Anharmonic NEGF for Thermal Transport

• Anharmonic NEGF is a perturbative framework that incorporates phonon-phonon interactions beyond the harmonic limit for detailed thermal transport analysis.
• It unifies Landauer-like, kinetic, and Green-Kubo approaches to derive corrections for ballistic energy current and current fluctuations using cubic and quartic interactions.
• The method is applied to nanoscale devices and interfaces with both microscopic self-consistent calculations and phenomenological Büttiker probes for disorder and inelastic scattering.

Searching arXiv for recent and foundational papers on anharmonic NEGF for thermal transport. I’ll gather papers spanning foundational NEGF/QKE derivations, microscopic anharmonic phonon NEGF, interface applications, and phenomenological Büttiker-probe approaches. Anharmonic non-equilibrium Green’s function (NEGF) denotes the application of NEGF perturbation theory to thermal transport in systems where phonon-phonon interactions are retained beyond the harmonic, purely ballistic limit. In this setting, vibrational heat flow is described through contour-ordered, retarded, advanced, lesser, and greater Green’s functions for a central device coupled to reservoirs, while anharmonicity enters through interaction self-energies or, in phenomenological variants, through scattering probes. Across the literature, anharmonic NEGF has been used to derive Landauer-like transport formulas with interaction corrections, to connect NEGF with kinetic and hydrodynamic descriptions, to quantify inelastic scattering in nanostructures and interfaces, and to construct self-consistent high-temperature schemes including cubic and quartic force constants (Leek, 2012).

1. Conceptual scope and relation to transport theory

A central claim in the formal development of the subject is that Non-equilibrium Green’s Function Perturbation Theory is “really the overarching perturbative transport theory.” In that formulation, NEGF is taken as the starting point and developed in three directions to obtain the usual transport-related expressions: Landauer-like theory, kinetic theory, and Green-Kubo linear response theory. Within that program, the emphasis falls on generalizing Landauer-like transport and kinetic theory to include phonon-phonon interaction effects (Leek, 2012).

The Landauer-like branch yields explicit anharmonic corrections to quantities that are ballistic in the harmonic limit. Using perturbation expansion, one can obtain anharmonic corrections to the ballistic energy current and to the noise associated to the energy current. The lowest 3-phonon interaction, the lowest and the second lowest 4-phonon interaction corrections to the ballistic energy current were obtained, as was the lowest 3-phonon interaction correction to the noise. This establishes that anharmonic NEGF is not confined to mean currents; it also addresses current fluctuations within the same perturbative framework.

The kinetic-theory branch uses Wigner coordinates together with gradient expansion to reproduce the usual phonon Boltzmann kinetic equation. In the same line of development, phonon-phonon correlation corrections to kinetic equations were derived, and kinetic equations were further linked to hydrodynamic balance equations, with phonon-phonon correlation corrections to the entropy, energy and momentum balance equations (Leek, 2012).

A further extension concerns disorder. High mass disorder can be incorporated into the ballistic energy current formula, and the coherent potential approximation (CPA) for treating high mass disorder was found to be compatible with the ballistic energy current expression. This places anharmonic NEGF within a broader transport-theory landscape that includes interaction effects, disorder averaging, and kinetic limits in a single perturbative language.

2. Green’s functions, Hamiltonians, and Dyson–Keldysh structure

In the microscopic formulations used for phonon transport, the basic object is the contour-ordered phonon Green’s function

$$G_{nm}(\tau,\tau') \equiv -i\,\langle T_C[u_n(\tau)u_m(\tau')]\rangle_0,$$

where $\tau,\tau'$ lie on the Keldysh contour $C$, $T_C$ is contour ordering, and $u_n(\tau)$ is the mass-scaled displacement operator. In steady state, the formalism is expressed in frequency space through

$$G^R(\omega) = [\omega^2 I-\Phi-\Sigma^R(\omega)]^{-1},\qquad G^A(\omega)=[G^R(\omega)]^\dagger,$$

together with

$$G^<(\omega)=G^R\Sigma^<(\omega)G^A,\qquad G^>(\omega)=G^R\Sigma^>(\omega)G^A.$$

Here the total self-energy contains both contact contributions and anharmonic scattering contributions (Guo et al., 2020).

For high-temperature non-equilibrium transport between Langevin thermostats, an explicitly anharmonic device Hamiltonian can be written up to quartic order as

$$H = \sum_i \frac{p_i^2}{2m_i} +\frac{1}{2!}\sum_{ij}\phi_{ij}u_i u_j +\frac{1}{3!}\sum_{ijk}\psi_{ijk}u_i u_j u_k +\frac{1}{4!}\sum_{ijk\ell}\chi_{ijk\ell}u_i u_j u_k u_\ell +H_{\rm leads}+H_{\rm coupling},$$

with bilinear device-lead coupling

$$H_{\rm coupling}=\sum_\alpha\sum_{i\in D}\sum_{l\in \alpha}W_{\alpha,il}\,u_i u_l.$$

The corresponding retarded Green’s function is

$$G^r(\omega)=\bigl[\omega^2\mathbf{I}-\mathbf{K}-\Sigma^r(\omega)\bigr]^{-1},$$

with $\tau,\tau'$0 (Esfarjani, 2021).

