Origin of superconductivity and giant phonon softening in TlInTe$_2$ under pressure
Sorb Yesudhas, N. Yedukondalu, Manoj K. Jana, Jianbo Zhang, Jie Huang, Bijuan Chen, Hongshang Deng, Raimundas Sereika, Hong Xiao, Stanislav Sinogeikin, Curtis K… 展开作者
Sorb Yesudhas, N. Yedukondalu, Manoj K. Jana, Jianbo Zhang, Jie Huang, Bijuan Chen, Hongshang Deng, Raimundas Sereika, Hong Xiao, Stanislav Sinogeikin, Curtis Kenney-Benson, Kanishka Biswas, John B. Parise, Yang Ding, Ho-kwang Mao
备注：
43 pages (27 pages for main text and 16 pages for SI), 20 figures (5 in main text and 15 in SI)
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 superconducting transition at 5.7 GPa (with Tc = 3.8 K) induced by a Lifshitz transition. The Lifshitz transition is initiated by the appearance of new electron pockets on the Fermi surface, which evolve with pressure and connect to the adjacent electron pockets forming an umbrella shaped Fermi surface at the top and bottom of the Brillouin zone. An unusual giant phonon softening (Ag mode) concomitant with a V-shaped Tc behavior appears at 10-12 GPa as a result of the interaction of optical phonons with the conduction electrons, resulting in Fano line shaped asymmetry in Ag mode. A prominent Tc anomaly concurrent with the Ag mode softening at 19-20 GPa is correlated to the semimetal to metal transition. The CSP calculations reveal that these transitions are not accompanied by any structural phase transitions up to the maximum pressure achieved, 33.5 GPa. Our findings on TlInTe$_2$ open up a new platform to study a plethora of unexplored high pressure novel phenomena in TlSe family induced by Lifshitz transition (electronic driven), phonon softening and electron-phonon coupling.
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(…
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…
Electron-phonon superconductors at high pressures have displayed the highest values of critical superconducting temperature $T_c$ on record, now rapidly approaching room temperature. Despite the importance of high-$P$ superconductivity in the quest for room-temperature superconductors, a mechanistic understanding of the effect of pressure and its complex interplay with phonon anharmonicity and superconductivity is missing, as numerical simulations can only bring system-specific details clouding out key players controlling the physics. Here we develop a minimal model of electron-phonon supercon…
Experimental evidence for a transient enhancement of the superconducting critical temperature in presence of an intense THz or IR pump pulse was ascribed to nonlinear phononics effects. Here I introduce a simple phenomenological Ginzburg-Landau model of this phenomenon, to explore further consequences and possible experimental tests of this interpretation. Upon cooling below Tc in the absence ofTHz pumping, both an abrupt softening of a Raman-active mode frequency and a spontaneous lattice distortion, growing linearly with (Tc - T), are predicted to occur. Numerical estimates for YBa2Cu3O6+x i…
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…
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…
Recently topological superconductors emerge as a platform to realise Majorana Fermions which has potential application in topological quantum computation. Superconductivity in these materials can be realised by chemical doping, high pressure or proximity effect. Herein we report the emergent superconductivity in single-crystals of Re doped type-II Weyl semimetal NiTe2. The magnetic and transport measurements highlight that Re substitution in Ni-site induces superconductivity at a maximum temperature of 2.36 K. Hall effect and specific heat measurements indicate that Re substitution is doping h…
We investigate the influence of a d-density wave (DDW) gap on the superconductivity-induced renormalization of phonon frequency and linewidth. The results are discussed with respect to Raman and inelastic neutron scattering experiments. It turns out that the DDW gap can enhance the range of frequencies for $q=0$ phonon softening depending on the underlying band structure. Moreover we show that an anisotropic 'd-wave' pseudogap can also contribute to the q-dependent linewidth broadening of the 340cm$^{-1}$ phonon in YBa$_2$Cu$_3$O$_7$.
We develop a theory for light-induced superconductivity in underdoped cuprates in which the competing bond-density wave order is suppressed by driving phonons with light. Close to a bond-density wave instability in a system with a small Fermi surface, such as a fractionalized Fermi liquid, we show that the coupling of electrons to phonons is strongly enhanced at the bond-density wave ordering wavevectors, leading to a strong softening of phonons at these wavevectors. For a model of classical phonons with anharmonic couplings, we show that the combination of strong softening and driving can lea…
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 physical properties and structural evolution of the 122-type solid solution BaNi2(Ge1-xPx)2 are reported. The in-plane X-X (X = Ge1-xPx) dimer formation present in the end member BaNi2Ge2, which results in a structural transition to orthorhombic symmetry, is completely suppressed to zero temperature on P substitution near x = 0.7, and a dome-shape superconducting phase with a maximum Tc = 2.9 K emerges. Clear indications of phonon softening and enhanced electron-phonon coupling are observed at the composition of the structural instability. Our findings show that dimer breaking offers new p…
The dispersion of the in-plane Cu-O bond-stretching LO phonon mode in the high-$T_{C}$ superconducting cuprates shows strong softening with doping near the zone boundary. We suggest that it can be described with a negative electronic dielectric function that results in overscreening of inter-site Coulomb interaction due to phonon-induced charge transfer and vibronic electron-phonon resonance. We propose that such a strong electron-phonon coupling of specific modes can form a basis for the phonon mechanism of high-temperature superconductivity. With the Eliashberg theory using the experimentall…
By realizing in thin films a tensile stress state, superconductivity of 13 K was introduced into FeTe, an non-superconducting parent compound of the iron pnictides and chalcogenides, with transition temperature higher than that of its superconducting isostructural counterpart FeSe. For these tensile stressed films, the superconductivity is accompanied by the softening of the first-order magnetic and structural phase transition; and also, the in-plane extension and out-of-plane contraction are universal in all FeTe films independent of sign of lattice mismatch, either positive or negative. More…
The main features in iron-pnictide superconductors are summarized as (i) the orthorhombic transition accompanied by remarkable softening of shear modulus, (ii) high-Tc superconductivity close to the orthorhombic phase, and (iii) stripe-type magnetic order induced by orthorhombicity. To present a unified explanation for them, we analyze the multiorbital Hubbard-Holstein model with Fe-ion optical phonons based on the orbital fluctuation theory. In the random-phase-approximation (RPA), a small electron-phonon coupling constant ($λ~ 0.2$) is enough to produce large orbital (=charge quadrupole) flu…
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…
Quantum phase transitions play an important role in shaping the phase diagram of high-temperature cuprate superconductors. These cuprates possess intertwined orders which interact strongly with superconductivity. However, the evidence for the quantum critical point associated with the charge order in the superconducting phase remains elusive. Here we show the short-range charge orders and the spectral signature of the quantum fluctuations in La$_{2-x}$Sr$_x$CuO$_4$ (LSCO) near the optimal doping using high-resolution resonant inelastic X-ray scattering. On performing calculations through a dia…
Due to competing long range ferromagnetic order, the transition metals Fe, Co and Ni are not superconductors at ambient pressure. While superconductivity was observed in a non-magnetic phase of Fe, stabilized under pressure, it is yet to be discovered in Co and Ni under any experimental conditions. Here, we report emergence of superconductivity in the recently discovered high-density nonmagnetic face centered cubic phase in Co thin films below a transition temperature (Tc) of ~5.4 K, as revealed in experiments based on point-contact spectroscopy and resistance, and four-probe measurements of r…
We investigate the 16-band d-p model for iron pnictide superconductors in the presence of the electron-phonon coupling g with the orthorhombic mode which is crucial for reproducing the recently observed ultrasonic softening. Within the RPA, we obtain the ferro-orbital order below TQ which induces the tetragonal-orthorhombic structural transition at Ts = TQ, together with the stripe-type antiferromagnetic order below TN. Near the phase transitions, the system shows the s++ wave superconductivity due to the orbital fluctuation for a large g case with TQ > TN, while the s+- wave due to the magnet…
