Phonon-induced pairing interaction in ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_7$ within the local-density approximation

The properties of the phonon-induced interaction between electrons are studied using the local-density approximation (LDA). Restricting the electron momenta to the Fermi surface we find generally that this interaction has a pronounced peak for large momentum transfers and that the interband contributions between bonding and antibonding band are of the same magnitude as the intraband ones. Results are given for various symmetry averages of this interaction over the Fermi surface. In particular, we find that the dimensionless coupling constant in the $d$-wave channel ${\lambda }^d$, relevant for superconductivity, is only 0.022, i.e., even about 10 times smaller than the small value of the $s$-wave channel. Similarly, the LDA contribution to the resistivity is about a factor 10 times smaller than the observed resistivity suggesting that phonons are not the important low-energy excitations in high-$T_c$ oxides.

Figure 1

Phonon-induced interaction $V\left(\mathbf{k}B,{\mathbf{k}}^{\prime }B,0\right)$ (upper part) and $V\left(\mathbf{k}A,{\mathbf{k}}^{\prime }B,0\right)$ (lower part) for a fixed momentum $\mathbf{k}$ at the nodal (left diagram) and antinodal (right diagram) Fermi point as a function of ${\mathbf{k}}^{\prime }$, encircling the Fermi surface around the point $S=\left(\pi ,\pi \right)$ in anticlock direction, starting from the antinodal point (see upper left inset). Insets show the considered mesh points near the Fermi surface described by the solid line.

Figure 2

Phonon-induced interaction $V\left(\mathbf{k}A,{\mathbf{k}}^{\prime }A,0\right)$ (upper part) and $V\left(\mathbf{k}B,{\mathbf{k}}^{\prime }A,0\right)$ (lower part) for a fixed momentum $\mathbf{k}$ at the nodal (left diagrams) and antinodal (right diagrams) Fermi point as a function of ${\mathbf{k}}^{\prime }$, encircling the Fermi surface similar as in Fig. 1. Insets show the considered mesh points near the Fermi surface described by the solid lines.

Figure 3

Phonon-induced interaction $V\left(\mathbf{k}B,{\mathbf{k}}^{\prime }C,0\right)$ (upper part) and $V\left(\mathbf{k}A,{\mathbf{k}}^{\prime }C,0\right)$ (lower part) for a fixed momentum $\mathbf{k}$ at the nodal (left diagrams) and antinodal (right diagrams) Fermi point as a function of ${\mathbf{k}}^{\prime }$, encircling the Fermi surface similar as in Fig. 1. Insets show the considered mesh points near the Fermi surface described by the solid lines.

Figure 4

(Color online) Coupling functions ${\lambda }_A\left(\mathbf{k}\right)$ (red squares joined by straight lines), ${\lambda }_B\left(\mathbf{k}\right)$ (black circles), and ${\lambda }_C\left(\mathbf{k}\right)$ (blue triangles) for $k_z=0.125$ (filled symbols) and $k_z=0.375$ (empty symbols). Angle $\alpha$ is the same as in Figs. 1, 2, 3.

Figure 5

(Color online) Coupling functions ${\lambda }^s\left(\omega \right)$ (black dashed line) and ${\lambda }^d\left(\omega \right)$ (red solid line) as a function of frequency using a width of 1 meV for the phonon modes.

Figure 6

(Color online) Upper panel: coupling functions ${\lambda }^s\left(\omega \right)$ (black dashed line) and ${\lambda }^{s^{\prime }}\left(\omega \right)$ (red solid line) as a function of frequency using a width of 1 meV for the phonon modes. Lower panel: coupling functions ${\lambda }^{p_x}\left(\omega \right)$ (blue dashed line) and ${\lambda }^{p_y}\left(\omega \right)$ (green solid line) as a function of frequency using a width of 1 meV for the phonon modes.

Figure 7

(Color online) Temperature dependence of the resistivity ${\rho }_i$ along the direction $i$.