Self-Consistent NEGF Framework

• Self-consistent NEGF is a rigorous framework that iteratively links Green's functions and self-energies to capture quantum transport in open systems.
• It utilizes the Dyson and Keldysh equations alongside iterative updates to account for charge redistribution, screening, and bias effects in nanoscale devices.
• This approach accurately predicts observables like current and spectral density, making it essential for modeling quantum dynamics and nanostructured optoelectronics.

A self-consistent Nonequilibrium Green's Function (NEGF) approach is a computationally and conceptually rigorous framework for quantum transport and nonequilibrium dynamics in open, interacting electronic systems. The self-consistency refers to a feedback loop where Green's functions and all relevant self-energies (from contacts, interactions, phonons, or disorder) are built from, and in turn determine, each other, ensuring all observables respond to the actual nonequilibrium distributions and electrostatic profiles in the device.

1. Formulation of the Self-Consistent NEGF Scheme

The NEGF formalism revolves around the retarded ($G^r$), advanced ($G^a$), lesser ($G^<$), and greater ($G^>$) single-particle Green's functions, which encode propagation, occupation, and spectral properties under nonequilibrium conditions. The core equations in energy representation are the Dyson equation and the Keldysh equation: $G^r(E) = \left[E I - H - \Sigma^r(E)\right]^{-1}$

$G^<(E) = G^r(E) \Sigma^<(E) G^a(E)$

where $H$ is the device Hamiltonian (possibly including a mean-field term), and $\Sigma^{r,<}$ are the retarded and lesser self-energies due to all couplings (contacts, electron-phonon, impurities, correlations) (Camsari et al., 2020, Talarico et al., 2018).

Each self-energy in turn depends on the Green's functions, e.g., via the self-consistent Born approximation (SCBA) for interactions or disorder: $$\Sigma_{\text{int}}^<(E) = \sum_q |M_q|^2 \left[ N_q G^<(E-\hbar\omega_q) + (1+N_q) G^<(E+\hbar\omega_q)\right]$$ and similarly for $\Sigma^r$, requiring $G$ within $\Sigma$, closing the self-consistent loop (Camsari et al., 2020, Kawamura et al., 7 Mar 2025). For correlated systems, a conserving (Baym–Kadanoff) $\Phi$-derivable self-energy functional is constructed, maintaining physical conservation laws through the self-consistency (Talarico et al., 2018, Cornean et al., 2017).

2. Self-Consistency Cycle: Algorithmic Structure and Numerical Considerations

The self-consistent NEGF solution is necessarily iterative:

1. Initialize $\Sigma^{r,<}$ (e.g., zero for interactions or from a noninteracting Green's function).
2. Solve the Dyson equation for $G^r$ (usually by direct sparse inversion or recursive algorithms for block-tridiagonal structures) (Chen et al., 2012, Mosallanejad et al., 8 Apr 2025).
3. Update $G^a=(G^r)^\dagger$, then solve for $G^<$ via the Keldysh equation.
4. Rebuild all self-energies using the updated Green's functions (including those for inelastic scattering, electron-electron correlation, disorder, or charge self-consistency).
5. Mix new and old quantities if needed (Pulay or linear mixing) to stabilize convergence.
6. Check the norm of the update or physical constraint (e.g., current conservation, local charge neutrality); repeat if not converged (Talarico et al., 2018, Kawamura et al., 7 Mar 2025).

Self-consistent coupling to Poisson's equation is essential for realistic modeling, as carrier densities from $G^<$ feed back into the electrostatic potential profile: $$n(\mathbf r) = -\frac{i}{2\pi} \int dE\, G^<(\mathbf r,\mathbf r; E)$$ which must be recomputed at each iteration and included in $H$ (or $H_{KS}$ in DFT-NEGF) (Yam et al., 2011, Mosallanejad et al., 8 Apr 2025). This coupling is critical for devices under strong bias, with realistic band bending and screening.

Parallelization strategies and optimal data distribution are important for scalable performance, with grid- or block-cyclic layouts for time/energy/orbital indices minimizing communication bottlenecks in large-scale calculations (Talarico et al., 2018, Mosallanejad et al., 8 Apr 2025).

3. Physical Role and Implementation of Self-Consistency

Physical self-consistency ensures that the out-of-equilibrium density, potential, and spectral function respond to the presence of applied bias, charge rearrangement, and dissipative processes. In transport through molecular devices or thin nanostructures, ignoring self-consistency leads to significant errors: for example, in DFT-NEGF, non-self-consistent (single-shot, frozen-density) approaches misestimate the screening, resulting in incorrect resonance energies and overestimation of current, particularly at low bias (Yam et al., 2011).

