Giant anharmonicity and non-linear electron-phonon coupling in MgB$_{2}$; A combined first-principles calculations and neutron scattering study
T. Yildirim, O. Gulseren, J. W. Lynn, C. M. Brown, T. J. Udovic, H. Z. Qing, N. Rogado, K. A. Regan, M. A. Hayward, J. S. Slusky, T. He, M. K. Haas, P. Khalifah… 展开作者
T. Yildirim, O. Gulseren, J. W. Lynn, C. M. Brown, T. J. Udovic, H. Z. Qing, N. Rogado, K. A. Regan, M. A. Hayward, J. S. Slusky, T. He, M. K. Haas, P. Khalifah, K. Inumaru, R. J. Cava
We report first-principles calculations of the electronic band structure and lattice dynamics for the new superconductor MgB$_{2}$. The excellent agreement between theory and our inelastic neutron scattering measurements of the phonon density of states gives confidence that the calculations provide a sound description of the physical properties of the system. The numerical results reveal that the in-plane boron phonons (with E$_{2g}$ symmetry) near the zone-center are very anharmonic, and are strongly coupled to the partially occupied planar B $σ$ bands near the Fermi level. This giant anharmonicity and non-linear electron-phonon coupling is key to explaining the observed high T$_{c}$ and boron isotope effect in MgB$_{2}$
The remarkable anharmonicity of the E_{2g} phonon in MgB_2 has been suggested in literature to play a primary role in its superconducting pairing. We investigate, by means of LDA calculations, the microscopic origin of such an anharmonicity in MgB_2, AlB_2, and in hole-doped graphite. We find that the anharmonic character of the E_{2g} phonon is essentially driven by the small Fermi energy of the sigma holes. We present a simple analytic model which allows us to understand in microscopic terms the role of the small Fermi energy and of the electronic structure. The relation between anharmonicit…
We consider a model Hamiltonian fitted on the ab-initio band structure to describe the electron-phonon coupling between the electronic $σ-$bands and the phonon E$_{2g}$ mode in MgB$_2$. The model allows for analytical calculations and numerical treatments using very large k-point grids. We calculate the phonon self-energy of the E$_{2g}$ mode along two high symmetry directions in the Brillouin zone. We demonstrate that the contribution of the $σ$ bands to the Raman linewidth of the E$_{2g}$ mode via the electron-phonon coupling is zero. As a consequence the large resonance seen in Raman experi…
The zone-center $E_{2g}$ modes play a crucial role in MgB$_2$, controlling the scattering mechanisms in the normal state as well the superconducting pairing. Here, we demonstrate via first-principles quantum-field theory calculations that, due to the anisotropic electron-phonon interaction, a $hot$-$phonon$ regime where the $E_{2g}$ phonons can achieve significantly larger effective populations than other modes, is triggered in MgB$_2$ by the interaction with an ultra-short laser pulse. Spectral signatures of this scenario in ultrafast pump-probe Raman spectroscopy are discussed in detail, rev…
We study the effects of pressure on the electron-phonon interaction in MgB_2 using density-functional-based methods. Our results show that the superconductivity in MgB_2 vanishes by 100 GPa, and then reappears at higher pressures. In particular, we find a superconducting transition temperature T_c \~ 2 K for mu*=0.1 at a pressure of 137 GPa.
We compute the anharmonic shift of the phonon frequencies in MgB2, using density functional theory. We explicitly take into account the scattering between different phonon modes at different q-points in the Brillouin zone. The shift of the E2g mode at the Gamma point is +5 % of the harmonic frequency. This result comes from the cancellation between the contributions of the four- and three-phonon scattering, respectively +10 % and -5 %. A similar shift is predicted at the A point, in agreement with inelastic X-ray scattering phonon-dispersion measurements. A smaller shift is observed at the M p…
We measure phonon dispersion and linewidth in a single crystal of MgB_2 along the Gamma-A, Gamma-M and A-L directions using inelastic X-Ray scattering. We use Density Functional Theory to compute the effect of both electron-phonon coupling and anharmonicity on the linewidth, obtaining excellent agreement with experiment. Anomalous broadening of the E_2g phonon mode is found all along Gamma-A. The dominant contribution to the linewidth is always the electron-phonon coupling.