Extreme pressure strongly affects the superconducting properties of ``simple'' elemental metals, like Li, K and Al. Pressure induces superconductivity in Li (as high as 17 K), while suppressing it in Al. We report first-principles investigations of the superconducting properties of dense Li, K and Al based on a recently proposed, parameter-free, method. Our results show an unprecedented agreement with experiments, assess the predictive power of the method over a wide range of densities and electron-phonon couplings, and provide predictions for K, where no experiments exist so far. More importa…
The electronic states and superconductivity in iron pnictides are studied on the basis of the 16 band $d$-$p$ model which includes both the onsite Coulomb interaction between Fe $d$ electrons and the intersite one between Fe $d$ and pnictogen $p$ electrons. The model well accounts for experimentally observed two fluctuations: the $d$-$d$ interaction-enhanced antiferromagnetic (AFM) fluctuation and the $d$-$p$ interaction-enhanced ferro-orbital (FO) fluctuation responsible for the $C_{66}$ elastic softening. The AFM fluctuation induces the repulsive pairing interaction for $\bm{q}\sim \bm{Q}_{\…
Using the full potential linear muffin-tin orbitals (FP-LMTO) method we examine the pressure-dependence of superconductivity in the two metallic phases of Boron: bct and fcc. Linear response calculations are carried out to examine the phonon frequencies and electron-phonon coupling for various lattice parameters, and superconducting transition temperatures are obtained from the Eliashberg equation. In both bct and fcc phases the superconducting transition temperature T$_c$ is found to decrease with increasing pressure, due to stiffening of phonons with an accompanying decrease in electron-phon…
Recently, Bi-based compounds have attracted attentions because of the strong spin-orbit coupling (SOC). In this work, we figured out the role of SOC in ABi$_{3}$ (A=Sr and Ba) by theoretical investigation of the band structures, phonon properties, and electron-phonon coupling. Without SOC, strong Fermi surface nesting leads to phonon instabilities in ABi$_{3}$. SOC suppresses the nesting and stabilizes the structure. Moreover, without SOC the calculation largely underestimates the superconducting transition temperatures ($T_{c}$), while with SOC the calculated $T_{c}$ are very close to those d…
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…
Experimental evidence suggests that superconductivity in SrTiO$_3$ is mediated by a soft transverse ferroelectric mode which, according to conventional theories, has negligible coupling with electrons. A phenomenological Rashba type coupling has been proposed on symmetry arguments but a microscopic derivation is lacking. Here we fill this gap and obtain a linear coupling directly from a minimal microscopic model of the electronic structure. We find that the effective electron-electron pairing interaction has a strong momentum dependence. This yields an unusual situation in which the leading $s…
Eliashberg's foundational theory of superconductivity is based on the application of Migdal's approximation, which states that vertex corrections to first order electron-phonon scattering are negligible if the ratio between phonon and electron energy scales is small. The resulting theory incorporates the first Feynman diagrams for electron and phonon self-energies. However, the latter is most commonly neglected in numerical analyses. Here we provide an extensive study of full-bandwidth Eliashberg theory in two and three dimensions, where we include the full back reaction of electrons onto the …
Superconductivity of transition metal dichalcogenide $1T$-TiTe$_2$ under high pressure was investigated by the first-principles calculations. Our results show that the superconductivity of $1T$-TiTe$_2$ exhibits very different behavior under the hydrostatic and uniaxial pressure. The hydrostatic pressure is harmful to the superconductivity, while the uniaxial pressure is beneficial to the superconductivity. Superconducting transition temperature $T_C$ at ambient pressure is 0.73 K, and it reduces monotonously under the hydrostatic pressure to 0.32 K at 30 GPa. While the $T_C$ increases dramati…
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）
The superconducting transition temperature (TC) in nanostructured Pb remains nearly constant as the particle size is reduced from 65 to 7nm, below which size the superconductivity is lost rather abruptly. In contrast, there is a large enhancement in the upper critical field (HC2) in the same size regime. We explore the origin of the unusual robustness of the TC over such a large particle size range in nanostructured Pb, by measuring the temperature dependence of the superconducting energy gap in planar tunnel junctions of Al/Al2O3/nano-Pb. We show that below 22nm, the electron phonon coupling …
We report a record high superconducting transition temperature ($T_{\rm c}$) up to 23.6 K under high pressure in the elemental metal Ti, one of the top ten most abundant elements in Earth's crust. The $T_{\rm c}$ increases monotonically from 2.3 K at 40.3 GPa to 23.6 K at 144.9 GPa, which surpasses all known records from elemental metals reported so far. With further compression, a robust $T_{\rm c}$ of ~23 K is observed between 144.9 and 183 GPa in the $δ$-Ti phase. The pressure-dependent $T_{\rm c}$ can be well described by the conventional electron-phonon coupling (EPC) mechanism. Density F…
The quest to govern the driving forces behind superconductivity and gain control over the superconducting transition temperature $T_{c}$ is as old as the phenomenon itself. Microscopically, this requires a proper understanding of the evolution of electron-lattice interactions in their parameter space. We report such a controlled study on $T_{c}$ in In$_{x}$Te via fine-tuning the In stoichiometry $x$. We find that increasing $x$ from 0.84 to 1 results in an enhancement of $T_{c}$ from 1.3 K to 3.5 K accompanied by an increase of the electron-phonon coupling constant from 0.45 to 0.63. Employing…
We have observed a softening of phonons and a structural phase transition in a superconducting Ba0.59K0.41BiO3 (Tc = 31 K) single crystal using elastic and inelastic neutron scattering measurements. The soft phonon occurs for the [111] transverse acoustic mode at the zone boundary. The phonon energies in this vicinity are found to continuously decrease with decreasing temperature from above room temperature to 200 K, where a structural phase transition from cubic to tetragonal symmetry occurs. The overall results are consistent with previous data that reported phonon softening and a (0.5, 0.5,…
The maximum critical temperature for superconductivity in pressurized hydrides appears at the top of superconducting domes in Tc versus pressure curves at a particular pressure, which is not predicted by standard superconductivity theories. The a high-order anisotropic van Hove singularity near the Fermi level observed in band structure calculations of pressurized sulfur hydride, typical of a supermetal, has been associated with the array of metallic hydrogen wires modules forming a nanoscale heterostructure at atomic limit called superstripes phase. Here we propose that pressurized sulfur hyd…
Structural and superconducting transitions of layered van der Waals (vdW) hydrogenated germanene (GeH) were observed under high-pressure compression and decompression processes. GeH possesses a superconducting transition at critical temperature (Tc) of 5.41 K at 8.39 GPa. A crystalline to amorphous transition occurs at 16.80 GPa while superconductivity remains. An abnormally increased Tc up to 6.1 K has been observed in the decompression process while the GeH remained amorphous. Thorough in-situ high-pressure synchrotron X-ray diffraction and in-situ high-pressure Raman spectroscopy with the d…
Laser control of solids was so far mainly discussed in the context of strong classical nonlinear light-matter coupling in a pump-probe framework. Here we propose a quantum-electrodynamical setting to address the coupling of a low-dimensional quantum material to quantized electromagnetic fields in quantum cavities. Using a protoypical model system describing FeSe/SrTiO$_3$ with electron-phonon long-range forward scattering, we study how the formation of phonon polaritons at the 2D interface of the material modifies effective couplings and superconducting properties in a Migdal-Eliashberg simula…
We performed temperature- and doping-dependent high-resolution Raman spectroscopy experiments on YBa$_2$Cu$_3$O$_{7-δ}$ to study $B$$_{\rm 1g}$ phonons. The temperature dependence of the real part of the phonon self-energy shows a distinct kink at $T=T_{\rm B1g}$ above $T$$_{\rm c}$ due to softening, in addition to the one due to the onset of the superconductivity. $T$$_{\rm B1g}$ is clearly different from the pseudogap temperature with a maximum in the underdoped region. The region between $T$$_{\rm B1g}$ and $T$$_{\rm c}$ resembles that of superconducting fluctuation or charge density wave o…
