Skip to content

Atomistic Quantum Transport#

Driven Quantum Systems#

In electronic structure calculations one is usually concerned with spectral properties (i.e. the eigenvalues and eigenvectors) of a quantum system in thermodynamic equilibrium. In quantum transport simulations, on the other hand, we are interested in properties of a system under non-equilibrium conditions.

In other words, instead of solving the eigenvalue problem

\[ \mathbf{H} \boldsymbol{\psi} = E \mathbf{S} \boldsymbol{\psi}, \]

where \(\mathbf{H}\) is the Hamiltonian and \(\mathbf{S}\) is the orbital overlap matrix, we are now interested in the system matrix

\[ \mathbf{M}(E) = E\mathbf{S} - \mathbf{H} \]

and how the system responds to external perturbations. The energy \(E\) is now an input parameter, entering the system matrix \(\mathbf{M}(E)\).

To drive the system away from equilibrium, we further couple it to external reservoirs that allow us to exert control over the particle flow through the system. Introducing these reservoirs leads to a change ("renormalization") of the original system's dynamics, which is typically expressed in terms of self-energies \(\mathbf{\Sigma}(E)\) that are subtracted from the system matrix:

\[ \mathbf{M}(E) = E\mathbf{S} - \mathbf{H} - \mathbf{\Sigma}(E) \]

A detailed description of how the self-energies \(\mathbf{\Sigma}(E)\) are computed for a given reservoir is provided in the section on open boundary conditions.

Besides boundary self-energies, one can also include other types of self-energies, such as those arising from scattering with different systems of particles, e.g. electron-electron or electron-phonon interactions. How these self-energies can be computed with quatrex, is discussed in the section on non-equilibrium Green's functions.

The self-energies \(\mathbf{\Sigma}(E)\) are usually non-Hermitian, which reflects the fact that the reservoirs and the interactions with other systems effectively act as sinks and sources of particles. The anti-Hermitian part of the self-energy is related to the lifetime of the states in the system, while the Hermitian part leads to a shift of the system's energy levels.

Atomistic Material Descriptions#

The matrices \(\mathbf{H}\) and \(\mathbf{S}\) are built from a set of localized basis functions \(\{\psi_i\}\), rather than from a plane-wave or real-space grid discretization. The matrix elements are given by the following integrals over the basis functions:

\[ \begin{aligned} H_{ij} &= \bra{\psi_i} \hat{H} \ket{\psi_j} \\ S_{ij} &= \braket{\psi_i \vert \psi_j} \end{aligned} \]

Depending on the level of theory, the basis functions \(\psi_i\) and the matrix elements \(H_{ij}\) and \(S_{ij}\) could be obtained from semi-empirical tight-binding models or from first-principles electronic structure calculations, e.g., using density functional theory (DFT). More information on how to provide this data to quatrex is provided in the section on electronic structure input data.

In any case, because the basis states are localized, \(\mathbf{H}\) and \(\mathbf{S}\) will be sparse, indicating that an orbital only has non-negligible overlap and only interacts with orbitals that are centered close by. This operator sparsity is what makes atomistic quantum transport computationally tractable for realistically-sized systems at all, and it is heavily exploited in quatrex, both in how all tensors are stored and in how the transport equations are solved.

In QTBM calculations, we use optimized sparse linear solvers to take advantage of the system's sparsity. In NEGF calculations, we can define a block-tridiagonal tiling of the matrices (if the atoms are ordered along the transport direction), which allows us to use the recursive Green's function (RGF) or related Schur-complement-based selected-inversion algorithms to compute the Green's functions without ever having to invert the full dense system matrix \(\mathbf{M}(E)\) explicitly.

Ordering of atoms and orbitals in NEGF calculations

The ordering of the atoms and orbitals in the input data is crucial for the performance of the transport calculations. In particular, the atoms should be ordered along the transport direction. This ensures that the system matrix \(\mathbf{M}(E)\) has a block-tridiagonal structure, which is necessary for the efficient use of the RGF algorithm in NEGF calculations.

While periodicity is always broken along the transport direction, it is often preserved in the transverse directions. This allows us to use Bloch's theorem to reduce the simulation domain size by solving the transport equations for a set of transverse wavevectors \(\mathbf{k}_\perp\). The Hamiltonian and overlap matrices for every transverse wavevector \(\mathbf{k}_\perp\) are obtained by Bloch-summation of the real-space matrices over the lattice vectors \(\mathbf{R}_\perp\) in the transverse directions:

\[ \begin{aligned} H_{ij}(\mathbf{k}_\perp) &= \sum_{\mathbf{R}_\perp} e^{i \mathbf{k}_\perp \cdot \mathbf{R}_\perp} H_{ij}(\mathbf{R}_\perp) \\ S_{ij}(\mathbf{k}_\perp) &= \sum_{\mathbf{R}_\perp} e^{i \mathbf{k}_\perp \cdot \mathbf{R}_\perp} S_{ij}(\mathbf{R}_\perp) \end{aligned} \]

Nanowire and nanoribbon structures have no periodicity in the transverse directions, so only the \(\Gamma\)-point (\(\mathbf{k}_\perp = 0\)) needs to be considered. For 2D materials, only the in-plane transverse direction is periodic, so a 1D grid of \(\mathbf{k}_\perp\)-points can be used. For 3D bulk materials, a 2D grid of \(\mathbf{k}_\perp\)-points would normally be used.