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The Fermi-Dirac distribution, given by

$$f(E) = \frac{1}{1 + \exp\left(\frac{E - E_{\text{F}}}{k_{\text{B}} T}\right)}$$


describes the probability that a state with energy $E$ is occupied by an electron at temperature $T$. $E_F$ is the Fermi energy, and $k_B$ is the Boltzmann constant.

The density of states $D(E)$ gives the number of states per unit energy interval. We hence can compute the total number of electrons in a material $N$ with

$$N = \int_{-\infty}^{\infty} D(E) f(E) \, dE$$

Now what confuses me is that I don't understand why for example $N$ could depend on the temperature. If the temperature changes, $f(E)$ will not be the same and hence we should expect something else from the integral which computes $N$. But $N$ is the fixed amount of electrons in materials and shouldn't depend on anything. So my guess would be to say that the density of states depend on the number of electrons, but this is not the case for example for the density of state of a free electron gas which is given by:

$$D(E) = \frac{m \cdot \sqrt{2 \cdot m \cdot E}}{\pi^2 \cdot \hbar^2}$$

So what is the mistake in my reasoning?

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    $\begingroup$ You are correct that $N$ does not vary with temperature. That particular equation is the normalization requirement on state occupancy. So that is not a definition of $N$, but a statement about how the occupancy of states changes with temperature through the Fermi function. $\endgroup$
    – Jon Custer
    Sep 21, 2023 at 14:53
  • $\begingroup$ You are pinning the chemical potential to the Fermi energy, i.e. to its value at zero temperature. In reality, for a fixed number of electrons, the chemical potential will depend on temperature (although typically rather weakly, so the equation you have written is typically a good approximation). Alternatively you can do as @TobiasFünke suggests and work in the grand canonical ensemble, in which case the chemical potential is now fixed but now $N$ is allowed to vary $\endgroup$ Sep 21, 2023 at 15:02

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To expand on my comment, in writing $$ f(E) = \frac{1}{e^{\frac{E-E_F}{k_BT}}+1} $$ you are pinning the chemical potential to the Fermi energy, i.e. to its value at zero temperature. A more correct expression would be $$ f(E,\mu) = \frac{1}{e^{\frac{E-\mu}{k_BT}}+1} $$ where $\mu$ is the chemical potential.

For a fixed number of electrons, your equation for $N$ $$ N = \int D(E) f(E,\mu)\,\mathrm{d}E $$ becomes an implicit equation determining $\mu$: the chemical potential will take whatever value it has to, given the other parameters of your system, to keep $N$ fixed. In particular it will depend on the temperature. In practice this temperature dependence is typically rather weak, so pinning the chemical potential to the Fermi energy is often a good approximation.

An alternative approach is to work in the grand canonical ensemble. In this case the chemical potential is now fixed externally but now the average value of $N$ is allowed to vary in accordance with the equation given

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