Aspects of electron-phonon self-energy revealed from angle-resolved photoemission spectroscopy

Lattice contribution to the electronic self-energy in complex correlated oxides is a fascinating subject that has lately stimulated lively discussions. Expectations of electron-phonon self-energy effects for simpler materials, such as Pd and Al, have resulted in several misconceptions in strongly correlated oxides. Here, we analyze a number of arguments claiming that phonons cannot be the origin of certain self-energy effects seen in high-$T_c$ cuprate superconductors via angle-resolved photoemission experiments, including the temperature dependence, doping dependence of the renormalization effects, the interband scattering in the bilayer systems, and impurity substitution. We show that in light of experimental evidences and detailed simulations, these arguments are not well founded.

Figure 1

(Color online) The calculated (a) real part $\mathrm{Re}\Sigma$, (b) imaginary part of the self-energy $\mathrm{Im}\Sigma$, and the corresponding spectral functions $A(k,\omega )$ in (c) normal state and (d) superconducting state. An extra $5\mathrm{meV}$ is added to the imaginary part of the self-energy in $120\mathrm{K}$ simulation to account for the thermal broadening of the quasiparticle lifetime. The location for this calculation is indicated in inset (a) by a red dot with a red curve representing the FS. Insets (c) and (d) are the data of optimally doped Bi2223 system $(T_c=110\mathrm{K})$ taken at the normal state $\left(25\mathrm{K}\right)$ and superconducting $\left(120\mathrm{K}\right)$ (Ref. 18), respectively.

Figure 2

(Color online) The intensity plots of the (a) spectral functions without resolution convolution and (b) resolution convoluted spectral functions in the superconducting state $\left(10\mathrm{K}\right)$ along the nodal direction, as indicated in inset (b) by the blue line. Black curves are the band dispersion extracted from the maximum positions of the momentum distribution curves, which cut the spectral functions at a fixed energy. The MDC-derived dispersions in (a) exhibit three sharp “subkinks” due to the coupling to the three phonon modes used in the model, while in (b) the subkinks are washed out by the finite instrument resolution effect leaving an apparent single kink in the band dispersion. The white dashed line illustrates the bare band for extracting $\mathrm{Re}\Sigma$ shown in Fig. 3a.

Figure 3

(Color online) (a) The $\mathrm{Re}\Sigma$ extracted from Fig. 2a (dashed lines) and Fig. 2b (solid lines) by subtracting a linear bare band [dashed line in Fig. 2a] from the band dispersion. The arrows indicate the maximum positions of the $\mathrm{Re}\Sigma$ where the “single” apparent kink in the band dispersion is usually defined. (b) Summary of the doping dependence of the apparent kink energy and the apparent mode energy extracted by assuming a single mode scenario.

Figure 4

(Color online) The illustration of the gerade and ungerade $c$-axis phonons. The eigenmode of the gerade (ungerade) phonons is even (odd) with respect to the mirror plane between two $\mathrm{Cu}{\mathrm{O}}_2$ layers at $q_z=0$, while their definition swapped at $q_z=\pi ∕c$. The black, gray, and white circles represent the Cu, Ca, and O atoms, respectively.