Properties of the Fermi function

In a system of non interacting fermions the Fermi function $f(E)$ gives the propability that an energy level $E$ is occupied at the absolut temperature $T$. For low temperatures almost all fermions are in states with energies lower than the chemical potential $\mu$. For theses temperatures the Fermi function looks like a step function.

$$f(E) = \frac{1}{\exp{\big(\frac{E-\mu}{k_BT}\big)}+1}$$

 $\begin{equation*} \phantom{[eV^-1]} f(E)\; \end{equation*}$ T=300 K $\begin{equation*} \quad{} (E - \mu) \; [\text{eV}] \end{equation*}$

It should be noted, that the Fermi function does not actually apply to fermion systems like metals as there is interaction between the fermions in these systems. However, it is still a good first approximation, when calculating their thermodynamical properties.

Derivative of the Fermi function

As the Fermi function mostly changes only in the close vicinity of the chemical potential, its derivative is mostly zero. Close to the chemical potential the derivative shows a negative peak, which in the limit of $T \rightarrow 0$ becomes a dirac delta function.

$$\frac{df(E)}{dE} = -\frac{1}{k_BT}\frac{\exp{\big(\frac{E-\mu}{k_BT}\big)}}{\big(\exp{\big(\frac{E-\mu}{k_BT}\big)}+1\big)^2}$$

 $\begin{equation*} \frac{df(E)}{dE}\; [\text{eV}^{-1}] \end{equation*}$ T=300 K $\begin{equation*} \quad{} (E - \mu) \; [\text{eV}] \end{equation*}$

Because of this behaviour, thermodynamic integrals that contain the Fermi function are often first integrated by parts.

$$\int\limits_{-\infty{}}^{\infty{}}H(E)f(E)\;{}dE = K(\infty)f(\infty) - K({-\infty})f({-\infty}) - \int\limits_{-\infty{}}^{\infty{}}K(E)\frac{df}{dE}\;{}dE\; ,$$

with $K(E)$ being $$K(E) = \int\limits_{-\infty{}}^{E}H(E')\;{}dE'\; .$$

The border terms vanish as the Fermi function becomes zero for $E\rightarrow \infty$ and $K(-\infty)=0$. One ends up with an integral which only needs to be evaluated numerically for a finite interval around $\mu$.

$$\int\limits_{-\infty{}}^{\infty{}}H(E)f(E)\;{}dE = - \int\limits_{-\infty{}}^{\infty{}}K(E)\frac{df}{dE}\;{}dE\; \approx - \int\limits_{\mu - 8k_BT}^{\mu + 8k_BT}K(E)\frac{df}{dE}\;{}dE\;.$$

Integral of the Fermi function

In the case of some thermodynamic properties like the grand potential $\phi$ or the entropy $s$ the integrals that need to be calculated contain the integral of the Fermi function $F(E)$.

$$F(E) = -k_BT\ln{\Bigg[\exp{\bigg({-\frac{E-\mu}{k_BT}}\bigg)}+1\Bigg]}$$

 $\begin{equation*} \phantom{^-1}F(E)\; [\text{eV}] \end{equation*}$ T=300 K $\begin{equation*} \quad{} (E - \mu) \; [\text{eV}] \end{equation*}$

One can often convert the integrals involving the integral of the Fermi function $F(E)$ into one involving the Fermi function by integrating by parts. The border terms vanish as $F(E)$ also goes to zero rather quickly for energies above $\mu$.

$$\int\limits_{-\infty{}}^{\infty{}}H(E)F(E)\;{}dE = - \int\limits_{-\infty{}}^{\infty{}}K(E)f(E)\;{}dE\;$$ $$K(E) = \int\limits_{-\infty{}}^{E}H(E')\;{}dE'\; .$$

One can now apply a similar scheme to the one shown in the section above. For example, the grand potential $\phi$ was calculated in this way (Calculate $\phi$).

Second integral of the Fermi function

There is an interesting property regarding the grand potential $\phi$ in the case of a 2D fermi gas. A solution can be found in terms of a dilogarithm fuction which can be used to check the numerical calculation.

$$\phi = -k_BT\int\limits_{-\infty{}}^{\infty{}}D(E)\ln{\Bigg[\exp{\bigg({-\frac{(E-\mu)}{k_BT}}\bigg)}+1\Bigg]}\;{}dE = \int\limits_{-\infty{}}^{\infty{}}D(E)F(E)\;{}dE$$

As the density of states is a constant function starting from some E0 the integral reduces to

$$\phi = D_0\int\limits_{E_0}^{\infty{}}F(E)\;{}dE = D_0\bigg[F_2(E)\bigg]_{E_0}^\infty \; ,$$

with $F_2(E)$ being the second integral of the Fermi function. In constrast to the first integral, the second integral does not have a closed form:

$$F_2(E) = -(k_BT)^2\text{Li}_2(x)$$ $$x = -\exp{\bigg(-\frac{E-\mu}{k_BT}}\bigg)$$

Here Li2(x) denotes the dilogarithm.

 $\begin{equation*} F_2(E)\; [\text{eV}^{2}] \end{equation*}$ T=300 K $\begin{equation*} \quad{} (E - \mu) \; [\text{eV}] \end{equation*}$

The dilogarithm and therefore $F_2(E)$ vanishes for $E\rightarrow\infty$. The grand potential then takes the form:

$$\phi{}(T) = -D_0F_2(E_0,T)$$

Usually the lowest occupied energy $E_0$ in case of a free fermi gas is chosen as the zero point and the chemical potential $\mu$ has a positive value. In the plots below, like in the cases before, a presentation with $\mu$ as the zero point is better suited. Therefore $E_0$ is negative and a change of the chemical potential will result in a shift of $E_0$.

On the left the negative second integral of the Fermi function is plotted against the energy $E$. The value of the chemical potential (and therefore the position of $E_0$ relative to it) depends in case of the 2D fermi gas just on the electron density $n$ and the mass of the particle (we choose free electrons $m=m_e$). The same goes for the constant density of states $D_0$.

$$\mu \approx E_F = \frac{\pi{\hbar{}}^2}{m}n$$ $$D_0 = \frac{m}{\pi{\hbar{}}^2}$$

The value of $-F_2(E_0)$ at the energy $E_0$ is marked with a cursor. When regulating the temperature $T_0$, $F_2$ changes its shape. The point for a fixed electron density aka fixed $E_0$ moves then up and down. When looking only at the movement of this point, one obtains the temperature dependancy of the grand potential density $\phi(T)$ of a 2D fermi gas with exactly that electron density.

 $\begin{equation*} -F_2(E) \; [\text{eV}^2] \end{equation*}$ $\begin{equation*} \phi(T)\; [\text{eV}] \end{equation*}$ $\begin{equation*} \quad{} (E - \mu) \; [\text{eV}] \end{equation*}$ $\begin{equation*} \quad{} T \; [\text{K}] \end{equation*}$ n = 1.0e+20 m-2 T0 = 600 K

For a fixed temperature $T_0$, when changing the electron density, the position of $E_0$ on the $-F_2$ curve changes. This results in a increase/decrease of $-F_2(E_0)$ and therfore of $\phi$. The increase/decrease depends hereby on the slope of $F_2$, which itself depends on the temperature $T_0$. Therefore the points of $\phi$ on the right change differently depending on their temperatures (small change for low temperatures and big change for high temperatures → shift + compressing/streching of graph).