The Helmholtz free energy density $f$ is the grand potential density $\phi$ plus the electron density $n$ times the chemical potential $\mu$. (See Thermodynamic properties of non-interacting fermions).
$$f = \phi + n\mu.$$Previously we discussed how to calculate the chemical from the electron density of states and to calculate the grand potential energy from the electron density of states. These to quantities can be combined to determine the Helmholtz free energy density from the density of states. The integral that is solved is,
$$f = \int\limits_{-\infty{}}^{\infty{}}D(E)\left( \frac{\mu}{\exp\left(\frac{E-\mu}{k_BT}\right) +1}-k_BT\ln{\left[\exp{\left({-\frac{(E-\mu)}{k_BT}}\right)}+1\right]}\right)\,dE.$$The form below calculates the Helmholtz free energy 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 → f' button is pressed, the Helmholtz free energy density is plotted as a function of temperature.
| D(E) [eV-1 m-5] | f [eV m-5] | |||
E [eV] | T [K] | |||
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Free electron model:
Tight binding: