Band-Edge Quasiparticles from Electron-Phonon Coupling and Resistivity Saturation

We address the problem of resistivity saturation observed in materials such as the $A$-15 compounds. To do so, we calculate the resistivity for the Hubbard-Holstein model in infinite spatial dimensions to second order in on-site repulsion $U\le D$ and to first order in (dimensionless) electron-phonon coupling strength $\lambda \le 0.5$, where $D$ is the half bandwidth. We identify a unique mechanism to obtain two parallel quantum conducting channels: low-energy and band-edge high-energy quasi-particles. We identify the source of the hitherto unremarked high-energy quasiparticles as a positive slope in the frequency dependence of the real part of the electron self-energy. In the presence of phonons, the self-energy grows linearly with the temperature at high $T$, causing the resistivity to saturate. As $U$ is increased, the saturation temperature is pushed to higher values, offering a mechanism by which electron correlations destroy saturation.

Figure 1

The dc resistivity vs temperature for the Hubbard-Holstein model for three sets of parameters: [$(U/D)=0.5$; $\lambda =0.25$], [$(U/D)=0.5$; $\lambda =0.5$], and [$(U/D)=1$; $\lambda =0.5$], with $D$ estimated as 2000 K. The prolonged region of negative curvature found in the middle set is observed in the $A$-15 compounds ${\mathrm{Nb}}_3\mathrm{Sn}$ and ${\mathrm{Nb}}_3\mathrm{Sb}$ [2].

Figure 2

The LDOS at $T=1000\mathrm{K}$ for the same sets of parameters as in Fig. 1. The peaks located beyond the edges of the bare band are signatures of the high-energy quasiparticles, and are a prediction of our theory which can be tested using STM. Figures 4, 8 display the $T$ variation of $A$.

Figure 3

${\rho }_{\mathrm{dc}}$ vs $T$ for $\lambda =0.25$, 0.5, and $U=0$. As $\lambda$ increases, $T_{\max }$ decreases, while the height of the peak increases. In the $T\to \infty$ limit, the resistivity curves collapse onto a straight line.

Figure 4

The LDOS for $\lambda =0.5$ and $U=0$ at $T=0.1$, 0.4, 1, 3. For $T\lesssim T_{\max }$, the central peak is a signature of the low-energy quasiparticles, while for $T\gtrsim T_{\max }$, the two high-energy peaks are signatures of the high-energy quasiparticles. The high-energy peaks get pushed to higher energies with increasing temperature. As the scattering rate of the high-energy quasiparticles decreases, so does the resistivity (Fig. 3).

Figure 5

${\rho }_{\mathrm{\Sigma }}$ and $\Re e\mathrm{\Sigma }$ for $\lambda =0.5$ and $U=0$ at $T=0.04$. This relatively low temperature is already in the semiclassical regime ($T\gtrsim {\omega }_0$), where Eqs. (8) and (9) are excellent approximations. The positive slope of $\Re e\mathrm{\Sigma }$ is responsible for the high-energy quasiparticles (Fig. 4).

Figure 6

The resistivity vs temperature curve for $\lambda =0$, 0.25, 0.5, and $U=1$. For $\lambda \ne 0$, the peak shifts to the left and becomes sharper with increasing $\lambda$. Note that when $T$ is restricted to experimentally relevant values, i.e., $T\lesssim 0.5D$, only the left half of the peak appears, mimicking saturation (see Fig. 1).

Figure 7

The resistivity vs temperature curve for $\lambda =0.5$ and $U=0.1$, 0.5, 1.0. $T_{\max }$ increases with increasing $U$.

Figure 8

The LDOS for $\lambda =0.5$ and $U=1$ at $T=0.1$, 0.4, 1, 3. For $T\lesssim T_{\max }$, the central peak is a signature of the low-energy quasiparticles, while for $T\gtrsim T_{\max }$, the high-energy peaks are a signature of the high-energy quasiparticles.