In strong-coupling superconductors with a short electron mean free path the self-energy effects in the superconducting order parameter play a major role in the phonon manifestation of the point-contact spectra at above-gap energies. We compare the expressions for the nonlinear conductivity of tunnel, ballistic, and diffusive point-contacts and show that these expression are similar and correspond to the measurements of the phonon structure in the point-contact spectra for the $π$-band of MgB$_{2}$.
We have investigated the effect of unconventional pairing mechanism in MgB2 using a two-band model within the framework of Bogoliubov-Valatin formalism. The approach incorporates the intra- band s-wave interaction in the s- and p-bands, as well as interband s-wave interaction between them. The analysis assumes the pairing interaction matrix comprises of attractive electron-phonon, charge fluctuation and repulsive electron-electron (Coulomb) interactions to account for superconductivity in MgB2. The model is used to estimate the transition temperature and isotope effect exponent as well as to e…
Room temperature Raman scattering measurements have been carried out on well characterized samples of MgB2-xCx. The Raman line corresponding to the E2g phonon mode shows progressive hardening from 620 cm-1 in pristine MgB2 to 775 cm-1 in the sample with carbon fraction x=0.2. The corresponding line width on the other hand, increases from a value of about 220 cm-1 to 286 cm-1 in samples with x = 0.1, beyond which it decreases to a value 167 cm-1 for x=0.2. From the average mode frequency and the line width obtained from Raman measurements and taking the values of N(0) obtained from the calculat…
In this work we explore the phase diagram of the binary Nb-S system from ambient pressures up to 250 GPa using $ab$ $initio$ evolutionary crystal structure prediction. We find several new stable compositions and phases, especially in the high-pressure regime, and investigate their electronic, vibrational, and superconducting properties. Our calculations show that all materials, besides the low-pressure phases of pure sulfur, are metals with low electron-phonon coupling strengths and critical superconducting temperatures below 15 K. Furthermore, we investigate the effects of phonon anharmonicit…
We show that the electron-phonon coupling in the transition-metal diborides NbB2 and TaB2 is dominated by the longitudinal acoustical (LA) mode, in contrast to the optical E_{2g} mode dominated coupling in MgB2. Our ab initio results, described in terms of phonon dispersion, linewidth, and partial electron-phonon coupling along Gamma to A, also show that (i) NbB2 and TaB2 have a relatively weak electron-phonon coupling, (ii) the E_{2g} linewidth is an order of magnitude larger in MgB2 than in NbB2 or TaB2, (iii) the E_{2g} frequency in NbB2 and TaB2 is considerably higher than in MgB2, and (iv…
Weak localization has a strong influence on both the normal and superconducting properties of metals. In particular, since weak localization leads to the decoupling of electrons and phonons, the temperature dependence of resistance (i.e., $λ_{tr}$) is decreasing with increasing disorder, as manifested by Mooij's empirical rule. In addition, Testardi's universal correlation of $T_{c}$ (i.e., $λ$) and the resistance ratio (i.e., $λ_{tr}$) follows. This understanding provides a new means to probe the phonon mechanism in superconductors including MgB$_{2}$. The merits of this method are its applic…
Two-dimensional (2D) van der Waals (vdW) materials have extraordinary thermal properties due to the effect of quantum confinement, making them promising for thermoelectric energy conversion and thermal management in microelectronic devices. In this review, the mechanism of phonon anharmonicity originating from three- and four-phonon interactions is derived. The phonon anharmonicity of 2D vdW materials, involving the Grüneisen parameter, phonon lifetime, and thermal conductivity, is summarized and derived in detail. The size-dependent thermal conductivity of representative 2D vdW materials is d…
We have studied the electron-phonon and superconducting properties of the Mg{1-x}AlxB2 and MgB{2(1-y)}C{2y} alloys within the framework of density functional theory using the self-consistent virtual-crystal approximation. For both alloys, the Eliashberg spectral functions and the electron-phonon coupling constants have been calculated in the two-band model for several concentrations up to x(Al)=0.55 and y(C)=0.175. We solved numerically the two-band Eliashberg gap equations without considering interband scattering. Using a single parameter for the Coulomb pseudopotential, which was determined …
Anharmonic lattice vibrations govern the thermal dynamics in materials and present how the atoms interact and how they conduct heat. An indepth understanding of the microscopic mechanism of phonon anharmonicity in condensed systems is critical for developing better functional and energy materials. In recent years, a variety of novel behaviors in condense matters are driven by phonon anharmonic effects in some way or another, such as soft mode phase transition, negative thermal expansion, multiferroicity, ultralow thermal conductivity or high thermal resistance, and high-temperature superconduc…
An approach to compute the anharmonic peaks of the phonon dispersion curves through the ab initio calculated Hellmann-Feynman forces from a series of supercells with realistic atomic displacements of all atoms, which correspond to a given temperature, is reported. Obtained phonon dispersion bands are able to represent the positions and shapes of the anharmonic peaks. As example, the approach to cubic PbTe and perovskite MgSiO3 crystals is applied.
