Thermal expansion and conductivity When thinking about how the lattice constant of silicon can be given up to eight decimal places without a remark for the temperature I realized that, it seems
most insulators and semiconductors seem to expand less than metals when exposed to heat.
Is there an intuitive connection between the band structure and the thermal expansion?
 A: Intuitive or not, a link between band structure and thermal expansion coefficient is hard to find. The reason is that the band structure is an equilibrium property of the un-distorted crystalline lattice and is directly connected to equilibrium positions, while thermal expansion coefficient is directly connected to anarmonic effects in the potential energy surface, i.e. to terms beyond quadratic approximation aroud the equilibrium structure. Said in a different way, expansion coefficient depends on  band structure changes in correspondence of large collective displacements. 
Even the metallic character of the bonds does not imply necessarily a larger thermal expansion coefficient. An example is Invar, a 36% nickel and 64% iron metallic alloy, which has a very low thermal expansion coefficient. 
A: I would say that your statement "most insulators and semiconductors seem to expand less than metals when exposed to heat" is simply false. Consider that rubber, plastics, and other organic insulators have huge thermal expansion coefficients compared to many metals. Ice, an insulator, has a 5x larger thermal expansion coefficient than platinum, a metal. Glass and titanium have roughly the same coefficient.
I think your observation is mostly limited to Silicon carbide, diamond, etc. where the lattice is just incredibly strong.
It is also worth mentioning that you can measure the average lattice spacing of any material at a finite temperature to many, many decimal places. All temperature does is broaden those positions, but you can still measure the center of the broadened peak to very high precision.
