Is current density independent of applied fields for Bloch electrons?

Question

Following Ashcroft-Mermin chapter 12 the semiclassical dynamics is governed by

$ \dot{\vec{r}} = \vec{v}_n(\vec{k}) = \frac{1}{\hbar}\frac{\partial \epsilon_n(\vec{k})}{\partial \vec{k}} $ and $ \hbar \dot{\vec{k}} = -e\vec{E} -\frac{e}{c} \vec{v}_n(\vec{k}) \times \vec{H} $

In which order one formally should attempt to solve the equations given band structure functions $\epsilon_n(\vec{k})$ and the fields $\vec{E}$ and $\vec{H}$.

Is the velocity independent of the fields and only depends on band structure? If so how one calculates resistances from $\vec{J} = -e \int \frac{d\vec{k}}{4 \pi^3} \frac{1}{\hbar}\frac{\partial \epsilon_n(\vec{k})}{\partial \vec{k}}$ [Eq. 12.16] as they will be independent of fields.

Answer 1

Finite resistance appears when we take into account the collision processes. There are different ways of doing it, e.g.,

• by inserting an ad-hoc dissipative term into the momentum equation: $$ \hbar\dot{\mathbf{k}}=-\frac{\hbar\mathbf{k}}{\tau}-e\mathbf{E}-\frac{e}{c}\mathbf{v}_n(\mathbf{k})\times\mathbf{H} $$
• by introducing a probability of resetting the momentum to zero, e.g., as $p(t)=e^{-t/\tau}$
• by integrating the velocity with the Fermi-Dirac distribution function
• by using the equations from the OP in kinetic/Boltzmann equation, e.g.,
  $$ \mathbf{j}(\mathbf{r},t)=-e\sum_{n,\sigma}\int\frac{d^3\mathbf{k}}{(2\pi)^3}f_{n\sigma}(\mathbf{r},\mathbf{k},t)\mathbf{v}_n(\mathbf{k}) $$ (The last equation is copied from here, but some of these are definitely covered in Ashkroft&Mermin or any other text on semiconductors.)

If no collisions are included, the, indeed, curious phenomena appear, such as Bloch oscillations. These are normally not observed in bulk semiconductors, but are easily obtainable in periodic semiconductor nanostructures (superlattices).

A very enlightening calculation can be found in the seminal paper by Esaki and Tsu, Superlattice and Negative Differential Conductivity in Semiconductors