Electronic thermal conductivity at high temperatures: Violation of the Wiedemann-Franz law in narrow-band metals

We study the electronic part of the thermal conductivity $\kappa$ of metals. We present two methods for calculating $\kappa$, a quantum Monte-Carlo method and a method where the phonons but not the electrons are treated semiclassically (SC). We compare the two methods for a model of alkali-doped ${\mathrm{C}}_{60}$, ${\mathrm{A}}_3{\mathrm{C}}_{60}$, and show that they agree well. We then mainly use the SC method, which is simpler and easier to interpret. We perform SC calculations for Nb for large temperatures $T$ and find that $\kappa$ increases with $T$ as $\kappa \left(T\right)=a+bT$, where $a$ and $b$ are constants, consistent with a saturation of the mean free path, $l$, and in good agreement with experiment. In contrast, we find that for ${\mathrm{A}}_3{\mathrm{C}}_{60}$, $\kappa \left(T\right)$ decreases with $T$ for very large $T$. We discuss qualitatively the reason for this in the limit of large $T$. We give a quantum-mechanical explanation of the saturation of $l$ for Nb and derive the Wiedemann-Franz law in the limit of $T⪡W$, where $W$ is the bandwidth. In contrast, due to the small $W$ of ${\mathrm{A}}_3{\mathrm{C}}_{60}$, the assumption $T⪡W$ can be violated. We show that this leads to $\kappa \left(T\right)\sim T^{-3∕2}$ for very large $T$ and a strong violation of the Wiedemann-Franz law.

Figure 1

(a) Thermal conductivity $\kappa \left(\omega \right)$ of ${\mathrm{A}}_3{\mathrm{C}}_{60}$ as a function of $\omega$ according to QMC (full line), SC (dashed line), and SCMEM (dot-dashed line). (b) The same as in (a) but with the renormalization of the phonon frequency suppressed. The parameters are ${\omega }_{\mathrm{ph}}=0.1\mathrm{eV}$, $\lambda =0.5$, and $T=0.1\mathrm{eV}$. In (a) ${\omega }_{\mathrm{ph}}$ is renormalized to ${\omega }_{\mathrm{eff}}=0.053\mathrm{eV}$. The model of ${\mathrm{A}}_3{\mathrm{C}}_{60}$ described in Sec. 2 was used. The calculations were done for a cluster with $3\times 4\times 4$ sites and averaged over 20 000 different random configurations. A Gaussian broadening with FWHM $0.03\mathrm{eV}$ was used. The figure illustrates that the QMC and SCMEM calculations agree very well and that the downturn at $\omega =0$ in the SC results is lost in the SCMEM results due to the MEM procedure.

Figure 2

Thermal conductivity of Nb metal according to the SC method (full line) and from experiments [“Peletskii1” (Ref. 16), “Moore” (Ref. 17), “Binkele” (Ref. 18), “Peletskii2” (Ref. 19] together with the line $\kappa \left(T\right)=47.5+0.013T$ adjusted to the experimental results. The calculated result was obtained for a $10\times 10\times 10$ lattice using a Gaussian broadening with the FWHM of $0.06\mathrm{eV}$. The results were averaged over ten different random displacements of the atoms. The model of Nb described in Sec. 2 was used. The figure shows that theory and experiment agree rather well and that $\kappa \left(T\right)$ increases approximately linearly with $T$.

Figure 3

Thermal conductivity of the ${\mathrm{C}}_{60}$ model according to the SC method for $\lambda =0.5$ and ${\omega }_{\mathrm{ph}}=0.1\mathrm{eV}$. The inset shows that ratio $\kappa ∕\left(\sigma LT\right)$, where $L$ is the Lorentz ratio. 400 random configurations for a $8\times 8\times 8$ cluster were generated, and a Gaussian broadening of $0.03\mathrm{eV}$ FWHM was used. The figure illustrates that $\kappa$ drops for large $T$, leading to a strong violation of the Wiedemann-Franz law.

Figure 4

Functions $g(\hslash \omega ,T)$ and $h(\hslash \omega ,T)$, defined in Eq. (37), as a function of $\omega$ for different values of $k_{B}T=0.25$, 0.5, 1.0, 1.5, and $2.0\mathrm{eV}$ calculated using the SC method. The parameters are $\lambda =0.5$ and ${\omega }_{\mathrm{ph}}=0.1\mathrm{eV}$. 400 random configurations for a $8\times 8\times 8$ cluster were generated, and a Gaussian broadening of $0.03\mathrm{eV}$ FWHM was used. If the approximations behind Eqs. (35, 36) were exact, the curves would fall on top of each other.