Why theory of energy bands works?

[jostpuur]

The standard explanation, for why some materials are conductors, and some are insulators, is that they have different band structure. An insulator has only the valence band filled, and no electrons on the conduction band. An conductor instead has electrons also on the conduction band.

The explanation looks like an explanation for a while, but I'm forced to ask a dumb question here: Why precisely is the lowest band a kind of band where electrons are not spatially mobile, and why are the electrons on the second band spatially mobile instead?

Does the electrons ability to move in the momentum space have something to do with their ability to move in the spatial space?

In fact this entire picture of conduction being explained by energy bands seems very strange. If we assume that the electrons are on some bands, we are already assuming that they are localized in the momentum space, and thus we are assuming the they are not localized in the spatial space. The conduction of electricity means the electrons move. How can phenomena like this be explained with a model, where electrons are assumed to be not spatially localized? The location of an electron cannot really move, if the electron is badly delocalized, since then the location doesn't exist.

 

[Riogho]

Are you familiar with electron configuration?

 

[jostpuur]

Are you familiar with electron configuration?

I'm not sure. If it has some specific well defined meaning in this context, then maybe I'm not.

I'm familiar with quantum mechanics, at least. I know something about the energy bands, to the extent that I was able to ask the above questions.

 

[BANG!]

Since you know a bit of quantum mechanics then you are probably pretty familiar with the "particle in a box" potential well. The solution to this problem reveals that the particle must absorb energy in discreet levels. If you solve for more difficult potential, ie: harmonic oscillator, finite well, coulomb potential, etc. you find that these also result in discreet energy levels. This implies that a very small push will often have no effect on the particle. However, all of these potentials assume that you have a single potential acting on the particle.

If you bring in another potential (introduce coupling) then you find that the energy levels can split into closely spaced levels. Indeed, if you continue to bring in more and more potentials then the original energy level splits into what, for all practical purposes, can be considered a continuous band of energies. Now, if an electon sits in the middle of a band then a small push will excite it. This is the basic idea of how a conduction band occurs in nature.

As far a models are concerned, you can begin with a fermi gas, which assumes that a solid is nothing but an empty box with non-interacting electrons zipping around. Quite amazingly, the solutions of this extremely crude 3D infinite well gives some very good approximations for things like conductivity and heat capacity. However, it is entirely useless for explaining why some elements are metals and others are insulators. But now if you introduce a periodic potential such as a repeating Delta potential then you now find that the solutions have the expected energy bands.

 

[BANG!]

If we assume that the electrons are on some bands, we are already assuming that they are localized in the momentum space, and thus we are assuming the they are not localized in the spatial space.

I'm not sure what you are driving at here. Why exactly are they localized in momentum space. At most, we can model the Fermi Surface in K-space which gives us all the levels at the ground state energy in terms of momentum quantum numbers. There are typically lots of k that satisfy the same energy. Moreover, in a band, thermal fluctions are sufficient to drive the particles off of the fermi surface.

The conduction of electricity means the electrons move. How can phenomena like this be explained with a model, where electrons are assumed to be not spatially localized? The location of an electron cannot really move, if the electron is badly delocalized, since then the location doesn't exist.

When modeling the motion of quantum particles you can calculate what is known as a probablility density current. The probablitiy wave moves through space with the passage of time and "carries" the electron with it.

 

[kanato]

Why precisely is the lowest band a kind of band where electrons are not spatially mobile, and why are the electrons on the second band spatially mobile instead?

It's not. Even in an insulator, electrons move, and their velocities can be quite high, like 10^6 cm/s. The difference between the valence and conduction bands isn't that the electrons move, it's that in the valence band, all available states are filled.

If you apply a small electric field in, say, the x-direction, then you will slightly lower the energy of states with momentum in the -x direction and slightly raise the energy of states with momentum in the +x direction. Now the thing is, in an insulator, this does nothing to create a current because all the states are already filled. But in a metal, the electrons in the highest energy states (+x) will shift to the states that had their energy slightly lowered by the field (-x). This is what creates the net current. And the bias is also quite small, which is why the drift velocity of electrons is much smaller than their average speed.

Also, I wouldn't say that we assume electrons are localized in momentum space. If you were to expand a Bloch wavefunction for a particular wavevector $$k$$ in planewaves, it would be composed of planewaves with wavevectors $$k + n_1 G_1 + n_2 G_2 + n_3 G_3$$ where the $$n_i$$ are integers and $$G_i$$ are the reciprocal lattice vectors. So in momentum space, the wave function is really a series of delta functions on a specific set of momentum values.

 

[jostpuur]

If you apply a small electric field in, say, the x-direction, then you will slightly lower the energy of states with momentum in the -x direction and slightly raise the energy of states with momentum in the +x direction. Now the thing is, in an insulator, this does nothing to create a current because all the states are already filled. But in a metal, the electrons in the highest energy states (+x) will shift to the states that had their energy slightly lowered by the field (-x). This is what creates the net current. And the bias is also quite small, which is why the drift velocity of electrons is much smaller than their average speed.

mhmhmhhmh... yeah...

 

[Gokul43201]

The standard explanation, for why some materials are conductors, and some are insulators, is that they have different band structure. An insulator has only the valence band filled, and no electrons on the conduction band. An conductor instead has electrons also on the conduction band.

Hmmm...this is a bad description of the difference between a conductor and an insulator - the second sentence is not saying the same thing as the first. It is vital that there is a difference in the band structure itself, not just in the filling of the bands. If you are looking for differences only in filling, then you should ask about Mott-insulators rather than bandgap insulators.

The explanation looks like an explanation for a while, but I'm forced to ask a dumb question here: Why precisely is the lowest band a kind of band where electrons are not spatially mobile, and why are the electrons on the second band spatially mobile instead?

The ability to produce a current has nothing to do with the band index, and everything to do with whether or not there exist low lying excitations (i.e., unfilled states within the same band) in the system. A small electric field can produce a current only if an electrons can be excited to slightly higher momenta. If all higher momentum states are filled, then electrons can only exchange states, leaving the <p> unchanged for that band. You need to have low-lying unoccupied states to produce a current.

Also, remember that the existence of the band structure is not postulated, but derived from some other ansatz (either a many-body Hamiltonian, like in the Hubbard model, or a wavefunction, like LCAO).

 

[jostpuur]

I had not understood the connection between electrons ability to get on higher excitation states, and the current, since I was thinking about the spatial mobility of individual electrons.

If all higher momentum states are filled, then electrons can only exchange states, leaving the <p> unchanged for that band.

This explains it. Your and kanato's replies made me remember some pictures of the fermi surface being translated slightly under external electric field. The average value of the all electrons' momentum is of course the relevant quantity.

 

[jVincent]

Why precisely is the lowest band a kind of band where electrons are not spatially mobile, and why are the electrons on the second band spatially mobile instead?

Simply put, full bands are immobile and empty bands are immobile. Thus, if the first band is full, and the second half full, you have a metal, since you have mobile electrons. If the second band is empty, you have an insulator. Now if you have a very small band gap, you can use energy to exite some electrons from the first band to the second band, and thus turn an insulator into a metal. this is when you have a semiconductor.

Why is it so? Well to have a current running, you need to have more electrons going in one direction then the other, thus you must have more electrons with positive k, then negative k.
This cannot be done in a full band, since all possible k's are occupied, and no two electrons can have the same k, that would violate the pauli principle. In an empty band it's easy to see why this is not possible.