# Conductors and Insulators from the point of view of Quantum Mechanics

So, I was watching this lecture of MIT 8.04 Quantum Mechanics course and at around 38:00, the instructor starts discussing periodic potentials to predict the properties of conductors and insulators.

In brief, if we have a 1-D periodic lattice, such that each well in the lattice corresponds to an atom with some integer number of electrons in each well, we end up with a band structure where states don't overlap as a consequence of non-degenerate states in 1-D and where bands are completely filled. To set the electrons in motion, we need to excite electrons to higher levels which is possible only if we provide energy $$E$$ greater than the band gap $$\Delta E_{gap}$$ i.e. $$E>\Delta E_{gap}$$. Such materials are called insulators.

But instead, if these wells were in 3-D, overlapping of bands would have been possible and this would have lead to presence of partially filled bands. So, exciting electrons to next highest energy states requires preposterously low energies. Such materials are called conductors.

Now, it is not as if insulators are 1-D lattices whereas conductors are 3-D lattices. Both are 3-D structures. So why is that insulators are related to 1-D periodic lattices while conductors are related to 3-D periodic lattices?

In some materials the topmost energy band with electrons is filled to the top, whereas the next band contains no electrons. Such materials are called insulators, and the two bands are called respectively valence band and conduction band. The energy difference between the lowest energy of the conduction band and the highest energy of the valence band is called *energy gap, usually written as $$E_g$$. In other materials the topmost band with electrons is only partially filled, and is called conduction band. These are metals.