Recursive Green's Function (RGF) Algorithm

• Recursive Green’s function (RGF) algorithm is a computational method that calculates retarded Green’s functions by iteratively exploiting a block-tridiagonal Hamiltonian structure.
• It employs forward and backward recursions to efficiently simulate quantum transport and thermal phenomena in mesoscopic and nanoscale systems.
• Extensions such as Büttiker probes, circular slicing, and defect decimation broaden its applicability to multi-terminal devices and complex disorder models.

The recursive Green’s function (RGF) algorithm is a computational framework central to the calculation of retarded Green's functions in quantum transport problems, particularly in mesoscopic and nanoscale systems described by sparse Hamiltonians. It leverages the block-tridiagonal structure of discretized Hamiltonians arising from tight-binding or finite-difference descriptions and is foundational in non-equilibrium Green’s function (NEGF) simulations for coherent and dissipative transport, phonon heat flow, and complex device modeling. Its efficiency derives from algorithmic recursion, which permits calculation of required Green’s function sub-blocks with computational and memory cost scaling as $O(N m^3)$ for $N$ principal layers of size $m \times m$, far outperforming direct matrix inversion for large-scale systems.

1. Mathematical Foundation and Block Structure

The central object in RGF is the retarded Green’s function $G^r(E) = [E I - H - \Sigma]^{-1}$, where $H$ is the device Hamiltonian and $\Sigma$ encodes all self-energies from attached leads or dissipative probes. Discretization (e.g., tight-binding, real-space grid) partitions the system into $N$ blocks or “slices” such that only nearest-neighbor blocks are coupled, rendering $H$ and thus the system matrix block-tridiagonal. Each semi-infinite lead is integrated out, contributing a boundary self-energy $\Sigma_j$ to the coupled sites, typically computed either analytically or by decimation algorithms (Thorgilsson et al., 2013, Lewenkopf et al., 2013, Nguyen et al., 2024, Sadasivam et al., 2016, Vaitkus et al., 2017).

Physically, the RGF approach is applicable wherever quantum transport or spectral properties of extended (quasi-)one-dimensional, planar, or multi-terminal geometries are of interest, and only selected matrix elements of $G^r$ (e.g., boundary-to-boundary transmission, local density) are necessary.

2. Core Recursion Algorithms

The standard RGF is constructed in two stages: a forward (left- or right-connected) recursion followed by a backward sweep to reconstruct specified Green’s function blocks.

• Forward Recursion: At each stage, the effective Green’s function $N$0 (left-connected) of the subsystem up to slice $N$1 is updated using a Dyson equation:

$N$2

where $N$3, and $N$4 is the inter-block coupling.

• Backward Recursion: After reaching the endpoint (including boundary self-energies), the full block-diagonal $N$5 is reconstructed by:

$N$6

Off-diagonal blocks such as $N$7 for $N$8 are propagated recursively when required.

This algorithm generalizes to multi-terminal or nonstandard geometries via appropriate block partitioning, such as spiral ring “circular slicing” for 2D devices with leads on all sides, ensuring block-tridiagonality is preserved for stable recursion (Thorgilsson et al., 2013).

3. Extensions: Dissipation, Inelasticity, and Multi-Terminal/Nonsimple Geometries

Büttiker Probes and Dissipative Processes

RGF’s modularity allows inclusion of Büttiker probes—local fictitious self-energies that enforce current conservation and encode inelastic/dissipative phenomena—directly into the recursion (Sadasivam et al., 2016, Vaitkus et al., 2017). This necessitates an extended recursion for both the lesser Green's function $N$9 and for Jacobians required in Newton–Raphson solvers for probe potentials or “temperatures”. The recursive structure extends to derivatives with respect to local chemical potentials and temperatures, enabling efficient, scalable inclusion of local scattering and energy relaxation phenomena.

Multi-Terminal and Circular Slicing

In systems with leads on multiple sides (multi-terminal), standard vertical or horizontal slicing leads to loss of block-tridiagonality. The circular slicing scheme partitions the device into concentric “rings” ordered by boundary distance, ensuring that all lead self-energies are absorbed into the outermost block and preserving tridiagonal coupling between adjacent rings (Thorgilsson et al., 2013).

Partial Periodicity and Defect Decimation

Quasi-one-dimensional systems such as defective carbon nanotubes admit a further optimization if most slices are periodic (ideal lattice) and only rare defects break periodicity. A renormalization-decimation algorithm (RDA) integrates out entire ideal segments recursively in $m \times m$0 steps, yielding a reduced defect-chain on which the standard RGF is performed, so that the total cost scales with the number of defects instead of the system length (Teichert et al., 2017).

