Non-Equilibrium Green’s functions (NEGF)
The retarded Green’s function
\[\mathbf{G}^{\mathrm{R}}=[E \mathbf{I}-\mathbf{H}-\mathbf{\Sigma}]^{-1}\]
is a function of energy \(E\) multiplied by the identity matrix \(\mathbf{I}\). The Hamiltonian \(\mathbf{H}\) and self-energy \(\mathbf{\Sigma}\) matrices are to be defined by the physical model.
Along with the advanced Green’s function
\[\mathbf{G}^{\mathrm{A}} = \left[ \mathbf{G}^{\mathrm{R}} \right]^\dagger\]
they provide the spectral function
\[\mathbf{A}=i\left[\mathbf{G}^{\mathrm{R}}-\mathbf{G}^{\mathrm{A}}\right]\]
and are used to solve for the “electron occupation” Green’s function
\[\mathbf{G}^{\mathrm{n}}=\mathbf{G}^{\mathrm{R}} \mathbf{\Sigma}^{\mathrm{in}} \mathbf{G}^{\mathrm{A}}\]
which gives the density matrix
\[\hat{\rho} = \mathbf{G}^{\mathrm{n}} / 2\pi .\]
The in-scattering term \(\mathbf{\Sigma}^{\mathrm{in}}\) is also defined by the physical model.
Both, the self-energy \(\mathbf{\Sigma}\) and the in-scattering term \(\mathbf{\Sigma}^{\mathrm{in}}\) are sums of the left contact \(\mathbf{\Sigma}_1\), right contact \(\mathbf{\Sigma}_2\) and an intrinsic term \(\mathbf{\Sigma}_0\), hence
\[\begin{split}\begin{align}
\mathbf{\Sigma} &= \mathbf{\Sigma}_1 + \mathbf{\Sigma}_2 + \mathbf{\Sigma}_0 , \\
\mathbf{\Sigma}^{\mathrm{in}} &= \mathbf{\Sigma}^{\mathrm{in}}_1 + \mathbf{\Sigma}^{\mathrm{in}}_2 + \mathbf{\Sigma}^{\mathrm{in}}_0 .
\end{align}\end{split}\]
NOTE: We use the (physically expressive) notation of S. Datta, where the self-energies and Green’s functions in relation to the standard notation (on the right) are defined as
\[\begin{split}\begin{align}
\mathbf{\Sigma} &\equiv \mathbf{\Sigma}^\mathrm{R} , \\
\mathbf{G}^\mathrm{n} &\equiv -i \mathbf{G}^< , \\
\mathbf{\Sigma}^\mathrm{in} &\equiv -i \mathbf{\Sigma}^< .
\end{align}\end{split}\]
Linear Chain Model
For the LinearChain model, the Hamiltonian
\[\begin{split}H_{ij} = \begin{cases}
\epsilon_0, & \text { if } i=j \\
t, & \text{ if } i \neq j
\end{cases}\end{split}\]
Impurity potential \(U\) can be added to the on-site energy as
\[\begin{split}\mathbf{H}=\left[
\begin{array}{ccccc}
\ddots & \vdots & \vdots & \vdots & \ddots \\
\cdots & \varepsilon & t & 0 & \cdots \\
\cdots & t & \varepsilon+U & t & \cdots \\
\cdots & 0 & t & \varepsilon & \cdots \\
\ddots & \vdots & \vdots & \vdots & \ddots
\end{array}
\right] .\end{split}\]
The self-energies
\[\begin{split}\mathbf{\Sigma}_1=\left[\begin{array}{ccccc}
\mathrm{te}^{i k a} & 0 & 0 & \cdots & 0 \\
0 & 0 & 0 & \cdots & 0 \\
0 & 0 & 0 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & 0
\end{array}\right], \quad \mathbf{\Sigma}_2=\left[\begin{array}{ccccc}
0 & \cdots & 0 & 0 & 0 \\
\vdots & \ddots & \vdots & \vdots & \vdots \\
0 & \cdots & 0 & 0 & 0 \\
0 & \cdots & 0 & 0 & 0 \\
0 & \cdots & 0 & 0 & \mathrm{te}^{i k a}
\end{array}\right] ,\end{split}\]
with the broadening functions \(\mathbf{\Gamma} \equiv i\left[ \mathbf{\Sigma} - \mathbf{\Sigma}^\dagger\right]\)
\[\begin{split}\mathbf{\Gamma}_1=\frac{\hbar v}{a}\left[\begin{array}{ccccc}
1 & 0 & 0 & \cdots & 0 \\
0 & 0 & 0 & \cdots & 0 \\
0 & 0 & 0 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & 0
\end{array}\right], \quad \mathbf{\Gamma}_2=\frac{\hbar v}{a}\left[\begin{array}{ccccc}
0 & \cdots & 0 & 0 & 0 \\
\vdots & \ddots & \vdots & \vdots & \vdots \\
0 & \cdots & 0 & 0 & 0 \\
0 & \cdots & 0 & 0 & 0 \\
0 & \cdots & 0 & 0 & 1
\end{array}\right] ,\end{split}\]
where \(v=\mathrm{d} E /(\hbar \mathrm{d} k) = -2 a t / \hbar \sin (k a)\) so that \(\frac{\hbar v}{a} = -2 t / \sin (k a)\).
The in-scattering terms
\[\mathbf{\Sigma}^\mathrm{in}_i = \mathbf{\Gamma}_i \cdot f_i(E) ,\]
where \(f_i(E)\) is the Fermi-Dirac distribution function for contact \(i \in \set{1, 2}\).
The self-energies describing the phase and phase-momentum relaxation are defined in terms of the Green’s functions themselves. Their strength is defined via the (scalar) coefficients \(D_0^\text{phase}\) and \(D_0^\text{phase-momentum}\), creating a “mask” matrix
\[\begin{split}\mathbf{D} = D_0^\text{phase}
\left[\begin{array}{ccccc}
1 & 1 & 1 & \cdots & 1 \\
1 & 1 & 1 & \cdots & 1 \\
1 & 1 & 1 & \cdots & 1 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
1 & 1 & 1 & \cdots & 1
\end{array}\right]
+ D_0^\text{phase-momentum}
\left[\begin{array}{ccccc}
1 & 0 & 0 & \cdots & 0 \\
0 & 1 & 0 & \cdots & 0 \\
0 & 0 & 1 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & 1
\end{array}\right] ,\end{split}\]
which is used for an element-wise multiplication \(\odot\) of the Green’s function matrices
\[\begin{split}\begin{align}
\mathbf{\Sigma}_0 &= \mathbf{D} \odot \mathbf{G}^\text{R}, \\
\mathbf{\Sigma}^\text{in}_0 &= \mathbf{D} \odot \mathbf{G}^\text{n} .
\end{align}\end{split}\]
Since the Green’s functions enter the definition of the self-energy, a self-consistent loop is performed, where \(\mathbf{G}^\text{R}\) and \(\mathbf{G}^\text{n}\) are initially set as zero matrices and iteratively updated, along with \(\mathbf{\Sigma}_0\) and \(\mathbf{\Sigma}^\text{in}_0\). About 70 iteration steps are usually enough to reach a convergence.