Solve the Non-Equillibrium Green's Functions (NEGF) transport on examples for educational purposes. Limited to a 1D linear chain for now.
See the documentation website.
If you find this package useful, please cite L. Vojáček, Multiscale Modeling of Spin-Orbitronic Phenomena at Metal, Oxide, and 2D Material Interfaces for Spintronic Devices, PhD thesis, Université Grenoble Alpes (2024).
- the
LinearChain class including the NEGF routines resides in src/fuNEGF/models.py
- a Jupyter notebook
examples/one-dimensional_channel.ipynb contains the linear chain case study with the underlying physics explained
- a Jupyter notebok
examples/time_complexity.ipynb contains a time complexity study of constructing the model $\mathcal{O}(N)$ and calculating the transmission coefficient $\mathcal{O}(N^2)$
A linear chain with a single or multiple on-site potential impurities will present a chemical potential (occupation) drop, which may not be apparent unless a phase relaxation is included, as shown below.
An additional momentum relaxation will cause a non-zero chemical potential slope in between the impurity regions.
The complete description and calculation are provided in the examples/one-dimensional_channel.ipynb notebook.
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
$$ \begin{align}
\mathbf{\Sigma} &= \mathbf{\Sigma}_1 + \mathbf{\Sigma}_2 + \mathbf{\Sigma}_0 , \\
\mathbf{\Sigma}^{\mathrm{in}} &= \mathbf{\Sigma}^{\mathrm{in}}_1 + \mathbf{\Sigma}^{\mathrm{in}}_2 + \mathbf{\Sigma}^{\mathrm{in}}_0 .
\end{align}
$$
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
$$
\begin{align}
\mathbf{\Sigma} &\equiv \mathbf{\Sigma}^\mathrm{R} , \\
\mathbf{G}^\mathrm{n} &\equiv -i \mathbf{G}^< , \\
\mathbf{\Sigma}^\mathrm{in} &\equiv -i \mathbf{\Sigma}^< .
\end{align}
$$
For the LinearChain model, the Hamiltonian
$$
H_{ij} = \begin{cases}
\epsilon_0, & \text { if } i=j \\
t, & \text{ if } i \neq j
\end{cases}
$$
Impurity potential $U$ can be added to the on-site energy as
$$
\mathbf{H}=\left[
\begin{array}{ccccc}
\ddots & \vdots & \vdots & \vdots & \ddots \\
\cdots & \varepsilon & t & 0 & \cdots \\
\cdots & t & \varepsilon+U & t & \cdots \\
\cdots & 0 & t & \varepsilon & \cdots \\
\ddots & \vdots & \vdots & \vdots & \ddots
\end{array}
\right] .
$$
The self-energies
$$
\mathbf{\Sigma}_1=\left[\begin{array}{ccccc}
\mathrm{te}^{i k a} & 0 & 0 & \cdots & 0 \\
0 & 0 & 0 & \cdots & 0 \\
0 & 0 & 0 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & 0
\end{array}\right], \quad \mathbf{\Sigma}_2=\left[\begin{array}{ccccc}
0 & \cdots & 0 & 0 & 0 \\
\vdots & \ddots & \vdots & \vdots & \vdots \\
0 & \cdots & 0 & 0 & 0 \\
0 & \cdots & 0 & 0 & 0 \\
0 & \cdots & 0 & 0 & \mathrm{te}^{i k a}
\end{array}\right] ,
$$
with the broadening functions $\mathbf{\Gamma} \equiv i\left[ \mathbf{\Sigma} - \mathbf{\Sigma}^\dagger\right] $
$$
\mathbf{\Gamma}_1=\frac{\hbar v}{a}\left[\begin{array}{ccccc}
1 & 0 & 0 & \cdots & 0 \\
0 & 0 & 0 & \cdots & 0 \\
0 & 0 & 0 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & 0
\end{array}\right], \quad \mathbf{\Gamma}_2=\frac{\hbar v}{a}\left[\begin{array}{ccccc}
0 & \cdots & 0 & 0 & 0 \\
\vdots & \ddots & \vdots & \vdots & \vdots \\
0 & \cdots & 0 & 0 & 0 \\
0 & \cdots & 0 & 0 & 0 \\
0 & \cdots & 0 & 0 & 1
\end{array}\right] ,
$$
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
$$
\mathbf{D} = D_0^\text{phase}
\left[\begin{array}{ccccc}
1 & 1 & 1 & \cdots & 1 \\
1 & 1 & 1 & \cdots & 1 \\
1 & 1 & 1 & \cdots & 1 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
1 & 1 & 1 & \cdots & 1
\end{array}\right]
+ D_0^\text{phase-momentum}
\left[\begin{array}{ccccc}
1 & 0 & 0 & \cdots & 0 \\
0 & 1 & 0 & \cdots & 0 \\
0 & 0 & 1 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & 1
\end{array}\right] ,
$$
which is used for an element-wise multiplication $\odot$ of the Green's function matrices
$$
\begin{align}
\mathbf{\Sigma}_0 &= \mathbf{D} \odot \mathbf{G}^\text{R}, \\
\mathbf{\Sigma}^\text{in}_0 &= \mathbf{D} \odot \mathbf{G}^\text{n} .
\end{align}
$$
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.