4. Numerical Optimizations and Performance

Efficient computation of large and/or atomistically detailed devices, such as Moiré superlattice systems, requires further optimization of self-energy calculations and storage:

• Surface Self-Energy Calculation: For large-periodic leads, splitting each supercell into a “surface” block and a “bulk” block enables a Dyson-type reduction, so only small-matrix recursions are required for semi-infinite lead self-energies. This yields speed-ups proportional to the square of the ratio of supercell to block size (Nguyen et al., 2024).
• Finite-Lead Recursion: Approximating semi-infinite leads by long but finite leads, and using forward recursion to extract converged self-energies, circumvents iterative mode-matching at large supercell sizes (Nguyen et al., 2024).
• Memory Efficiency: Only current recursion blocks and blocks needed for observable extraction are stored, with total memory per recursion step limited to $m \times m$1, independent of total device length (Lewenkopf et al., 2013, Thorgilsson et al., 2013, Nguyen et al., 2024).

Summary Table: RGF Complexity

Algorithmic Scenario	Time Complexity	Memory per Step
Standard RGF (N slices, m×m)	$m \times m$2	$m \times m$3
Direct inversion (N m × N m)	$m \times m$4	$m \times m$5
RGF+Defect Decimation (N_D)	$m \times m$6	$m \times m$7
Large-supercell α–β scheme	$m \times m$8	$m \times m$9

5. Applications in Electronic and Phononic Transport

RGF is the core engine in NEGF-based calculations of quantum coherent electronic transport, phonon-mediated thermal transport, and hybrid electron-phonon devices:

• Electronic Transport: Calculation of conductance, transmission, local density of states (LDOS), and bond currents in systems ranging from graphene ribbons to large 2D superlattices, under magnetic fields or disorder (Lewenkopf et al., 2013, Nguyen et al., 2024, Thorgilsson et al., 2013).
• Phonon and Thermal Transport: Integration of atomistic Green’s function (AGF) with RGF and Büttiker probes facilitates inclusion of anharmonic inelastic phonon scattering and direct computation of thermal interface conductance, matching experimental data across wide temperature ranges (Sadasivam et al., 2016).
• Perturbative Analyses: For finite-level systems, a recursive Green’s function construction yields effective propagators central to time-independent perturbation theory, removing repetition terms and generalizing Brillouin–Wigner and Feenberg series (Ishida, 2019).

Notable application benchmarks include twisted bilayer graphene superlattices, carbon nanotubes with realistic vacancy defects, and thermally inhomogeneous interfaces (Nguyen et al., 2024, Teichert et al., 2017, Sadasivam et al., 2016).

6. Algorithmic and Practical Considerations

Stability and accuracy require the inclusion of a small imaginary component to the energy ($G^r(E) = [E I - H - \Sigma]^{-1}$0) to ensure non-singular inversion at each step, and robust linear solvers (e.g., LU with partial pivoting). Block-tridiagonality must be preserved by correct system partitioning; “rings” are employed for multi-terminal cases versus slices for conventional two-terminal systems (Thorgilsson et al., 2013).

For dissipative systems, self-energies and Fermi/Bose functions may depend on self-consistent variables (chemical potentials, temperatures), and rapid convergence is typically obtained with Newton–Raphson iterations, for which the RGF efficiently delivers both Green’s functions and their parametric derivatives (Vaitkus et al., 2017, Sadasivam et al., 2016).

7. Impact and Future Perspectives

The RGF algorithm has established itself as a universal backbone for quantum transport simulation, enabling atomistic modeling of devices in the $G^r(E) = [E I - H - \Sigma]^{-1}$1 to $G^r(E) = [E I - H - \Sigma]^{-1}$2 atom regime (Nguyen et al., 2024), multi-terminal architectures, and inclusion of realistic disorder and dissipative processes. Recent progress in hierarchical decimation, efficient surface Green’s function calculations, and modular inclusion of inelasticity/baths continue to expand applications to more complex materials (Moiré patterns, twist-angle dependent superlattices) and multi-physics couplings (electron-phonon, non-equilibrium thermodynamics). The barrier to scaling is continually pushed further as RGF-based techniques are optimized in conjunction with emerging hardware and sparse linear algebra methods.

References:

• "Recursive Green's function method for multi-terminal nanostructures" (Thorgilsson et al., 2013)
• "Recursive Green's functions optimized for atomistic modelling of large superlattice-based devices" (Nguyen et al., 2024)
• "The recursive Green's function method for graphene" (Lewenkopf et al., 2013)
• "Thermal Transport Across Metal Silicide-Silicon Interfaces: First-Principles Calculations and Green's Function Transport Simulations" (Sadasivam et al., 2016)
• "Büttiker probes and the Recursive Green's Function; an efficient approach to include dissipation in general configurations" (Vaitkus et al., 2017)
• "Improved recursive Green's function formalism for quasi one-dimensional systems with realistic defects" (Teichert et al., 2017)
• "Recursive Green's function approach to Feenberg perturbation theory" (Ishida, 2019)