Elasticity of disordered binary crystals

In the adiabatic approximation the values of the spring constants of the springs contained in a molecule depend on its electronic state. We consider molecules with two electronic energy levels separated by Δ. For a crystal of such molecules, the phonon branches depend therefore on the electronic states of the molecules. One can ask if that dependence does not introduce a coupling between the molecules via the optical and the acoustic branches. It is known that for a one-dimensional chain of N identical diatomic molecules there are two phonon branches, an optical branch and an acoustic one. In this study we introduce in the hamiltonian of the chain two assumptions: (i) each molecule has two electronic energy levels separated by Δ and the spring constant of the spring contained in the molecule has a value which depends on its electronic state; (ii) the spring constant of the spring which links two molecules nearest neighbours has a value which depends on the electronic states of both molecules linked. One can show that phonons create on each molecule a field-like which favours the excited level and create between two molecules nearest neighbours an exchange-like interaction which can be ferro-like, antiferro-like and which can be equal to zero. For some values of T and Δ, the chain can display a first-order phase transition with the presence of a thermal hysteresis loop. The phase transition takes place between the phase where all the molecules are in the fundamental level and that where they are in the excited one. The parameters of the model can be expressed in function of the applied pressure and of the volume of the crystal.

For CuInP2S6 ferrielectrics by Brillouin spectroscopy, the temperature dependence of the longitudinal hypersound velocity was investigated for the acoustic phonons propagated in plane of the crystal layers. Pronounced softening of acoustic phonon branch is observed in the paraelectric phase below T* ≈ 330 K at cooling to temperature Tc ≈ 312 K of the first order phase transition into ferrielectric phase. Below Tc the hypersound velocity growth in correlation with spontaneous polarization rise and its temperature anomaly is similar to earlier observed temperature behavior of longitudinal ultrasound velocity for acoustic wave propagated normally to the structural layers. Determined anomalous part of the CuInP2S6 crystal thermal conductivity in the vicinity of the ferrielectric transition also demonstrates lowering of the heat transferring phonons group velocity with cooling from T* to Tc. Observed acoustic softening obviously is induced by flexoelectric coupling of relaxational soft polar optical and acoustic branches that is related to an in ingomogeneously polarized state appearing between the paraelectric and ferrielectric phases.

Rock masses consist of nesting rock blocks with various scales separated by weak structural layers, and their complex hierarchical structures play a significant role in dynamic deformation and stress wave propagation. Based on the Cosserat theory, a dynamic model of pendulum-type and rotational waves in blocks rock mass with complex hierarchical structures is established to determine the influence of hierarchical structures on dynamic deformation. Then, aiming to low-frequency and low-velocity characteristics of pendulum-type waves, dispersion equations of waves are determined and solved in different hierarchical structures based on the Bloch theorem, and furthermore, the dispersion relation and velocity characteristics of waves are investigated. Finally, mechanism of low-frequency characteristics of pendulum-type waves is revealed on the basis of solid energy band theory, and the possibility of pendulum-type and rotational waves inducing rock bursts is discussed based on the research results. It is indicated that ignoring higher-order hierarchical structures of rock masses may underestimate displacement and overall deformation of rock masses, resulting in unsafe numerical results. Under the action of long wave disturbance, for the first mode pendulum-type waves (the acoustic branches), the dispersion is not significant and propagation velocity decreases, and higher-order hierarchical structures inside rock masses hinder the wave propagation. However, the dispersion of other waves (the optical branches) is significant so that they hardly exist and propagate independently. The low-frequency pendulum-type waves are dominant, which have slower attenuation and longer propagation distance than the high-order mode waves and traditional P and S-waves.

The results of the 1D study of a multifluid model of nonlinear ion-acoustic solitons in a bi-ion plasma are presented. The model is based on gasdynamic equations for each plasma component. For the first time, the effect of taking into account Dalton's law on the characteristics of ion-acoustic solitons was assessed. The dispersion equation of ion-acoustic waves is derived both taking into account Dalton's law and without it. In both cases, the dispersion equation provides two branches—an optical branch and an acoustic branch. It is shown that taking into account Dalton's law leads to a change in the values of phase velocities for both branches in a short-wave part of the spectrum. The velocity of an ion sound is 2 times higher when Dalton's law is taken into account than without it. Profiles of physical quantities in ion-acoustic solitons were calculated in the framework of a nonlinear theory. It was found that taking into account Dalton's law leads to an amplitude increase in an electrostatic potential in a soliton and to an amplitude decrease in both concentrations of components in plasma; a concentration burst length in a soliton when taking into account Dalton's law is noticeably greater than without taking into account this law; some soliton profiles may take the form of solitons of a W-type at large Mach numbers.

