From WolfWikis

More realistic systems

The previous topic is rather amazing because it shows that bands, band gaps, effective masses, metals semiconductors even phonon bands can all be derived at from little more that the periodic nature (its translation symmetry) without going into any actual shapes of atoms, their wave functions their overlaps, their hybridization or anything physical. It almost too good to be true. Well actually, yes, our story is really far too simplistic to be true. Nevertheless, however much sophistication we wish to add to describe solids the above concepts are there to stay and will remain a powerful way to look at our systems.

Tight binding

So far all we took into account is the weak interactions between otherwise independent gas like entities if they happened to transform as the same irrep of our imposed periodic potential. Even for a metal like cesium this is an understatement of the interactions that are going on in this almost-free electron gas metal. In most materials the electrons experience far stronger interactions with the atoms in the lattice. Let us take hydrogen as an example. In such a cases it is better to start from the other end. Rather than assuming that electrons are completely free from the atom, let us assume they belong to a particular atom, and resides in the hydrogen 1s functions.

Let us start with a linear chain of hydrogen atoms at a unit distance of aÅ. If they are all at the same distance we once again have a system with one-dimensional translation symmetry and the size of the unit cell is a. If this distance is large we should have little overlap and making linear combinations of the atomic functions will not alter their energies. When a is small enough we could try to make linear combinations of the s functions and we best do that to comply with the translation symmetry. Thus we form new wave functions like:

Ψ = (...+ c₁s₁+ c₂s₂+ c₃s₃+ c₄s₄+ ... )/N

where we let the coefficients be the characters of each irrep of our translation group. We only need one Brillouin zone full because each atom (each unit cell) only contributes one s-function.

The combination at k=0 would look like:

Ψ_Γ = (...+ s₁+ s₂+ s₃+ s₄+ ... )/√N

At the (final) edge of Brillouin zone (Nyqvist) we would have

Ψ_M = (...+ s₁-s₂+ s₃- s₄+ ... )/√N

Halfway in between the combination would look like

Ψ_k=0.25 = (...+ s₁+ is₂- s₃-i.s₄+ ... )/√N

We can actually write these linear combinations rather compactly by multiplying the hydrogen 1s functions with a Bloch function:

Ψ_k = (Σexp(ikna).s(r-na))/√N

Note that we are now writing the s-functions as a function of the radial coordinate around a center point n.a. I.e. we are assuming a one-dimensional chain of three-dimensional hydrogen atoms. Such an object still has one-dimensional translation symmetry.

Due to differences in overlap between the neighbors in the chain, these wave functions will have different energies. Just look at the two atoms 1 and 2: The k=0 function has s₁+ s₂ which looks like the bonding MO for H₂, whereas for k=1/2 we have s₁ - s₂, which looks more like the antibonding MO! For the other combinations we will likely get intermediate energies and again we will get an energy band. If we get down to actually putting in the overlap integrals we can even calculate what is should look like, but this is an exercise in futility because our model is not stable.

Peierls' instability

Because each hydrogen atom only contributes one electron, the band we calculate is only half full, because each state can take two electrons, one with spin up the other down. This means that the Fermi level will be at k=+1/4 and k=-1/4 and in priciple we should have a one-dimensional metal. A much coveted quantum-wire!

Unfortunately, Peierls has shown that in such a case we can always lower the energy of the system by doubling the size of the unit cell. Doing so halves the Nyqvist frequency and a new Brillouin zone boundary will result at what was k=+1/4 and k=-1/4. (Notice again the reciprocal relationship between real and reciprocal space: doubling the periodicity in one will halve the Brillouin zone in the other space.)

Diagram of a Peierls' distortion

Once we have the new boundary we can lower the energy of the topmost filled states at the cost of the lowest empty ones in much the same way we described for our free electron gas: by mixing states that differ by a whole (new!) Q vector.

Of course the question is: how do we actually lower the symmetry? The answer is by shifting the atoms or, putting it differently by making a phonon go soft. (More about that in the next topic). If we shift atoms closer together pairwise so that the distance now alternates as short-long-short-long etc. our unit cell will now consist of the sum of a short and a long distance, and this is on the average twice the old H-H distance.

The problem is that we have now created a row of H₂ molecules. Sadly, they will all fly away to turn our quantum wire into a puff of hydrogen gas.