This paper reports the first experimental observation of phonons and their softening on single crystalline LaPt$_2$Si$_2$ via inelastic neutron scattering. From the temperature dependence of the phonon frequency in close proximity to the charge-density wave (CDW) $q$-vector, we obtain a CDW transition temperature of T$_{CDW}$ = 230 K and a critical exponent $β$ = 0.28 $\pm$ 0.03. This value is suggestive of a non-conventional critical behavior for the CDW phase transition in LaPt$_2$Si$_2$, compatible with a scenario of CDW discommensuration (DC). The DC would be caused by the existence of two CDWs in this material, propagating separately in the non equivalent (Si1-Pt2-Si1) and (Pt1-Si2-Pt1) layers respectively, with transition temperatures T$_{CDW-1}$ = 230 K and T$_{CDW-2}$ = 110 K. A strong $q$-dependence of the electron-phonon coupling has been identified as the driving mechanism for the CDW transition at T$_{CDW-1}$ = 230 K while a CDW with 3-dimensional character, and Fermi surface quasi-nesting as a driving mechanism, is suggested for the transition at T$_{CDW-2}$ = 110 K. Our results clarify some aspects of the CDW transition in LaPt$_2$Si$_2$, which have been so far misinterpreted by both theoretical predictions and experimental observations, and give direct insight into its actual temperature dependence.
A softening of phonon-dispersion has been observed experimentally in under-doped cuprate superconductors at the charge-density wave (CDW) ordering wave vector. Interestingly, the softening occurs below the superconducting (SC) transition temperature T$_{c}$, in contrast to the metallic systems, where the softening occurs usually below the CDW onset temperature T$_{\text{CDW}}$. An understanding of the `anomalous' nature of the phonon-softening and its connection to the pseudo-gap phase in under-doped cuprates remain open questions. Within a perturbative approach, we show that a complex interpl…
Unique Charge-density-wave Order and Fluctuation-induced Phonon Softening in the Optimally-doped Trilayer Cuprate Bi$_2$Sr$_2$Ca$_2$Cu$_3$O$_{10+δ}$
University of Hyogo（兵库大学）；National Institutes for Quantum Science and Technology (QST)（量子科学技术研究开发机构）；National Synchrotron Radiation Research Center（国家同步辐射研究中心）；Institute of Multidisciplinary Research for Advanced Materials (IMRAM), Tohoku University（东北大学先进物质多学科研究所）；University of Tokyo（东京大学）；National Institute of Advanced Industrial Science and Technology（产业技术综合研究所）
BaNi$_2$As$_2$ is a nonmagnetic analogue of the iron pnictide superconductors, and in its tetragonal state exhibits an incommensurate charge-density-wave (IC-CDW) and a sizable elastoresistance. In this work, phonons in BaNi$_2$As$_2$ associated with the IC-CDW and uniform in-plane lattice distortions are investigated using high-resolution inelastic X-ray scattering. The in-plane transverse acoustic phonons reveal no softening at temperatures where the elastoresistance increases strongly, indicating the latter to be electronically driven. Systematic phonon measurements reveal the IC-CDW occurs…
We present a high energy-resolution inelastic x-ray scattering data investigation of the charge-density-wave (CDW) soft phonon mode upon entering the superconducting state in $2H$-NbSe$_2$. Measurements were done close to the CDW ordering wavevector $\mathbf{q}_{CDW}$ at $\mathbf{q}=\mathbf{q}_{CDW}+(0,0,l)$,$0.15\leq l \leq 0.5$, for $T=10\,\rm{K}$ (CDW order) and $3.8\,\rm{K}$ (CDW order + superconductivity). We observe changes of the phonon lineshape that are characteristic for systems with strong electron-phonon coupling in the presence of a superconducting energy gap $2Δ_c$ and from which…
We use diffuse and inelastic x-ray scattering to study the formation of an incommensurate charge-density-wave (I-CDW) in BaNi$_2$As$_2$, a candidate system for charge-driven electronic nematicity. Intense diffuse scattering is observed around the modulation vector of the I-CDW, $Q_{I-CDW}$. It is already visible at room temperature and collapses into superstructure reflections in the long-range ordered state where a small orthorhombic distortion occurs. A clear dip in the dispersion of a low-energy transverse optical phonon mode is observed around $Q_{I-CDW}$. The phonon continuously softens u…
We investigate carrier and collective mode dynamics in 2H-NbSe$_2$ using time-resolved optical pump-probe spectroscopy and compare the results with first-principle calculations. Broadband ultrafast reflectivity studies of 2H-NbSe$_2$ in a wide temperature interval covering the normal, charge density wave (CDW) and superconducting phase were performed. Spectral features observed in the transient reflectivity experiment were associated with specific optical transitions obtained from band structure calculations. Displacive excitation of coherent phonons showed CDW-associated coherent oscillations…
