Reaffirming the $d_{x^2-y^2}$ Superconducting Gap Using the Autocorrelation Angle-Resolved Photoemission Spectroscopy of ${\mathrm{Bi}}_{1.5}{\mathrm{Pb}}_{0.55}{\mathrm{Sr}}_{1.6}{\mathrm{La}}_{0.4}{\mathrm{CuO}}_{6+\delta }$

Knowledge of the gap function is important to understand the pairing mechanism for high-temperature ($T_c$) superconductivity. However, Fourier transform scanning tunneling spectroscopy (FT STS) and angle-resolved photoemission spectroscopy (ARPES) in the cuprates have reported contradictory gap functions, with FT-STS results deviating strongly from a canonical $d_{x^2-y^2}$ form. By applying an “octet model” analysis to autocorrelation ARPES, we reveal that a contradiction occurs because the octet model does not consider the effects of matrix elements and the pseudogap. This reaffirms the canonical $d_{x^2-y^2}$ superconducting gap around the node, which can be directly determined from ARPES. Further, our study suggests that the FT-STS reported fluctuating superconductivity around the node at far above $T_c$ is not necessary to explain the existence of the quasiparticle interference at low energy.

Figure 1

Image plots in $\mathbf{k}$ and $\mathbf{q}$ spaces of nearly optimally doped Pb-Bi2201. (a)–(c) $\mathbf{k}$-space ARPES intensities at 0 meV ($E_F$) for three temperatures, 8 K ($<T_c$), 45 K ($T_c<T\ll T^{*}$), and 172 K ($>T^{*}$), respectively. (d)–(o) $\mathbf{q}$-space AC-ARPES intensities at various energies for the three temperatures as denoted. All the intensities are integrated within $\pm 0.5\mathrm{meV}$. ${\mathbf{q}}_1–{\mathbf{q}}_7$ in $\mathbf{q}$ space are exemplified in (d) in appropriate colors, which are the $\mathbf{q}$ vectors expected for the octet model. Corresponding $\mathbf{q}$ vectors in $\mathbf{k}$ space are indicated in (a). Definitions of ${\mathbf{q}}_3^{*}$ and ${\mathbf{q}}_5^{*}$ at higher binding energy in $\mathbf{q}$ space are shown in (m).

Figure 2

Energy dispersions of the $\mathbf{q}$ vectors from AC ARPES. (a)–(f) Energy dependences of AC-ARPES intensities along high-symmetry $\mathbf{q}$ directions at 8 K ($<T_c$), 45 K ($T_c<T\ll T^{*}$), and 172 K ($>T^{*}$), as denoted. The intensities at each energy are normalized to the height of ${\mathbf{q}}_5$ (or ${\mathbf{q}}_5^{*}$) in (a), (c), and (e), and ${\mathbf{q}}_3$ (or ${\mathbf{q}}_3^{*}$) in (b), (d), and (f), respectively, for visualization purposes. (g),(h) Summary of the extracted dispersions of the $\mathbf{q}$ vectors at 8 K ($<T_c$) and 45 K ($T_c<T\ll T^{*}$), respectively. Solid curves in (g) are the dispersions of ${\mathbf{q}}_1–{\mathbf{q}}_7$ for a $d_{x^2-y^2}$ gap ${\Delta }_{\mathrm{sc}}(\mathbf{k})={\Delta }_{\max }|\cos (k_x)-\cos (k_y)|/2$ (${\Delta }_{\max }=19\mathrm{meV}$). The same colors as Fig. 1 are used for the symbols and labels of the $\mathbf{q}$ vectors.

Figure 3

Deviation from the $d_{x^2-y^2}$ gap anisotropy in the octet-model analysis of AC ARPES. (a) Comparison of the FS shape obtained from the two different analysis methods applied to the same data at various temperatures. Dashed curve and dotted line are guides for the eye for the FS shape and the antiferromagnetic Brillouin zone boundary, respectively. (b) Comparison of the gap anisotropy between the two methods. The definition of the FS angle is shown in (c). Dotted curve in (b) is for the $d_{x^2-y^2}$ gap function (${\Delta }_{\max }=19\mathrm{meV}$). The energy ranges for the FS shape from AC ARPES in (a) are the same as those for the gap functions in (b).