Unusual electronic ordering in the pseudogap phase of underdoped cuprate superconductors

TL;DR

The paper addresses how the pseudogap phase in underdoped cuprate superconductors modulates electronic states and LDOS, and how these features relate to the superconducting state. It develops a $T$-matrix formalism based on the inverse impurity-scattering matrix, derived from a microscopic homogeneous propagator in a kinetic-energy-driven superconductivity framework, to compute LDOS in both superconducting and pseudogap phases for various impurities. A key finding is that the quasiparticle interference octet, observed in the superconducting state, persists in the pseudogap phase with dispersive, particle-hole-symmetric ${\bf q}_i(\omega)$; in addition, the pseudogap enables a non-dispersive checkerboard LDOS modulation at ${\bf Q}\approx[\pm0.36\pi,0]$ and $[0,\pm0.36\pi]$ due to residual antinodal spectral weight. The results connect STM/ARPES observations to a unified microscopic picture where the normal-state pseudogap organizes the LDOS spectrum, coexisting and competing with superconductivity.

Abstract

The pseudogap phase of the underdoped cuprate superconductors harbours diverse manifestations of different ordered electronic-states, and then these ordered electronic-states coexist or compete with superconductivity. Here starting from the microscopic electron propagator, the nature of the ordered electronic-states in the pseudogap phase is investigated within the $T$-matrix approach. This $T$-matrix is derived in terms of the inverse of matrix for various kinds of a single impurity, and then is used to evaluate the local density of states (LDOS) by the involvement of all the quasiparticle excitations and scattering processes. It is shown that a number of the anomalous properties in the underdoped cuprate superconductors is directly correlated to the opening of the normal-state pseudogap: (i) the structure of the microscopic octet scattering model generated by the normal-state pseudogap is essentially the same both in the superconducting (SC)-state and pseudogap phase, which naturally leads to that the quasiparticle scattering interference octet phenomenology observed in the SC-state exists in the pseudogap phase; (ii) however, the spectral weight at around the antinodal region in the SC-state is gapped out completely by both the SC gap and normal-state pseudogap, while it in the pseudogap phase is suppressed partially by the normal-state pseudogap, this directly leads to that the non-dispersive checkerboard charge ordering with a finite wave vector ${\bf Q}$ appears in the pseudogap phase only. The theory therefore also shows that the electronic-states affected by the normal-state pseudogap exhibit the LDOS modulation spectrum organization.

Figures (7)

• Figure 1: (Color online) (a) The superconducting gap $2\bar{\Delta}$ (black-line) and normal-state pseudogap $2\bar{\Delta}_{\rm PG}$ (red-line) as a function of doping with temperature $T=0.002J$. (b) The superconducting transition temperature $T_{\rm c}$ (black-line) and the normal-state pseudogap crossover temperature $T^{*}$ (red-line) as a function of doping.
• Figure 2: (Color online) The constant energy contour map in the $[k_{x},k_{y}]$ plane at doping $\delta=0.09$ for the binding-energy $\omega=0.06J$ in (a) the pseudogap phase with temperature $T=0.07J$ and (b) the superconducting-state with temperature $T=0.002J$. The eight tips of the Fermi arcs determine the scattering within the octet scattering model, while the scattering wave vectors which connect the tips of the Fermi arcs are shown as arrows labeled by the designation of each scattering wave vector ${\bf q}_{i}$.
• Figure 3: (Color online) Upper panel: the intensity maps of (a) the local density of states in real-space, (b) the amplitude of the local density of states in momentum-space, and (c) the phase of the momentum-space local density of states in the pseudogap phase for an in-plane single impurity at doping $\delta=0.09$ and the binding-energy $\omega=0.06J$ with temperature $T=0.07J$ for the impurity-scattering strength $V_{s}=1.0J$ and screening length $L=5.0a$. Lower panel: the corresponding intensity maps of (d) the local density of states in real-space, (e) the amplitude of the local density of states in momentum-space, and (f) the phase of the momentum-space local density of states in the superconducting-state with temperature $T=0.002J$.
• Figure 4: (Color online) The evolution of the scattering wave vectors (a) ${\bf q}_{1}$ and (b) ${\bf q}_{7}$ with energy in the pseudogap phase for an in-plane single impurity at doping $\delta=0.09$ (red-line) and doping $\delta=0.12$ (black-line) with temperature $T=0.07J$ for the impurity-scattering strength $V_{s}=1.0J$ and screening length $L=5.0a$.
• Figure 5: The magnitude of the checkerboard charge-order wave vector as a function of energy for an in-plane single impurity at doping $\delta=0.09$ with temperature $T=0.07J$ for the impurity-scattering strength $V_{s}=1.0J$ and screening length $L=5.0a$.