# Why thermal conductivity increases with temperature?

what is the molecular mechanism with which thermal conductivity increases by increasing temperature? at least for metals? I know that heat increases the oscillations of the atoms in the crystal. But how that explains the increase in thermal conductivity?

• In metals conductivity is primarily due to free electrons. Following the Wiedemann–Franz law, thermal conductivity of metals is approximately proportional to the absolute temperature (in kelvin) times electrical conductivity. In pure metals the electrical conductivity decreases with increasing temperature and thus the product of the two, the thermal conductivity, stays approximately constant. In alloys the change in electrical conductivity is usually smaller and thus thermal conductivity increases with temperature, often proportionally to temperature. – jim May 23 '16 at 21:03
• you said "In metals conductivity is primarily due to free electrons", are you refering for thermal conductivity? I understand that for electrical conductivity this is true, but for thermal I am not sure. Any reference? – ergon May 23 '16 at 22:53

The mechanism for increasing the thermal conductivity is phonon assisted hopping. For a disordered system, one which do not preserve the long range order, the electronic wave function becomes localized. The wave function extent is typically much smaller than the system size and is characterized by the localization length $\xi$, a parameter in theory. In this case, electron propagates by hopping events along the applied electric field. The tunneling rate for an electron to hop from a localized state $i$ to another state $j$ is proportional (exponentially) to the distance between those states $r_{ij}$ and temperature $$\Gamma_{ij}=\gamma_0\exp(-\frac{r_{ij}}{\xi}-\frac{|E_i-E_j|}{T})$$ where $\gamma_0$ is another parameter of the theory, that depends of phonon DOS and electron-phonon coupling. From this equation you may see how $\Gamma$ (and hence conductivity) depends on $T$.
Yes, heat increases atomic displacements, but this apples to "normal" (not disordered) metal. As a result, electron scattering increases and conductivity decreases. Note that this is related to different conduction mechanism. If a system studied shows increasing conductivity with $T$ - it is a hopping conduction.