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I am currently looking at changes in DOS when sampling recipocal space finely. More precisely, I am looking at the expressions

$$\rho_\text{1D}(E)\text{d}E = \frac{m}{\pi \hbar} \sum_i \text{H}(E-E_i)\text{d}E, \\ \rho_\text{2D}(E)\text{d}E = \frac{1}{\pi}\sqrt{\frac{2m}{\hbar^2}} \sum_i \frac{n_i\text{H}(E-E_i)}{\sqrt{E-E_i}}\text{d}E$$

where $E_i=\frac{\hbar^2k^2}{2m}$ is taken from the free-electron model. Here $H$ is the Heaviside step-function, and $n_i$ is a degeneracy factor (set to unity).

Let's now say that I can sample $k$-space arbitrarily fine. I want to see how the 1D and 2D cases converge. I can easily see that 2D converges to 3D (i.e. $\sqrt{E}$-behaviour) (see figure below), but what can one expect for the 1D system?

Usually, DOS explanations in books and online only focus on the actual expression and not what happens when you sum up all channels. At least it's not something I have seen.

Any ideas?

2D DOS

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Unfortunately, the premise is wrong. In general, lower dimension DOS can't converge to higher dimension DOS. That's one primary reason why 2D devices are interesting.

For example, the van Hove singularities of the DOS can diverge in 2D, but only their derivatives can diverge in 3D.

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