Pair-density-wave pseudogap and superconducting states in cuprates

Bogoliubov quasiparticle interference and localized high-energy excitations observed in cuprates in nodal and antinodal regions of the momentum space, respectively, would lead to a conclusion that only the nodal region might give rise to superconductivity whereas the antinodal one might be associated with the pseudogap. We argue that both pseudogap and superconducting states arise exactly in the antinodal region with pronounced nesting feature of the Fermi contour as spatially inhomogeneous incoherent and coherent states of pairs with large momentum. The nodal region gives rise to conventional superconducting pairing with zero momentum which, together with the pairing with large momentum in the antinodal region, forms a biordered superconducting state in the whole of the Brillouin zone. This coherent state with complicated momentum dependence of the order parameter manifests itself as a pair-density wave that can exist without any driving insulating order. We believe that quasiparticle interference, other than observed in the nodal region, should be observable in the antinodal region as well.     

高温超导体中的赝隙与超导位相涨落

高温超导体赝隙态与超导态之间的关系一直是研究的焦点.交流电导和能斯特（Nernst）效应测量相继探测到超导转变温度Tc0以上温区一定范围内存在磁通涡旋激发,利用力矩技术的磁化率测量则探测到超导涨落引起的弱抗磁性.这些发现都支持了高温超导体赝隙相中存在有限的超导序参量振幅和强烈的位相涨落的图像,说明Tc0处的相变是由库珀对之间长程位相关联的消失所驱动的.文章首先简短地介绍高温超导体的电子态相图和赝隙态,以及能斯特效应的原理和测量方法,然后对能斯特效应的测量结果作一评述性介绍,还讨论了相关的理论模型.     

Superconductivity in the pseudogap state in the hot-spot model: Ginzburg-Landau expansion

Peculiarities of the superconducting state (s and d pairing) are considered in the model of the pseudogap state induced by short-range order fluctuations of the dielectric (AFM (SDW) or CDW) type, which is based on the model of the Fermi surface with “hot spots.” A microscopic derivation of the Ginzburg-Landau expansion is given with allowance for all Feynman diagrams in perturbation theory in the electron interaction with short-range order fluctuations responsible for strong scattering in the vicinity of hot spots. The superconducting transition temperature is determined as a function of the effective pseudogap width and the correlation length of short-range order fluctuations. Similar dependences are derived for the main parameters of a superconductor in the vicinity of the superconducting transition temperature. It is shown, in particular, that the specific heat jump at the transition point is considerably suppressed upon a transition to the pseudogap region on the phase diagram.     

The “pair” Fermi contour and high-temperature superconductivity

Superconducting pairing of holes with a large (on the order of doubled Fermi) total pair momentum and small relative motion momenta is considered taking into account the quasi-two-dimensional electronic structure of high-T _c cuprates with clearly defined nesting of the Fermi contour situated in an extended neighborhood of the saddle point of the electronic dispersion law (the momentum space region with a hyperbolic metric) and the arising of a spatially inhomogeneous (stripe) structure as a result of the redistribution of current carriers (holes) that restores regions with antiferromagnetic ordering. The superconducting energy gap and condensation energy were determined, and their dependences on the doping level were qualitatively studied. The energy gap was shown to exist in some hole concentration region limited on both sides. The superconducting state with a positive condensation energy appears in a narrower range of doping within this region. The reason for the arising of the superconducting state at a repulsive screened Coulomb interaction between holes is largely the redistribution of hole pairs in the momentum space related to the special features of the hyperbolic metric, which is responsible for the formation of the “pair” Fermi contour, and the renormalization of the kinetic energy of holes when the chemical potential changes because of the condensation of pairs. Hole pairs of the type under consideration exist not only in the condensate but also in the form of quasi-stationary states with very weak decay at temperatures substantially exceeding the superconducting transition temperature. The pseudogap region of the phase diagram of high-T _c cuprates is related to such states. The pairing mechanism under consideration allows not only the principal characteristics of the phase diagram but also key experimental data on high-T _c cuprate materials to be qualitatively explained.     