The highly distinctive phonon self-energy dependences on the wave vector, calculated respectively for the static and dynamic stripe phase models of the underdoped cuprates are presented. The negative values of the real part of the on-stripe holes polarization loop lead to appearance of localized vibration states in the presence of well-separated stripes, relevant for underdoped cuprates. The localized modes split below the bare optical phonon frequencies exhibiting strong ``softening effect''. The calculated gap between localized and propagating phonon frequencies is practically doping indepen…
Soft Phonon Charge-Density Wave Formation in the Kagome Metal KV$_3$Sb$_5$
Zhejiang University（浙江大学）；Hangzhou Normal University（杭州师范大学）；University of Science and Technology of China（中国科学技术大学）；Zhejiang University of Technology（浙江工业大学）；Hefei National Research Center for Physical Sciences at the Microscale, University of Science and Technology of China（中国科学院微观尺度物理科学合肥国家研究中心，中国科学技术大学）
Charge-density-wave (CDW) order has long been interpreted as arising from a Fermi-surface instability in the parent metallic phase. While phonon-electron coupling has been suggested to influence the formation of CDW order in quasi-two-dimensional (quasi-2D) systems, the presumed dominant importance of Fermi-surface nesting remains largely unquestioned in quasi-1D systems. Here we show that phonon-electron coupling is also important for the CDW formation in a model quasi-1D system ZrTe$_3$. Our joint experimental and computational study reveals that particular lattice vibrational patterns posse…
The combination of non-trivial band topology and symmetry breaking phases gives rise to novel quantum states and phenomena such as topological superconductivity, quantum anomalous Hall effect and axion electrodynamics. Evidence of intertwined charge density wave (CDW) and superconducting order parameters has recently been observed in a novel kagome material AV3Sb5 (A=K,Rb,Cs) that features a Z2 topological invariant in the electronic structure. However, the origin of the CDW and its intricate interplay with topological state has yet to be determined. Here, using hard x-ray scattering, we demon…
Polarized Raman scattering measurements have been performed on Na0.5CoO2 single crystal from 8 to 305 K. Both the A1g and E1g phonon modes show a softening below Tc1 ~ 83 K. Additionally, the A1g phonon mode, which is forbidden in the scattering geometry of cross polarization for the triangular CoO2 layers, appears below Tc1. In contrast, the metal-insulator transition at Tc2 ~ 46 K has only secondary effect on the Raman spectra. The phonon softening and the ``forbidden'' Raman intensity follow closely magnetic order parameter and the gap function at the Fermi surface, indicating that the dist…
In this work we present the results of an inelastic x-ray scattering experiment detailing the behavior of the transverse acoustic [110] phonon in BaFe2As2 as a function of temperature. When cooling through the structural transition temperature, the transverse acoustic phonon energy is reduced from the value at room temperature, reaching a maximum shift near inelastic momentum transfer q=0.1. This softening of the lattice results in a change of the symmetry from tetragonal to orthorhombic at the same temperature as the transition to long-range antiferromagnetic order. While the lattice distorti…
While charge density wave (CDW) instabilities are ubiquitous to superconducting cuprates, the different ordering wavevectors in various cuprate families have hampered a unified description of the CDW formation mechanism. Here we investigate the temperature dependence of the low energy phonons in the canonical CDW ordered cuprate La$_{1.875}$Ba$_{0.125}$CuO$_{4}$. We discover that the phonon softening wavevector associated with CDW correlations becomes temperature dependent in the high-temperature precursor phase and changes from a wavevector of 0.238 reciprocal space units (r.l.u.) below the o…
Phonons in nearly optimally doped HgBa$_2$CuO$_{4+δ}$ were studied by inelastic X-ray scattering. The dispersion of the low energy modes is well described by a shell model, while the Cu-O bond stretching mode at high energy shows strong softening towards the zone boundary, which deviates strongly from the model. This seems to be common in the hole-doped high-$T_\mathrm{c}$ superconducting cuprates, and, based on this work, not related to a lattice distortion specific to each material.
