From WolfWikis

Metals

The fib-in-a-box model

The simplest quantum mechanical model of a metal is essentially that of the infamous particle-in-a-box. I.e. we assume that the electrons can move freely as an electron gas inside the (say cubic) box of the metal (say crystal).

In a 1D 'box' the wave functions are pretty Bloch-like until we put the confinement by the size of the box into consideration. The walls force the waves to extinguish and that reduces the Bloch function ψ= exp(ikx) = cos(kx)+ i.sin(kx) to be zero at the edges. That reduces them to sine waves.

The energies are interestingly quadratic in k.: E~k²

We can therefore make a nice picture of energy in reciprocal space. It is a parabola. Such pictures are known as dispersion curves. We should note that it looks like a continuous curve, but it really consists of discrete energy levels (like the pixels of an image). The number of energy levels is finite if the box is finite, but N can be very large, like the number of Avogadro or so. We refer to such sets of energy states as bands and in the case of the particle-in-the-box this band is parabolic'.

Looking a little more closely at the math is pretty revealing. The complete expression of E(k), our dispersion curve is:

E(k) = [ħ²/2m]k²

and the momentum p is:

p =ħ.k

Here we see yet another meaning of our spatial frequency aka resolution k: it also symbolizes momentum.

If we take the curvature of the E curve -its second derivative versus k- we get a constant:

∂²E/∂k²= 2[ħ²/2m]=[ħ²/m]

Thus the mass of the particle is:

m=[ħ²/∂²E/∂k²]

And this is a constant over the whole curve. Maybe this looks like a rather academic exercise at this point but that will change soon enough.

Stop fibbing please

Of course all this is quite a fib. Maybe if we could put a gas of neutrons in a box this might apply but electrons are charged particles that would interact strongly with each other and we are ignoring that completely. It is only because the metal atoms are present that those interactions are compensated. Partly that is. The closest to a free electron gas is probably the single valence electron metal cesium, but even there the presence of the Cs⁺ is noticed by electrons in ways other than just to compensate their charges.

The periodic potential

As the cesium ions are stacked in a regular lattice with translation symmetry we should examine what their contribution to the potential energy V to the Hamiltonian H is.

If we assume the contribution is weak we really only need to consider what it does to the symmetry that the electrons experience. In the empty box symmetry is uniform, i.e. the potential is same everywhere.

We could also say that we have translation symmetry with an infinitesimally small unit cell. You can apply any translation you want, even tiny ones and you still have the same potential. In terms of sampling frequency this sends the Nyqvist frequency to infinity! Thus, we have only one giant first Brillouin zone within which all wave functions are independent, because within a BZ each k-value represents its own irrep.

The presence of the cesium ions causes the symmetry to become much lower. Their potential is periodic rather than uniform. This means that the smallest translation vector is now finite in size and so is our Nyqvist frequency. This imposes Brillouin zone boundaries on the reciprocal space and we know that any wave function differing by a whole Q vector from another now transforms are the same irrep. This is why we can now move them over into the first BZ. Any states related vertically can now mix. That means there is energy to be gained by making bonding and antibonding combinations of them!

This is particularly true if two wave functions are close in energy such as the wave functions for k=-1/2 and k=+1/2 at the edges of the first Brillouin zone. Without periodic potential their energies would be identical. With the periodic symmetry we should combine them as

φ₊= ψ_-1/2 + ψ_+1/2

φ_-= ψ_-1/2 - ψ_+1/2

These new wave functions should have different energies. Much in the way of the hydrogen molecule one goes up the other down: the states split.

Band splitting due to periodic potential
Notice the alias shift denoted by the red arrow

For functions close to the zone edge the same applies but less so because the functions at k and k+Q increasingly differ in energy when moving away from the edge. The mixing will lead to less splitting.

The result is that the single parabolic energy band of the free electron box splits up in a number of sub bands separated by band gaps.

The curvature in relation to the effective mass

If you examine the bottom band, you will see that around the Γ₀ irrep at k=0 (often simply denoted as the Γ point) the band still has more or less the original parabolic shape, because any states that ended up vertically above it by the +Q shift are too far away in energy to really interact efficiently. Towards the edge of the Brillouin zone however the curvature flips from convex to concave because the mixing with the upper states gets stronger and stronger . The change in curvature affects our 'calculation' of the mass of the electron. Above we found one and the same constant mass from:

m=[ħ²/∂²E/∂k²]

Now this value will depend on the value of k and at the edge of the Brillouin it will actually be negative for the top of the lower band (usually called the valence band). Phyiscally it means that an electron in such a state behaves abit like a donkey: pull its tail backwards and it moves forward. We could also treat it as a particle with opposite positive charge if the force we apply is electrical -which it usually is- and therefore it is often called a hole h⁺.

Band filling

If the particles we stuff into the band of allowed energy states are fermions as in the case of electrons, we must apply the exclusion principle. Each state on the ribbon can only take up two electrons, one spin up one spin down. In that case it is possible to fill up the band all the way to the top (depending on how many valence electrons are available. The HOMO and the LUMO are then separated by the gap created by the periodic potential. A material where the Fermi level (up to where states are filled) is in the band gap is a semiconductor. If the Fermi level is somewhere in the middle of a band we have a metal.

For bosons the situation is quite different. There each state can be occupied many times simultaneously. A good example of such a case are phonons. They are the vibrational modes of our periodic lattice rather than the electronic ones. Although now we are dealing with the motions of sluggish big atoms rather than fleet-footed electrons, the symmetry of our problem is still the same and so their energy sates also form bands in very similar ways. Of course the energies are very different and the states can be multiply occupied.