Doping-Dependent Nodal Fermi Velocity of the High-Temperature Superconductor ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathbf{O}}_{8+\delta }$ Revealed Using High-Resolution Angle-Resolved Photoemission Spectroscopy

The improved resolution of laser-based angle-resolved photoemission spectroscopy (ARPES) allows reliable access to fine structures in the spectrum. We present a systematic, doping-dependent study of a recently discovered low-energy kink in the nodal dispersion of ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ (Bi-2212), which demonstrates the ubiquity and robustness of this kink in underdoped Bi-2212. The renormalization of the nodal velocity due to this kink becomes stronger with underdoping, revealing that the nodal Fermi velocity is nonuniversal, in contrast with assumed phenomenology. This is used together with laser ARPES measurements of the gap velocity ($v_2$) to resolve discrepancies with thermal conductivity measurements.

Figure 1

(a)–(c) False-color image plots of the nodal dispersion of UD55, UD65, and UD92, measured at 10 K. Solid curves indicate band dispersions derived from MDC peak positions. Inset of (a) Brillouin zone schematic with Fermi surface indicated by the red curve. The $x$ axes in (a)–(c) correspond to momentum along diagonal blue line (“nodal cut”) in inset. (d) EDCs (solid lines) and symmetrized EDCs (dashed lines) at $k_F$. A single peak at $E_F$ in the latter confirms that spectra in (a)–(c) are ungapped. $k_F$ determined from Fermi crossing of dispersion [vertical dashed lines in (a)–(c)].

Figure 2

(a) Band dispersions from Fig. 1, offset for clarity. Dashed line accompanying UD55 denotes assumed linear bare band. Black dotted lines are linear fits 30–40 meV ($v_{\mathrm{mid}}$), extrapolated to $E_F$. This differs from $v_F$ (fit 0–7 meV), indicated on UD65 dispersion by orange dashed line. Lower (pink) shaded bar marks 70 meV kink and upper (gray) bar marks low-energy kink, where $v_F$ deviates from $v_{\mathrm{mid}}$. Inset: UD92 dispersion below 20 meV. (b) $\mathrm{Re}\Sigma$, approximated by subtracting a linear bare band from dispersions in (a), is peaked at the position of the 70 meV kink and changes slope near the energy position of the low-energy kink. (c) Detail of the low-energy portion of $\mathrm{Re}\Sigma$. Thick dotted lines are fits near $E_F$. All curves deviate from these lines between 6–10 meV, as highlighted by the shaded bar. For the two most underdoped samples, the slope of $\mathrm{Re}\Sigma$ evolves between the low-energy kink and 20–30 meV, suggesting an additional kink. (d) $2\mathrm{Im}\Sigma$, with dotted lines as guides to the eye. All curves decrease more rapidly near $E_F$, but most markedly in UD92. For more underdoped samples, the effect may be obscured by a larger linewidth and possible additional kink.

Figure 3

(a) Doping dependence of $v_F$ and $v_{\mathrm{mid}}$. Open red (blue) squares denote average $v_F$ ($v_{\mathrm{mid}}$) for each doping. Boundary of shaded region denotes doping dependence of $T_c$. $v_{\mathrm{mid}}$ has little systematic doping dependence, while $v_F$ decreases with underdoping. (b) Doping dependence of $v_{\mathrm{mid}}/v_F$, which is related to the renormalization coupling strength.

Figure 4

(a) Cutout showing Fermi surface (FS) (top) and dispersion perpendicular ($v_F$, left) and tangential ($v_2$, right) to FS at node, from measurement on UD92 at 10 K. (b),(c) Image plots showing measured $v_F$ and $v_2$ directly. The latter image consists of EDCs at $k_F$ from many parallel cuts near the node. (d) Comparison of synchrotron- and laser-based ARPES measurements of the superconducting gap of UD92 around the FS. When the data in (d) are fit to a simple $d$-wave form, $\Delta (\theta )={\Delta }_0\cos (2\theta )$ close to the node, $v_2\approx 2{\Delta }_0/k_F$, where $k_F$ is the distance from the node to $(\pi ,\pi )$. (e) $v_2$ from synchrotron- and laser-based ARPES experiments. (f) Comparison between $v_F/v_2$ from laser ARPES and thermal conductivity from Ref. 5. Doping dependence is consistent, though absolute values differ. Open squares indicate average laser ARPES values.