Enhanced superconductivity in atomically thin noble metals: From quantum confinement to interface-induced Lifshitz transition
School of Physical Science and Technology, Inner Mongolia University（内蒙古大学物理科学与技术学院）；Key Laboratory of Semiconductor Photovoltaic Technology and Energy Materials at Universities of Inner Mongolia Autonomous Region, Inner Mongolia University（内蒙古自治区高校半导体光伏技术与能源材料重点实验室）；Research Center for Quantum Physics and Technologies, Inner Mongolia University（内蒙古大学量子物理与技术研究中心）；Inner Mongolia Key Laboratory of Microscale Physics and Atomic Manufacturing, Inner Mongolia University（内蒙古微尺度物理与原子制造重点实验室）
Effects of the choice of the exchange-correlation functional on high-temperature superconductivity predictions in covalent-cage M(BN)5 family
City University of Hong Kong（香港城市大学）；Shenzhen Institute of Advanced Technology, Chinese Academy of Sciences（中国科学院深圳先进技术研究院）；University of Chinese Academy of Sciences（中国科学院大学）；Pritzker School of Molecular Engineering, The University of Chicago（芝加哥大学普里茨克分子工程学院）；Shenzhen Polytechnic University（深圳职业技术大学）；Zhejiang University（浙江大学）
The structural, electronic, lattice dynamics, electron-phonon coupling, and superconducting properties of the alkali-metal hydride RbH, metalized through electron-doping by the construction of the solid-solution Rb$_{1-x}$Sr$_x$H, are systematically analyzed as a function of Sr-content within the framework of density functional perturbation and Migdal-Eliashberg theories, taking into account the effect of zero-point energy contribution by the quasi-harmonic approximation. For the entire studied range of Sr-content, steady increments of the electron-phonon coupling constant and the superconduct…
In superconducting copper oxides some Cu-O bond-stretching phonons around 70meV show anomalous giant softening and broadening of electronic origin and electronic dispersions have large renormalization kinks near the same energy. These observations suggest that phonon broadening originates from quasiparticle excitations across the Fermi surface and the electronic dispersion kinks originate from coupling to anomalous phonons. We measured the phonon anomaly in underdoped (x=0.05) and overdoped (x=0.20,0.25) La$_{2-x}$Sr$_x$CuO$_4$ by inelastic neutron and x-ray scattering with high resolution. Co…
The main features in iron-based superconductors would be (i) the orthorhombic transition accompanied by remarkable softening of shear modulus, (ii) high-Tc superconductivity close to the orthorhombic phase, and (iii) nematic transition in the tetragonal phase. In this paper, we present a unified explanation for them, based on the orbital fluctuation theory, considering both the e-ph and the Coulomb interaction. It is found that a small e-ph coupling constant ($λ~ 0.2$) is enough to produce large orbital (=charge quadrupole $O_{xz/yz}$) fluctuations, which causes the s-wave superconductivity wi…
SrPt3P has recently been reported to exhibit superconductivity with Tc = 8.4 K. To explore its superconducting mechanism, we have performed electron and phonon band calculations based on the density functional theory, and found that the superconductivity in SrPt3P is well described by the strong coupling phonon-mediated mechanism. We have demonstrated that superconducting charge carriers come from pdπ-hybridized bands between Pt and P ions, which couple to low energy (~ 5 meV) phonon modes confined on the ab in-plane. These in-plane phonon modes, which do not break antipolar nature of SrPt3P, …
Search for stable high-pressure compounds in the Ti--H system reveals the existence of titanium hydrides with new stoichiometries, including Ibam-Ti$_2$H$_5$, I4/m-Ti$_5$H$_{13}$, I$\bar{4}$-Ti$_5$H$_{14}$, Fddd-TiH$_4$, Immm-Ti$_2$H$_{13}$, P$\bar{1}$-TiH$_{12}$, and C2/m-TiH$_{22}$. Our calculations predict I4/mmm $\rightarrow$ R$\bar{3}$m and I4/mmm $\rightarrow$ Cmma transitions in TiH and TiH$_2$, respectively. Phonons and the electron--phonon coupling of all searched titanium hydrides are analyzed at high pressure. It is found that Immm-Ti$_2$H$_{13}$ rather than the highest hydrogen con…
A strong electron-phonon interaction arises from the modulation of the superexchange interaction by lattice vibrations. It is responsible for the softening of the half-breathing modes around (pm pi/a,0) and (0, pm pi/a) in the two-dimensional Brillouin zone, with a being the lattice constant of CuO_2 planes, as is studied in Phys. Rev. B70, 184514 (2004). Provided that antiferromagnetic spin fluctuations are developed around Q=(pm 3 pi/4a, pm pi/a) and (pm pi/a, pm 3 pi/4a), the electron-phonon interaction can also cause the softening of Cu-O bond stretching modes around 2Q, or around (pm pi/2…