The discovery of superconductivity in Magnesium Diborate (MgB$_2$) has stimulated great interest in the search of new superconductors with similar lattice structures. Unlike cuprate or iron-based superconductors, MgB$_2$ is indisputably a phonon-mediated high temperature superconductor. The emergence of high temperature superconductivity in this material results from the strong coupling between the boron $σ$-bonding electrons around the Fermi level and the bond-stretching optical phonon modes. Here we show, based on the first-principles calculations, that Li$_2$B$_3$C is such a good candidate …
We show that a recently predicted layered phase of lithium monoboride, Li2B2, combines the key mechanism for strong electron-phonon coupling in MgB2 (i.e., interaction of covalent B sigma bands with B bond-stretching modes) with the dominant coupling mechanism in CaC6 (i.e., interaction of free-electron-like interlayer states with soft intercalant modes). Yet, surprisingly, the electron-phonon coupling in Li2B2 is calculated to be weaker than in either MgB2 or CaC6. We demonstrate that this is due to the accidental absence of B pi states at the Fermi level in Li2B2. In MgB2, the pi electrons p…
Opposite impact of thermal expansion and phonon anharmonicity on the phonon-limited resistivity of elemental metals from first principles
European Theoretical Spectroscopy Facility, Institute of Condensed Matter and Nanosciences, Université catholique de Louvain（欧洲理论光谱设施，凝聚态物质与纳米科学研究所，天主教鲁汶大学）；Scientific Computing Department, Science and Technology Facilities Council, UK Research and Innovation, Daresbury Laboratory（科学与工程设施理事会英国研究与创新达尔斯伯里实验室科学计算部）；Département de Physique et Institut Courtois, Université de Montréal（蒙特利尔大学物理系与库瓦托瓦研究所）；Department of Molecular Chemistry and Materials Science, Weizmann Institute of Science（魏茨曼科学研究所分子化学与材料科学系）；WEL Research Institute（WEL研究所）
A series of calculations on MgB2 and related isoelectronic systems indicates that the layer of Mg(2+) ions lowers the non-bonding B "pi" (p_z) bands relative to the bonding "sigma" (sp_xp_y) bands compared to graphite, causing sigma --> pi charge transfer and sigma band doping of 0.13 holes/cell. Due to their two dimensionality the sigma bands contribute strongly to the Fermi level density of states. Calculated deformation potentials of Gamma point phonons identify the B bond stretching modes as dominating the electron-phonon coupling. Superconductivity driven by sigma band holes is consistent…
Lattice vibrations in CH$_3$NH$_3$PbI$_3$ are strongly interacting, with double well instabilities present at the Brillouin zone boundary. Analysis within a first-principles lattice dynamics framework reveals anharmonic potentials with short phonon quasi-particle lifetimes and mean-free paths. The phonon behaviour is distinct from the inorganic semiconductors GaAs and CdTe where three-phonon interaction strengths are three orders of magnitude smaller. The implications for the applications of hybrid halide perovskites arising from thermal conductivity, band-gap deformation, and charge-carrier s…
The partition function of an oscillator disturbed by a set of electron particle paths has been computed by a path integral method which permits to evaluate at any temperature the relevant cumulant terms in the series expansion. The time dependent source current peculiar of the semiclassical Su-Schrieffer-Heeger model induces large electron-phonon anharmonicities on the phonon subsystem. As a main signature of anharmonicity the phonon heat capacity shows a peak whose temperature location strongly varies with the strength of the {\it e-ph} coupling. High energy oscillators are less sensitive to …
The variational stochastic self-consistent harmonic approximation is combined with the calculation of third-order anharmonic coefficients within density-functional perturbation theory and the "$2n+1$" theorem to calculate anharmonic properties of crystals. It is demonstrated that in the perturbative limit the combination of these two methods yields the perturbative phonon linewidth and frequency shift in a very efficient way, avoiding the explicit calculation of fourth-order anharmonic coefficients. Moreover, it also allows calculating phonon lifetimes and inelastic neutron scattering spectra …
Analysis of the point-contact spectroscopy (PCS) data on the new dramatic high-T$_c$ superconductor MgB$_2$ reveals quite different behavior of two disconnected $σ$ and $π$ electronic bands, deriving from their anisotropy, different dimensionality, and electron-phonon interaction. PCS allows direct registration of both the superconducting gaps and electron-phonon-interaction spectral function of the two-dimensional $σ$ and three-dimensional $π$ band, establishing correlation between the gap value and intensity of the high-T$_c$ driving force -- the $E_{2g}$ boron vibration mode. PCS data on so…
