NEGF Formalism in Quantum Transport

• Non Equilibrium Green's Function formalism is a comprehensive quantum-field framework for simulating open, interacting systems out of equilibrium, capturing both coherent and dissipative processes.
• It employs contour-ordered Green's functions and self-energy corrections via Dyson and Keldysh equations to model transport, scattering, and reservoir interactions in nanoscale devices.
• NEGF enables precise quantitative analysis of electronic, thermal, and optoelectronic transport while leveraging advanced numerical methods for large-scale and time-dependent simulations.

The non-equilibrium Green’s function (NEGF) formalism is a comprehensive quantum-field-theoretic framework for simulating open, interacting systems driven out of equilibrium. It provides unified methods for describing transport, dissipation, coherence, and correlation effects in nanoscale devices, including electronic, phononic, and optoelectronic applications. The NEGF approach encodes all relevant physical observables—such as densities, currents, and entropy production—in terms of real- and complex-time correlation functions (Green’s functions) and their associated self-energies, capturing both coherent evolution and irreversible scattering processes. Over the past three decades, NEGF has become the core methodology for quantum transport theory in nanoelectronics, mesoscopic physics, thermal conduction, photovoltaics, and beyond.

1. Fundamentals: Green’s Functions, Keldysh Formalism, and Core Equations

The central objects of NEGF are the contour-ordered Green’s functions, typically defined on the Keldysh (non-equilibrium) time contour. For fermionic fields, the principal components are:

• Retarded Green’s function:

$$G^r(1,2)\equiv -i\Theta(t_1-t_2)\langle\{\Psi(1), \Psi^\dagger(2)\}\rangle$$

• Advanced Green’s function: $$G^a(1,2)\equiv +i\Theta(t_2-t_1)\langle\{\Psi(1), \Psi^\dagger(2)\}\rangle$$
• Lesser (correlation) Green’s function: $$G^<(1,2)\equiv i\langle\Psi^\dagger(2)\Psi(1)\rangle$$
• Greater Green’s function: $$G^>(1,2)\equiv -i\langle\Psi(1)\Psi^\dagger(2)\rangle$$

The Dyson equation relates the full Green’s function to the noninteracting one and the self-energy $\Sigma$:

$G = G_0 + G_0 \Sigma G$

or in matrix notation, in energy representation:

$G^r(E) = [EI - H - \Sigma^r(E)]^{-1}$

with the advanced and lesser/greater components related by Keldysh equations such as:

$G^< = G^r \Sigma^< G^a$

The self-energy $\Sigma$ encodes all effects of reservoirs, interactions, and inelastic scattering. The structure of these equations enables a unified treatment of coherent evolution and source/sink processes.

2. Open Quantum Systems: Boundary Conditions, Self-Energies, and Particle/Boson Baths

Open boundary conditions are represented via retarded and lesser self-energies, derived from coupling the device region to external reservoirs. For lead $\alpha$, typical reservoir self-energies in the wide-band approximation are:

$$\Sigma_\alpha^r(E) = -\frac{i}{2}\Gamma_\alpha, \quad \Sigma_\alpha^<(E) = i f_\alpha(E)\Gamma_\alpha$$

where $f_\alpha(E)$ is the Fermi function of reservoir $\alpha$.

This formalism extends to bosonic (e.g., phonon, photon) baths through analogous self-energies, permitting coupled electron-photon-phonon dynamics essential for quantum optoelectronics and nonequilibrium thermodynamics (Aeberhard, 2012, Michelini et al., 2016). Coupling to Lindblad dissipators (dissipative, possibly non-Markovian operations) can be incorporated by further extending the NEGF contour structure and self-energy definitions (Stefanucci, 2024).

3. Interactions, Scattering, and Approximations

Interparticle (e.g., electron-electron, electron-phonon, or electron-photon) interactions are systematically included via diagrammatic expansions (e.g., self-consistent Born, $GW$, $T$-matrix, or ladder approximations), generating nonlocal, energy-dependent, and history-dependent self-energies.

For elastic and inelastic scattering, the SCBA yields self-energies coupling Green’s functions at energies $E$ and $E\pm\hbar\omega$ (for phonon or photon exchange), and the lesser/greater self-energies are functionals of the occupation and mode populations (Pal et al., 2012, Wang et al., 2013, Sano, 28 Jan 2025). The Hubbard-operator approach allows for exact treatment of strong on-site interaction with perturbative system-bath coupling (Chen et al., 2016).

Impurity scattering is rigorously nonlocal. For discrete dopants, the total impurity potential is partitioned into a long-range Hartree field and a short-range scattering part, resulting in self-energies that, formulated in Wigner–Fourier coordinates, are non-diagonal and position-dependent (Sano, 28 Jan 2025). The position-dependent scattering rate extracted in these coordinates yields a rigorous, spatially resolved generalization of Fermi’s golden rule.