铁基超导体系基于电子关联强度的统一相图

铁基超导和铜基超导具有诸多相似性,这为建立统一的高温超导机理图像提供了可能性.然而,对铁基超导体系中无论是进行电荷掺杂、还是等价掺杂来改变化学压力,都能产生定性上类似、而细节上纷繁复杂的相图,这对建立统一的图像造成了困难.研究化学掺杂效应如何在微观上影响电子结构和超导电性,区分主导超导电性演化的主要因素和次要因素,对建立统一图像和揭示高温超导机理至关重要.本文综述了对铁基超导体系中化学掺杂效应的一系列角分辨光电子能谱研究,涵盖了基于FeAs和FeSe面的多种代表性铁基超导体系,包括异价掺杂、等价掺杂、在元胞不同位置的化学掺杂,及其对电子体系在费米面结构、杂质散射、电子关联强度等方面的影响.实验结果表明：电子关联性或能带宽度是多个铁基超导相图背后的普适参数,不同的晶格和杂质散射效应导致了并不重要的复杂细节,而费米面拓扑结构与超导电性的关联并不强.这些结果对弱耦合机理图像提出了挑战,并促使人们通过局域反铁磁交换作用配对图像在带宽演化层面上统一地理解铁基超导.     

The vortex-like excitations detected by Nernst signals in high-T c superconductors

The nature of the pseudogap state and its relation to the d-wave superconductivity in high-T _c superconductors is still an open issue. The vortex-like excitations detected by the Nernst effect measurements exist in a certain temperature range above superconducting transition temperature T _c, which strongly support that the pseudogap phase is characterized by finite pairing amplitude with strong phase fluctuations and imply that the phase transition at T _c is driven by the loss of long-range phase coherence. We first briefly introduce the electronic phase diagram and pseudogap state of high-T _c superconductors, and then review the results of Nernst effect for different high-T _c superconductors. Related theoretical models are also discussed.     

Topology of the Fermi surface and the coexistence of orbital antiferromagnetism and superconductivity in cuprates

It has been shown that the strong coupling model taking into account a rise in the spin antiferromagnetic insulating state explains the doping dependence of the topology and shape of the Fermi contour of superconducting cuprates. Hole pockets with shadow bands in the second Brillouin zone form the Fermi contour with perfect ordinary and mirror nesting, which ensures the coexistence of orbital antiferromagnetism and superconductivity with a large pair momentum for T < T_C. The weak pseudogap region (T_* < T < T*) corresponds to the orbital antiferromagnetic ordering, which coexists with the incoherent state of superconducting pairs with large momenta in the strong pseudogap region (T_C < T < T_*).     

Phase fluctuations and pseudogap phenomena

This article reviews the current status of precursor superconducting phase fluctuations as a possible mechanism for pseudogap formation in high-temperature superconductors. In particular we compare this approach which relies on the two-dimensional nature of the superconductivity to the often used T-matrix approach. Starting from simple pairing Hamiltonians we present a broad pedagogical introduction to the BCS–Bose crossover problem. The finite temperature extension of these models naturally leads to a discussion of the Berezinskii–Kosterlitz–Thouless superconducting transition and the related phase diagram including the effects of quantum phase fluctuations and impurities. We stress the differences between simple Bose–BCS crossover theories and the current approach where one can have a large pseudogap region even at high carrier density where the Fermi surface is well-defined. Green's function and its associated spectral function, which explicitly show non-Fermi liquid behavior, is constructed in the presence of vortices. Finally different mechanisms including quasi-particle–vortex and vortex–vortex interactions for the filling of the gap above T_c are considered.     

The vortex-like excitations detected by Nernst signals in high-Tc superconductors

The nature of the pseudogap state and its relation to the d-wave superconductivity in high-T _c superconductors is still an open issue. The vortex-like excitations detected by the Nernst effect measurements exist in a certain temperature range above superconducting transition temperature T _c, which strongly support that the pseudogap phase is characterized by finite pairing amplitude with strong phase fluctuations and imply that the phase transition at T _c is driven by the loss of long-range phase coherence. We first briefly introduce the electronic phase diagram and pseudogap state of high-T _c superconductors, and then review the results of Nernst effect for different high-T _c superconductors. Related theoretical models are also discussed.     

Superconducting clusters in the pseudogap region

The upper boundary of the pseudogap state has been derived as a function of the doping within the impurity mechanism of high-T _c superconductivity under the assumption that, in the region bordering the superconducting phase on the side of the doping less than the optimum level (the pseudogap region), there exist finite superconducting clusters and the boundary of the superconducting phase corresponds to the threshold of the existence of an infinite superconducting cluster. The position of this boundary is in agreement with experiment. The condition imposed on the doping level at which the giant proximity effect should be observed has been derived. It means essentially that the thickness of the nonsuperconducting layer should be small compared to the average size of the superconducting clusters.     