These formulations differ in regime and approximation but share the same structural content: open boundaries are encoded by embedding self-energies, while anharmonicity is encoded by interaction self-energies. A plausible implication is that “anharmonic NEGF” is best understood not as a single algorithm but as a family of Keldysh-based transport theories distinguished by the representation of $\tau,\tau'$1 and by the degree of self-consistency imposed.

3. Microscopic anharmonic self-energies

For cubic anharmonicity, the leading microscopic treatment in quantum phonon NEGF is a second-order diagrammatic expansion, commonly implemented in the self-consistent Born approximation (SCBA). With

$\tau,\tau'$2

the contour self-energy emerges as

$\tau,\tau'$3

and after analytic continuation and Fourier transform,

$\tau,\tau'$4

In 3D periodic slabs, Fourier representation of harmonic and cubic force constants makes transverse-momentum conservation automatic in the Dyson equation, while the $\tau,\tau'$5 integration enforces energy conservation (Guo et al., 2020).

A distinct high-temperature development retains both cubic and quartic interactions within SCBA. In that framework,

$\tau,\tau'$6

with cubic “bubble” terms and a quartic “tadpole” or Hartree contribution. The quartic term is written as

$\tau,\tau'$7

and the static Hartree part renormalizes the harmonic matrix according to

$\tau,\tau'$8

This quartic contribution was shown to be important “in order to properly describe thermal expansion and temperature dependence to leading order in anharmonicity” (Esfarjani, 2021).

The self-consistent solution loop reflects the nonlinearity of the problem. One initializes with harmonic Green’s functions, computes $\tau,\tau'$9 and $C$0, updates cubic bubble and quartic tadpole self-energies, redefines $C$1, and iterates until convergence. The same formulation states that current conservation is only guaranteed in the fully self-consistent solution; partial or one-shot implementations may violate Kirchhoff’s law (Esfarjani, 2021).

4. Heat current, transmission, and fluctuation observables

The heat current in interacting phonon NEGF is commonly written in Meir–Wingreen form. For a lead $C$2,

$C$3

In the classical limit of the high-temperature multiprobe formalism, one typically writes

$C$4

where the first term is the elastic Landauer form and $C$5 collects the contributions involving anharmonic lesser and greater self-energies (Esfarjani, 2021).

At interfaces and in linear response, one may define an effective transmission function by expressing the current as

$C$6

which reduces in the harmonic limit to

$C$7

The thermal boundary conductance per unit area is then

$C$8

This representation makes explicit how inelastic scattering enters both the spectral function and the non-equilibrium populations through $C$9 (Guo et al., 2021).

The formalism also admits thermally relevant observables beyond conductance. In the perturbative Landauer-like development of NEGF, anharmonic corrections were obtained not only for the ballistic energy current but also for the noise associated to the energy current, including the lowest 3-phonon interaction correction to the noise (Leek, 2012). This indicates that anharmonic NEGF naturally extends to fluctuation diagnostics, not only average transport coefficients.

5. Interfaces, interfacial mode decay, and phenomenological scattering models

A major application of anharmonic NEGF concerns solid/solid interfaces, where inelastic scattering of thermal phonons remains an open question. In a fully quantum theoretical scheme based on a real-space scattering rate matrix, the interfacial spectral energy exchange is decomposed into contributions from local and non-local anharmonic interactions. Within that decomposition, the local contribution was shown to be predominant for high-frequency phonons, whereas both local and non-local contributions are important for low-frequency phonons. The anharmonic decay of interfacial phonon modes was further shown to play a crucial role in bridging the bulk modes across the interface, while the overall quantitative contribution of anharmonicity to thermal boundary conductance was found to be moderate (Guo et al., 2021).

The spectral energy exchange due to anharmonic scattering inside the interface is written as

$T_C$0

and layer-resolved decompositions separate on-site and non-local terms. This provides an intuitive interpretation of how interfacial regions exchange energy spectrally through anharmonic processes.

A different route to inelasticity uses Büttiker probes rather than microscopic Born self-energies. In that approach, each atom and polarization is assigned a fictitious probe with retarded self-energy

$T_C$1

and lesser component

$T_C$2

with an empirical isotropic Umklapp rate

$T_C$3

For Si, the constants were fitted to molecular-dynamics data as $T_C$4 and $T_C$5, after which the same parameters were used for Si/heavy-Si interface calculations (Chu et al., 2019).

