Prepairing and the “filling” gap in the cuprates from the tomographic density of states

We use the tomographic density of states (TDOS), which is a measure of the density of states for a single slice through the band structure of a solid, to study the temperature evolution of the superconducting gap in the cuprates. The TDOS provides accuracy in determining both the superconducting pair-forming strength $\Delta$ and the pair-breaking rate $\Gamma$. In both optimally and underdoped Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+\delta }$, we find the near-nodal $\Delta$ smoothly evolves through the superconducting transition temperature—clear evidence for the existence of preformed pairs. Additionally, we find the long-observed “filling” of the superconducting gap in the cuprates is due to the strongly temperature-dependent $\Gamma$.

Figure 1

Creating and fitting the tomographic density of states (TDOS). (a) Tight-binding model of the Bi2212 band structure slices of which ARPES samples. A nodal slice (orange) and an off-nodal slice (red) are shown. The location of each slice is characterized by the Fermi surface angle ${\theta }_{\text{FS}}$ measured from the node about the $Y$ point. (b) Nodal and off-nodal ARPES spectra with corresponding coherent spectral weights (color) and incoherent backgrounds (black) from an optimally doped ($T_C=91$ K) Bi2212 sample at $T=40$ K. (c) Tomographic density of states created by normalizing the off-nodal spectral weight by the nodal spectral weight as well as the fit to the Dynes formula. (d) Two TDOS ($T=50$ K) and the respective fits for $T=50$ K and angles ranging from less than $2^{\circ }$ from the node to 40$\%$ of the way to the antinode. (e) Two TDOS from ${\theta }_{\text{FS}}=7.8^{\circ }$ and the respective fits for temperatures ranging from deep in the superconducting state until just below $T_C$.

Figure 2

Temperature dependence of the TDOS. (a) Temperature evolution of ARPES spectra for two angles in the near-nodal region for lightly underdoped ($T_C=85$ K) Bi2212. (b) Temperature evolution for TDOS for 12${}^{\circ }$ from the node [marked in red in the inset Brillouin zone and the antinodal pseudogap region (Ref. 22) is marked in green]. The inset shows a zoom of the 90 K TDOS (orange) and its fit to the Dynes formula (black). (c) Temperature evolution for TDOS for 18${}^{\circ }$ from the node. The inset shows a zoom of the 100 K TDOS (red) and its fit to the Dynes formula (black). (d) Fit results of TDOS to Dynes formula showing a strongly temperature-dependent $\Gamma$ and slowly evolving $\Delta$, which continues to exist above $T_C$. The error bars represent twice the statistical uncertainties, as returned from our least-squares fitting procedures, to incorporate systematic errors as well ($\pm$$2\sigma$). (e) Temperature evolution of symmetrized EDCs for ${\theta }_{\text{FS}}={12}^{\circ }$ showing the formation of a resolvable peak away from $E_F$ only exists below $T_C$ [the same color scale for temperature is used in (b), (c), and (e)]. The EDCs have been rescaled to emphasize the evolution of the line shape.

Figure 3

The “filling” gap of Bi2212. (a) Temperature evolution of near-nodal (${\theta }_{\text{FS}}=7.8^{\circ }$) TDOS for optimally doped ($T_C=91$ K) Bi2212. (b) Recreation of observed filling gap by only increasing the pair-breaking rate $\Gamma$ while holding the pairing strength $\Delta$ fixed. (c) Recreation of Dynes's original results showing the temperature dependence of the gap for a strongly coupled conventional superconductor. Dashed curves are extrapolations based on the reported temperature dependence. (d) Comparison of fit results of Dynes formula to the original Pb${}_{0.9}$Bi${}_{0.1}$ to the TDOS of the optimally doped Bi2212 shown in (a). Here we have normalized both energy scales $\Gamma$ and $\Delta$ to the maximum $\Delta$ as found at low temperature and, for Bi2212, the antinode. The error bars represent twice the statistical uncertainties ($\pm$$2\sigma$), as returned from our least-squares fitting procedures, to incorporate systematic errors as well.