Pseudogap-state superconductivity in a “hot-point” model: Gor’kov equations

The specific features of the superconducting state (with s and d pairing) are considered in terms of a pseudogap state caused by short-range order fluctuations of the “dielectric” type, namely, antiferromagnetic (spin density wave) or charge density wave fluctuations, in a model of the Fermi surface with “hot points.” A set of recurrent Gor’kov equations is derived with inclusion of all Feynman diagrams of a perturbation expansion in the interaction between an electron and short-range order fluctuations causing strong scattering near hot points. The influence of nonmagnetic impurities on superconductivity in such a pseudogap state is analyzed. The critical temperature for the superconducting transition is determined, and the effect of the effective pseudogap width, correlation length of short-range-order fluctuations, and impurity scattering frequency on the temperature dependence of the energy gap is investigated.     

Self-Consistent Consideration of Fluctuations in Singlet Superconducting Phases with s and d Symmetry

We analyze the behavior of thermal fluctuations of the superconducting order parameter with extended s-wave and $${{d}_{{{{x}^{2}} - {{y}^{2}}}}}$$-wave symmetry. For this purpose, we develop a method of self-consistent consideration of the order parameter fluctuations and charge carrier scatterers by fluctuations of coupled electron pairs using the theory of functional integration. The study is performed based on the quasi-two-dimensional one-band model with attraction between electrons located at neighboring sites. We obtain the distribution functions of the phase fluctuation probabilities depending on temperature, charge carrier concentrations, and model parameters. It is shown that the phase of the order parameter in the superconducting region is coherent, and the density of states has a dip at the Fermi level. In approaching the incoherent region of the phase diagram, the dip in the density of states disappears simultaneously with the loss of phase coherence. At the same time, the order parameter amplitude averaged over fluctuations remains finite at any temperature and concentration of charge carriers. Our results show that the pseudogap state cannot be explained in the frames of this scenario.     

Thermodynamics and phase diagram of high temperature superconductors

Thermodynamic quantities are derived for superconducting and pseudogap regimes by taking into account both amplitude and phase fluctuations of the pairing field. In the normal (pseudogap) state of the underdoped cuprates, two domains have to be distinguished: near the superconducting region, phase correlations are important up to temperature T(phi). Above T(phi), the pseudogap region is determined only by amplitudes, and phases are uncorrelated. Our calculations show excellent quantitative agreement with specific heat and magnetic susceptibility experiments on cuprates. We find that the mean field temperature T0 has a similar doping dependence as the pseudogap temperature T(*), whereas the pseudogap energy scale is given by the average amplitude above T(c).     

Phase fluctuations and the pseudogap in YBa(2)Cu(3)O (x)

The thermodynamics of the superconducting transition is studied as a function of doping using high-resolution expansivity data of YBa(2)Cu(3)O (x) single crystals and Monte Carlo simulations of the anisotropic 3D- XY model. We directly show that T(c) of underdoped YBa(2)Cu(3)O (x) is strongly suppressed from its mean-field value (T(MF)(c)) by phase fluctuations of the superconducting order parameter. For overdoped YBa(2)Cu(3)O (x) fluctuation effects are greatly reduced and T(c) approximately T(MF)(c). We find that T(MF)(c) exhibits a similar doping dependence as the pseudogap energy, naturally suggesting that the pseudogap arises from phase-incoherent Cooper pairing.     

Pseudogap and symmetry of superconducting order parameter in cuprates

The phase diagram, nature of the normal state pseudogap, type of the Fermi surface, and behavior of the superconducting gap in various cuprates are discussed in terms of a correlated state with valence bonds. The variational correlated state, which is a band analogue of the Anderson (RVB) states, is constructed using local unitary transformations. Formation of valence bonds causes attraction between holes in the d-channel and corresponding superconductivity compatible with antiferromagnetic spin order. Our calculations indicate that there is a fairly wide range of doping with antiferromagnetic order in isolated CuO₂ planes. The shape of the Fermi surface and phase transition curve are sensitive to the value and sign of the hopping interaction t′ between diagonal neighboring sites. In underdoped samples, the dielectrization of various sections of the Fermi boundary, depending on the sign of t′, gives rise to a pseudogap detected in photoemission spectra for various quasimomentum directions. In particular, in bismuth-and yttrium-based ceramics (t′>0), the transition from the normal state of overdoped samples to the pseudogap state of underdoped samples corresponds to the onset of dielectrization on the Brillouin zone boundary near k=(0,π) and transition from “large” to “small” Fermi surfaces. The hypothesis about s-wave superconductivity of La-and Nd-based ceramics has been revised: a situation is predicted when, notwithstanding the d-wave symmetry of the superconducting order parameter, the excitation energy on the Fermi surface does not vanish at all points of the phase space owing to the dielectrization of the Fermi boundary at k _x=± k _y. The model with orthorhombic distortions and two peaks on the curve of T _c versus doping is discussed in connection with experimental data for the yttrium-based ceramic. Zh. éksp. Teor. Fiz. 115, 649–674 (February 1999)     