The wave-propagation characteristics of dense granular materials have been studied from the micromechanical viewpoint, in which relationships are sought between properties at the micro-scale of particles and interparticle contacts and properties at the continuum, macro-scale. The dispersion and damping characteristics have been determined from a three-dimensional lattice analysis in which the particle interaction is modeled with linear elastic springs and linear viscous dashpots. Due the presence of rotational degrees of freedom of the particles, optical branches are observed in the dispersion and damping characteristics, besides the acoustical branches. The influence of the micromechanical characteristics on the macroscopic dispersion and damping characteristics has been determined for a face-centered cubic lattice and a body-centered cubic lattice. For small wave numbers (large wave lengths) the damping of the optical branches is very large. This means that the optical branches will not be observed in conditions where a continuum-mechanical description is appropriate.

By employing the Green's-function expansion method, the optical spectrum rearrangement in the quasi-one-dimensional (1D) impurity vibrational systems has been investigated. Based on the validity criteria for the conventional coherent potential approximation (CPA), the dispersion of phononlike modes and main parameters for the rearranged spectra are determined. The most interesting rearrangement occurs when the bandwidth exceeds the limiting frequency of the optical branch. In this case, the spectrum rearrangement exhibits features characteristic of both the acoustic branch and high-frequency optical branch. Formations of the impurity optical branch and a well-defined limiting frequency are described.

We obtain the lattice wave solution and the dispersion relation of the lattice vibration equation of the two identical coupled Frenkel-Kontorova (FK) chains, and we study the lattice vibration characteristics of the coupled FK chains. The results show that the dispersion relation of coupled FK chain contains an acoustic branch and an optical branch. The spectral range and frequency band gap are related to the coefficient of restoring force of each chain and the inter-chain coupling strength, and it is also related to the depth of the on-site potential for the low temperature case and finite temperature case. Moreover, it is found that there is no frequency band gap for weak inter-chain coupling. The frequency gap appears when the inter-chain coupling strength exceeds a critical value, and the frequency band gap will become bigger with the inter-chain interaction increasing. This is because the optical branch moves towards high frequency region with the inter-chain coupling increasing. We also find that the critical inter-chain coupling strength of frequency band gap is always twice the restoring force coefficient of FK chain, and it does not depend on temperature. In addition, we study the effect of temperature on the dispersion relationship of coupled FK chain with a fixed inter-chain coupling strength. These results provide a theoretical basis for analyzing the effects of inter-chain coupling and temperature on the vibrational characteristics and physical properties of lattice, and thus providing an important guide for the energy transport, thermal management and other practical applications.

Các nghiên cứu gần đây đã chỉ ra rằng graphene có những tính chất đặc biệt và có nhiều tiềm năng ứng dụng trong các lĩnh vực công nghệ khác nhau. Bài báo này trình bày các kết quả về phổ kích thích tập thể, một đặc tính quan trọng của vật liệu, trong một cấu trúc ba lớp graphene lớp kép trên nền điện môi không đồng nhất trong gần đúng pha ngẫu nhiên ở nhiệt độ không tuyệt đối. Kết quả giải số cho thấy, một nhánh quang học và hai nhánh âm học tồn tại trong hệ khảo sát. Nhánh quang nhanh chóng biến mất do bị hấp thụ mạnh còn hai nhánh âm học tiếp tục kéo dài hơn cho đến khi tiếp xúc với đường biên vùng kích thích đơn hạt của hệ. Sự không đồng nhất của điện môi nền làm giảm tần số các nhánh quang phổ tần số cao nhưng lại có thể làm tăng tần số của nhánh phổ tần số thấp. Ảnh hưởng của khoảng cách giữa các lớp lên các nhánh phổ cũng không khác nhiều so với các cấu trúc nhiều lớp đã được khảo sát trước đây. Các kết quả thu được có thể có cung cấp thêm thông tin, góp phần hoàn thiện lý thuyết về vật liệu mới graphene.

