Very often we care about the electrical properties of materials. Conceptually, we imagine hooking the positive terminal of a battery up to one end of a material, hooking the negative terminal up to the other end, and checking to see if any current is flowing. We broadly lump solids into two groups, those that conduct electricity and those that don't. Materials in the latter category are known as insulators, and it turns out that there are at least three different kinds.
- Band insulators. One useful way of thinking about electrons in solids is to think about the electrons as filling up single-particle states (typically with two electrons per state). This is like what you learn in high school chemistry, where you're taught that there are certain orbitals within atoms that get filled up, two electrons per orbital. Helium has two electrons in the 1s orbital, for example. In solids, there are many, many states, each one with an associated energy cost for being occupied by an electron, and the states are grouped into bands separated by intervals of energy (band gaps) with no states. (Picture a ladder with groups of closely spaced rungs, and each rung has two little divots where marbles (the electrons) can sit.) Now, in clean materials, you can think of some states as corresponding to electrons moving to the left. Some states correspond to electrons moving to the right. In order to get a net flow of electrons when a battery is used to apply a voltage difference across a slab of material, there have to be transitions that, for example, take electrons out of left-moving states and put them into right-moving states, so that more electrons are going one way than the other. For this to happen, there have to be empty states available for the electrons to occupy, and the net energy cost of shifting the electrons around has to be low enough that it's supplied by the battery or by thermal energy. In a band insulator, all of the states in a particular band (usually called the valence band) are filled, and the energetically closest empty states are too far away energetically to be reached. (In the ladder analogy, the next empty rung is waaay far up the ladder.) This is the situation in materials like diamond, quartz, and sapphire.
- Anderson insulators. These are materials where disorder is responsible for insulating behavior. In the ladder analogy above, each rung of the ladder corresponded to what we would call an "extended" state. To get a picture of what this means, consider looking at a smooth, grooved surface, like a freshly plowed field, and filling it partially with water. Each furrow would be an extended state, since on a level field water would extend along the furrow from one end of the field to the other. Now, a disordered system in this analogy would look more like a field pockmarked with hills and holes. Water (representing the electrons) would pool in the low spots rather than forming a continuous line from one end of the field to the other. These local low spots are defects, and the puddles of water correspond to localized states. In the real quantum situation things are a bit more complicated. Because of the wavelike nature of electrons, even weak disorder (shallow dips rather than deep holes in the field) can lead to reflections and interference effects that can cause states to be localized on a big enough "field". Systems like this are insulating (at least at low temperatures) because it takes energy to hop electrons from one puddle to another puddle. For small applied voltages, nothing happens (though clearly if one imagines tilting the whole field enough, all the water will run down hill - this would correspond to applying a large electric field.). Examples of this kind of insulating behavior include doped polymer semiconductors.
- Mott insulators. Notice that nowhere in the discussion of band or Anderson insulators did I say anything at all about the fact that electrons repel each other. Electron-electron interactions were essentially irrelevant to those two ways of having an insulator. To understand Mott insulators, think about trying to pack ping-pong balls closely in a 2d array. The balls form a triangular lattice. Now the repulsion of the electrons is represented by the fact that you can't force two ping-pong balls to occupy the same site in the 2d lattice. Even though you "should" be able to put two balls (electrons) per site, the repulsion of the electrons prevents you from doing so without comparatively great energetic cost (associated with smashing a ping-pong ball). The result is, for exactly 1 ball (electron) per site ("half-filled band") in this situation dominated by ball-ball interactions ("on-site repulsion"), no balls are able to move in response to an applied push (electric field). To get motion (conduction) in this case, one approach is to remove some of the balls (electrons) to create vacancies in the lattice. This can be done via chemical doping. Examples of Mott insulators are some transition metal oxides like V2O3 and the parent compounds of the high temperature superconductors.