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Typically materials which are good conductor of electricity are also good conductor of heat, and vice versa. Are there notable exceptions?

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Beryllium oxide is a very good electrical insulator but at the same time the best non-metal (except diamond) thermal conductor.

So to summarize. In general, good thermal conductivity is correlated with good electrical conductivity, but it is not a strict relationship.

For example, there is the empirical Wiedemann-Franz law for metals which states that the ratio of thermal conductivity $k$ and electrical conductivity $\sigma$ is proportional to temperature $T$.

$$ \frac{k}{\sigma} =c_0T $$

The proportionality constant is $c_0$. But as the wiki article points out there are exceptions to this empirical relationship.

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For metals, good electrical conductivity does indeed imply good thermal conductivity. This is known from the Wiedemann–Franz law, which gives the ratio between electronic contribution of thermal conductivity ($\lambda$) and electrical conductivity ($\sigma$) and is proportional to the temperature ($T$).

$$\frac{\lambda}{\sigma} = LT $$

This gives the empirical constant $L$ known as the Lorenz number.

$$ L = \frac{\lambda}{\sigma T} = \frac{\pi^2}{3}(\frac{\lambda_B}{e})^2 = 2.44 \times 10^{-8} W\Omega K^{-2} $$

As stated, this law applies to metals. Diamond, for example, is an excellent thermal conductor because of its structure, but at the same time it has a very high resistance to electrical current.

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    $\begingroup$ And, naturally, it gets a little more interesting when playing with semimetals :-) . By which I just mean complicated. Electrons are pretty good at carrying energy. $\endgroup$ – Carl Witthoft May 14 '17 at 11:29
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As alluded to in the other answers, a most notable exception is diamond. Diamond is an excellent thermal conductor. The thermal conductivity of natural diamond is around 22 W/(cm·K), which makes diamond five times better than copper at conducting heat. At the same time, the electrical resistivity of most diamonds is on the order of 10E11 to 10E18 Ω·m.

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