Room temperature neutron inelastic scattering measurements of the polar transverse optic (TO) phonon mode in the cubic relaxor Pb(Mg1/3Nb2/3)O3 (PMN) reveal anomalous behavior, similar to that recently observed in Pb(Zn1/3Nb2/3)0.92Ti0.08O3, in which the optic branch appears to drop precipitously into the acoustic branch at a finite value of the momentum transfer q=0.23 Å−1, measured from the zone center. By contrast, a recent neutron study indicates that PMN exhibits a normal TO phonon branch at much higher temperature (800 K). We thus speculate that this unusual feature is common to all relaxor materials at low temperatures, and is the result of the presence of nanometer-scale polarized domains in the crystal that form below a temperature Td, which effectively prevent the propagation of long wavelength (q=0) phonons.

The van der Waals (vdW) heterostructures of bilayer transition metal dichalcogenide obtained by vertically stacking have drawn increasing attention for their enormous potential applications in semiconductors and insulators. Here, by using the first-principles calculations and the phonon Boltzmann transport equation (BTE), we studied the phonon transport properties of WS2/WSe2 bilayer heterostructures (WS2/WSe2-BHs). The lattice thermal conductivity of the ideal WS2/WSe2-BHs crystals at room temperature (RT) was 62.98 W/mK, which was clearly lower than the average lattice thermal conductivity of WS2 and WSe2 single layers. Another interesting finding is that the optical branches below 4.73 THz and acoustic branches have powerful coupling, mainly dominating the lattice thermal conductivity. Further, we also noticed that the phonon mean free path (MFP) of the WS2/WSe2-BHs (233 nm) was remarkably attenuated by the free-standing monolayer WS2 (526 nm) and WSe2 (1720 nm), leading to a small significant size effect of the WS2/WSe2-BHs. Our results systematically demonstrate the low optical and acoustic phonon modes-dominated phonon thermal transport in heterostructures and give a few important guidelines for the synthesis of van der Waals heterostructures with excellent phonon transport properties.

The electronic structure and phonon transport properties of HfSe2 under different in-plane strains at finite temperatures are systematically investigated by combining first-principles calculations with machine learning force field molecular dynamics simulations. Within a strain range of -3% to 3%, the electronic band gap value of HfSe2 varies between 0.39 and 0.87 eV. Under compressive strain, the conduction band minimum moves towards the Fermi level, with the distribution of electrons near the valence band maximum becoming more delocalized. This will reduce the scattering of electrons during the transport process, helping to improve the carrier mobility. Under tensile strain, the localization of the density of states near the valence band maximum is strengthened, accompanied by enhanced metallic properties of the Hf-Se bonds, which facilitates the enhancement of the thermoelectric power factor. Both compressive and tensile strains intensify the coupling of phonon normal modes with phonon scattering, and elevating the temperature amplifies this impact. The anharmonicity-induced reduction in phonon frequencies is especially pronounced for modes in the vicinity of the Debye frequency. This not only curtails the phonon lifetimes but also diminishes the lattice thermal conductivity through the enhancement of vibrational coupling among optical branches and the reduction of the group velocity in acoustic branches. These results demonstrate the synergistic effects of strain and temperature on the electronic structure and phonon transport of HfSe2.

Recent studies have shown that although black phosphorene ($\ensuremath{\alpha}$-P) has promising electrical, thermal, and mechanical properties, its poor stability in the air makes its applications challenging. Very recently, monolayer puckered-phase nitrophosphorene ($\ensuremath{\alpha}$-NP) has emerged as a new two-dimensional material with improved stability compared to $\ensuremath{\alpha}$-P, while it preserves the appealing electronic and mechanical properties of $\ensuremath{\alpha}$-P. However, the phonon transport properties of $\ensuremath{\alpha}$-NP remain poorly understood. In this study, we systematically compare phonon transport and thermal conductivity of $\ensuremath{\alpha}$-NP and $\ensuremath{\alpha}$-P by means of an iterative solution for the Boltzmann transport equation and first-principles calculations. We pay particular attention to careful enforcement of the Born-Huang sum rules on the interatomic force constants so that the flexural phonon branch of $\ensuremath{\alpha}$-NP becomes perfectly quadratic close to the center of the Brillouin zone. We find that $\ensuremath{\alpha}$-NP has a less anisotropic thermal conductivity compared to $\ensuremath{\alpha}$-P, with the room-temperature values of 37.65 and $18.66\phantom{\rule{4pt}{0ex}}{\mathrm{W}\phantom{\rule{0.16em}{0ex}}\mathrm{m}}^{\ensuremath{-}1}\phantom{\rule{0.16em}{0ex}}{\mathrm{K}}^{\ensuremath{-}1}$ in the armchair and zigzag directions, respectively. Interestingly, the anisotropy in the thermal conductivity of $\ensuremath{\alpha}$-NP is found to be reversed compared to that of $\ensuremath{\alpha}$-P, which has a higher thermal conductivity in the zigzag direction than the armchair direction. We found that as opposed to the usual understanding, considering only the directional dependence of the phonon group velocity fails to explain this reversed anisotropic thermal conductivity, and other phonon behaviors, such as avoided crossing of the acoustic and optical phonon branches, play a key role in determining the phonon relaxation time and thermal conductivity in $\ensuremath{\alpha}$-NP. Our results may provide theoretical guidance for future fundamental studies on the anisotropic phonon thermal transport in low-dimensional materials, and they may also enable efficient heat management of nanoelectronic devices based on $\ensuremath{\alpha}$-NP.