Essence of Pseudogap and Oscillation in High-Temperature Cuprate Superconductor Junction of Pseudogap State

This paper gives methods to calculate the pairing temperature T*,at which a pseudogap is opened,and the superconducting temperature Tc,at which superconductivity appears,in the high-Tc cuprates,and demonstrates directly that at Tc ＜ T ＜ T* the pseudogap is the gap of Cooper pair without long-range phase coherence,and at T ＜ Tc there is long-range phase coherence between Cooper pairs.Based on the above clear physical picture on the pseudogap state and our mechanism for the ac Josephson effect,this paper proposes that there should be a novel oscillatory current in P-I-P junction,induced by a constant bias on the junction.Here,P represents the high-Tc curates in the pseudogap state,where Cooper pairs do not have long-range phase coherence,and I represents the thin insulating barrier.This paper conjectures that there is a possible high-temperature superconductivity in the heavily underdoped high-Tc cuprates.     

Crossover from a pseudogap state to a superconducting state

This paper deduces that the particular electronic structure of cuprate superconductors confines Cooper pairs to be first formed in the antinodal region which is far from the Fermi surface,and these pairs are incoherent and result in the pseudogap state.With the change of doping or temperature,some pairs are formed in the nodal region which locates the Fermi surface,and these pairs are coherent and lead to superconductivity.Thus the coexistence of the pseudogap and the superconducting gap is explained when the two kinds of gaps are not all on the Fermi surface.It also shows that the symmetry of the pseudogap and the superconducting gap are determined by the electronic structure,and non-s wave symmetry gap favours the high-temperature superconductivity.Why the high-temperature superconductivity occurs in the metal region near the Mott metal-insulator transition is also explained.     

Coexistence of ferromagnetism and superconductivity close to a quantum phase transition: the Heisenberg- to Ising-type crossover

A microscopic mean-field theory of the phase coexistence between ferromagnetism and superconductivity in the weakly ferromagnetic itinerant electron system is constructed, while incorporating a realistic mechanism for superconducting pairing due to the exchange of critical spin fluctuations. The self-consistent solution of the resulting equations determines the superconducting transition temperature which is shown to depend strongly on the exchange splitting. The effect of phase crossover from isotropic (Heisenberg-like) to uniaxial (Ising-like) spin fluctuations near the quantum phase transition is analyzed and the generic phase diagram is obtained. This scenario is then applied to the case of itinerant ferromagnet ZrZn2, which sheds light on the proposed phase diagram of this compound. A possible explanation of superconductivity in UGe2 is also discussed.     

Nodal superconductivity and antinodal pseudogap in cuprate superconductors

According to recent experimental findings the leading pairing resides in the nodal (FS arcs) momentum region of hole doped cuprates. The pseudogap is an antinodal feature. A corresponding multiband model of the electronic background evolving with doping serves the usually presented phase diagram. The pairing is due by the pair-transfer between overlapping nodal defect (polaron) band and the itinerant band. A bare gap vanishing with extended doping between the antinodal defect subband and the itinerant band top leads to the formation of the pseudogap as a perturbative band-structure effect. The calculated behaviour of two superconducting gaps and of the pseudogap on the whole doping scale is in qualitative agreement with the observations. Arguments to include cuprates into the class of multiband-multigap superconductors are given by these results.     

Strong pairing fluctuations and lattice anomalies in the pseudogap phase of the cuprates

Abstract

The pseudogap state in underdoped cuprates shows many very anomalous features. Among them are an extended temperature region of pairing fluctuations above the superconducting transition temperature and an unusual giant phonon anomaly in the same temperature and hole density range. A recent theoretical proposal that ascribes these anomalies to the presence of strong phase fluctuations related to a Leggett mode, is summarised.