Unmasking the nodal quasiparticle dynamics in cuprate superconductors using low-energy photoemission

Nodal quasiparticles of ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+\delta }$ have been studied by angle-resolved photoemission spectroscopy with high momentum and energy resolution. Low-energy tunable photons have enabled us to resolve a small nodal bilayer splitting, unmasking the intrinsic single-particle scattering rate. The nodal scattering rate is abruptly suppressed upon the superconducting transition, and shows a linear energy dependence at low energies, indicating the nontrivial effect of elastic scatterings on the quasiparticles. With increasing energy, the antibonding-band scattering rate becomes higher than the bonding one. The observations imply the character of the scatterers dominant at low energies.

Figure 1

(Color) Overview of low-energy ARPES result at $h\nu =7.57\mathrm{eV}$, taken in the nodal direction of superconducting ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+\delta }$ at $T=9\mathrm{K}$. (a) Spectral-intensity map in the energy-momentum space. (b) Energy distribution curves (EDCs) at the momenta denoted by purple triangles. (c) Momentum distribution curves (MDCs) at the energies denoted by green triangles. The intensity of the MDC at $\omega =0$ is multiplied by 2.

Figure 2

(Color) Enlarged view of Fig. 1 around the Fermi-surface crossings. (a) Spectral-intensity map. Red and blue circles denote the peak positions of the MDCs. (b) EDCs at each $0.0012{\mathrm{\AA }}^{-1}$. (c) MDCs at each $2\mathrm{meV}$. (d) Bilayer-resolved scattering rates, determined from the momentum widths (FWHM), $\Delta k_b$ (red), and $\Delta k_a$ (blue), of the bonding and antibonding peaks, respectively. Thin solid lines show the linear increasing rates, $\sim 0.11$ and $\sim 0.21{\mathrm{\AA }}^{-1}{\mathrm{eV}}^{-1}$, of $\Delta k_b$ and $\Delta k_a$, respectively.

Figure 3

(Color) Single-particle spectral function in the superconducting ($T=9$ and $50\mathrm{K}$) and normal ($T=95$ and $200\mathrm{K}$) states, obtained by dividing the ARPES intensity by the Fermi-Dirac distribution function. Dotted and thin dashed lines denote the weighted center of the bonding and antibonding peaks in the MDCs, and the hypothetical unrenormalized dispersion ${ϵ}_k^{0^{\prime }}=v_0^{\prime }(k-k_F)$ for $v_0^{\prime }=2.8\mathrm{eV}\mathrm{\AA }$, respectively.

Figure 4

(Color) Self-energy $\Sigma \left(\omega \right)$ in the nodal direction, obtained by regarding the splitting parameters as constants. (a) Real part of self-energy, determined from the quasiparticle dispersion by $\mathrm{Re}{\Sigma }^{\prime }\left({\omega }_k\right)={\omega }_k-{ϵ}_k^{0^{\prime }}$. (b) Imaginary part of self-energy, determined from the quasiparticle momentum width $\Delta k$ (FWHM) by $\mathrm{Im}\Sigma \left(\omega \right)=-\frac{1}{2}{v}_0\Delta k$, where the scaling factor is $v_0=2.8\mathrm{eV}\mathrm{\AA }$ (Ref. 18). Fivefold magnified view shows the $\omega$-linear fit (black line) of $\mathrm{Im}\Sigma \left(\omega \right)$ at $T=9\mathrm{K}$. Inset shows the inverse lifetimes, deduced from ARPES by $1∕\tau \left(\omega \right)=v_k\Delta k$ (open circles) and from optical conductivity (line) (Ref. 4). (c) and (d) Comparison between the width-derived $\mathrm{Im}\Sigma \left(\omega \right)$ (filled circles) and the Kramers-Krönig transformation $\mathrm{Im}{\Sigma }_{\mathrm{KK}}^{\prime }\left(\omega \right)$ (solid lines) of the dispersion-derived $\mathrm{Re}{\Sigma }^{\prime }\left(\omega \right)$ for $T=95$ and $9\mathrm{K}$. Black dotted lines denote the difference, $\mathrm{Im}\Sigma \left(\omega \right)-\mathrm{Im}{\Sigma }_{\mathrm{KK}}^{\prime }\left(\omega \right)$.

Figure 5

(Color) (a) Temperature dependence of the MDC at $\omega =0$. (b) Temperature dependence of the scattering rate, $-\mathrm{Im}\Sigma \left(\omega \right)=\frac{1}{2}{v}_0\Delta k$, determined from the momentum width $\Delta k$ (FWHM) at $\omega =0$ (filled circles) and $15\mathrm{meV}$ (open circles). Inset shows the scattering rate as a function of energy for the superconducting (SC) and normal (N) states.