# Energy of Bloch electrons

The Bloch Theorem says, that in a periodic potential $$V(\vec r) = V(\vec r + \vec R)$$ the wavefunction is
$$\psi(\vec r) = e^{i\vec k \cdot \vec r} u_k(\vec r)$$

For the energies, there is always graphs like that below, showing the splitting of the energy bands at the zone boundaries, when switching from the free electron to one in a crystal.

So my question is: Do these Bloch states only exist above the potential in the crystal? Because otherwise they would have to tunnel through the potential in order to be delocalized in the crystal. But on the other hand in the derivation of the Bloch theorem there is no information about the strength of the potential as far as i know. So as shown below there could be some states like the lowest one, that does not lose energy when tunneling through the potential. So i could make the potential really high and still have a delocalized electron in there.

Recall that for a solid the work function is the amount of energy required to remove an electron from the solid. So for a solid that is modeled to have a constant potential in its interior and bounded by a potential related to the work function would look something like figure 2.3.1 in the following link:

Semiconductor Fundamentals

A comparable, and better, schematic can be constructed by imagining bringing together many atomic potentials and would look like figure 1c here:

Energy Bands in Solids and Their Calculations

Note how between each atom, the potential bends over to connect to its neighbor's potential when added, thereby flattening the potential between them. So the electrons in the conduction band of a metal have energies that obviously sit below the work function level, but above the atomic potentials. The potential they see is very flat.

In general when atoms are brought together to construct a solid, the valence electrons will be enter into Bloch states which extend throughout the crystal, while the core electrons remain essentially localized. The core electrons do not show Bloch characteristics.

In a metal, simple calculations, such as the Wigner-Seitz method, show that the bottom of the conduction band sits a little below the energy barrier that exists between atoms. Note how the image in figure 1c referenced above has the bottom of the conduction band slightly below the energy barrier. But the state at k=0, which would be exactly at the bottom of the band, still has Bloch structure.

At the other end of the spectrum, crystals that have high ionic character, such as NaCl, the valence electrons do not show Bloch characteristics. The electronic structure of the ions is essentially a closed shell.

The valence electrons in solids that have more covalent bonding character will also have Bloch character, just like the valence electrons in a metal. Where these bands actually sit in relation to the barrier that exists between the atomic cores will likely vary from solid to solid. As in the metallic case, some part of the valence band will likely sit below the barrier between atomic cores.

An old text, but a good one, that discusses these points is J.M. Ziman's Principles of the Theory of Solids. Chapters 3 and 4.

• But still in that picture 1c of the second link there are bands with energies lower than the potential. For those there must occur tunneling in order for the electrons to be delocalized that decreases the probability amplitude after every potential-peak. So how can there be Bloch wave with perfect periodicity in that case? – Peter Jul 11 '20 at 7:11
• Hi Peter. I have expanded my answer. Hope this helps! – CGS Jul 11 '20 at 12:27
• Thank you! I also looked into the book you mentioned and also found that really nice. – Peter Jul 11 '20 at 13:41