Formulation	Anharmonic treatment	Representative emphasis
Microscopic quantum NEGF	Second-order cubic self-energy in SCBA	1D and 3D nanostructures
Interface NEGF	Real-space scattering-rate matrix with local/non-local split	Interfacial spectral energy exchange and mode decay
High-temperature multiprobe NEGF	Cubic bubble plus quartic tadpole, self-consistent	Large temperature differences, thermal expansion
Büttiker-probe NEGF	Frequency-dependent phenomenological linewidths	Thermal boundary resistance at finite temperatures

The Büttiker-probe results show that with scattering parameters tuned against MD in homogeneous Si, the NEGF-predicted thermal boundary resistance quantitatively agrees with MD for wide mass ratios, and artificial resistances produced by the unaltered Landauer approach at virtual interfaces in homogeneous systems are absent. The same study reports that scattering between different phonon modes plays a crucial role in thermal transport across interfaces (Chu et al., 2019).

6. Numerical realization and validation

Microscopic anharmonic NEGF has been developed into a parallelized computational framework with first-principle force constants input for large-scale quantum heat transport simulation. In the formulation for 1D and 3D nanostructures, Fourier representation is introduced for both harmonic and anharmonic terms, and several implementation approximations are investigated “to ensure the balance between numerical accuracy and efficiency” (Guo et al., 2020).

The reported numerical strategy includes SCBA iteration until heat-current conservation is better than 1%, mode-space truncation retaining only diagonal and nearest-neighbor blocks in slab decomposition, finite-range force-constant truncation, recursive Green’s-function scaling of order $T_C$6, and MPI parallelization over $T_C$7 and $T_C$8. Harmonic force constants were taken from Phonopy on a $T_C$9 supercell, cubic force constants from Thirdorder on $u_n(\tau)$0, with cubic interactions truncated at first shell (Guo et al., 2020).

A quantitative validation was demonstrated for cross-plane heat transport through silicon thin film. The model used 1–24 unit cells of Si, each 5.4 Å, between two Si reservoirs at $u_n(\tau)$1 and $u_n(\tau)$2. Mesh convergence employed $u_n(\tau)$3 with approximately 100 points and a $u_n(\tau)$4 $u_n(\tau)$5 grid. The anharmonic phonon-phonon scattering was shown to be appreciable and to introduce about 20% reduction of thermal conductivity at room temperature even for a film thickness around 10 nm. Good agreement with MC-BTE using identical DFT force constants was reported, local temperature profiles from NEGF and MC matched, and spectral heat flow indicated that optical and LA scattering dominate the conductivity drop (Guo et al., 2020).

The phenomenological Büttiker-probe route has also been benchmarked quantitatively. There, the probe parameters were calibrated so that the NEGF-computed bulk Si thermal conductivity exactly matches the thermal conductivity extracted from direct molecular-dynamics simulations using the same interatomic potential, and the calibrated model was then transferred to interface calculations without further adjustment (Chu et al., 2019).

7. Limitations, methodological distinctions, and significance

The literature makes clear that “anharmonic NEGF” is not methodologically uniform. Fully microscopic schemes based on cubic or cubic-plus-quartic force constants differ sharply from probe-based approaches that mimic inelastic scattering by a fitted lifetime. The latter are computationally efficient and provide direct spectral access, but are explicitly described as phenomenological, with no explicit momentum-conservation tracking and no straightforward diagrammatic extension to higher-order processes (Chu et al., 2019).

Even within microscopic schemes, approximation structure is central. In the high-temperature self-consistent formalism, the classical limit replaces Bose factors by $u_n(\tau)$6, excludes zero-point motion and low-temperature quantum effects, keeps only second-order cubic bubbles and first-order quartic tadpoles, neglects vertex corrections and higher loops, assumes moderate anharmonicity, and becomes numerically heavy for large devices because of dense frequency integrals and matrix inversions at each frequency. Leads are treated in a single-particle Langevin approximation, which enforces instantaneous equilibration but ignores finite lead anharmonicity (Esfarjani, 2021).

A recurrent misconception is that the harmonic Landauer expression is the defining content of phonon NEGF. The documented developments show instead that NEGF has been used to derive conserving self-energy approximations for phonon-phonon interaction, to recover the usual phonon Boltzmann kinetic equation by Wigner coordinates plus gradient expansion, and to derive phonon-phonon correlation corrections to kinetic and hydrodynamic balance equations (Leek, 2012). Another misconception is that quartic interactions are necessarily subleading in practice; in the high-temperature multiprobe setting, quartic terms are specifically reported to be important for thermal expansion and temperature dependence to leading order in anharmonicity (Esfarjani, 2021).

Taken together, these results place anharmonic NEGF at the intersection of quantum transport theory, many-body perturbation theory, and phonon kinetics. The framework supports both microscopic and phenomenological treatments of inelastic phonon scattering, extends from nanostructures to interfaces and multiprobe devices, and provides a route to studying the transition from coherent to incoherent heat transport in systems where broken translational symmetry, large temperature bias, or interfacial mode conversion are central (Guo et al., 2020).