Unmasking the Origin of Kinks in the Photoemission Spectra of Cuprate Superconductors

The origin of a ubiquitous bosonic coupling feature in the photoemission spectra of high-$T_c$ cuprates, an energy-momentum dispersion “kink” observed at $\sim 70\mathrm{meV}$ binding energy, remains a two-decade-old mystery. Understanding this phenomenon requires an accurate description of the coupling between the electron and some collective modes. We report here ab initio calculations based on $GW$ perturbation theory and show that correlation-enhanced electron-phonon interaction in cuprates gives rise to the strong kinks, which not only explains quantitatively the observations but provides new understanding of experiments. Our results reveal it is the electron density of states being the predominant factor in determining the doping dependence of the kink size, manifesting the multiband nature of the cuprates, as opposed to the prevalent belief of it being a measure of the mode-coupling strength.

Figure 1

(a),(b) Contribution of electron-phonon coupling to the real part of the nodal electron self-energy $\mathrm{Re}{\overline{\mathrm{\Sigma }}}_{n\mathbf{k}}^{e\text{-ph}}$ for (a) optimally doped ($x=0.15$) and (b) overdoped ($x=0.30$) LSCO at $T=20\mathrm{K}$, calculated using $e\text{-ph}$ matrix elements from $GW$PT (red solid dots) and DFPT (blue empty dots). The Fermi energy $E_F$ is set to zero. (c),(d) Imaginary part of the phonon-induced contribution ($T=20\mathrm{K}$) to the electron self-energy $\mathrm{Im}{\mathrm{\Sigma }}_{n\mathbf{k}}^{e\text{-ph}}$, the magnitude of which continuously rises until the binding energy reaches the maximum phonon frequency ${\omega }_{\mathrm{ph}}^{\max }\sim 87\mathrm{meV}$.

Figure 2

(a) Spectral function $A_{n\mathbf{k}}(\omega )$ (plotted in color scale) calculated with ${\mathrm{\Sigma }}_{n\mathbf{k}}^{e\text{-ph}}(\omega )$ using $GW$PT for optimally doped ($x=0.15$) LSCO at $T=20\mathrm{K}$, along the nodal direction in the Brillouin zone. (b) Momentum-distribution curves (MDCs) at each energy slice between $E_F$ and $-0.2\mathrm{eV}$ with 0.005 eV spacing. Each MDC is fitted to a Lorentzian function to get the peak position and linewidth. The MDCs at the extremal energies are indicated by the blue lines. The red line is the MDC at $E_1=-0.12\mathrm{eV}$ whose peak position is used to determine the bare-band straight reference line (connecting the peaks at $E_F$ and $E_1$ on a given dispersion relation). (e) Comparison of MDC-derived dispersion relation between experiment (open circles) [3] and theory from $GW$PT (red line) and DFPT (blue line), after aligning the reference line (black dashed line). ${\mathbf{k}}_F$ is the Fermi wave vector and ${\mathbf{k}}_1$ is the wave vector corresponding to $E_1$. (g) The $e\text{-ph}$ contribution to the full-width-at-half-maximum (FWHM) of MDC peaks extracted from experiments (open circles) [22] and from $GW$PT (red line) and DFPT (blue line). (c),(d),(f),(h) Similar to (a),(b),(e),(g) but for overdoped ($x=0.30$) LSCO.

Figure 3

(a),(b) Kink size as a function of quasiparticle-state energy, extracted as the deviation of the MDC-derived dispersion relation from the straight reference line (see Fig. 2). Data shown are from various experiments [3, 21, 23, 24, 25, 26] on LSCO and Bi2201 (open symbols) and from the calculations of LSCO using $GW$PT (red line) and DFPT (blue line), at the optimally doped and overdoped regimes. (c),(d) Total kink area, defined as the area under the curve in (a) and (b), as a function of temperature.

Figure 4

(a) Doping dependence in the total apparent kink area of LSCO from experiments [3, 23], $GW$PT, and DFPT calculations. (b) Direct $GW$PT calculations of $\mathrm{Re}{\overline{\mathrm{\Sigma }}}_{n\mathbf{k}}^{e\text{-ph}}$ at $x=0.15$ (red solid dot) and $x=0.30$ (blue solid squares) at $T=20\mathrm{K}$. The orange empty squares represent a rigid-band approximation calculation with a rigid shift of $E_F$ from that of $x=0.15$ to that of $x=0.30$ in constructing ${\mathrm{\Sigma }}_{n\mathbf{k}}^{e\text{-ph}}$ (i.e., the band structure, phonon dispersions, and $e\text{-ph}$ matrix elements remain unchanged from the $x=0.15$ level), resulting in the elevation of the self-energy tail, as indicated by the arrow. (c) Band structure of LSCO at $x=0.15$ and $x=0.30$. The black arrow points to the position of the kink along the $\mathrm{\Gamma }\text{−}X$ nodal direction. (d),(e) DOS of LSCO at the indicated two doping levels. The scatterings (indicated by arrow-headed dashed lines) of quasihole from $E_1$ to higher-binding-energy hole final states with significant DOS have larger energy separations [thus contributing less to $\mathrm{Re}{\overline{\mathrm{\Sigma }}}_{n{\mathbf{k}}_1}^{e\text{-ph}}(E_1)$] at $x=0.15$ than at $x=0.30$.