Investigating the competition between ferroelectric ordering and quantum fluctuations is essential to tailor the desired functionalities of mixed ferroelectric and incipient ferroelectric systems, like, (Ba,Sr)TiO3 and (Eu,Ba)TiO3. Recently, it has been shown that suppression of quantum fluctuations increases ferroelectric ordering in (Eu,Ba)TiO3 and since these phenomena are coupled to crystallographic phase transitions it is essential to understand the role of phonons. Here, we observe that the unusual temperature dependence of phonons in BaTiO3 gets suppressed when Ba2+ is replaced by Eu2+.…
The microscopic origin of the large negative thermal expansion of cubic scandium trifluorides (ScF3) is investigated by performing a set of anharmonic free-energy calculations based on density functional theory. We demonstrate that the conventional quasiharmonic approximation (QHA) completely breaks down for ScF3 and the quartic anharmonicity, treated nonperturbatively by the self-consistent phonon theory, is essential to reproduce the observed transition from negative to positive thermal expansivity and the hardening of the R4+ soft mode with heating. In addition, we show that the contributio…
Materials with ultralow thermal conductivity are of great interest for efficient energy conversion and thermal barrier coating. Copper-based semiconductors such as copper chalcogenides and copper halides are known to possess extreme low thermal conductivity, whereas the fundamental origin of the low thermal conductivity observed in the copper-based materials remains elusive. Here, we reveal that s-d coupling induced giant phonon anharmonicity is the fundamental mechanism responsible for the ultralow thermal conductivity of copper compounds. The symmetry controlled strong coupling of high-lying…
A theory of superconductivity is presented where the effect of anharmonicity, as entailed in the acoustic, or optical, phonon damping, is explicitly considered in the pairing mechanism. The gap equation is solved including diffusive Akhiezer damping for longitudinal acoustic phonons or Klemens damping for optical phonons, with a damping coefficient which, in either case, can be directly related to the Gruneisen parameter and hence to the anharmonic coefficients in the interatomic potential. The results show that the increase of anharmonicity has a strikingly non-monotonic effect on the critica…
Analytical theory of large size intrinsic localized modes (ILMs) in anharmonic two-atomic chain is presented. It is shown that ILMs with frequencies close to the borders of phonon gap are govern by quartic anharmonicity while the effect of cubic anharmonicity is essentially reduced. This reduction effect is a consequence of small amplitude of vibrations of every second atom. As a result, an ILM cannot split down from the optical phonon band, in contradiction with the commonly accepted point of view. But it can spit up from the top of the acoustic phonon band making possible the existence of th…
The binary compound SnSe exhibits record high thermoelectric performance, largely because of its very low thermal conductivity. The origin of the strong phonon anharmonicity leading to the low thermal conductivity of SnSe is investigated through first-principles calculations of the electronic structure and phonons. It is shown that a Jahn-Teller instability of the electronic structure is responsible for the high-temperature lattice distortion between the Cmcm and Pnma phases. The coupling of phonon modes and the phase transition mechanism are elucidated, emphasizing the connection with hybrid …
The unusual Raman spectrum of MgB$_2$ and its formidable temperature dependence are successfully reproduced by means of a parameter-free \emph{ab initio} nonadiabatic theory that accounts for the electron-hole pair scattering mechanisms with the system phonons. This example turns out to be a prototypical case where a strong nonadiabatic renormalization of the phonon frequency is partially washed out by the aforementioned scattering events, bringing along a characteristic temperature dependence. Both electron-hole pair lifetime and energy renormalization effects due to dynamical electron-phonon…
