Quartic Anharmonic Terms Research Articles

Self-consistent phonon calculations and quartic anharmonic lattice thermal conductivity in 2D InS monolayer

Low lattice thermal conductivity (κl) is a crucial factor for higher figure-of merit and hence the efficiency of thermoelectric generators. There are several reports on intrinsically low κl values in two-dimensional (2D) van der Waals materials using density functional theory and molecular dynamics simulations. In general, phonon dispersions are studied at absolute zero temperature using the finite-displacement approach within harmonic approximations. In addition, the κlis calculated using the third-order cubic interatomic force constants (IFCs) by solving the Boltzmann transport equation for phonons. In these calculations, we use quartic IFCs to solve self-consistent phonon equations to obtain dynamical dispersion relations at a finite temperature of 300 K using finite-temperature Green’s function for Bosonic systems in 2D indium sulfide (InS) monolayer. The cubic and quartic IFCs are calculated using machine learning algorithms, namely, the ordinary least square fitting and the least absolute shrinkage and selection operator. It was found that there is a lowering of the renormalized anharmonic phonon frequencies in the dispersion relations at 300 K upon the inclusion of quartic IFCs and anharmonic terms in the case of the InS monolayer. Thus, the κlvalue is reduced to 0.6 W/mK as compared to 0.9 W/mK obtained using cubic IFCs.

Anharmonic phonon renormalization and thermoelectric properties of CsPbX3 (X = Cl, Br, and I): first-principles calculations.

Halide perovskites with ultralow thermal conductivity have emerged as promising candidates for thermoelectric materials. We study the lattice dynamics and thermoelectric properties of cubic all-inorganic lead halide perovskites CsPbX3 (X = Cl, Br, and I) through first-principles calculations. Combined with self-consistent phonon theory, we have successfully renormalized the phonon frequency using a quartic anharmonic term, allowing us to accurately reproduce the phonon dispersion of the high-temperature cubic phase of CsPbX3 without any imaginary frequencies. Cubic CsPbX3 exhibit ultralow lattice thermal conductivities (0.61-1.71 Wm-1 K-1) at room temperature. Because of the strong quartic anharmonic renormalization and hardening of the soft modes, the lattice thermal conductivities of cubic CsPbX3 all exhibit weak temperature dependence. Notably, CsPbCl3 exhibits remarkably high thermal conductivity and a long phonon lifetime. This can be attributed to the smallest atomic mean square displacement and the weakest tilting and distortions of PbCl6 octahedra, resulting from the strongest Pb-Cl covalent bonding. Furthermore, the maximum ZT value of 0.63 at 900 K is obtained for the n-type CsPbBr3.

Temperature Dependence of Raman Frequency Shift in SrWO4 Crystal Studied by Lattice Dynamical Calculations

The frequency shift of the Raman modes in strontium tungstate (SrWO4) was investigated in the temperature range from 15 to 295 K. The experimental temperature dependence of the shift was analyzed using both the lattice dynamical calculations and the lattice perturbative approach. We found that the quartic anharmonic term of the first-order perturbation and the cubic term of the second-order perturbation, as well as the thermal expansion, contribute to the temperature shift of the highest-frequency Ag(ν1) mode. The values of the temperature sensitivity of the frequency shift of the Raman modes at room temperature were measured, which is important for developing high-power crystalline Raman lasers and frequency shifters.

Investigation of band gap effect on Raman shift of scheelite PbWO4 studied by Raman spectrometry and first-principles calculations

The Raman frequency shifts of the Bg (58cm−1) and Ag (906cm−1) in the PbWO4 were measured in the temperature range from 4K to 500K. We found that the frequency shift of Ag (906cm−1) shows interesting characteristic of positive change with increasing temperature. The temperature dependence of the Raman shift of these modes was analyzed using both the first-principles calculations and lattice perturbative approach. We found that the quartic anharmonic term of the first-order perturbation, as well as the second order one, and thermal expansion contribute to the temperature dependence of the frequency shift of Ag (906cm−1).

On the interaction of propagating intrinsic localized modes with an isolated impurity in a linear atomic chain

We study the interaction between an intrinsic localized mode and a substitutional defect in a linear chain of atoms interacting with their nearest neighbors through a translational invariant potential function with a harmonic and a quartic anharmonic terms. The classical equations of motion are solved by means of a numerical iterative procedure and moving solutions are sought. The frequencies and atomic displacements from the equilibrium positions are calculated and the interaction between the excitation and the impurity is observed through the changes of the mode profile and the frequency as the breather moves towards and through the impurity.