The Holstein Model (HM) describes the interaction between fermions and a collection of local (dispersionless) phonon modes. In the dilute limit, the phonon degrees of freedom dress the fermions, giving rise to polaron and bipolaron formation. At higher densities, the phonons mediate collective superconducting (SC) and charge density wave (CDW) phases. Quantum Monte Carlo (QMC) simulations have considered both these limits, but have not yet focused on the physics of more general phonon spectra. Here we report QMC studies of the role of phonon dispersion on SC and CDW order in such models. We qu…
Monoclinic crystal structure of the undoped BaBiO3 can be described as a cubic perovskite distorted by a frozen breathing and tilting phonon modes of BiO6 octahedra. The phonon mode softening is experimentally observed [M. Braden et al., Europhysics Letters (EPL) 34, 531 (1996)] in Ba1-xKxBiO3 through potassium doping followed by a transition into an ideal cubic perovskite structure at x = 0.37 close to the appearance of superconductivity. In our previous paper [D. Korotin et al., Journal of Physics: Condensed Matter 24, 415603 (2012)] we demonstrated that it is necessary to take into account …
The Kagome superconductors AV3Sb5 (A=K, Rb, Cs) have received enormous attention due to their nontrivial topological electronic structure, anomalous physical properties and superconductivity. Unconventional charge density wave (CDW) has been detected in AV3Sb5. High-precision electronic structure determination is essential to understand its origin. Here we unveil electronic nature of the CDW phase in our high-resolution angle-resolved photoemission measurements on KV3Sb5. We have observed CDW-induced Fermi surface reconstruction and the associated band folding. The CDW-induced band splitting a…
We report the first measurement of the optical phonon dispersion in optimally doped single layer Bi2Sr1.6La0.4Cu2O6+delta using inelastic x-ray scattering. We found a strong softening of the Cu-O bond stretching phonon at about q=(0.25,0,0) from 76 to 60 meV, similar to the one reported in other cuprates. A direct comparison with angle-resolved photoemission spectroscopy measurements taken on the same sample, revealed an excellent agreement in terms of energy and momentum between the ARPES nodal kink and the soft part of the bond stretching phonon. Indeed, we find that the momentum space where…
Quasiparticle dynamics of FeSe single crystals revealed by dual-color transient reflectivity measurements (ΔR/R) provides unprecedented information on Fe-based superconductors. The amplitude of fast component in ΔR/R clearly tells a competing scenario between spin fluctuations and superconductivity. Together with the transport measurements, the relaxation time analysis further exhibits anomalous changes at 90 K and 230 K. The former manifests a structure phase transition as well as the associated phonon softening. The latter suggests a previously overlooked phase transition or crossover in FeS…
We have studied the temperature dependence of the optical spectra for the CDW gap and optical phonons in a single crystal of BaBiO_3 and a film deposited by laser ablation. The integrated optical conductivity for optical phonons decreases rapidly with increasing temperature above 150 K, while the single-particle excitation across the CDW gap remains constant.
