Atomistic Green's Function Method

• Atomistic Green's Function Method is a rigorous framework that models quantum and classical transport using atomistic Hamiltonians or force-constant matrices.
• It partitions systems into finite device regions and semi-infinite leads, accurately capturing interface effects, defect scattering, and open-boundary conditions.
• The method enables efficient first-principles simulations of phonon and electron transport, leveraging block-recursive algorithms for large-scale systems.

The atomistic Green's function (AGF) method provides a mathematically rigorous and computationally efficient framework for modeling quantum and classical transport, elastic wave scattering, and surface and interface properties in materials, starting from atomistic Hamiltonians or force-constant matrices. AGF is formulated as a real-space implementation of the non-equilibrium Green's function (NEGF) formalism, enabling the treatment of systems coupled to reservoirs or infinite leads and supporting first-principles modeling from density functional theory (DFT), tight-binding, or empirical interatomic force constants. AGF is central in the quantitative description of ballistic and incoherent phonon transport, quantum electron transport, surface state engineering, defect scattering, dynamics at interfaces, and the analysis of more complex open-boundary systems.

1. Theoretical Foundations and Formalism

AGF partitions physical space into a finite device, surface, or defect region, and one or more semi-infinite bulk leads or reservoirs, which serve as both sources/sinks for particles and enforcers of open boundary conditions. At the core is the retarded Green's function for the central (device or surface) region, $G^r(E)$ (electronic) or $G^r(\omega)$ (phononic), incorporating both the local Hamiltonian/dynamical matrix and self-energies due to coupling to the semi-infinite environment. For electronic systems in DFT-based approaches, starting from the projected Kohn–Sham Hamiltonian and overlap matrices, the surface Green's function method replaces the infinite bulk by a frequency-dependent self-energy, enabling a formally exact treatment of the semi-infinite domain (Smidstrup et al., 2017, Li et al., 2016).

For lattice dynamics and phonon or acoustic/elastic wave problems, the AGF formalism operates in the harmonic approximation, constructing Green's functions from atomistic force-constant matrices, and partitioning the system into leads and a device region (Khodavirdi et al., 2023, Bachmann et al., 2011, Gu et al., 2015). Open-system boundary conditions are enforced through lead self-energies, which capture the correct outgoing/incoming wave physics.

Dyson's equation relates the full system Green’s function $G$ to the uncoupled (lead/device) Green’s functions and the coupling matrices, allowing for exact inclusion of arbitrary localized perturbations such as defects, interfaces, or heterostructures (Ong, 2024). Non-equilibrium effects, bias, or time-periodic drive are included through the Keldysh NEGF extension for steady-state or transient response (Mucciolo et al., 8 Sep 2025).

2. Basis-Set and Matrix Representations

AGF is most computationally efficient when constructed in a localized, finite-support atomic basis (e.g., LCAO or pseudoatomic orbitals for electrons, real-space site displacements for phonons). In this basis, the Hamiltonian (electronic) or harmonic force-constant matrix (phononic) becomes sparse and can be arranged into block-tridiagonal form when the system has short-range couplings or is built from repeated principal layers (Smidstrup et al., 2017, Nguyen et al., 2024).

Key quantities:

• $$H_{ij} = \langle \phi_i | \hat{H}^{KS} | \phi_j \rangle$$ and $S_{ij} = \langle \phi_i | \phi_j \rangle$ for electrons
• $K_{ij}$ and mass-weighted dynamical matrices for phonons
• Coupling blocks $V_{SB}, V_{BB}$ for device-lead circuits

This enables block-recursive or decimation algorithms to efficiently access only the relevant matrix blocks, avoiding cubic-scaling full inversions even for very large supercells.

3. Central Equations and Computational Algorithms

The core AGF equations in the device or surface region S (omitting frequency/energy arguments) are: $G^r = [E\,S_S - H_S - \Sigma^r]^{-1}$ for electrons, and

$$G^r(\omega) = [( \omega + i\eta )^2 I_S - D_S - \Sigma_L(\omega) - \Sigma_R(\omega)]^{-1}$$

for phonons (Bachmann et al., 2011, Sadasivam et al., 2017). The self-energy

$\Sigma^r = V_{SB} g_B^r V_{BS}$

encapsulates the full effect of the semi-infinite leads (with $g_B^r$ the surface Green’s function of the uncoupled bulk, typically computed via Lopez-Sancho–type decimation).

The spectral function, local density of states (LDOS), and transmission are given as

$A(E) = -\frac{1}{\pi}\Im[G^r(E)]$

$T(E) = \Tr[\Gamma_L G^r \Gamma_R G^{r\dagger}]$

$$\Gamma_{L/R} = i [\Sigma_{L/R} - \Sigma_{L/R}^\dagger]$$

Landauer-type integral expressions yield physical observables such as current or thermal conductance.

For large superlattice devices, memory and speed are optimized using recursive algorithms exploiting principal-layer decomposition, backward/forward sweeps for connected Green's functions, and block-LU elimination to target only the small subset of relevant active orbitals (Nguyen et al., 2024).

