Introduction to NEGF-DFT

This section provides an overview of the Non-Equilibrium Green’s Function (NEGF) method combined with Density Functional Theory (DFT) for quantum transport calculations.

Historical Context

The gauNEGF package builds upon the foundational work of ANT.Gaussian [Palacios2002], which pioneered the implementation of NEGF-DFT calculations using Gaussian basis sets. This approach has proven particularly effective for molecular electronics and nanoscale transport calculations.

Core Functionality

The gauNEGF package provides:

  • Energy-independent and energy-dependent NEGF calculations [Damle2002]

  • Multiple contact models (Bethe lattice [Jacob2011], 1D chain)

  • Transmission and current calculations

  • Spin-dependent transport [Zollner2020]

  • Temperature and voltage effects

Mathematical Framework

Key Equations

The central quantities in NEGF-DFT are:

  1. Green’s Functions

    (1)\[G^r(E) = [(E+i\eta)S - F - \Sigma(E)]^{-1}\]

    where:

    • \(G^r(E)\) is the retarded Green’s function

    • \(E\) is the energy

    • \(\eta\) is a small broadening parameter

    • \(S\) is the overlap matrix

    • \(F\) is the Fock matrix

    • \(\Sigma(E)\) is the self-energy

    The lesser Green’s function is given by:

    (2)\[G^\lt(E) = G^r(E) [\Gamma_L f(E-\mu_L) + \Gamma_R f(E-\mu_R)] G^{r\dagger}(E)\]

    where:

    • \(G^\lt(E)\) is the lesser Green’s function

    • \(\Gamma_{L,R}\) are the broadening matrices for left and right contacts

    • \(f(E)\) is the Fermi-Dirac distribution

    • \(\mu_{L,R}\) are the chemical potentials of the contacts

  2. Self-Energies

    \[\Sigma(E) = \sum_i \tau_i g_{s,i}(E) \tau_i^\dagger\]

    where:

    • \(\tau_i\) is the coupling matrix to contact i

    • \(g_{s,i}(E)\) is the surface Green’s function of contact i

    • The sum runs over all contacts

  3. Density Matrix

    \[\rho = -\frac{1}{2\pi} \int_{-\infty}^{\infty} G^\lt(E) dE\]

    This gives the electron density used in the self-consistent cycle.

Contact Models and Testing

The package includes several contact models:

  1. Bethe Lattice (surfGBethe.py)

    • Ideal for metallic contacts

    • Energy-dependent self-energy

    • Realistic density of states

  2. 1D Chain (surfG1D.py)

    • Perfect for molecular wires

    • Periodic boundary conditions

    • Band structure effects

  3. Constant Self-Energy (surfGTester.py)

    • Performance benchmarking

    • Adding Temperature dependence

    • Future development: Energy-dependent decoherence

Next Steps

Continue to Self-Consistent NEGF-DFT for details on the self-consistent procedure.

References

[Palacios2002]

Palacios, J. J., Pérez-Jiménez, A. J., Louis, E., SanFabián, E., & Vergés, J. A. (2002). First-principles approach to electrical transport in atomic-scale nanostructures. Physical Review B, 66(3), 035322. https://doi.org/10.1103/PhysRevB.66.035322

[Damle2002]

Damle, P., Ghosh, A. W., & Datta, S. (2002). First-principles analysis of molecular conduction using quantum chemistry software. Chemical Physics, 281(2-3), 171-187. https://doi.org/10.1016/S0301-0104(02)00496-2

[Jacob2011]

Jacob, D., & Palacios, J. J. (2011). Critical comparison of electrode models in density functional theory based quantum transport calculations. The Journal of Chemical Physics, 134(4), 044118. https://doi.org/10.1063/1.3526044

[Zollner2020]

Zöllner, M. S., Varela, S., Medina, E., Mujica, V., & Herrmann, C. (2020). Insight into the Origin of Chiral-Induced Spin Selectivity from a Symmetry Analysis of Electronic Transmission. Journal of Chemical Theory and Computation, 16(5), 2914-2929. https://doi.org/10.1021/acs.jctc.9b01078