PHY.K02UF Molecular and Solid State Physics
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PHY.K02UF Molecular and Solid State Physics | ||||
The thermal conductivity describes how heat flows through a solid. The heat current $\vec{j}_Q$ is related to the gradient of the temperature by the thermal conductivity $K$,
$$\vec{j}_Q = -K\nabla T.$$Both the electrons and phonons can contribute to the thermal conductivity. They carry the heat as they diffuse from the hot side to the cold side of the sample. In semiconductors and insulators, the electrical contribution is negligible, but for metals, the electron contribution can dominate the thermal conductivity. The Einstein relation states that the mobility is proportional to the diffusion constant, so a metal with a high mobility will have a large diffusion constant. Consequently, good electrical conductors are good thermal conductors. This is known as the Wiedemann-Franz law,
$$\frac{K_e}{\sigma} = LT.$$Here $K_e$ is the electron contribution to the thermal conductivity and $L$ is called the Lorentz number. In the quantum theory of free electrons,
$$L = \frac{\pi^2 k_B^2}{3e^2} = 2.44\times 10^{-8}\,\text{W}\,\Omega\,\text{K}^{-2}.$$Generally, the thermal conductivity goes to zero as $T\rightarrow 0$. There is a peak in thermal conductivity at low temperature, and the thermal conductivity decreases as $1/T$ near room temperature. Below, the thermal conductivity of silicon, diamond, copper, and aluminum can be plotted.
| $K$ [W/cm K] | |
$T$ [K] |
The thermal conductivity of silicon.[1]