Density of states → grand potential density for electrons in metals

The grand potential density $\phi$ can be calculated from the grand canonical partition function $Z_{gr}$ (See Thermodynamic properties of non-interacting fermions).

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

Here μ is the chemical potential which can be calculated from the density of states and the electron density, (Calculate μ) and $F(E)$ is the integral of the Fermi function.

The numerical calculation of the grand potential density can be numerically handled better if we first integrate by parts,

$$ \phi = -k_BT\Bigg[M(\infty)F(\infty) - M({-\infty})F({-\infty}) - \int\limits_{-\infty{}}^{\infty{}}M(E)\frac{dF}{dE}\;{}dE\Bigg].$$

Here the function $M(E)$ is,

$$ M(E) = \int\limits_{-\infty{}}^{E}D(E')\;{}dE'$$

The boundary terms vanish because $M(-\infty )= 0$ and the weight factor becomes zero for $E\rightarrow\infty$. The grand potential density can then be written as,

$$\phi = k_BT\int\limits_{-\infty{}}^{\infty{}}\frac{-M(E)}{k_BT}\frac{\exp{\big(\frac{\mu{}-E}{k_BT}\big)}}{\exp{\big(\frac{\mu{}-E}{k_BT}\big)}+1}\;{}dE = \int\limits_{-\infty{}}^{\infty{}}\frac{-M(E)}{\exp{\big(\frac{E-\mu}{k_BT}\big)}+1}\;{}dE.$$

Now we arrived at a form similar to that of the internal energy $u$ with an integral of a function times the fermi-factor. We can therefore apply a scheme similar to the one used in (Calculate u).

The only difference between our expression and the one used in the programm for the internal energy is that we have the function $-M(E)$ instead of $ED(E)$ in the integrand. Following the scheme of the internal energy calculation we arrive at the analogous expression:

$$\phi \approx \sum\limits_{\mu{}-7k_BT<{}E_i<{}\mu{}+7k_BT} \left[\frac{\frac{(K_2(E_{i+1})-K_2(E_i))}{(E_{i+1}-E_i)}\Big[\exp{(\frac{E-\mu{}}{k_BT})}(E-\mu{})+(E_i-\mu{})\Big]-K_2(E_i)}{\exp{(\frac{E-\mu{}}{k_BT})}+1} - k_BT\frac{(K_2(E_{i+1})-K_2(E_i))}{(E_{i+1}-E_i)}\ln{\Bigg[\exp{\bigg(\frac{E-\mu{}}{k_BT}\bigg)}+1\Bigg]}\right]_{E_i}^{E_{i+1}}.$$

with $K_2(E)$ being the integral of $-M(E)$:

$$ K_2(E) = \int\limits_{-\infty{}}^{E}-M(E')\;{}dE' = \int\limits_{-\infty{}}^{E}\int\limits_{-\infty{}}^{E'}-D(E'')\;{}dE''\;{}dE'.$$

This formula for $\phi$ reduces the necessary calculation to a sum over a finite number of discrete points $E_i$ in the close vacinity of $\mu$.

The form below calculates the grand potential density numerically from the density of states. The density of states is input as two columns of text in the textbox below. The first column is the energy in eV. The second column is the density of states in unit of eV-1 m-d, where d is the dimensionality (1,2, or 3). The electron density can be calculated from the number of electrons per unit cell and the volume of the unit cell. After the 'DoS → phi' button is pressed, the grand potential density is plotted as a function of temperature.

D(E) [eV-1 m-5
ϕ [eV m-5

E [eV]

T [K]


dimensionality: 1  2  3

number of electrons per primitive unit cell:  volume of a primitive unit cell:  m2 electron density =  m-5

Tmin:  K  Tmax:  K 


Density of states: E [eV], D(E) [eV-1 m-5]

Grand potential density: T [K], ϕ(T) [eV m-5]

Free electron model:
Tight binding: