Momentum Dependence of the Electron-Phonon Coupling and Self-Energy Effects in Superconducting ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_7$ within the Local Density Approximation

Using the local density approximation and a realistic phonon spectrum we determine the momentum and frequency dependence of ${\alpha }^{2}F(\mathbf{k},\omega )$ in ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_7$ for the bonding, antibonding, and chain band. The resulting self-energy $\Sigma$ is rather small near the Fermi surface. For instance, for the antibonding band the maximum of $\mathrm{Re}\Sigma$ as a function of frequency is about 7 meV at the nodal point in the normal state and the ratio of bare and renormalized Fermi velocities is 1.18. These values are a factor of 3–5 too small compared to the experiment showing that only a small part of $\Sigma$ can be attributed to phonons. Furthermore, the frequency dependence of the renormalization factor $Z(\mathbf{k},\omega )$ is smooth and has no anomalies at the observed kink frequencies which means that phonons cannot produce well-pronounced kinks in stoichiometric ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_7$, at least, within the local density approximation.

Figure 1

Coupling function ${\lambda }_{\nu }(\mathbf{k},\omega )$ integrated over $\omega$ between 0 and 50 meV (left diagrams) and $\omega$ larger than 50 meV (right diagrams). Shown are $xy$ cuts through the irreducible Brillouin zone ($0\le k_x,k_y\le 0.5$) for $k_z=0.125$. The upper plots refer to the bonding, the middle ones to the antibonding, and the lower ones to the chain band. The latter two are hybridized in the $(0.5,0)$ direction. The gray areas represent states with energies far away ($\ge 0.8\mathrm{eV}$) from the Fermi energy.

Figure 2

Momentum and frequency-resolved coupling function $\lambda (\mathbf{k},\omega )$ in $1/\mathrm{meV}$ of the antibonding band near the Fermi surface in the nodal (upper diagram) and the antinodal (lower diagram) direction. Solid and dotted lines correspond to $k_z=0.375$ and 0.125, respectively.

Figure 3

Real and imaginary parts of the self-energy $\Sigma$ of the antibonding band near the Fermi points along the nodal and antinodal directions at the temperature $T=1\mathrm{meV}$.

Figure 4

Renormalization function $Z(\mathbf{k},\omega )$ of the antibonding band near the Fermi points in nodal and antinodal directions for $k_z=0.125$ at the temperature $T=1\mathrm{meV}$.

Figure 5

Left: Electronic dispersion; right: spectral function of the antibonding band for different momenta $k_x=k_y$ along the nodal direction using $T=1\mathrm{meV}$ and $k_z=0.125$.