PHY.K02UF Molecular and Solid State Physics

Electron contribution to the thermodynamic properties of a solid

The contribution of the electrons to any of the thermodynamic properties of a material can be calculated from the electron density of states. The density of states for a material can be determined by a band-structure calculation. It is possible to calculate a band structure yourself using a program like Quantum Espresso, or many densities of states can be found at The Materials Project. Some example electron densities of states are listed below.

Al fcc, Au fcc, Cu fcc, Cr bcc, Li bcc, Na bcc, Pt fcc, W bcc, Si diamond, Fe bcc, Ni fcc, Co fcc, Mn bcc, Cr bcc, Gd hcp, Pd fcc, Pd3Cr, Pd3Mn, PdCr, PdMn, GaN, 6H SiC, GaAs, GaP, Ge, InAs, V bcc

Other thermodynamic quantities can then be calculated using the standard formulas of thermodynamics. Some of these quantities are listed below. Each quantity is expressed as an integral over the density of states. On the right are buttons that link to programs that will numerically calculate the thermodynamic quantities given the electron density and a tabulated density of states. The numerical calculation proceeds by first calculating the chemical potential from the electron density and then using this value of the chemical potential to determine the other thermodynamic quantities.

Chemical potential
(implicitly defined by):

 $n=\int\limits_{-\infty}^{\infty}D(E)f(E)dE=\large \int\limits_{-\infty}^{\infty}\frac{D(E)}{1+\exp\left(\frac{E-\mu}{k_BT}\right)}dE$ 

Energy spectral density:

$\large u(E)= \frac{ED(E)}{1+\exp\left(\frac{E-\mu}{k_BT}\right)}$

 

 

Internal energy density:

$\large u= \int\limits_{-\infty}^{\infty}\frac{ED(E)}{1+\exp\left(\frac{E-\mu}{k_BT}\right)}dE$

 

Grand potential density:

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

 

Helmholz free
energy density:

$\large f=\int\limits_{-\infty}^{\infty}D(E)\left[\frac{\mu}{1+\exp\left(\frac{E-\mu}{k_BT}\right)}-k_BT\ln\left(\exp\left(-\frac{E-\mu}{k_BT}\right)+1\right)\right]dE$

 

Entropy density:

 \begin{equation} \large s=\frac{1}{T}\int\limits_{-\infty}^{\infty}D(E)\left[\frac{E-\mu}{1+\exp\left(\frac{E-\mu}{k_BT}\right)}+k_BT\ln\left(\exp\left(-\frac{E-\mu}{k_BT}\right)+1\right)\right]dE \end{equation}

 

Specific heat:

$\large c_v= \int\limits_{-\infty}^{\infty}ED(E)\frac{df(E)}{dT}dE= \large \int\limits_{-\infty}^{\infty}\frac{ED(E)(E-\mu)\exp\left(\frac{E-\mu}{k_BT}\right)}{k_BT^2\left(1+\exp\left(\frac{E-\mu}{k_BT}\right)\right)^2}dE$