In interacting systems, non-self-consistent (first-order or non-renormalized) evaluation of correlation or disorder effects violates conservation laws and yields unphysical results (e.g., non-conserved currents or negative occupation probabilities). Self-consistency, e.g., via SCBA for phonons, restores proper redistribution of spectral weight, energy flow, and phase coherence (Camsari et al., 2020, Kawamura et al., 7 Mar 2025).

For disorder averaging, the coherent potential approximation (CPA) is solved self-consistently for the disorder-averaged Green's function, enforcing correct statistical properties without explicit configuration sampling. The nonequilibrium CPA (NECPA) introduces coupled self-consistency loops in both retarded and lesser sectors that precisely account for nonequilibrium vertex corrections (Zhu et al., 2013).

4. Specialized Self-Consistent NEGF Implementations

DFT–NEGF and TDDFT–NEGF Comparison

In combined DFT–NEGF, the Kohn–Sham Hamiltonian $H_{KS}[\rho]$ depends self-consistently on the density extracted from $G^<$. Numerical studies show that time-dependent DFT (propagating the density matrix under bias until the system reaches a steady-state) and fully self-consistent DFT–NEGF yield identical steady-state currents for realistic molecular junctions, both well within and well beyond linear response. The principal source of quantitative discrepancy in non-self-consistent approximations is the failure to capture the screening of the device region, manifest through the density–density susceptibility difference between noninteracting ($\chi_0$) and fully interacting ($\chi$) forms (Yam et al., 2011).

Quantum Optoelectronics and Reaction Kinetics

In photoelectrochemical systems, NEGF must be self-consistently solved with Poisson's equation and the kinetic constraints imposed by surface electrochemistry (e.g., Butler–Volmer boundary conditions). Here, self-consistency extends to reaction-induced Fermi level shifts and photon/phonon self-energies, with convergence criteria including current–overpotential matching (Hällström et al., 2023).

Finite-Volume and Three-Dimensional Discretizations

Practical solution of self-consistent NEGF equations in 3D or for complex device geometries leverages cell-centered finite-volume discretization. This approach guarantees charge conservation, unit consistency, and robust handling of material interfaces, and is essential for nanostructure modeling. The self-consistent NEGF–Poisson scheme in this framework incorporates block-tridiagonal sparse solves, adaptive energy meshes, and accurate treatment of contact self-energies (Mosallanejad et al., 8 Apr 2025).

Large-Scale and Long-Time Dynamics

Emerging computational techniques incorporate tensor-train (QTT) compression and advanced predictor-corrector iteration in the self-consistency loop, vastly extending accessible time and system size for correlated dynamics. Dynamic mode decomposition (DMD) extrapolation supplies accurate QTT initial guesses, and causality-preserving block-time-stepping ensures stable convergence for, e.g., GW-level self-energies on $32 \times 32$ lattices (Środa et al., 26 Sep 2025).

5. Applications, Observables, and Limitations

Self-consistent NEGF enables ab initio quantum transport (DFT–NEGF), correlated dynamics (GW, second Born, T-matrix), ultrafast response, optoelectronic device simulation, and quantum thermodynamics. Physical observables, such as current, local spectral density, and occupation, are computed from $G^<$ and $G^r$, using formulas like the Landauer–Büttiker current,

$$I = \frac{2e}{h} \int dE\, \mathrm{Tr} \left[\Gamma_L G^r \Gamma_R G^a \right](f_L - f_R)$$

and its generalizations (Yam et al., 2011, Chen et al., 2012).

In strongly correlated, nanoscale, or optoelectronic systems, self-consistent NEGF captures phenomena inaccessible by semiclassical or non-self-consistent approaches (e.g., bias-induced level shifts, nonlocal impurity scattering, energy-resolved carrier injection, and non-equilibrium phase transitions) (Sano, 28 Jan 2025, Hällström et al., 2023).

Nonetheless, the approach faces limits from computational scaling, especially with two-time memory kernels in time-dependent problems. Recent algorithmic innovations—including time-local reformulations (G1–G2), memory compression (QTT), and rigorous convergence criteria—have substantially mitigated these challenges for both equilibrium and far-from-equilibrium regimes (Bonitz et al., 2023, Balzer et al., 2022, Środa et al., 26 Sep 2025).

6. Mathematical Validation and Physical Consistency

The self-consistent NEGF structure can be placed on rigorous mathematical grounds. For finite or open systems with retarded self-energies meeting Volterra-type causality and boundedness requirements, existence and uniqueness of solutions are ensured. The iterative scheme guarantees convergence under suitable initial conditions, preserving physical causality and dissipation (Cornean et al., 2017). The absence of an explicit contour-ordering in real-time treatments is not an approximation; Dyson and Langreth identities are constructed in operator form and hold equally for transient and steady-state regimes.

Self-consistent NEGF thus offers a complete, conserving description of nonequilibrium quantum transport and correlated dynamics, essential for predictive modeling of modern nanoscale and mesoscopic systems.