Atomic-scale Electronic Structure of the Cuprate d-Symmetry Form Factor Density Wave State

Abstract

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 $\hat{I}{”}_1$. Moreover, we demonstrate that the dFF-DW modulations at $E=-\hat{I}{”}_1$ (filled states) occur with relative phase $\ddot{I}€$ compared to those at $E=\hat{I}{”}_1$ (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.