# Difference of the Transmission Coefficient between Thermal and Charge Conductance by Nonequilibirum Green Function Method

The equation 57 in the reference [Jian-Sheng Wang, Jian Wang and J. T. Lu, Quantum thermal transport in nanostructures, Eur. Phys. J. B 62, 381 (2008)] explains the the transmission coefficient for the thermal conductance, as follows: $$T_{\omega}={Tr}\left[G^{R}\Gamma_\text{left}G^{A}\Gamma_\text{right}\right]\tag{1}$$ The equation 12 in the reference [A. R. Rocha, V. M. García-Suárez, S. Bailey, C. Lambert, J. Ferrer and S. Sanvito, Spin and molecular electronics in atomically generated orbital landscapes, Phys. Rev. B, 73, 085414 (2006)] explains the transmission coefficient for the charge conductance, as follows: $$G=\frac{2e^{2}}{h}\mathrm{Tr}\left[\Gamma_\text{left}G_{M}^{A}\Gamma_\text{right}G_{M}^{R}\right]\tag{2}$$

It seems that these two quantities are same; except the coefficient $$\frac{2e^{2}}{h}$$. This means that the following thermal conductance and charge conductance also seems same.

Would anyone explain to me the difference between them; particularly, the difference in the calculation procedures for these two quantities (the transmission for the thermal and charge conductance / the thermal and charge conductance)?

• Tr($\Gamma_L G^A \Gamma_R G^R$) = Tr($G^R \Gamma_L G^A \Gamma_R$). If the matrices are small enough (and diagonalizable), you can evaluate the products and diagonalize. Then the trace is simply the sum of the eigenvalues! The difference is basically units. Transmission is the (roughly) the probability of a carrier crossing the system. Conductance is basically the probability weighted by the charge the carriers take with them. Thermal conductance is just the probability weighted by the heat the carriers take with them. The essential physics is encoded in the transmission. Commented Feb 2 at 0:33
• @Ty Sterling Thank you for the reply. In other words, the charge conductance and thermal conductance are same, if the coefficient ($\frac{2e^{2}}{h}$) in the charge conductance is ignored. Is my understanding correct or not? Commented Feb 2 at 4:44
• I think you mean that the transmission and conductance are the same if the coefficient is ignored (which I think is correct). I changed fields ~5 years ago, but IIRC I think what you are looking for is called 'Landauer formalism'. See e.g. here: gianlucafiori.org/qpc/node7.html $G \sim T$ with $G$ the conductance and $T$ transmission probability. The difference between heat and charge conductance is a constant prefactor with the whatever are the right dimensions. I think it is called 'thermal transmission' above because of the context; transmission is more general than that. Commented Feb 2 at 18:42
• @Ty Sterling Thank you for the link. I have one more question. When computing the thermal transmission coefficient, the green function needs to be computed first; for example, $G^{R}_{scattering\:region}=\left[\left(\omega+i0^{+}\right)^{2}I-D\right]^{-1}$, where D is the dynamic constant matrix. I compare it with the retarded green function in charge conductance. I found that this dynamic constant matrix is just the off-diagonal matrix in the Hamiltonian, which is the interaction matrix between the lead and the central scattering region. Is my understanding correct? Commented Feb 3 at 9:36