Fermi surface fluctuations and lattice instabilities in the 2D metallic kagome superconductor CsV$_3$Sb$_5$ are elucidated via polarization-resolved Raman spectroscopy. The presence of a weak electronic continuum in high-quality samples marks the cross-over into the charge-density-wave (CDW) ordered phase, while impurity-rich samples promote strong defect-induced electronic scattering processes that affect the coherence of the CDW phase. CDW-induced phonon anomalies appear below $T_{\mathrm{CDW}}$, with emergent $C2$ symmetry for one of the CDW amplitude modes, alluding to nematicity. In conju…
The cuprates exhibit a prominent charge-density-wave (CDW) instability with wavevector along [100], i.e., the Cu-O bond direction. Whereas CDW order is most prominent at moderate doping and low temperature, there exists increasing evidence for dynamic charge correlations throughout a large portion of the temperature-doping phase diagram. In particular, signatures of incipient charge order have been observed as phonon softening and/or broadening near the CDW wavevector approximately half-way through the Brillouin zone. Most of this work has focused on moderately-doped cuprates, for which the CD…
The electron-phonon interaction is a major factor influencing the competition between collective instabilities in correlated-electron materials, but its role in driving high-temperature superconductivity in the cuprates remains poorly understood. We have used high-resolution inelastic x-ray scattering to monitor low-energy phonons in YBa$_2$Cu$_3$O$_{6.6}$ (superconducting $\bf T_c = 61$ K), which is close to a charge density wave (CDW) instability. Phonons in a narrow range of momentum space around the CDW ordering vector exhibit extremely large superconductivity-induced lineshape renormaliza…
Charge-density-wave (CDW) instability and pressure-induced superconductivity in bulk 1T-NbS2 are predicted theoretically by first-principles calculations. We reveal a CDW instability towards the formation of a stable commensurate CDW order, resulting in a sqart(13)*sqart(13) structural reconstruction featured with star-of-David clusters. The CDW phase exhibits one-dimensional metallic behavior with in-plane flat-band characteristics, and coexists with an orbital-density-wave order predominantly contributed by 4d_(z^2-r^2 ) orbital from the inner Nb atoms of the star-of-David cluster. By doubli…
Phonon softening is a ubiquitous phenomenon in condensed matter systems which is often associated with charge density wave (CDW) instabilities and anharmonicity. The interplay between phonon softening, CDW and superconductivity is a topic of intense debate. In this work, the effects of anomalous soft phonon instabilities on superconductivity are studied based on a recently developed theoretical framework that accounts for phonon damping and softening within the Migdal-Eliashberg theory. Model calculations show that the phonon softening in the form of a sharp dip in the phonon dispersion relati…
We observe strong softening of optical phonon modes in superconducting (Tc = 4.2 K) boron doped diamond near the Brillouin zone center using inelastic x-ray scattering from a CVD-grown highly oriented sample. The magnitude of the softening, and our observation that it becomes stronger approaching zone center, supports theoretical models suggesting a phonon-mediated pairing mechanism via coupling of optical phonon modes to Fermi surfaces around the zone center. The electron-phonon coupling parameter is determined as approximately lambda= 0.33.
The intriguing coexistence of the charge density wave (CDW) and superconductivity in SrPt$_2$As$_2$ and LaPt$_2$Si$_2$ has been investigated based on the {\it ab initio} density functional theory band structure and phonon calculations. We have found that the local split distortions Pt atoms in the [As-Pt-As] layers play an essential role in driving the five-fold supercell CDW instability as well as the phonon softening instability in SrPt$_2$As$_2$. By contrast, the CDW and phonon softening instabilities in LaPt$_2$Si$_2$ occur without split distortions of Pt atoms, indicating that the driving…
Inelastic x-ray scattering is used to investigate charge density wave (CDW) formation and the low-energy lattice dynamics of the underdoped high temperature superconductor ortho-II YBa2Cu3O6.54. We find that, for a temperature ~1/3 of the CDW onset temperature (~155 K), the CDW order is static within the resolution of the experiment, that is the inverse lifetime is less than 0.3 meV. In the same temperature region, low-energy phonons near the ordering wavevector of the CDW show large increases in their linewidths. This contrasts with the usual behavior in CDW systems where the phonon anomalies…
Determinant quantum Monte Carlo (DQMC) simulations are used to study non-linear electron-phonon interactions in a two-dimensional Holstein-like model on a square lattice. We examine the impact of non-linear electron-lattice interactions on superconductivity and on Peierls charge-density-wave (CDW) correlations at finite temperatures and carrier concentrations. We find that the CDW correlations are dramatically suppressed with the inclusion of even a small non-linear interaction. Conversely, the effect of the non-linearity on superconductivity is found to be less dramatic at high temperatures; …
Mode-selective electron-phonon coupling drives charge density waves in the kagome metals YRu$_3$Si$_2$ and LaRu$_3$Si$_2$
Key Laboratory of Materials Physics, Institute of Solid State Physics, HFIPS, Chinese Academy of Sciences（中国科学院合肥物质科学研究院固体物理研究所材料物理重点实验室）；University of Science and Technology of China（中国科学技术大学）；High Magnetic Field Laboratory, HFIPS, Chinese Academy of Sciences（中国科学院合肥物质科学研究院高磁场实验室）；Collaborative Innovation Center of Microstructures, Nanjing University（南京大学微结构协同创新中心）
The dispersion of phonons in Ba1-xKxBiO3 along the (3+q, 0, 0) direction in reciprocal space was determined for x=0, 0.30, 0.37, 0.52 using high-resolution inelastic X-ray scattering. The observed phonon energies near Γwere in good agreement with published optical data. It was found that two high-energy modes strongly soften near q=0.25 when the system becomes metallic (x >0.35). There was no softening in the insulating phase, even at x=0.30, very near the metal-insulator transition. This suggests that the phonon softening is linked to the appearance of the Fermi-surface.
Charge density wave (CDW) correlations are prevalent in all copper-oxide superconductors. While CDWs in conventional metals are driven by coupling between lattice vibrations and electrons, the role of the electron-phonon coupling (EPC) in cuprate CDWs is strongly debated. Using Cu $L_3$ edge resonant inelastic x-ray scattering (RIXS), we study the CDW and Cu-O bond-stretching phonons in the stripe-ordered cuprate La$_{1.8-x}$Eu$_{0.2}$Sr$_{x}$CuO$_{4+δ}$. We investigate the interplay between charge order and EPC as a function of doping and temperature, and find that the EPC is enhanced in a na…
Monolayer 2H-NbSe2 has recently been shown to be a 2-dimensional superconductor, with a coexisting charge-density wave (CDW). As both phenomena are intimately related to electron-lattice interaction, a natural question is how superconductivity and CDW are interrelated through electron-phonon coupling (EPC), which is important to the understanding of 2-dimensional superconductivity. This work investigates the superconductivity of monolayer NbSe2 in CDW phase using the anisotropic Migdal-Eliashberg formalism based on first principles calculations. The mechanism of the competition between and coe…
Experimental evidence on high-Tc cuprates reveals ubiquitous charge density wave (CDW) modulations, which coexist with superconductivity. Although the CDW had been predicted by theory, important questions remain about the extent to which the CDW influences lattice and charge degrees of freedom and its characteristics as functions of doping and temperature. These questions are intimately connected to the origin of the CDW and its relation to the mysterious cuprate pseudogap. Here, we use ultrahigh resolution resonant inelastic x-ray scattering (RIXS) to reveal new CDW character in underdoped Bi…
The discovery of charge-density wave (CDW)-related effects in the resonant inelastic x-ray scattering (RIXS) spectra of cuprates holds the tantalizing promise of clarifying the interactions that stabilize the electronic order. Here, we report a comprehensive RIXS study of La2-xSrxCuO4 (LSCO) finding that CDW effects persist up to a remarkably high doping level of x = 0.21 before disappearing at x = 0.25. The inelastic excitation spectra remain essentially unchanged with doping despite crossing a topological transition in the Fermi surface. This indicates that the spectra contain little or no d…
Lattice-charge coupling in a trilayer nickelate with intertwined density wave order
Argonne National Laboratory（阿贡国家实验室）；Institute of High Energy Physics, Chinese Academy of Sciences（中国科学院高能物理研究所）；Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所）；Brookhaven National Laboratory（布鲁克海文国家实验室）；Center for Computational Quantum Physics, Flatiron Institute（平顿研究院计算量子物理中心）；Arizona State University（亚利桑那州立大学）；Southeast University（东南大学）；Shandong University（山东大学）；Northwestern University（西北大学）
We report optical spectra of Lu$_5$Ir$_4$Si$_{10}$ and Er$_5$Ir$_4$Si$_{10}$, exhibiting the phenomenon of coexisting superconductivity or antiferromagnetism and charge density wave (CDW) order. We measure the maximum value of the charge density wave gap present on part of the Fermi surface of Lu5Ir4Si10, corresponding to a ratio 2Δ/k_B T_CDW \approx 10, well above the value in the limit of weak electron-phonon coupling. Strong electron-phonon coupling was confirmed by analyzing the optical conductivity with the Holstein model describing the electron-phonon interactions, indicating the couplin…
Anharmonic Strong-Coupling Effects at the Origin of the Charge Density Wave in CsV$_3$Sb$_5$
Stanford University（斯坦福大学）；SLAC National Accelerator Laboratory and Stanford University（SLAC国家加速器实验室和斯坦福大学）；Chinese Academy of Sciences（中国科学院）；University of Chinese Academy of Sciences（中国科学院大学）
The effects of phosphorus doping on the structural and superconducting phase transitions of BaNi$_2$(As$_{1-x}$P$_x$)$_2$ were studied. The specific heat, resistivity, and magnetic susceptibility were measured. The results revealed an abrupt increase in the superconducting transition temperature ($T_c$) from 0.6 K in the triclinic phase (space group P$\bar{1}$) with less phosphorus (0 $\leq$ $x$ $\leq$ 0.067) to 3.3 K in the tetragonal phase (space group I4/mmm) with more phosphorus ($x$ $\geq$ 0.067). Our data analysis suggests that a doping-induced softening related to an in-plane Ni and As(…
Enhanced Phonon-Phonon Interactions and Weakened Electron-Phonon Coupling in Charge-Density-Wave Topological Semimetal EuAl4 with a Possible Intermediate Electronic State
We report measurements of the phonon density-of-states in iron oxypnictide superconductors by inelastic x-ray scattering. A good agreement with ab-initio calculations that do not take into account strong electronic correlations is found, and an unpredicted softening of phonon branches under F doping of these compounds is observed. Raman scattering experiments lead us to conclude that this softening is not related to zone center phonons, and consequently imply an important softening of the relevant phonon branches at finite momentum transfer Q.
