Conductors and Insulators from the point of view of Quantum Mechanics

Question

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?

Answer 1

Periodic structure leads to appearance of energy bands - this is equally true in 1-D, 2-D or 3-D. Note that these are energy bands - overlap of the states is not exactly part of the description here (However, since we are talking about the eigenstates of the Hamiltonian, the states corresponding to different energies are orthogonal. One also speaks of overlap in the context of the tight-binding approach - one of the methods for calculating the energy bands.).

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.

This distinction is not related to 1-D vs. 3-D - indeed, all real materials are three-dimensional. Rather it has to do with the crystal structure, valence of the atoms making the solid, etc. Predicting whether a particular substance crystallizes into a metal or an insulator is a rather non-trivial task. In 3-D however the picture may be far more complicated than in 1-D models, e.g., the energy bands may overlap in the energy space, but not in momentum space, which breaks the neat distinction into metals and insulators described above.

Electric current indeed involves exciting electrons to higher momentum states. It is easy to understand: accelerating an electron means that its momentum increases. In insulators this is not possible, since the states with higher momenta are not available, unless the field is sufficiently strong to excite the electrons across the energy gap (i.e. to the conduction band). The gap width is usually several electron-Volts and such an excitation requires extremely strong electric fields. In practice electrons are usually excited across the gap with light. Photodetectors and solar cells are literally the devices where light excites electrons to the conduction band and thus makes the material conducting.