Zn(CN)2 is known to have an isotropic negative thermal expansion (NTE) coefficient (about -51 x 10-6 K-1) over 10-370 K that is twice as large as that of ZrW2O8. We have measured the pressure dependence of the phonon spectra up to 30 meV from a polycrystalline sample of Zn(CN)2 at pressures of 0, 0.3, 1.9 and 2.8 kbar at temperatures of 165 and 225 K. The measurements enabled us to estimate the energy dependence of the ratios Gamma/B (Gamma are Gruneisen parameters as a function of phonon energy Ei at ambient pressure and B is the bulk modulus), which reflect the anharmonicity of phonons. We c…
The phonon-mode decomposition of the electron-phonon coupling in the MgB2-like system Li_{1-x}BC is explored using first principles calculations. It is found that the high temperature superconductivity of such systems results from extremely strong coupling to only ~2% of the phonon modes. Novel characteristics of E_2g branches include (1) ``mode lambda'' values of 25 and greater compared to a mean of $\sim 0.4$ for other modes, (2) a precipitous Kohn anomaly, and (3) E_2g phonon linewidths within a factor of ~2 of the frequency itself, indicating impending breakdown of linear electron-phonon t…
Despite the widespread use of silicon in modern technology, its peculiar thermal expansion is not well-understood. Adapting harmonic phonons to the specific volume at temperature, the quasiharmonic approximation, has become accepted for simulating the thermal expansion, but has given ambiguous interpretations for microscopic mechanisms. To test atomistic mechanisms, we performed inelastic neutron scattering experiments from 100-1500K on a single-crystal of silicon to measure the changes in phonon frequencies. Our state-of-the-art ab initio calculations, which fully account for phonon anharmoni…
The layered semiconductor SnSe is one of the highest-performing thermoelectric materials known. We demonstrate, through a first-principles lattice-dynamics study, that the high-temperature Cmcm phase is a dynamic average over lower-symmetry minima separated by very small energetic barriers. Compared to the low-temperature Pnma phase, the Cmcm phase displays a phonon softening and enhanced three-phonon scattering, leading to an anharmonic damping of the low-frequency modes and hence the thermal transport. We develop a renormalisation scheme to quantify the effect of the soft modes on the calcul…
In this work, we performed temperature-dependent Raman spectroscopy studies on three layered metal thiophosphates: CuInP2S6, CuCrP2S6, and CuInP2S6-In4/3P2S6 heterostructures. These materials are emerging multiferroics, with CuInP2S6 and CuInP2S6-In4/3P2S6 exhibiting room temperature ferrielectric polarization while CuCrP2S6 is both anti-ferroelectric and antiferromagnetic at low temperatures (<145 and <30 K, respectively). We studied two prominent peaks in the Raman spectra of these compounds, namely their P-P and P-S stretching modes. First order linear fits to the mode softening yielded coe…
The anharmonic phonon properties of SnSe in the Pnma phase were investigated with a combination of experiments and first-principles simulations. Using inelastic neutron scattering (INS) and nuclear resonant inelastic X-ray scattering (NRIXS), we have measured the phonon dispersions and density of states (DOS) and their temperature dependence, which revealed a strong, inhomogeneous shift and broadening of the spectrum on warming. First-principles simulations were performed to rationalize these measurements, and to explain the previously reported anisotropic thermal expansion, in particular the …
Motivated by the excellent electronic and optoelectronic properties of two-dimensional (2D) tin oxides, we systematically investigated the thermal conductivity of monolayer SnO and SnO2by the first-principles calculations. The room-temperature thermal conductivity of monolayer SnO and SnO2reaches 9.6 W/(mK) and 98.8 W/(mK), respectively. The size effect is much weaker for monolayer SnO than for monolayer SnO2, due to the coexistence of size dependent and independent component in the thermal conductivity of monolayer SnO. The large difference between the thermal conductivity of 2D tin oxides ca…
The thermal properties of materials are critically important to various technologies and are increasingly the target of materials design efforts. However, it is only relatively recent advances in first-principles computational techniques that have enabled researchers to explore the microscopic mechanisms of thermal properties, such as thermal expansion. We use the Grüneisen theory of thermal expansion in combination with density functional calculations and the quasiharmonic approximation to uncover mechanisms of thermal expansion in PbTiO$_3$ thin-films in terms of elastic and vibrational cont…