Anharmonicity and Disorder Effect on the Free Energy of Crystalline Solids

Dealing with the anharmonic phonon-electron problem with the help of double time temperature dependent Green’s Function theory of quantum dynamics the general and explicit expressions for the lattice energy, partition function, free energy have been derived for an impurity induced anharmonic crystalline solid. The effects of mass and force constant change terms between the impurity and host lattice atoms are taken into account, and the cubic and quartic anharmonic terms are also retained in an almost complete Hamiltonian to develop the many body theory. The general trend of low and high temperature contributions to Helmholtz free energy is obtained via three dimensional graphics.

Collective librations of water molecules in the crystal lattice of rubidium bromide: experiment and simulation

Terahertz spectroscopy of RbBr reveals four prominent absorption lines at room temperature and a further 15 lines at 10 K. Via density-functional-theory (DFT) numerical modelling using the PBE0 hybrid GGA functional, all the absorptions are identified as correlated librations of water molecules in the RbBr lattice. Each libration mode is a combination of rocking, wagging and twisting motions of the water molecules. The number of libration lines and numerical modelling show that the C2v symmetry of water in RbBr is broken. Our modelling shows that the distribution of libration amplitudes and phases for different water molecules in the RbBr unit cell varies greatly between the different modes. All librational lines red-shift with increasing temperature. The rate of change for most lines is in the range 60-90 MHz K(-1) (or (2-3) × 10(-3) cm(-1) K(-1)). Two lines shift more rapidly with temperature, at rates of 240 and 300 MHz K(-1) (or (8 and 10) × 10(-3) cm(-1) K(-1)), respectively. Furthermore, the temperature dependence of the linewidth distinguishes two groups of lines. For one group, with weak linear temperature dependence of linewidth, cubic anharmonic terms in the RbBr crystal field are significant. This group is mainly associated with fully symmetric correlated librations. For the second group, with strong non-linear temperature dependence of the linewidth, quartic anharmonic terms in the RbBr crystal field are significant. However, the distribution of libration amplitudes, as well as the type of libration modes, influence the temperature dependence of the red shift, the linewidth, and the intensity, as well. Our combined experimental and theoretical investigation confirms the necessity of obtaining low-temperature data to observe all the calculated modes; moreover, the richness of detail in the temperature dependence of the data invites further modelling spanning a range of temperatures.

Calculation of spectroscopic parameters and vibrational overtones of methanol

A high-level quartic ab initio potential energy surface of methanol has been used to calculate spectroscopic constants of the 12CH3OH molecule. These include coefficients of quartic anharmonic resonance terms, Darling-Dennison constants, for stretching states. A model expressed in terms of dimensionless normal coordinates has been employed in the calculation of O—H and C—H stretching vibrational states in high-overtone regions. Both cubic Fermi and quartic Darling-Dennison anharmonic coupling terms have been included in the model in order to take into account strong resonances between different states. The nonlinear least-squares method has been used to optimize some of the model parameters employing experimental term values of 12CH3OH as data. Vibrational assignments are suggested for the first C—H stretching overtone region.

Lattice dynamics and temperature dependence of the first‐order Raman spectra for α‐SiO2 crystals

AbstractThe phonon dispersion curves for α‐SiO2 crystals were calculated on the basis of a rigid‐ion model. The first‐order Raman spectra of the A1 mode of the α‐SiO2 crystal were measured in the temperature range from 303.1 to 773.1 K. The temperature dependence of the linewidth was analyzed by using the phonon dispersion curves, and the results showed that it was caused approximately by the cubic anharmonic term in the crystal potential energy. The temperature dependence of the frequency shift of the A1 mode was analyzed by using the lattice dynamic perturbative treatment. We found that the quartic anharmonic term of the first‐order perturbation and also the cubic term of the second‐order perturbation contribute to the temperature dependence of the frequency shift of the A1 mode in the α‐SiO2 crystal. Copyright © 2003 John Wiley & Sons, Ltd.

