Pseudogap and the superconducting energy in hole-doped high-temperature superconductors

The pseudogap measured by different techniques in the high-temperature superconductors is interpreted as a two-particle screened Zhang-Rice singlet type state. The so-called (neutron) resonance energy measures the energy lowering of the superconducting state relative to the normal one and should be called the superconducting energy or the order parameter. The superconductivity in the high-temperature superconductors is in the crossover regime between a Bose-Einstein condensate and a BCS state, similar in physics to the situation for the fermionic cold atoms. The superconducting transition temperature is given as the product of the pseudogap energy $\left(E_{\text{pg}}\right)$ and the carrier density, leading to its parabolic dependence on the doping. The formation of the Zhang-Rice singlet and the accompanying creation of the pseudogap are instrumental for the occurrence of the high-temperature superconductivity, as also for the large magnetoresistance in the manganites. The lowest ionization state in transition-metal compounds therefore constitutes a promising tool for the engineering of materials.

Figure 1

(Color online) Pseudogap energy $E_{\text{pg}}=2{\Delta }_{\text{pg}}$ and resonance energy $E_r={\Omega }_r=2{\Delta }_{\text{sc}}$ for Bi 2212, Tl 2201, and Y 123, as measured by ARPES, tunneling [ superconductor/insulator/superconductor (SIS) and superconductor/insulator/normal (SIN) metals, STM, and AR-Andreev–Saint-James reflection), Raman scattering (RS), and neutron scattering (INS), as well as heat conductivity (HC) as a function of hole doping $x$. The data fall on two universal curves with $E_{\text{pg}}=152\text{meV}/0.22\cdot \left(0.27-x\right)$ and $E_{\text{sc}}=E_{\text{sc}}\left(x=0.16\right)\cdot \left[1-82.6\cdot {\left(0.16-x\right)}^2\right]$ with $E_{\text{sc}}\left(x=0.16\right)=42\text{meV}$ and $E_{\text{sc}}=\left(5.1\pm 1\right)\cdot k_B{T}_c$. pg stands for pseudogap state; the other area designations are self explanatory (see Ref. 9).

Figure 2

(Color online) (a) Schematic spectral function of CuO and of ${\text{La}}_{2-x}{\text{Sr}}_x{\text{CuO}}_4$ as an example of a hole-doped HTSC for $T_c<T<T^{\ast }$. (b) Low doping insulating state. (c) Optimal doping at the antinode $\left(\pi ,0\right)$, showing the pseudogap below $T^{\ast }$. (d) Same as panel (c) at the node, showing no gap. (e) Situation for doping above 0.3, showing the normal metallic state.

Figure 3

Schematic showing the pseudogap energy $E_{\text{pg}}{E}_{\text{pg}}\left(x=0.05\right)/0.22\cdot \left(0.27-x\right)$, the carrier density $D=D\left(x=0.27\right)/0.22\cdot \left(x-0.05\right)$, and the superconducting transition temperature $T_c\sim E_{\text{pg}}\cdot D\sim \left[1-82.6\cdot {\left(0.16-x\right)}^2\right]$. $\text{sc}=\text{superconducting}$ phase, $\text{pg}=\text{pseudogap}$ phase, and $\text{nm}=\text{normal}$ metal phase.

Figure 4

Schematic phase diagram for the HTSC as derived from Ref. 66. The charge-carrier density and the potential between the electrons were given as parameters. The diagram shows the same structure as that in Fig. 3.