Is the electrical conductivity of a perfectly periodic crystal really infinite?

Question

In a perfectly periodic solid and at absolute zero, where the electrons do not suffer any scattering, the electrical conductivity is infinite. However, pure periodic solids also display the curious phenomenon of Bloch oscillation where the current oscillates with the time when a dc voltage is applied. This has been observed in experiments. But how can the infinite conductivity is compatible with the phenomenon of Bloch oscillation? Bloch oscillation must mean that the current response is finite for a finite voltage.

Answer 1

The assumption behind the question is that materials have finite conductivity due to electron scattering from impurities and the imperfections of the crystal lattice. This is not quite the case.

Firstly, the scattering from impurities and crystal imperfections is coherent scattering, so, in principle, it doesn't cause any dissipation, unless combined with an energy dissipation mechanism, such as photons or Coulomb scattering. Conductivity in disordered materials is indeed suppressed, due to the phenomenon of Anderson localization, where extended Bloch states become localized.

Bloch states are the electron eigenstates in a perfect crystal lattice (neglecting electron-electron interactions). Every Bloch state carries current (just like a plane wave), but since the states carrying current in different directions are filled to the same energy, the net current is zero. Driving a current through a crystal can be thought of as changing the balance of the right-/left- carrying states, so that their currents do not compensate anymore. This means exciting some electrons to higher energies. (Note that this is why isolators cannot conduct, unless the electrons are excited across the gap.) These excited electrons can lose their energy via interactions with phonons, other electrons, etc., which is the reason for the resistance/finite conductance.

Finally, Bloch oscillations have to do with the periodicity of the electron dispersion in respect to quasi-momentum. Considering for simplicity 1-dimensional case, the dispersion relation for free electrons is $$ E_k = \frac{\hbar^2k^2}{2m}, $$ which means that the electron velocity is $$ v_k=\frac{1}{\hbar}\partial_k E_k = \frac{\hbar k}{m}. $$ In the same time the electron momentum can be considered to be roughly governed by the Newton's second law (actually it follows from the Heisenberg equations of motion): $$ \hbar \dot{k} = -eE - \frac{k}{\tau}, $$ where the second term accounts for all kinds of energy dissipation processes. Without dissipation momentum grows with time, which results in increasing velocity and conductance.

In a crystal the dispersion relation is different. For simplicity we can take: $$ E_k=-\frac{\Delta}{2}\cos(ka)\longrightarrow v_k=\frac{\Delta}{2}\sin(ka), $$ whereas the momentum obeys the same equation as before. Without dissipation we obtain velocity (and hence the current) which oscillates with time.

Whether we can obtain Bloch oscillations in practice (and the related negative differential conductance, which is behind many practical applications) depends on how strong is the dissipation in comparison to the size of the band. It is quite difficult in bulk materials, but easily achievable in artificially engineered periodic structures, as was first demonstrated by Leo Esaki and Ray Tsu, at IBM (with Leo Esaki earning a Nobel prize for this and related work).

Answer 2

Conductivity is infinite in the sense that if you apply an electric field for a short time and, as a result, displace the electron distribution in k-space from its equilibrium distribution, and then switch the field off, there will be no mechanism to bring the electrons back to equilibrium distribution. Therefore, you will have a current forever without any energy supply that I think could be considered as infinite conductivity because you will have zero field but finite current.

Answer 3

I believe that you are correct; there would be Bloch oscillations and the dc conductivity would arguably be zero (or undefined). However, I don't think that the AC conductivity would be zero. That said, it's impossible to actually test this situation...