The effects of copper doping on the structural and superconducting phase transitions of Ba(Ni$_{1-x}$Cu$_{x}$)$_{2}$As$_{2}$ were studied by examining the resistivity, magnetic susceptibility, and specific heat. We found an abrupt increase in the superconducting transition temperature $T_{\rm c}$ from 0.6 K in the triclinic phase with less copper ($x$ $\leq$ 0.16) to 2.5-3.2 K in the tetragonal phase with more copper ($x$ $>$ 0.16). The specific-heat data suggested that doping-induced phonon softening was responsible for the enhanced superconductivity in the tetragonal phase. All of these obse…
Phonon fluctuation diagnostics: Origin of charge order in AV$_3$Sb$_5$ kagome metals
Universität Würzburg（维尔茨堡大学）；University of Bremen（不来梅大学）；University of Hamburg（汉堡大学）；Massachusetts Institute of Technology（麻省理工学院）；University of Bologna（博洛尼亚大学）
The lattice dynamics and the effect of pressure on superconducting LiFeAs in both nonmagnetic (NM) and striped antiferromagnetic (SAF) phases are investigated using the plane-wave pseudopotential, density-functional-based method. While the obtained electron-phonon coupling $λ$ is very small for the NM calculation, the softening of phonon in the SAF phase may lead to a large increase in $λ$. In the SAF phase, strong anisotropy of the phonon softening in the Fe plane is found to arise from different spin orders in the $x$ and $y$ directions, indicating that the phonon softening is of spin-phonon…
Analogous to 2D layered transition metal dichalcogenides, the TlSe family of 1D chain materials with Zintl-type structure exhibits exotic phenomena under high-pressure. In the present work, we have systematically investigated the high-pressure behavior of TlInTe 2 using Raman spectroscopy, synchrotron X-ray diffraction, and transport measurements, in combination with crystal structure prediction (CSP) based on the evolutionary approach and first principles calculations. We found that TlInTe$_2$ undergoes a pressure driven semiconductor to semimetal transition at 4 GPa, followed by a supercondu…
We present Raman measurements on the iron-pnictide superconductors CeFeAsO_{1-x}F_{x} and NdFeAsO{1-x}F_{x}. Modeling the Fe-As plane in terms of harmonic and a cubic anharmonic Fe-As interaction we calculate the temperature dependence of the energy and lifetime of the Raman active Fe B_{1g} mode and fit to the observed energy shift. The shifts and lifetimes are in good agreement with those measured also in other Raman studies which demonstrate that the phonon spectrum is well represented by phonon-phonon interactions without any significant electronic contribution. We also estimate the anharm…
We report inelastic neutron scattering measurements of the phonon density-of-states in superconducting Sr0.6K0.4Fe2As2 (Tc=30 K) and Ca0.6Na0.4Fe2As2 (Tc=18 K). Compared with the parent compound BaFe2As2 doping affects mainly the lower and intermediate frequency part of the vibrations. Mass effects and lattice contraction cannot explain these changes. A lattice dynamical model has been used to identify the character of the various phonon bands. Softening of phonon modes below 10 meV has been observed in both samples on cooling from 300 K to 140 K. In the Ca doped compound the softening amounts…
