For a 2D electron gas of spinless fermions we can easily compute the density profile $n(x,y)$. If now I add a series of square barriers along only one direction, say $V(y)$, I can factories my density into $n(x,y) = n^0_xn(y)$. Under what conditions am I allowed to use the Thomas-Fermi approximation to compute $n(y)$? How about I consider a series of Gaussian potentials?
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2 Thomas-Fermi model gives expression for the charge density in terms of electric potential (external + generated by the charge density itself). This charge density then solved self-consistently with the Poisson equation for the potential: $$ \nabla^2\phi_{ind}(\mathbf{r})=-\frac{en(\mathbf{r})}{\epsilon} $$ In 2D case this is indeed particularly simple to deal with, since the charge density is linear in the potential: $$ n(\mathbf{r})=n_0\left[1-\phi_{ind}(\mathbf{r})/\mu-\phi_{ext}(\mathbf{r})/\mu\right] $$ Now, if we plug this into the Poisson equation, and the external potential is dependent only on one coordinate, the variables separate - there is no need for potentials to be square or anything like that.
The limitations come from the applicability of the approximation itself - the Fermi wavelength should remain much shorter than potential features, so that we can treat the charges as continuum. In terms of the quantities above this means that $$|\phi_{ind}(\mathbf{r})+\phi_{ext}(\mathbf{r})|/\mu\ll 1$$
The problem with square barriers is that:
- If they are really square on the atomic level, we have boundary charge accumulation, which violates Thomas-Fermi model.
- On the other hand, if they are square in the sense that they change rapidly compared to the external potential and Fermi wavelength, but still slow on the atomic scale, this poses no problem.
- Finally, even if the barriers change sharply on the atomic scale, but the scale of interest is much longer, we still can use Thomas-Fermi model. The boundary effects then simply modify the form of the external potential (i.e., the barriers potential).
Finally, it is important to keep in mind that all these technical details are of little importance, if you are dealing with calculating something like optical spectra or other properties, which are affected by hundreds of factors and parameters beyond the experimental control. On the other hand, if you go with an AFM or STM to study the potential at the barrier surface - that's a different story.
- $\begingroup$ Thanks for the detailed answer. What I have in mind is a cold atoms setups. For instance if I use the Thomas-Fermi approximation for particles that are confined with harmonic traps, than it results that in the middle of the trap (minimum of the trap) I have the highest density - even tho I remember that this comes with some caveats. My question was: can I safely apply the TF approx to the case where I start with atoms in a box trap (non density inhomogeneity) and then I shine a series of square potential barriers (or Gaussians)? Or is there I have to be very careful with? $\endgroup$Derrick Rossi– Derrick Rossi2025-10-17 06:15:00 +00:00Commented Oct 17 at 6:15
- 1$\begingroup$ @DerrickRossi there are always some boundary effects, but I think, as long as you have large regions with high density, where the density only slightly perturbed by the potential, boundary effects can be neglected. I suggest googling "thomas fermi cold atoms" - there are quite a few results coming up. I also found this question in Physics SE: physics.stackexchange.com/q/233070/247642 $\endgroup$Roger V.– Roger V.2025-10-17 06:25:31 +00:00Commented Oct 17 at 6:25