms for eg electrons in Mn ions. Among these states, in the case of SNMO, the C-, A-and CE-type antiferromagnetic states were reported for 0.20 ≤ x ≤ 0.38, for 0.38 ≤ x ≤ 0.48, and for 0.48 ≤ x ≤ 0.52, respectively. The points to note here are that these antiferromagnetic states are directly associated with corresponding orbital orderings, and that the CE-type state also accompanies charge ordering. Because of these features, we were interested in the (A → CE) state change in SNMO. The crystallographic features of prepared SNMO samples with 0.46 ≤ x ≤ 0.50 have thus been investigated mainly by means of a transmission electron microscope equipped with a low-temperature holder. As a result, the state around 100 K for x = 0.48 was first understood to be identified as the Imma state, which includes a large number of orbital-modulated (OM) regions with an average size of about 10 nm. The feature of such regions is that the orbital modulation has an incommensurate periodicity and a charge modulation is absent in them. On the other hand, the CE-type state having the commensurate orbital and charge modulations was also confirmed to be present for x = 0.50. In addition to these two states, the state around 100 K for x = 0.46 was found to be characterized by the coexistence state consisting of the C-type orbital-ordered state and the Imma states including OM regions. In other words, the presence of the A-type orbital-ordered state could not be confirmed in the temperature range between 300 K and about 100 K for 0.46 ≤ x ≤ 0.50 in this study.

Chemical Pressure-Driven Incommensurability in CaPd5: Clues to High-Pressure Chemistry Offered by Complex Intermetallics.

While composition and pressure are generally considered orthogonal parameters in the synthesis and optimization of solid state materials, their distinctness is blurred by the concept of chemical pressure (CP): microscopic pressure arising from lattice constraints rather than an externally applied force. In this article, we describe the first cycle of an iterative theoretical/experimental investigation into this connection. We begin by theoretically probing the ability of physical pressure to promote structural transitions in CaCu5-type phases that are driven by CP in other systems. Our results point to the instability of the reported CaCu5-type CaPd5 phase to such a transition even at ambient pressure, suggesting that new structural chemistry should arise at only modest pressures. We thus attempted to synthesize CaPd5 as a starting material for high-pressure experiments. However, rather than obtaining the expected CaCu5-type phase, we encountered crystals of an incommensurately modulated variant CaPd5+q/2, whose composition is related to its satellite spacing, q = qbbasic* with q ≈ 0.44. Its structure was solved and refined in the (3 + 1)D superspace group Cmcm(0β0)s00, revealing CaCu5-type slabs separated by distorted Pd hexagonal nets with an incommensurate periodicity. DFT-CP analysis on a commensurate model for CaPd5+q/2 indicates that the new Pd nets serve to relieve intense negative CPs that the Ca atoms would experience in a CaCu5-type CaPd5 phase but suffer from a desire to contract relative to the rest of the structure. In this way, both the Pd layer substitution and incommensurability in CaPd5+q/2 are anticipated by the CP schemes of simpler model systems, with CP quadrupoles tracing the paths of the favorable atomic motions. This picture offers predictions for how elemental substitution and physical pressure should affect these structural motifs, which could be applicable to the magnetic phase Zr2Co11 whose previously proposed structures show close parallels to CaPd5+q/2.

Observation of nonequilibrium behavior near the Lifshitz point in ferroelectrics with incommensurate phase

We have investigated non-equilibrium properties of proper uniaxial Sn$_2$P$_2$(Se$_x$S$_{1-x}$)$_6$ ferroelectrics with the Type II incommensurate phase above Lifshitz point $x_{\rm LP} \sim 0.28$. We measured dielectric susceptibility with cooling and heating rate ranging 0.002-0.1~K/min, and high-resolution ultrasound experiments and hypersound Brillouin scattering. For samples with $x \geqslant 0.28$ clear anomalies were observed at incommensurate second order transition ($T_i$) and at first order lock-in transition ($T_c$) in the regime of very slow cooling rate, whereas the intermediate IC phase is not observed when the rate is faster then 0.1~K/min. In general, increasing the cooling rate leads to smearing the anomaly at $T_c$. We explain this effect in terms of Kibble-Zurek model for non-equilibrium second order phase transitions. In the ferroelectrics with strongly nonlinear local potential cooling rate defines concentration of domain walls and their size: domain width decreases when cooling rate increases. At certain conditions the size of domain is comparable to the incommensurate phase modulation period, which lies in micrometer scale in the vicinity of Lifshitz point and leads to pinning of the modulation period by domain wall.