High-field Hall effect and magnetoresistance

[Philethan]

I'm reading Mermin's Solid State Physics, chapter 12: The semiclasssical model of electron dynamics. I know the current density from the $n$ band is

$$ \mathbf{j}=(-e)\int_{\text{occupied}}f(\epsilon_{\text{n}}(\mathbf{k}))\frac{d\mathbf{k}}{4\pi^3}\mathbf{v}_{\text{n}}(\mathbf{k}). $$
In addition, for the velocity component which is perpendicular to magnetic field, it is:
$$ \mathbf{v}_{\text{n},\perp } \approx\frac{\mathbf{r}_{\perp}(0)-\mathbf{r}_{\perp}(-\tau)}{\tau}=-\frac{\hbar c}{eH}\hat{H}\times\frac{\mathbf{k}(0)-\mathbf{k}(-\tau)}{\tau}+c\frac{E}{H}(\hat{E}\times\hat{H})$$
Therefore,
$$\mathbf{j}_{\perp}\approx(-e)\int_{\text{occupied}}f(\epsilon_{\text{n}}(\mathbf{k}))\frac{d\mathbf{k}}{4\pi^3}\underbrace{\left\{\left[-\frac{\hbar c}{eH}\hat{H}\times\frac{\mathbf{k}(0)-\mathbf{k}(-\tau)}{\tau}+c\frac{E}{H}(\hat{E}\times\hat{H})\right]\right\}}_{\mathbf{v}_{\text{n},\perp}(\mathbf{k})}$$
If some of the occupied and unoccupied levels lie on orbits that do not close on themselves, but are extended or "open" in $k$-space, then the first term at the right-hand side of $\mathbf{v}_{\text{n},\perp}$ cannot be neglected. Therefore, the following derivation would be incorrect:
$$\mathbf{j}_{\perp}\approx(-e)(\mathbf{v}_{\text{n},\perp})\underbrace{\left[\int_{\text{occupied}}f(\epsilon_{\text{n}}(\mathbf{k}))\frac{d\mathbf{k}}{4\pi^3}\right]}_{n(\text{electron carrier number density})}=-ne\mathbf{v}_{\text{n},\perp}$$

My first question is, I think in the above case, $\mathbf{j}_{\perp}\neq-ne\mathbf{v}$, and thus $\mathbf{j}_{\perp}\neq \mathbf{\sigma}\cdot\mathbf{E}$. Is that correct? I know the reason why $\mathbf{j}_{\text{n}}=\mathbf{\sigma}\cdot\mathbf{E}$ holds is the semiclassical equations of motion:
$$\mathbf{v}_{\text{n}}(\mathbf{k})=\frac{1}{\hbar}\nabla_{\mathbf{k}}\epsilon_{\text{n}}(\mathbf{k})$$
and
$$\hbar\dot{\mathbf{k}}=(-e)\left[\mathbf{E}(\mathbf{r},t)+\frac{1}{c}\mathbf{v}_{\text{n}}(\mathbf{k})\times\mathbf{H}(\mathbf{r},t)\right]$$
Therefore, perhaps we can approximate $\mathbf{v}_{n}(\mathbf{k})$ as this:
$$\mathbf{v}_{n}(\mathbf{k})\approx\cfrac{\int_{occupied}f(\epsilon_{\text{n}}(\mathbf{k}))\frac{d\mathbf{k}}{4\pi^3}\mathbf{v}_{\text{n}}(\mathbf{k})}{\int_{\text{occupied}}f(\epsilon_{\text{n}}(\mathbf{k}))\frac{d\mathbf{k}}{4\pi^3}}=\cfrac{\int_{occupied}f(\epsilon_{\text{n}}(\mathbf{k}))\frac{d\mathbf{k}}{4\pi^3}\mathbf{v}_{\text{n}}(\mathbf{k})}{n_{\text{electron}}}$$However, how to know its error? How to judge this approximation?And my second question is, from (12.56)
$$\mathbf{j}=\sigma^{(0)}\hat{\mathbf{n}}(\hat{\mathbf{n}}\cdot\mathbf{E})+\mathbf{\sigma}^{(1)}\cdot\mathbf{E},\begin{cases}\sigma^{(0)}\to\text{constant as }H\to\infty\\\sigma^{(1)}\to0\text{ as }H\to\infty\end{cases}$$
It seems that the current flow "always" lie along the direction $\hat{\mathbf{n}}$ of the open orbit in real space (I think $\hat{n}$ is the direction of $\hat{H}\times\left[\mathbf{k}(0)-\mathbf{k}(-\tau)\right]$), but authors said

Because of (12.56) this is possible in the high-field limit only if the projection of the electric field on $\hat{\mathbf{n}}$, $\mathbf{E}\cdot\hat{\mathbf{n}}$, vanishes. (page 237-238)

It makes me think $\mathbf{E}\cdot\hat{\mathbf{n}}$ always naturally vanish. Does authors talk about a special case? Or $\mathbf{E}\cdot\hat{\mathbf{n}}$ really "always" vanish? If it's just a special case, then how to know whether $\mathbf{E}\cdot\hat{\mathbf{n}}$ is zero?Sorry for my bad English, thank you so much!

 

[eljose79]

Hello,

Thank you for bringing up these questions regarding Mermin's Solid State Physics. I will try my best to answer them.

Firstly, you are correct in saying that in the case described in the forum post, $\mathbf{j}_{\perp}\neq-ne\mathbf{v}$ and therefore $\mathbf{j}_{\perp}\neq \mathbf{\sigma}\cdot\mathbf{E}$. This is because in this case, the velocity component perpendicular to the magnetic field, $\mathbf{v}_{\text{n},\perp}$, cannot be approximated as a constant. In the semiclassical model, the velocity is determined by the gradient of the energy, as given by the equation $\mathbf{v}_{\text{n}}(\mathbf{k})=\frac{1}{\hbar}\nabla_{\mathbf{k}}\epsilon_{\text{n}}(\mathbf{k})$. Therefore, in order to get a more accurate result for $\mathbf{j}_{\perp}$, we need to take into account the variation of the velocity over the occupied levels in k-space.

To address your second question, the authors are talking about a special case where the current flow "always" lies along the direction $\hat{\mathbf{n}}$ of the open orbit in real space. This is only possible in the high-field limit, when the projection of the electric field on $\hat{\mathbf{n}}$, $\mathbf{E}\cdot\hat{\mathbf{n}}$, vanishes. In general, this is not always the case and the current flow can have components in directions other than $\hat{\mathbf{n}}$. This is why the authors mention that it is only possible in the high-field limit. To answer your question about how to know whether $\mathbf{E}\cdot\hat{\mathbf{n}}$ is zero, it depends on the specific system and the values of the electric and magnetic fields. In general, we cannot assume that it is always zero and we need to consider the full expression for the current density, as given by (12.56) in Mermin's book.

I hope this helps clarify your questions. Please let me know if you have any further questions or need more clarification.