4. Applications: Quantum Transport, Thermal Conductance, and Spintronics

The NEGF framework underpins virtually all advanced quantum transport simulations in modern nanoelectronics and mesoscopic physics. Examples include:

• Electronic conduction and device physics: NEGF is the standard for simulating quantum dots, nanowires, molecular electronics, MOSFETs, and spintronic devices under nonequilibrium conditions, incorporating coherent effects, dissipation, and complex boundary conditions (Pal et al., 2012, Cauley et al., 2011, Camsari et al., 2014).
• Thermal transport: Quantum heat flow by phonons is naturally handled via analogous NEGF machinery, with “ballistic” and “diffusive” regimes accessible by including or neglecting anharmonic self-energies. Thermal boundary conductance and spectral energy exchange are obtained via Meir–Wingreen-type formulas (Wang et al., 2013, Guo et al., 2021).
• Optoelectronics and photovoltaics: Full coupled nonequilibrium quantum kinetics of electrons, photons, and phonons is available, including photogeneration, recombination, electron-photon-phonon cascades, and entropy production (Aeberhard, 2012, Michelini et al., 2016).
• Spintronic transport: The formalism is generalized to treat charge, spin, and spin-transfer torque currents in coherent or partially dephased devices, via $4\times 4$ conductance matrices and spin-resolved self-energies (Camsari et al., 2014, 0807.1709).

The framework allows current, density, and local density of states calculations at arbitrary bias, doping, disorder, or time-dependent drives. In particular, the generalized Landauer–Büttiker and Meir–Wingreen formulas are encoded in the NEGF structure, naturally incorporating interaction, correlation, and dissipation with rigorous sum rules and conservation laws (Cornean et al., 2017).

5. Numerical Implementation, Algorithmic Scaling, and Large-Scale Simulations

Practical simulation of NEGF models is computationally demanding due to the need to solve for Green’s functions that depend on two space and time (or energy) arguments, and often require self-consistent solution of Dyson–Keldysh–Poisson systems. Algorithmic innovations include:

• Recursive Green’s Function (RGF) and semiseparable representations: These allow $O(N)$ (or $O(N^2)$ for multi-terminal) scaling for systems with tridiagonal/block-tridiagonal structure (Cauley et al., 2011, Sajjad et al., 2013).
• Mode-space and dimensional reduction: In devices with periodicity or symmetry, mode summation or effective 1D reductions drastically reduce computational complexity without significant loss of accuracy (Pal et al., 2012).
• Distributed algorithms and parallelization: Block and domain decomposition enables the simulation of systems with $>10^5$ atoms and efficient computation of $G^r$, $G^<$ for large device geometries under realistic scattering (Cauley et al., 2011).
• Time-linear algorithms: The generalized Kadanoff–Baym ansatz (GKBA) reduces memory and CPU requirements from $O(t_{\mathrm{f}}^3)$ or $O(t_{\mathrm{f}}^2)$ to $O(t_{\mathrm{f}})$, allowing simulation of nonequilibrium evolution over long timescales (Pavlyukh et al., 2021).

These computational tools enable ab initio simulation of mesoscopic devices, multi-terminal quantum networks, and systems with many degrees of freedom.

6. Recent Extensions: Strong Correlations, Dissipation, and Floquet-Driven Dynamics

NEGF has been extended to treat strongly correlated systems using Hubbard operators, in which intra-system interactions are captured exactly, and coupling to reservoirs is perturbative (Chen et al., 2016). Dissipative systems governed by Lindblad-type operators are incorporated by modifying the self-energy structure on the Keldysh contour, maintaining the diagrammatic consistency and conservation laws (Stefanucci, 2024).

For periodically driven (Floquet) systems, the NEGF formalism is reformulated in extended Floquet space, with time-periodic Hamiltonians handled via block-matrix Dyson–Keldysh equations, enabling the study of photon-assisted transport, spin currents under optical driving, and light–matter coupling in quantum dots and molecular junctions (Mosallanejad et al., 2023).

7. Physical Interpretation, Limitations, and Future Directions

NEGF provides a unifying, formally exact structure for modeling nonequilibrium quantum systems. It enforces all fundamental conservation laws, rigorously incorporates open-system boundary conditions, accommodates quantum statistics of fermions and bosons, and captures both mean-field and beyond-mean-field correlation phenomena. The formalism’s critical strengths include its flexibility for treating mixed particle-boson systems, rigorous connection to measurable observables, and systematic inclusion of arbitrary interactions and dissipative processes (Ridley et al., 2022, Aeberhard, 2012).

Its principal limitations are the steep scaling with system size and complexity for full two-time (or two-frequency) propagators, the need for judicious approximations in self-energy computation for strong correlations, and the numerical effort for full self-consistency, particularly in time-dependent or multi-component cases. Recent algorithmic improvements—such as time-linear scaling, semiseparable algorithms, and hybrid quantum-classical reductions—are rapidly expanding the tractable domain.

NEGF remains foundational for quantum transport, open-system quantum dynamics, and microscopic nonequilibrium thermodynamics, with ongoing advances in mathematical rigor, computational algorithms, and application breadth (Cornean et al., 2017, Pavlyukh et al., 2021, Sano, 28 Jan 2025).