# Electron Transport in Carbon Nanotubes with Adsorbed Chromium Impurities
## 5 Free Energy  
We first focus on the Gibbs free energy (also called the thermodynamic potential) of the system which satisfies [32]:  
\[\Omega=-\Theta\,\ln\,\operatorname{Tr}(e^{-H/\Theta}). \tag{109}\]  
The Hamiltonian \(H\) is defined in Equation (1). To perform the trace, we need to sum over all of the band states, but we also need to take into account the disorder averaging. The latter is commonly handled via a configurational average [31]. As for the correlations, we employ Equation (40) for the interaction picture, which allows us to re-express the Gibbs free energy as  
\[\Omega=\langle\delta\Phi\rangle-\Theta\,S_{c}+\Omega^{(0)}_{e}+\Omega^{(0)}_{ ph}+\Omega^{\prime} \tag{110}\]where \(\Omega_{e}^{(0)}\) and \(\Omega_{ph}^{(0)}\) are the thermodynamic potentials for the electrons and the phonons in the field of the ionic cores, respectively. As before, the equilibrium ion core positions are chosen to be the same as those of the crystal lattice for the pure ordered crystal, even when we introduce disorder and change the type of some of the atoms. The symbol \(\Omega^{\prime}\) is the component of the thermodynamic potential that is caused by the mutual scattering of electrons and phonons; it is defined by  
\[\Omega^{\prime}=-\Theta\,\ln\left\langle\left\langle\sigma(1/\Theta)\right\rangle _{0}\right\rangle \tag{111}\]  
with \(\sigma\) given in Equation (40) for the interaction picture. In addition, \(S_{c}=-\langle\ln P_{c}\rangle\) is the configurational entropy, where \(P_{c}\) denotes the distribution function for atoms with a specific \(z\)-component of the magnetic moment on a given lattice site. The angular brackets \(\langle...\rangle\) denote the configurational averaging over different disorder configurations for a given density of disorder. Next, we use the \"integration over the coupling constant\" method to simplify the results further. By replacing the interacting Hamiltonian \(H_{\text{int}}\) (defined in Equation (5)) by \(H_{\text{int}}(\lambda)=\lambda H_{\text{int}}\), differentiating the expression for the piece of the thermodynamic potential \(\Omega^{\prime}(\lambda)\) in Equation (111) with respect to \(\lambda\), and then integrating them (with the boundary conditions \(\Omega^{\prime}(0)=0,\Omega^{\prime}(1)=\Omega^{\prime}\)), we obtain the following after a long derivation:  
\[ \begin{split}&\Omega^{\prime}=-\frac{1}{\pi vN}\text{Im}\int \limits_{0}^{1}\frac{d\lambda}{\lambda}\int\limits_{-\infty}^{\infty}d\varepsilon \bigg{[}f(\varepsilon)\\ &\times\text{Tr}\left\langle\left(w(\lambda)+\Sigma_{eph}( \varepsilon,\lambda)+\Sigma_{ee}(\varepsilon,\lambda)\right)G^{aa^{+}}( \varepsilon,\lambda)\right\rangle\\ &+\frac{1}{2}\coth\left(\frac{\varepsilon}{2\Theta}\right)\text{ Tr}\Big{\langle}\Delta M^{-1}(\lambda)G^{PP}(\varepsilon,\lambda)\\ &+\,\left(\Delta\Phi(\lambda)+\Sigma_{phph}(\varepsilon,\lambda) \right)G^{uu}(\varepsilon,\lambda)\Big{\rangle}\bigg{]}.\end{split}  \tag{112}\]  
This expression can be immediately evaluated, because we know all the Green's functions and self-energies. The contribution to the thermodynamic potential from the electrons (in the field of the ionic cores) is also simple to find. It is given by  
\[\Omega_{e}^{(0)}=-\Theta\int\limits_{-\infty}^{\infty}\ln\left(1+e^{(\mu_{e}- \varepsilon)/\Theta}\right)g_{e}^{(0)}(\varepsilon)\,d\varepsilon. \tag{113}\]  
Similarly, the contribution to the thermodynamic potential from the phonons (in the field of the ionic cores) is given by  
\[\Omega_{ph}^{(0)}=\Theta\int\limits_{-\infty}^{\infty}\ln\left(1-e^{-\varepsilon /\Theta}\right)g_{ph}^{(0)}(\varepsilon)\,\varepsilon. \tag{114}\]  
The densities of states \(g_{e}^{(0)}(\varepsilon)\), \(g_{ph}^{(0)}(\varepsilon)\) in Equations (113) and (114) are given by the results in Equations (93) and (94); note that the Green's functions \(G^{aa^{+}}(\varepsilon)\) and \(G^{uu}(\varepsilon)\) are replaced by Green's functions \(G_{0}^{aa^{+}}(\varepsilon)\) and \(G_{0}^{uu}(\varepsilon)\) for the zeroth-order approximation.