Uneisen Constant Research Articles

Phonon properties, thermal expansion, and thermomechanics of silicene and germanene

We report a hierarchical first-principles investigation on the entangled effects of lattice dimensionality and bond characteristics in the lattice dynamics of silicene and germanene. It is found that bond bending (stretching) negatively (positively) contributes to Gr\"uneisen constant, which results in the negative acoustic (positive optical) Gr\"uneisen constant. The layer thickening (bond weakening) caused by chemical functionalization tends to increase (decrease) the acoustic (optical) Gr\"uneisen constant, due to the increased (decreased) bond-stretching effect. The excitation of the negative-$\gamma$ modes results in negative thermal expansion, while mode excitation and thermal expansion compete with each other in thermomechanics. The sensitive structural and electronic responses of silicene and germanene to functionalization help us to derive a generic physical picture for two-dimensional lattice dynamics.

Correlation between structure, phonon spectra, thermal expansion, and thermomechanics of single-layerMoS2

Using first-principles simulation, the correlation between structure, phonon spectra, thermal expansion, and thermomechanics of single-layer ${\mathrm{MoS}}_{2}$ is established. The laminar structure results in the low-dimension ZA mode with a parabolic dispersion and negative Gr\"uneisen constants ($\ensuremath{\gamma}$), while the nonorthogonal covalent Mo--S bonds (or intralayer thickness) result in the interatom and interdirection vibrational hybridizations, which tend to increase $\ensuremath{\gamma}$. There is a negative-positive crossover in thermal expansion coefficient at 20 K, because of the competition between the modes with negative and positive $\ensuremath{\gamma}$. Although the phononic activation at finite temperatures has a stiffening effect on the bulk modulus, the dominant effect from thermal expansion softens the lattice upon heating. The intralayer thickness results in the similarity between the thermal expansions of SL and bulk ${\mathrm{MoS}}_{2}$. Our numerical results explicitly support that the experimentally measured thermal shifts of the Raman modes are dominated by multiphonon scattering, but not thermal expansion.

Neutron diffraction study of gallium nanostructured within a porous glass

Neutron diffraction studies of structure and atomic vibrations in gallium nanoparticles with the size of 13 nm embedded into a porous glass were performed. At crystallization of gallium, which at room temperature is in a liquid state, the texture effects were observed. The modeling of texture by generalized spherical harmonics allowed us to measure the temperature dependence of the mean-square displacement. It was shown that the contribution of acoustic vibrations in the phonon spectrum of conned gallium is dominant. The Debye temperature of nanostructured gallium appeared to be close to that for the bulk. The Gr\"uneisen constant was found to be strongly reduced with respect to the bulk, as well as the thermal expansion coefficient. It was demonstrated that texture affects physical properties, in particular, superconductivity through inner stresses.

Lattice dynamics of rhenium trioxide from the quasiharmonic approximation

The quasiharmonic theory is applied to study the lattice dynamics and thermal properties of rhenium trioxide, a material exhibiting the negative thermal-expansion phenomenon. Phonons are calculated at several external pressures. The pressure dependence of the $M$, $R$, and zone-center phonon modes is analyzed. Relying on the Gr\uneisen formalism an influence of temperature on the $M$ phonon mode is investigated. The calculated free energy of the system provides predictions for the temperature dependence of such quantities as equilibrium volume and crystal thermal expansivity. The mean-squared vibrations of rhenium-trioxide atoms are investigated versus temperature and pressure. Results of the calculations indicate a softening of the $M$ and $R$ phonons with pressure and a large negative Gr\uneisen constants for these modes. A decrease in the crystal volume is observed up to 170 K. The thermal-expansion coefficient shows a minimum in the vicinity of 100 K which corresponds to a minimum of the anisotropy in the oxygen thermal vibrations perpendicular to the Re-O bond and along the bond. Anisotropy in the oxygen Debye-Waller factors follows from the anisotropy of the force constants at the oxygen site. Inelastic neutron-scattering measurements performed at room temperature are reported together with the generalized phonon densities of states. A comparison between present calculations and the experimental data as well as other theoretical studies is provided.

Origin of Low Thermal Conductivity in Nuclear Fuels

Using a novel many-body approach, we report lattice dynamical properties of UO2 and PuO2 and uncover various contributions to their thermal conductivities. Via calculated Grüneisen constants, we show that only longitudinal acoustic modes having large phonon group velocities are efficient heat carriers. Despite the fact that some optical modes also show their velocities which are extremely large, they do not participate in the heat transfer due to their unusual anharmonicity. Ways to improve thermal conductivity in these materials are discussed.

Ab initiocalculations of third-order elastic constants and related properties for selected semiconductors

We present theoretical studies for the third-order elastic constants ${C}_{ijk}$ in zinc-blende nitrides AlN, GaN, and InN. Our predictions for these compounds are based on detailed ab initio calculations of strain-energy and strain-stress relations in the framework of the density functional theory. To judge the computational accuracy, we compare the ab initio calculated results for ${C}_{ijk}$ with experimental data available for Si and GaAs. We also underline the relation of the third-order elastic constants to other quantities characterizing anharmonic behavior of materials, such as pressure derivatives of the second-order elastic constants ${c}_{ij}^{\ensuremath{'}}$ and the mode Gr\"uneisen constants for long-wavelength acoustic modes $\ensuremath{\gamma}(\mathbf{q},\mathbf{j})$.

Physical momentum versus crystal momentum of acoustic phonons in a crystal lattice

From the statistical mechanics together with the kinetic theory of the pressure of phonons in a crystal we have established the general relation of ${\mathbit{p}}_{\mathbit{q}}=n\ensuremath{\gamma}\ensuremath{\hbar}\mathbit{q}$ for the physical momentum (or anharmonic momentjm) of an acoustic phonon of wave vector $\mathbit{q}$ in $n$ dimensions, $\ensuremath{\gamma}$ being the Gr\"uneisen constant. This is only valid under the condition of broken translational symmetry. For then the anharmonic interaction would enable a $\mathbit{q}\ensuremath{\ne}0$ phonon to couple with and hence acquire a mate of acoustic phonon of $\mathbit{q}=0$, resulting in a composite of the same $\mathbit{q}$. This composite can then carry a physical momentum relative to the wall confining the crystalline medium via its momentum-carrying, $\mathbit{q}=0$ mate. On the other hand, if translational invariance is maintained, a $\mathbit{q}\ensuremath{\ne}0$ phonon is incapable of any physical momentum relative to the inertial wall frame, with or without anharmonic coupling; it can only carry the more familiar ``crystal momentum'' which is the kind that is transported like sound intensity within and hence relative to the equilibrium crystalline medium. An elementary, semiclassical derivation of the ``crystal momentum'' is presented and discussed versus the ``anharmonic momentum.'' Some applications of the latter are presented.

Theory of the thermal properties ofδ-plutonium

The thermal properties of $\ensuremath{\delta}$-plutonium are studied in terms of the elementary electronic structure, and contrasted to those of aluminum. In both metals the thermal expansion can be understood as from the pressure of the gas of phonons, a negative expansion in the case of plutonium because of negative Gr\"uneisen constants for some transverse modes. Thermal expansion in aluminum is found positive, in agreement with experiments. This phonon gas also has a bulk modulus, negative for plutonium, which dominates the observed negative temperature dependence of the bulk modulus for plutonium, with a smaller negative contribution arising from the thermal expansion (in contrast to contraction for pure plutonium) of the alloys on which it is measured. For aluminum, the phonon-gas contribution is negligible and thermal expansion causes the bulk modulus to decrease with temperature, as in experiments. The Debye-Waller temperature, another measure of rigidity, contains a small positive contribution to the temperature dependence from thermal expansion in plutonium and a negative contribution in aluminum. There is also a much larger negative contribution to the temperature dependence of the Debye-Waller temperature in both (in accordance with experiments for plutonium) arising from anharmonicity of some transverse modes. The nearest-neighbor, central-force interactions used here inevitably lead to sizable errors in some elastic constants, and replacement of Brillouin-zone averages by averages along the $[100]$ and $[111]$ directions is very questionable for some properties. However, the simplification is major and the results appear to reveal which contributions are important.

Temperature dependence of lattice vibrations and analysis of the specific heat of graphite