Effect of anharmonicities in the critical number of trapped condensed atoms with an attractive two-body interaction

We determine the quantitative effect, in the maximum number of particles and other static observables, due to small anharmonic terms added to the confining potential of an atomic condensed system with negative two-body interaction. As an example of how a cubic or quartic anharmonic term can affect the maximum number of particles, we consider the trap parameters and the results given by Roberts et al. [Phys. Rev. Lett. 86, 4211 (2001)]. However, this study can be easily transferred to other trap geometries to estimate anharmonic effects.

Nonlinear Gap Modes in a 1D Alternating Bond Monatomic Lattice with Anharmonicity

We analytically study the nonlinear localized gap modes in a one-dimensional atomic chain with uniform atomic mass but two periodically alternating force constants between the nearest neighbors by means of a quasi-continuum approximation. This model simulates a row of atoms in the direction of a diamond-structure type of crystals or molecular crystals with alternating double and single bonds. For this lattice system, we find that the harmonic plus quartic anharmonic terms of inter-site potential produce a new type of nonlinear localized gap modes with a slightly asymmetry distribution of atomic displacements. These localized gap modes are somewhat different from widely studied localized gap modes with a symmetry atomic displacement distribution in diatomic ion lattices.

Asymmetry Localized Modes in an Anharmonic Sphalerite-Structure Lattice

We numerically study the intrinsic localized vibrational modes in a diatomic chain with different masses and alternating force constants between nearest neighbors. This model simulates a row of atoms in the ⟨111⟩ direction of sphalerite-structure crystal. We found that the harmonic and quartic anharmonic terms in the nearest-neighbor interaction potential produce the intrinsic localized modes with frequencies above the optical branch or in the gap of the linear spectrum, the distribution patterns of atom amplitudes are asymmetry with a form of quasi-even- or quasi-odd-parity, and the inclusion of cubic term in the potential lowers the frequencies of the modes and introduces static displacements for the atoms.

Two-dimensional vibrational spectroscopy. VI. Higher-order contributions to the two-dimensional vibrational response functions

The temperature-dependent first-order quantum correction contributions to the two-dimensional vibrational response functions, that are linearly proportional to ℏ, are theoretically investigated. By carrying out the semiclassical expansion of the nonlinear response function in the limit of weak anharmonicity of the potential-energy surface, seventy nine quantum correction terms are obtained. Although it is the cubic anharmonic couplings that were found to be important at the level of classical calculation of the associated nonlinear response functions, it is found that the quartic anharmonic terms can play a role as the quantum contributions. For a three-Brownian-oscillator model system, various numerical calculations of the 2D (two-dimensional) spectra are carried out to show how these quantum correction terms depend on temperature as well as on relative amplitudes of the quartic and cubic anharmonic coupling constants.

Vibration anharmonicity and fast relaxation in the region of the glass transition

It is shown that the anharmonicity of vibrations leads to the quasielastic scattering in glasses and supercooled liquids. The vibrational self-energy term which arises due to the anharmonic interaction provides the one-phonon quasielastic response. Estimations show that in the glass transition region the contribution of this mechanism to the quasielastic spectrum is dominant. The underlying fast relaxation process corresponds to the fluctuations of the vibration occupation numbers. For the boson peak vibrations the respective relaxation time is of the order of a picosecond. The spectral shape of this fast relaxation is found. The amplitude of the quasielastic scattering intensity and its temperature dependence is estimated within the framework of the model and compared with experimental data on light scattering for various materials. The strength of the fast relaxation which is the integral ratio of the quasielastic to vibrational contribution was found to be proportional to the squared Gr\"uneisen parameter. It is shown that at high temperatures the quartic anharmonic term suppresses the contribution of the third-order anharmonicity to the quasielastic scattering. As a result, a crossover temperature appears in the model; above this temperature the intensity of the fast relaxation does not increase anymore. This result is in good agreement with the analysis of the Raman scattering data in ${\mathrm{B}}_{2}{\mathrm{O}}_{3}$ [A. Brodin et al., Phys. Rev. B 53, 11 511 (1996)]. Within the framework of the model, the ratio of the crossover temperature to that of the glass transition is proportional to the inverse fourth-order anharmonic coefficient.

Localized modes in an anharmonic diamond-structure chain

In this paper we study localized vibrational modes in a chain of atoms with the same mass, but with two alternating force constants between nearest neighbors. This model simulates a row of atoms in the 〈111〉 direction of a diamond or silicon crystal. Cubic and quartic anharmonicity are included in the potential. The harmonic and quartic anharmonic terms produce intrinsic localized modes with frequencies above the optical branch and in the gap. The inclusion of cubic terms with negative coefficients reduces the frequencies, leads to new localized modes in the gap, and introduces static distortions of the chain. Modes of both even and odd parity occur. Surface modes occur when the localization is near a free end.

Temperature Dependence of the Ag+Bg-Mode of Raman Shift for CaWO4Crystal

The first-order Raman frequency shifts of the A g (2) + B g (2) mode in a CaWO 4 crystal were measured in the temperature range of 77∼771 K. The frequency shift for this mode changes very slowly with increasing temperature. The temperature dependence of the frequency shift of the A g (2) + B g (2) mode was analyzed using the lattice dynamical perturbative treatment. We found that the quartic anharmonic term of the first-order perturbation, as well as the cubic term of the second-order one, and thermal expansion contribute to the temperature dependence of the frequency shift of the A g (2) + B g (2) mode in a CaWO 4 crystal.

On the refinement of the anharmonic force field of BrNO

The geometry and the anharmonic force field of BrNO have been investigated using multireference configuration interaction (MRCI) theory and, for comparison, multireference perturbation theory to second and third order (CASPT2, CASPT3) to include the dynamic correlation energy of valence electrons. The importance for the equilibrium geometry of the correlation energy terms for inner electrons, 3s, 3p, 3d of Br, has been verified using a new second order algorithm based on determinants and on the Epstein-Nesbet partition of the Hamiltonian matrix. For BrNO, compared to triatomic systems with simpler electronic shells structure, the accuracy of the MRCI treatment, restricted to the valence orbitals and electrons, is somewhat decreased with main effects on the computed equilibrium geometry. The MRCI force field and geometry have been refined empirically by least squares adjustments of the force constants against all available experimental spectroscopic parameters. Additional constraints had to be enforced to the force fields obtained with the other methods, which have been refined fixing the molecular geometry and the quartic anharmonic terms. All the refined force fields are very similar up to cubic terms and reproduce all spectral data and parameters to a good accuracy. The main differences have been found in the anharmonic quartic interactions, which, we conclude, are correctly described only by MRCI.

Raman study of the NH +4 libration—phonon coupling in NH 4MnF 3

The temperature dependence of the bandwidth of the librational modes of the NH + 4 rotator on the orthorhombic (low-temperature) structure of the NH 4MnF 3 perovskite has been obtained from Raman spectra. This dependence has been analysed in the framework of libration—phonon coupling giving rise to cubic and quartic anharmonic terms in the interaction potential of the atoms in the crystal. Evidence is given that the librational modes at 342 and 355 cm −1 are coupled to phonon modes which involve rotations of MnF 6 octahedra and translational motions of the NH + 4 group itself.

The temperature dependence of the A1g-mode Raman line of MgF2

The first-order Raman spectrum for the A1g mode of an MgF2 single crystal was measured at various temperatures from 301 to 973 K. The frequency shift and linewidth change significantly with increasing temperature. In order to analyse the temperature dependence of the frequency shift, first the authors calculated the anharmonic force constants of an MgF2 crystal. Then applying the lattice-dynamical perturbative treatment to the third- and fourth-order proper self-energy, they evaluated each contribution from the cubic and quartic anharmonic terms to the whole shift and could explain the frequency shift throughout the whole range of temperatures. They found that the cubic as well as quartic anharmonic terms play an important role in the temperature dependence of the Raman frequency shift for the A1g mode.

Temperature dependence of the linewidth of the first-order Raman spectrum of a MgF2 crystal.

The first-order Raman spectrum of a ${\mathrm{MgF}}_{2}$ crystal was measured at various temperatures from 300 to 903 K. In addition, the linewidth was calculated theoretically with inclusion of the cubic and quartic anharmonic terms of the repulsive part in the crystal-potential energy. These theoretical results are found to be in reasonably good agreement with experiment. The temperature dependence of the linewidth of the Raman spectrum for ${\mathrm{MgF}}_{2}$ was found to arise from both the quartic and the cubic anharmonic terms; the contribution of the former was seen to become dominant at higher temperatures.