Simulation Output#

NEGF Simulation Output#

Charge carrier densities#

The charge carrier densities are computed from the lesser and greater Green's functions and saved as numpy arrays (electron_density.npy / hole_density.npy) in the output_directory. These quantities are both orbital-resolved and have shapes of (num_energies, *num_kpoints, num_orbitals). *num_kpoints is only present if the simulation is performed with a transverse k-point grid (see kpoint_grid).

\[ \rho^{e/h}_i(E, \mathbf{k}_\perp) = \frac{2_{\mathrm{spin}}}{2\pi} \mathrm{Im}\left\{ G^{\lessgtr}_{ii}(E, \mathbf{k}_\perp) \right\} \]

Local density of states (LDOS)#

Similar to the charge carrier densities, the local density of states (LDOS) is computed from the retarded Green's function and saved as a numpy array (ldos.npy) in the output_directory. The LDOS is also orbital-resolved and has a shape of (num_energies, *num_kpoints, num_orbitals).

\[ g_i(E, \mathbf{k}_\perp) = \frac{2_{\mathrm{spin}}}{2\pi} \mathrm{Im}\left\{ G^{R}_{ii}(E, \mathbf{k}_\perp) \right\} \]

Spectral current#

Two types of spectral device current can be computed with quatrex:

The Meir-Wingreen current, which is computed from the lesser and greater Green's functions and the self-energies, and saved as a numpy array (current_meir_wingreen.npy). This quantity is resolved per transport cell and has a shape of (num_energies, *num_kpoints, num_transport_cells + 1). It includes the contribution from the reservoirs into / out of the device (hence the + 1). It is only output if compute_current flag is set to true. In block-distributed simulations, only the contact currents are computed, while the remainder will be set to NaN.

\[ j_{n-1 \to n}(E, \mathbf{k}_\perp) = \mathrm{tr}\left[ \mathbf{\widetilde{\Sigma}}^{>}_{nn}(E, \mathbf{k}_\perp) \mathbf{G}^{<}_{nn}(E, \mathbf{k}_\perp) - \mathbf{G}^{>}_{nn}(E, \mathbf{k}_\perp) \mathbf{\widetilde{\Sigma}}^{<}_{nn}(E, \mathbf{k}_\perp) \right] \]

The spectral current computed from the commutator of the Hamiltonian and the lesser Green's function (quantum Liouville equation). This quantity is also resolved per transport cell and has a shape of (num_energies, *num_kpoints, num_transport_cells - 1). This only includes the current flowing between the transport cells (not the reservoirs).

\[ j_{n \to n+1}(E, \mathbf{k}_\perp) = \mathbf{H}_{n, n+1} \odot \mathbf{G}^{<}_{n+1, n}(E, \mathbf{k}_\perp) - \mathbf{G}^{<}_{n, n+1}(E, \mathbf{k}_\perp) \odot \mathbf{H}_{n+1, n} \]

QTBM Simulation Output#

Transmission function#

The transmission function is the main output of a QTBM simulation. It is written for every combination of leads and saved as a numpy array transmission_<xy>.npy, where <xy> indicates the direction of transport denoted by the contact name initials (e.g., lr for two contacts named "left" and "right"). The transmission function has a shape of (*num_kpoints, num_energies).

Current#

The current output from a QTBM simulation is already integrated over energy and the transverse k-points and saved as current_<xy>.npy, where <xy> is again the direction of transport denoted by the contacts.

Local density of states (LDOS) per contact#

The LDOS per contact contains the contribution of each contact to the total LDOS. These are saved as numpy arrays dos_<x>.npy, where <x> is the contact name. They are orbital-resolved and have a shape of (*num_kpoints, num_orbitals, num_energies).

Self-consistent Schrödinger-Poisson Simulation Output#

Besides the regular transport outputs, self-consistent Schrödinger-Poisson simulations will produce orbital-centered potential (potential.npy) and "real-space" excess charge density (real_space_charge_density.npy) and potential (real_space_potential.npy) files for each iteration of the self-consistent loop. Real-space here means that the quantities are given on the finite-element mesh used for the Poisson solver.

Profiling and Timing Information#

Every simulation run will produce a quatrex_times.out file in the directory where quatrex was invoked. This file contains timing information for different parts of the simulation. The file contents vary depending on the simulation type and configuration, but they typically look something like this:

SCBA: Sparsity Pattern : 0.0006s
SCBA: Sparsity Pattern all : 0.0007s
SCBA: Sparsity Pattern : 0.0006s
SCBA: Sparsity Pattern all : 0.0006s
      ElectronSolver: Assemble : 0.2277s
      ElectronSolver: Assemble all : 0.2277s
      ElectronSolver: Band edges : 0.0236s
      ElectronSolver: Band edges all : 0.0236s
      ElectronSolver: OBC : 4.7030s
      ElectronSolver: OBC all : 4.7030s
      ElectronSolver: Solve : 1.1370s
      ElectronSolver: Solve all : 1.1370s
      ElectronSolver: Filter : 0.0017s
      ElectronSolver: Filter all : 0.0017s
    ElectronSolver : 6.0935s
    ElectronSolver all : 6.0935s
    SCBA: G observables : 0.0244s
    SCBA: G observables all : 0.0244s
    SCBA: stack->nnz transpose : 0.1931s
    SCBA: stack->nnz transpose all : 0.1931s
...

The indentation indicates the hierarchy of the different parts of the simulation, i.e., the line ElectronSolver: Assemble : 0.2277s is accounted for in the total time of ElectronSolver : 6.0935s. The word all means that the timing occurs after synchronization across all MPI processes, while the lines without all indicate the timing for only rank 0.