4. Extensions: Scattering, Interface, and Embedding Problems

The AGF formalism has been extended to treat scattering by isolated defects, interfaces between mismatched crystals, and embedding problems for local regions in infinite media.

• Defect scattering is formulated via the $T$-matrix, fully capturing the scattering amplitude across all modes and permitting exact computation of mode-resolved cross sections (Ong, 2024).
• Interface Green's functions are constructed using partial Fourier transforms and Schur-complement block matrix techniques, explicitly handling bicrystal and interface regions with connections to elastic theory (Ghazisaeidi et al., 2010).
• Embedding approaches (e.g., PEXSI-$\Sigma$) employ Schur complements and Green's function matching to impose open boundary conditions and efficiently relax interior defect structures (Li et al., 2016).

Specialized eigenspectrum-based AGF formulations allow for direct, mode-by-mode decomposition of interfacial conductance and atomistic elucidation of mode-conversion processes at disordered interfaces (Sadasivam et al., 2017).

Enhanced AGF schemes incorporate anharmonic interfacial self-energies to account for inelastic phonon processes, using third-order force-constants and Keldysh NEGF techniques to resolve both elastic and inelastic channels in thermal transport (Dai et al., 2019, Lemus et al., 2020).

5. Computational Implementation and Numerical Strategies

A typical AGF workflow for first-principles transport or surface calculations consists of:

• First-principles or tight-binding computation of Hamiltonians or force-constant matrices for device and leads.
• Assembly of principal-layer matrices and appropriate coupling blocks.
• Recursive or decimation solution for lead surface Green's functions (Lopez-Sancho, Sancho–Rubio, or mode space methods).
• Construction of energy/frequency-dependent self-energies for the device.
• Block inversion or recursive sweep solution for the device region Green's function at each frequency/energy.
• Calculation of transmission, spectral functions, and population observables (density matrix, current, Landauer conductance).
• For non-periodic or embedded systems, construction of the Schur-complement self-energy from reference Green's functions.
• For inelastic or anharmonic cases, self-consistent Born approximation and Kramers–Kronig/Hilbert transforms for the real part of scattering self-energies.

Recursive approaches reduce the scaling from $O(N^3)$ in full inversion (for $N$ orbitals) to $O(N_SM_S^3)$ for $N_S$ slices of size $M_S$ per slice, allowing treatment of supercells with up to $10^5-10^6$ orbitals (Nguyen et al., 2024).

6. Applications, Advantages, and Comparative Performance

AGF's primary strength is its treatment of open quantum and classical systems at the atomistic level, bypassing artificial boundary conditions and periodic images. Notable applications include:

• Surface electronic structure and band alignment in semiconductors and metals, yielding work functions and interface states converged in a few layers compared to many-layer periodic slabs (Smidstrup et al., 2017).
• Ab initio thermal conductance and phonon transmission at heterointerfaces, interfaces, and disordered alloys, with parameter-free accuracy and direct mapping to experimental observables (Bachmann et al., 2011, Gu et al., 2015).
• Quantum electron/phonon transport in nanowires, superlattices, and 2D materials with disorder, including inelastic effects and large-scale systems (Nguyen et al., 2024, Lemus et al., 2020).
• Accurate extraction of scattering cross-sections for complex defects and quantitative input to Boltzmann or Monte Carlo transport simulations (Ong, 2024).
• Supercurrent computations in Josephson junctions at the atomistic scale, including both DC and AC components (Mucciolo et al., 8 Sep 2025).

Advantages over traditional slab or periodic supercell methods include:

• Elimination of spurious quantum-well states and image interactions.
• Rapid convergence with slab/surface thickness and accuracy in electric-field or charge-transfer problems.
• Linear or near-linear scaling in system dimension.
• Natural open-system relaxation and charge exchange with reservoirs.

7. Limitations and Current Developments

Standard AGF is limited to harmonic or linear response unless explicit many-body corrections are included. The inclusion of anharmonicity, inelastic processes, or strong electron-electron correlations requires additional approximations (self-consistent Born, Keldysh, Floquet extensions). The cubic scaling in slice size or device width remains a challenge for very large cross sections, although mode-space and selected-inversion strategies provide major improvements (Lemus et al., 2020).

Emerging extensions address:

• Full mode and polarization-resolved analysis for understanding interface mode conversion and scattering mechanisms (Sadasivam et al., 2017).
• Advanced embedding for QM/MM, structural relaxation, and non-periodic entry/exit of defect excitations (Li et al., 2016).
• Inelastic and nonlinear response (phonon-phonon, electron-phonon) via Keldysh NEGF and high-order self-energies (Dai et al., 2019, Lemus et al., 2020).
• Large supercell and moiré pattern systems (e.g., twisted bilayer graphene) (Nguyen et al., 2024).

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AGF, as a unifying paradigm, underpins a spectrum of computational materials science research, from quantum transport in devices to interface and defect engineering, providing both theoretical fidelity and computational scalability grounded in first-principles or empirical input.