Calculation of charge transport with the non-orthogonal basis Hamiltonian based on the non-equilibrium Green's function method

Question

I am thinking about one question. If the Hamiltonian matrix is based on the non-orthogonal basis, how to compute the charge conductance with the Non-Equilibrium Green's Function (NEGF) method.

Suppose the Hamiltonian of the system is denoted by $H$ and the overlap matrix is denoted by $S$. They are written below.

\begin{align}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\left(\varepsilon_{F}+i\right)S-H&=\varepsilon^{'}S-H\tag{1}\\ &=\begin{pmatrix} \varepsilon^{'}S_{ll}-H_{ll}&\varepsilon^{'}S_{ls}-H_{ls}&0\tag{2}\\ \varepsilon^{'}\times S_{sl}-H_{sl}&\varepsilon^{'} S_{ss}-H_{ss}&\varepsilon^{'} S_{sr}-H_{sr}\\0&\varepsilon^{'} S_{rs}-H_{rs}&\varepsilon^{'} S_{rr}-H_{rr} \end{pmatrix}\\ &=\begin{pmatrix}H^{'}_{ll}&H^{'}_{ls}&0\\H^{'}_{sl}&H^{'}_{ss}&H^{'}_{sr}\\0&H^{'}_{rs}&H^{'}_{rr}\end{pmatrix}.\tag{3} \end{align}

Here, $\varepsilon_{F}$ is the Fermi energy level of the central scattering zone and $i$ is the imaginary number.

\begin{align} G_{ll}&=\left[H^{'}_{ll}\right]^{-1}\tag{4} \\ G_{rr}&=\left[H^{'}_{rr}\right]^{-1}\tag{5} \\ \Gamma_\text{left}&=H^{'}_{sl} G_{ll} H^{'}_{ls}\tag{6} \\ \Gamma_\text{right}&=H^{'}_{sr} G_{ll} H^{'}_{rs}\tag{7} \end{align}

Self energy for the left and right leads and the green function for the scattering zone are written below: \begin{align} \Sigma_\text{left}&=i \times \left[\Gamma_\text{left}-\left(\Gamma_\text{left}\right)^{+}\right] \tag{8} \\ \Sigma_\text{right}&=i \times \left[\Gamma_\text{right}-\left(\Gamma_\text{right}\right)^{+}\right]\tag{9} \\ G_{ss}^{R}&=\left[H^{'}_{ss}-\Gamma_\text{left}-\Gamma_\text{right}\right]^{-1}\tag{10} \\ G_{ss}^{A}&=\left(G_{ss}^{R}\right)^{+}\tag{11} \end{align}

Then, the charge conductance is written below:

$$G_{k}=\frac{2e^{2}}{h}\mathrm{Tr}\left[\Sigma_\text{left}G_{ss}^{A}\Sigma_\text{right}G_{ss}\right]\tag{12}$$

This conductance is only for one single k point in the reciprocal space. The total charge conductance is the integral of this single-k-point-conductance over the whole first Brillouin zone.

$$G_{\textrm{total}}=\frac{e}{h}\int G_{k}\times \left[f_{(\varepsilon_{F}-\mu_\text{left})}-f_{(\varepsilon_{F}-\mu_\text{right})}\right]d_{k}\tag{13}$$

where, $f_{(\varepsilon_{F}-\mu_\text{left})}$ and $f_{(\varepsilon_{F}-\mu_\text{right})}$ are Fermi-Dirac distribution function for left and right leads with the chemical potential $\mu_\text{left}$ and $\mu_\text{right}$ for the left and right leads.

To calculate the surface green function in equation (4) and (5), I used the Sancho's method.^[1]

The final conductivity value, $G_{\textrm{total}}$, is only correct at the $\Gamma$ point in the reciprocal space, but wrong at other k points.

I checked the internal output data and found that the calculation of the Surface Green's function was not right, since some data was NaN type. I guess this is because I did not use the correct way to avoid the singular matrix in the Surface Green's function calculation step.

Would anyone please give me some suggestions on the solution?

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References

1. M. P. L. Sancho, J. M. L. Sancho and J, Rubio, Highly convergent schemes for the calculation of bulk and surface Green functions, J. Phys. F: Met. Phys. 15 (1985) 851-858.