In the semicontinuum model of lattice vibrations in graphite proposed by Komatsu and Nagamiya, the expressions of the dispersion relation contain the elastic constants ${C}_{11},$ ${C}_{12},$ ${C}_{13},$ ${C}_{33},$ ${C}_{44},$ and \ensuremath{\kappa} and the interlayer spacing c as parameters, where $c\ensuremath{\rho}{\ensuremath{\kappa}}^{2}$ is the bending elastic constant of a graphite layer and \ensuremath{\rho} is the density. We improve this semicontinuum model by taking these parameters as a function of temperature. For the parameters except \ensuremath{\kappa}, we use the experimental data already known and the relations derived from them. \ensuremath{\kappa} is determined by fitting the calculated specific heat to the experimental one. The experimental specific heat for single-crystal graphite is derived by evaluating the data reported in the literature. With the value of \ensuremath{\kappa} thus determined, the improved semicontinuum model can explain the experimental specific heat well in the temperature range below 350 K. Then, \ensuremath{\kappa} decreases more rapidly with increasing temperature than the other elastic constants; this result means that softening of the out-of-plane vibrations occurs. It is suggested that the softening is closely related to the negative thermal expansion parallel to the layer planes. The calculated Gr\uneisen constants are compared with the experimental ones. The second derivative of the specific heat curve with respect to temperature gives information on the frequency distribution function of lattice vibrations. From the analysis of the low-temperature specific heat, the value of ${C}_{44}$ at room temperature is determined to be $0.425\ifmmode\times\else\texttimes\fi{}{10}^{11}{\mathrm{d}\mathrm{y}\mathrm{n}/\mathrm{c}\mathrm{m}}^{2}.$

Estimation of the isotope effect on the lattice thermal conductivity of group IV and group III-V semiconductors

The isotope effect on the lattice thermal conductivity for group IV and group III-V semiconductors is calculated using the Debye-Callaway model modified to include both transverse and longitudinal phonon modes explicitly. The frequency and temperature dependences of the normal and umklapp phonon-scattering rates are kept the same for all compounds. The model requires as adjustable parameters only the longitudinal and transverse phonon Gr\"uneisen constants and the effective sample diameter. The model can quantitatively account for the observed isotope effect in diamond and germanium but not in silicon. The magnitude of the isotope effect is predicted for silicon carbide, boron nitride, and gallium nitride. In the case of boron nitride the predicted increase in the room-temperature thermal conductivity with isotopic enrichment is in excess of 100%. Finally, a more general method of estimating normal phonon-scattering rate coefficients for other types of solids is presented.

Nonequivalence of chemical and hydrostatic pressures: A1gandEgfrequencies and Stokes shift ofMn2+-doped perovskites

The frequency of ${A}_{1g}$ and ${E}_{g}$ modes of ${\mathrm{MnF}}_{6}^{4\ensuremath{-}}$ in cubic $\mathrm{AB}{\mathrm{F}}_{3}$ (A: K; B: Mg, Zn and A: Rb, Cs; B: Cd, Ca) perovskites has been derived through density functional calculations on ${\mathrm{MnF}}_{6}{A}_{8}{B}_{6}^{16+}$ clusters which reproduce the experimental impurity-ligand distance ${R}_{e}.$ Both frequencies are found to experience a drastic decrement on passing from ${\mathrm{KMgF}}_{3}:{\mathrm{Mn}}^{2+}$ $(\ensuremath{\Elzxh}{\ensuremath{\omega}}_{A}=556{\mathrm{cm}}^{\ensuremath{-}1},$ $\ensuremath{\Elzxh}{\ensuremath{\omega}}_{E}=456{\mathrm{cm}}^{\ensuremath{-}1})$ to ${\mathrm{CsCdF}}_{3}:{\mathrm{Mn}}^{2+}$ $(\ensuremath{\Elzxh}{\ensuremath{\omega}}_{A}=317{\mathrm{cm}}^{\ensuremath{-}1},$ $\ensuremath{\Elzxh}{\ensuremath{\omega}}_{E}=239{\mathrm{cm}}^{\ensuremath{-}1})$ despite the fact that ${R}_{e}$ increases only by 5%, leading to effective Gr\uneisen constants (called ${\ensuremath{\gamma}}_{A}^{c}$ and ${\ensuremath{\gamma}}_{E}^{c})$ around 3.0 along the series. This figure is 60% higher than the usual Gr\uneisen constant ${\ensuremath{\gamma}}_{A}$ calculated for a given system like ${\mathrm{KMgF}}_{3}:{\mathrm{Mn}}^{2+}$ or ${\mathrm{KZnF}}_{3}:{\mathrm{Mn}}^{2+}$ when hydrostatic pressure is applied. For supporting this relevant result the value of ${\ensuremath{\gamma}}_{A}$ for ${\mathrm{CrF}}_{6}^{3\ensuremath{-}}$ in a fluoroelpasolite has been calculated as well. The obtained value ${\ensuremath{\gamma}}_{A}=2.1$ is close to the experimental one $({\ensuremath{\gamma}}_{A}=1.9)$ derived in ${\mathrm{K}}_{2}{\mathrm{NaGaF}}_{6}:{\mathrm{Cr}}^{3+}$ by combined optical and Raman measurements. As a salient feature, the increase of the Stokes shift when ${R}_{e}$ increases observed along the $\mathrm{AB}{\mathrm{F}}_{3}:{\mathrm{Mn}}^{2+}$ series is now well explained through the bigger variations of ${A}_{1g}$ and ${E}_{g}$ frequencies induced by the chemical pressure in comparison to those coming from an hydrostatic pressure on a given system. The difference between ${\ensuremath{\gamma}}_{A}^{c}$ and ${\ensuremath{\gamma}}_{A}$ is discussed through a simple model that emphasizes the role played by the coupling of the ${\mathrm{MnF}}_{6}^{4\ensuremath{-}}$ complex to the lattice. The influence of chemical and hydrostatic pressures upon the luminescence efficiency is also briefly discussed. Attention is addressed to the method of calculating impurity-associated Gr\uneisen constants using clusters. Recent results on ${\mathrm{Cr}}^{3+}$ in several cubic chloroelpasolites that support the present conclusions are briefly discussed as well.

Effect of hydrostatic pressure on the Raman spectrum ofGenSimmultiple quantum wells withn<~4andm<~7

We report a Raman study of the effects pressure has on the vibrational structure of ${\mathrm{Ge}}_{n}{\mathrm{Si}}_{m}$ multiple quantum wells (MQW's) with $n<~4$ and $m<~7.$ Three primary phonon bands are studied: Ge-Ge within the germanium layers, Si-Si within the silicon layers, and the Ge-Si interface mode. Pressure shifts each of these bands consistent with a mode-Gr\"uneisen constant of unity for all samples and laser excitations used. We observe resonance effects with the confined Ge-like ${E}_{1}$ transition for the ${\mathrm{Ge}}_{4}{\mathrm{Si}}_{5}$ sample. The transition is near 2.4 eV at ambient pressure and blueshifts at $\ensuremath{\approx}4\ifmmode\pm\else\textpm\fi{}1$ meV/kbar. This pressure coefficient is smaller than the corresponding quantity in bulk germanium. This is attributed to the fact that silicon dictates the in-plane contraction of the Ge layer that is at a smaller rate than the corresponding quantity in bulk germanium. We see no evidence of resonance enhancement in samples with thinner Ge layers in each MQW period. This implies that at least four Ge atoms are necessary to form the states producing the ${E}_{1}$ transition, consistent with previous studies. An additional feature seen in the spectra near $310 {\mathrm{cm}}^{\ensuremath{-}1}$ is identified by the pressure study to be 2TA Raman scattering from silicon.

Electronic topological transitions in Mo-Re random alloys

Electronic spectra and bulk properties of Mo-rich bcc Mo-Re random alloys have been studied in the framework of density-functional theory. We show that the Fermi surface topology changes with rhenium concentration, i.e., so-called electronic topological transitions take place. We have found two electronic topological transitions, the formation of an electronic void at about 2 at. % of Re and a formation of a neck at about 6 at. % of Re. The influence of these transitions on the thermodynamic (lattice parameter, bulk modulus, and Gr\"uneisen constant) and kinetic (thermoelectric power) properties is analyzed.

Extended Mie-Grüneisen theory applied to C60 in the disordered fcc phase.

The large difference between the interatomic forces within a ${\mathrm{C}}_{60}$ molecule and the intermolecular forces allows the intermolecular contributions to thermodynamic properties to be treated by a simple pairwise central force theory when the molecules are freely rotating. This circumstance was utilized to compute the intermolecular contribution to the properties of the disordered phase. The anharmonicity was treated by an extension of Mie-Gr\"uneisen theory in which the Gr\"uneisen constant was replaced by a Gr\"uneisen parameter calculated as a function of volume from the intermolecular potential. The equation of state, compressibility, and coefficient of thermal expansion, as well as the ratio of the constant pressure to constant-volume heat capacity were computed in the quasiharmonic approximation using the calculated Gr\"uneisen parameter. The agreement with experiment was satisfactory on all counts.

Acoustic properties of gallium selenide.

