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Register a new userAtomic-scale Electronic Structure of the Cuprate Pair Density Wave State Coexisting with Superconductivity
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The defining characteristic of hole-doped cuprates is $d$-wave high temperature superconductivity. However, intense theoretical interest is now focused on whether a pair density wave state (PDW) could coexist with cuprate superconductivity (D. F. Agterberg et al., Annual Review of Condensed Matter Physics 11, 231 (2020)). Here, we use a strong-coupling mean-field theory of cuprates, to model the atomic-scale electronic structure of an eight-unit-cell periodic, $d$-symmetry form factor, pair density wave (PDW) state coexisting with $d$-wave superconductivity (DSC). From this PDW+DSC model, the atomically-resolved density of Bogoliubov quasiparticle states N(r,E) is predicted at the terminal BiO surface of Bi$_2$Sr$_2$CaCu$_2$O$_8$ and compared with high-precision electronic visualization experiments using spectroscopic imaging STM. The PDW+DSC model predictions include the intra-unit-cell structure and periodic modulations of N(r,E), the modulations of the coherence peak energy $Delta_p$ (r), and the characteristics of Bogoliubov quasiparticle interference in scattering-wavevector space (q-space). Consistency between all these predictions and the corresponding experiments indicates that lightly hole-doped Bi$_2$Sr$_2$CaCu$_2$O$_8$ does contain a PDW+DSC state. Moreover, in the model the PDW+DSC state becomes unstable to a pure DSC state at a critical hole density p*, with empirically equivalent phenomena occurring in the experiments. All these results are consistent with a picture in which the cuprate translational symmetry breaking state is a PDW, the observed charge modulations are its consequence, the antinodal pseudogap is that of the PDW state, and the cuprate critical point at p* ~ 19% occurs due to disappearance of this PDW.
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Extensive research into high temperature superconducting cuprates is now focused upon identifying the relationship between the classic pseudogap phenomenon$^{1,2}$ and the more recently investigated density wave state$^{3-13}$. This state always exhibits wave vector $Q$ parallel to the planar Cu-O-Cu bonds$^{4-13}$ along with a predominantly $d$-symmetry form factor$^{14-17}$ (dFF-DW). Finding its microscopic mechanism has now become a key objective$^{18-30}$ of this field. To accomplish this, one must identify the momentum-space ($k$-space) states contributing to the dFF-DW spectral weight, determine their particle-hole phase relationship about the Fermi energy, establish whether they exhibit a characteristic energy gap, and understand the evolution of all these phenomena throughout the phase diagram. Here we use energy-resolved sublattice visualization$^{14}$ of electronic structure and show that the characteristic energy of the dFF-DW modulations is actually the pseudogap energy $Delta_{1}$. Moreover, we demonstrate that the dFF-DW modulations at $E=-Delta_{1}$ (filled states) occur with relative phase $pi$ compared to those at $E=Delta_{1}$ (empty states). Finally, we show that the dFF-DW $Q$ corresponds directly to scattering between the hot frontier regions of $k$-space beyond which Bogoliubov quasiparticles cease to exist$^{31,32,33}$. These data demonstrate that the dFF-DW state is consistent with particle-hole interactions focused at the pseudogap energy scale and between the four pairs of hot frontier regions in $k$-space where the pseudogap opens.
When very high magnetic fields suppress the superconductivity in underdoped cuprates, an exceptional new electronic phase appears. It supports remarkable and unexplained quantum oscillations and exhibits an unidentified density wave (DW) state. Although generally referred to as a charge density wave (CDW) because of the observed charge density modulations, theory indicates that this could actually be the far more elusive electron-pair density wave state (PDW). To search for evidence of a field-induced PDW in cuprates, we visualize the modulations in the density of electronic states $N(bf{r})$ within the halo surrounding Bi$_2$Sr$_2$CaCu$_2$O$_8$ vortex cores. This reveals multiple signatures of a field-induced PDW, including two sets of $N(bf{r})$ modulations occurring at wavevectors $bf{Q}_P$ and $2bf{Q}_P$, both having predominantly $s$-symmetry form factors, the amplitude of the latter decaying twice as rapidly as the former, along with induced energy-gap modulations at $bf{Q}_P$ . Such a microscopic phenomenology is in detailed agreement with theory for a field-induced primary PDW that generates secondary CDWs within the vortex halo. These data indicate that the fundamental state generated by increasing magnetic fields from the underdoped cuprate superconducting phase is actually a PDW with approximately eight CuO$_2$ unit-cell periodicity ($lambda = 8a_0$) and predominantly $d$-symmetry form factor.
An unidentified quantum fluid designated the pseudogap (PG) phase is produced by electron-density depletion in the CuO$_2$ antiferromagnetic insulator. Current theories suggest that the PG phase may be a pair density wave (PDW) state characterized by a spatially modulating density of electron pairs. Such a state should exhibit a periodically modulating energy gap $Delta_P(pmb r)$ in real-space, and a characteristic quasiparticle scattering interference (QPI) signature $Lambda_P(pmb q)$ in wavevector space. By studying strongly underdoped Bi$_2$Sr$_2$CaDyCu$_2$O$_8$ at hole-density ~0.08 in the superconductive phase, we detect the $8a_0$-periodic $Delta_P(pmb r)$ modulations signifying a PDW coexisting with superconductivity. Then, by visualizing the temperature dependence of this electronic structure from the superconducting into the pseudogap phase, we find evolution of the scattering interference signature $Lambda(pmb q)$ that is predicted specifically for the temperature dependence of an $8a_0$-periodic PDW. These observations are consistent with theory for the transition from a PDW state coexisting with d-wave superconductivity to a pure PDW state in the Bi$_2$Sr$_2$CaDyCu$_2$O$_8$ pseudogap phase.
When the Mott insulating state is suppressed by charge carrier doping, the pseudogap phenomenon emerges, where at the low-temperature limit, superconductivity coexists with some ordered electronic states. Within the framework of the kinetic-energy-driven superconductivity, the nature of the pair-density-wave order in cuprate superconductors is studied by taking into account the pseudogap effect. It is shown that the onset of the pair-density-wave order does not produce an ordered gap, but rather a novel hidden order as a result of the interplay of the charge-density-wave order with superconductivity. As a consequence, this novel hidden pair-density-wave order as a subsidiary order parameter coexists with the charge-density-wave order in the superconducting-state, and is absent from the normal-state.
The mysterious pseudo-gap (PG) phase of cuprate superconductors has been the subject of intense investigation over the last thirty years, but without a clear agreement about its origin. Owing to a recent observation in Raman spectroscopy, of a precursor in the charge channel, on top of the well known fact of a precursor in the superconducting channel, we present here a novel idea: the PG is formed through a Higgs mechanism, where two kinds of preformed pairs, in the particle-particle and particle-hole channels, become entangled through a freezing of their global phase. Remarkably, this entanglement is equivalent to fractionalizing a Cooper pair density wave (PDW) into its elementary parts; the particle-hole pair, giving rise to both density modulations and current modulations, and the particle-particle counterpart, leading to the formation of Cooper pairs. From this perspective, the fractionalized PDW becomes the central object around the formation of the pseudo-gap. The locking of phases between the charge and superconducting modes gives a unique explanation for the unusual global phase coherence of short-range charge modulations, observed below $T_{c}$ on phase sensitive scanning tunneling microscopy (STM). A simple microscopic model enables us to estimate the mean-field values of the precursor gaps in each channel and the PG energy scale, and to compare them to the values observed in Raman scattering spectroscopy. We also discuss the possibility of a multiplicity of orders in the PG phase and give an overview of the phase diagram.
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