Phosphorene, an emerging elemental two-dimensional (2D) direct band gap semiconductor with fascinating structural and electronic properties distinctively different from other 2D materials such as graphene and MoS2, is promising for novel nanoelectronic and optoelectronic applications. Phonons, as one of the most important collective excitations, are at the heart for the device performance, as their interactions with electrons and photons govern the carrier mobility and light-emitting efficiency of the material. Here, through a detailed first-principles study, it is demonstrated that monolayer …
We devise a framework to compute accurate phonons in molecular crystals even in case of strong quantum anharmonicity. Our approach is based on the calculation of the static limit of the phononic Matsubara Green's function from path integral molecular dynamics simulations. Our method enjoys a remarkably low variance, which allows one to compute accurate phonon frequencies after a few picoseconds of nuclear dynamics, and it is further stabilized by the use of appropriate constrained displacement operators. We applied it to solid hydrogen at high pressure. For phase III, our predicted infrared (I…
The effect of biaxial strain on the superconducting properties of MgB2 thin films was studied by first-principles calculations. The stability analyses by phonon dispersions show that biaxial strain as much as 7% can be applied onto MgB2 without inducing any imaginary frequency. The superconducting property calculations based on the frame of Migdal-Eliashberg theory successfully reproduce the two-gap superconductivity of MgB2. The results show that the tensile biaxial strain can increase the critical temperature of MgB2 while the compressive biaxial strain would decrease the critical temperatur…
The ultrafast dynamics of electrons and collective modes in systems out of equilibrium is crucially governed by the energy transfer from electronic degrees of freedom, where the energy of the pump source is usually absorbed, to lattice degrees of freedom. In conventional metals such process leads to an overall heating of the lattice, usually described by an effective lattice temperature $T_{\rm ph}$, until final equilibrium with all the degrees of freedom is reached. In specific materials, however, few lattice modes provide a preferential channel for the energy transfer, leading to a non-therm…
Knowledge of lattice anharmonicity is essential to elucidate distinctive thermal properties in crystalline solids. Yet, accurate \textit{ab initio} investigations of lattice anharmonicity encounter difficulties owing to the cumbersome computations. Here we introduce the phonon quasiparticle approach and review its application to various materials. This method efficiently and reliably addresses lattice anharmonicity by combining \textit{ab initio} molecular dynamics and lattice dynamics calculations. Thus, in principle, it accounts for full anharmonic effects and overcomes finite-size effects t…
We have carried out temperature-dependent x-ray diffraction and Raman scattering experiments on powder $Cs_3Bi_2Br_9$. Trigonal to monoclinic structural transition at around 95 K is discussed and shown to be driven by the softening of the soft mode. We propose a model to describe the dynamics of the incomplete soft-mode. Raman scattering experiments demonstrate the origin of the soft mode to the rocking motions of Br atoms that participate to form $BiBr_6$ octahedra, which correlates the reported theoretical calculations. Some of the Raman mode frequencies exhibit anomalous temperature depende…
We examine anharmonic contributions to the optical phonon modes in bulk $T_d$-MoTe$_2$ through temperature-dependent Raman spectroscopy. At temperatures ranging from 100 K to 200 K, we find that all modes redshift linearly with temperature in agreement with the Grüneisen model. However, below 100 K we observe nonlinear temperature-dependent frequency shifts in some modes. We demonstrate that this anharmonic behavior is consistent with the decay of an optical phonon into multiple acoustic phonons. Furthermore, the highest frequency Raman modes show large changes in intensity and linewidth near …
All phonons in a single crystal of NaBr were measured by inelastic neutron scattering at temperatures of 10, 300 and 700 K. Even at 300 K the phonons, especially the longitudinal-optical (LO) phonons, showed large shifts in frequencies, and showed large broadenings in energy owing to anharmonicity. Ab initio computations were first performed with the quasiharmonic approximation (QHA), in which the phonon frequencies depend only on $V$, and on $T$ only insofar as it alters $V$ by thermal expansion. This QHA was an unqualified failure for predicting the temperature dependence of phonon frequenci…