We investigate the effect of illumination on the velocity of the ${\mathit{L}}_{\mathrm{\ensuremath{\perp}}}$ and ${\mathrm{FT}}_{\mathrm{\ensuremath{\perp}}}$ acoustic-wave propagation along layers of \ensuremath{\varepsilon}-GeSe [in the (0001) plane]. Frequency and temperature dependences and also relative changes of various acoustic-wave velocities under hydrostatic pressure in \ensuremath{\varepsilon}-GaSe are used to show that photosensitive absorption and the change of the ${\mathit{L}}_{\mathrm{\ensuremath{\perp}}}$ and ${\mathrm{FT}}_{\mathrm{\ensuremath{\perp}}}$ wave-propagation velocity are associated with the piezoelectric interaction of the acoustic wave with free charge carriers. High anisotropy of anharmonic properties at compressive and shearing strain of the layers relative to each other is a characteristic feature of lamellar crystals and is related to anisotropy of the crystal elasticity. The velocity of piezoactive acoustic waves in P6\ifmmode\bar\else\textasciimacron\fi{}m2 crystals has been calculated theoretically. Piezoelectric characteristics and Gr\"uneisen constants for the acoustic branches of the phonon spectrum in \ensuremath{\varepsilon}-GaSe have been determined.

Ab initio calculation of structural and lattice-dynamical properties of silicon carbide.

The plane-wave pseudopotential approach to density-functional theory (DFT) in the local-density approximation has been applied to investigate a variety of ground-state properties of the 3C, 2H, and 4H polytypes of silicon carbide. The linear-response theory within DFT has been used to obtain lattice-dynamical properties of cubic SiC such as the phonon-dispersion curves, phonon eigenvectors, elastic and Gr\uneisen constants, as well as the thermal expansion coefficient and specific heat within the quasiharmonic approximation. Finally, we present some results for phonon-dispersion curves in the hexagonal 2H (wurtzite) and 4H structure. These results are analyzed and discussed in view of further applications to temperature-dependent properties.

Theoretical study of the anharmonic decay of nonequilibrium LO phonons in semiconductor structures.

Using a model cubic anharmonic crystal potential and Fermi's golden rule, an expression for the anharmonic relaxation time for a nonequilibrium long-wavelength LO phonon is rigorously derived. Decay times for various bulk, alloy, and quantum-well semiconductor systems are calculated. Comparison with time-resolved Raman measurements indicates that a relatively high value of the Gr\"uneisen constant (1.8 for GaAs) gives the best agreement between experiment and theory.

Transferable tight-binding models for silicon.

A transferable tight-binding model for silicon is found by fitting the energies of silicon in various bulk crystal structures and examining functional parametrizations of the tight-binding forms. The model has short-range radial forms similar to the tight-binding Hamiltonian of Goodwin, Skinner, and Pettifor but can be utilized in molecular dynamics with a fixed radial cutoff for all structural configurations. In addition to a very good fit to the energy of Si in different bulk crystal structures the model describes very well the elastic constants, defect-formation energies for vacancies and interstitials in crystalline silicon, the melting of Si, and short-range order in liquid silicon. Results for phonon frequencies and Gr\"uneisen constants in c-Si are also presented.

Anharmonic thermal resistivity of dielectric crystals at low temperatures.

The intrinsic thermal resistance of dielectric crystals at low temperatures due to three-phonon Umklapp processes has been treated theoretically, taking account of the restrictions imposed by wave-vector selection and frequency conservation rules, as well as the known phonon-dispersion curves. Since low-frequency phonons cannot interact directly with zone-boundary phonons, the transfer of excess momentum takes place in two steps: a normal process with modes of intermediate frequency, followed by an Umklapp process of intermediate-frequency phonons with phonons near the zone boundary of two different polarization branches with different frequencies. The intermediate frequency is given by that frequency difference. The second step can also be scattering by point defects, so that the apparent Umklapp resistance can be modified by point defects, which scatter the intermediate-frequency phonons much more strongly than the thermal phonons. The important processes in different frequency ranges depend on the dispersion curves and on the residual point defects; various cases are discussed. The theory has only one adjustable parameter, the Gr\uneisen constant. The point-defect scattering strength is deduced from the departures of the thermal conductivity in the boundary scattering regime at very low temperatures from a ${\mathit{T}}^{3}$ dependence. The theory has been compared to measurements on LiF, NaF, diamond, Si, and enriched $^{74}\mathrm{Ge}$. Reasonably good agreement with experimental data can be obtained without invoking an adjustable exponent in the Umklapp resistance.

Atomic vibrations in a self-consistent-field atom-in-jellium model of condensed matter.

The electronic structure of many of the elemental solids is fairly well represented by a single atom embedded in a degenerate electron gas. This is particularly true for bulk properties such as the equation of state of highly compressed matter. Because the atom-in-jellium model is spherically symmetric, it is simpler than band-structure models. We study what happens when the nucleus is moved off center in the atomic cell to form a nonspherical system. This forms the basis of an Einstein model of atomic vibrations. The model is used to calculate Einstein temperatures and Gr\"uneisen constants of simple solids by self-consistent-field electronic-structure methods.