Phonon dispersion relations in ${\mathrm{Cu}}_{2}$O have been studied at 20\ifmmode^\circ\else\textdegree\fi{}C using inelastic neutron scattering. Seven acoustic branches and twelve optical branches have been studied in detail in the three symmetry directions [$00\ensuremath{\zeta}$], [$\ensuremath{\zeta}\ensuremath{\zeta}0$], and [$\ensuremath{\zeta}\ensuremath{\zeta}\ensuremath{\zeta}$] of the cubic lattice. Four of the six zone-center phonons have been observed and the assignments and energies are confirmed as ${\ensuremath{\Gamma}}_{25}=87\ifmmode\pm\else\textpm\fi{}2$ ${\mathrm{cm}}^{\ensuremath{-}1}$, $\ensuremath{\Gamma}_{12}^{}{}_{}{}^{\ensuremath{'}}=105\ifmmode\pm\else\textpm\fi{}3$ ${\mathrm{cm}}^{\ensuremath{-}1}$, ${\ensuremath{\Gamma}}_{15}=146\ifmmode\pm\else\textpm\fi{}1$ ${\mathrm{cm}}^{\ensuremath{-}1}$, and $\ensuremath{\Gamma}_{2}^{}{}_{}{}^{\ensuremath{'}}\ensuremath{\approx}347$ ${\mathrm{cm}}^{\ensuremath{-}1}$. The dispersion relations agree only qualitatively with the rigid-ion-model calculations. It is suggested that more detailed calculations may be performed in the light of the present measurements. The elastic constants obtained from the slopes of the acoustic branches are (units of ${10}^{11}$ dyn/${\mathrm{cm}}^{2}$) ${c}_{11}=12.61\ifmmode\pm\else\textpm\fi{}0.12$, ${c}_{44}=1.36\ifmmode\pm\else\textpm\fi{}0.04$, and ${c}_{12}=10.86\ifmmode\pm\else\textpm\fi{}1.14$, which are in close agreement with those obtained by the sound-echo technique.

Lattice dynamics of zinc

Vibration frequencies have been calculated for finite one-dimensional lattices in which the point masses alternately have the values ${m}_{\ensuremath{\alpha}}$ and ${m}_{\ensuremath{\beta}}$, ${m}_{\ensuremath{\alpha}}l{m}_{\ensuremath{\beta}}$. Only nearest neighbor Hooke's law interactions are considered. The end atoms are assumed to interact only with their nearest neighbors on the interior of the lattice and are otherwise free. If the numbers of atoms having masses ${m}_{\ensuremath{\alpha}}$ and ${m}_{\ensuremath{\beta}}$ are equal, there exists a single mode whose frequency squared lies at the middle of the forbidden gap between the optical and acoustical branches. For this surface mode the displacements of the atoms from their equilibrium positions decrease roughly exponentially from the end having the lighter atom. For the case of $N$ atoms of mass ${m}_{\ensuremath{\beta}}$ and $N+1$ atoms of mass ${m}_{\ensuremath{\alpha}}$ there exist two modes whose frequencies lie in the forbidden gap provided $(\frac{{m}_{\ensuremath{\alpha}}}{{m}_{\ensuremath{\beta}}})l\frac{N}{(N+1)}$. These modes correspond to symmetric and antisymmetric displacements. The displacements are largest for the end atoms and decrease roughly exponentially toward the center of the lattice.