Source-linked AI summary
Perturbo: a software package for ab initio electron-phonon interactions, charge transport and ultrafast dynamics
Jin-Jian Zhou, Jinsoo Park, I-Te Lu, Ivan Maliyov, Xiao Tong, Marco Bernardi
TL;DR
Perturbo addresses the need for quantitative first-principles tools for electron interactions, charge transport, and ultrafast dynamics in complex materials. It combines DFT and DFPT inputs with Wannier interpolation, transport and time-dependent Boltzmann-equation solvers, and tested SOC and polar corrections. The resulting package computes electron-phonon-limited properties and nonequilibrium carrier dynamics across metals, semiconductors, insulators, and 2D materials.
Problem
Quantitative tools are needed to compute electron interactions, charge transport, and ultrafast dynamics across increasingly complex materials and devices.
Method
Perturbo combines DFT and DFPT inputs, Wannier interpolation, Boltzmann-equation solvers, and electron-phonon treatments for transport and dynamics.
Results
Perturbo computes electron-phonon scattering, transport coefficients, and nonequilibrium electron dynamics, with SOC and polar corrections supported and tested.
Takeaways & Limitations
Perturbo provides a broadly applicable ab initio tool for quantitative electron-phonon interactions and carrier dynamics in bulk and 2D materials.
Takeaways & Limitations
The interpolation approach requires short-ranged electron-phonon interactions whose matrix elements decay rapidly in a localized basis.
Abstract
from arXiv · showhide
Perturbo is a software package for first-principles calculations of charge transport and ultrafast carrier dynamics in materials. The current version focuses on electron-phonon interactions and can compute phonon-limited transport properties such as the conductivity, carrier mobility and Seebeck coefficient. It can also simulate the ultrafast nonequilibrium electron dynamics in the presence of electron-phonon scattering. Perturbo uses results from density functional theory and density functional perturbation theory calculations as input, and employs Wannier interpolation to reduce the computational cost. It supports norm-conserving and ultrasoft pseudopotentials, spin-orbit coupling, and polar electron-phonon corrections for bulk and 2D materials. Hybrid MPI plus OpenMP parallelization is implemented to enable efficient calculations on large systems (up to at least 50 atoms) using high-performance computing. Taken together, Perturbo provides efficient and broadly applicable ab initio tools to investigate electron-phonon interactions and carrier dynamics quantitatively in metals, semiconductors, insulators, and 2D materials.
PROGRAM SUMMARY
Perturbo computes first-principles transport properties and simulates ultrafast nonequilibrium electron dynamics driven by phonon interactions.
- Perturbo computes electrical conductivity, carrier mobility, and Seebeck coefficients from first-principles calculations.
- Perturbo simulates relaxation of excited carriers through interactions with phonons.
1. Introduction
Perturbo is designed as a unified first-principles platform for quantitative electron interactions, charge transport, and ultrafast dynamics across diverse materials. Its current release focuses on electron-phonon processes while supporting broad material classes and scalable computation.
- Perturbo uses DFT and DFPT outputs to compute electron-phonon interactions, charge transport, and ultrafast dynamics.
- The current release computes phonon-limited conductivity, mobility, and Seebeck coefficients and evolves electron distributions under electron-phonon scattering.
- Transport and ultrafast-dynamics calculations cover metals, semiconductors, insulators, and two-dimensional materials.
- Perturbo supports long-range polar electron-phonon interactions, spin-orbit coupling, norm-conserving and ultrasoft pseudopotentials.
- Hybrid MPI and OpenMP parallelization supports systems with at least 50 atoms and scales to thousands of CPU cores.
- The software targets first-principles researchers, experimental teams, and laboratories studying transport, ultrafast spectroscopy, and functional materials.
2.1. Boltzmann transport equation
Perturbo solves the semiclassical Boltzmann transport equation using first-principles electron-phonon scattering. It supports time-dependent carrier dynamics and steady-state transport through relaxation-time or iterative approaches.
- 2.1.1. Ultrafast carrier dynamics: The current dynamics implementation assumes homogeneous materials, slowly varying fields, fixed phonon occupations, and E = 0.
- Perturbo computes electron-phonon scattering from DFT electronic structures and DFPT lattice dynamics using the first-principles Boltzmann transport formalism.
- The collision integral includes phonon emission and absorption through electron-phonon matrix elements and energy-conservation terms.
- 2.1.1. Ultrafast carrier dynamics: Perturbo evolves nonequilibrium electron occupations from an initial distribution using numerical time stepping, typically with approximately 1 fs steps.
- 2.1.1. Ultrafast carrier dynamics: The ultrafast solver uses fourth-order Runge-Kutta by default, while Euler integration is faster but only first-order accurate in time.
- 2.1.2. Charge transport: For steady-state transport, Perturbo solves the linearized BTE either with the relaxation-time approximation or an iterative method.
- 2.1.2. Charge transport: The computed transport distribution yields conductivity, mobility, and Seebeck coefficients, with mobility obtained by dividing conductivity by carrier concentration.
2.2. Electrons, phonons, and e-ph interactions
Perturbo combines DFT and DFPT data with Wannier-based interpolation to obtain electronic, phonon, and electron–phonon quantities on fine Brillouin-zone grids. The approach relies on localized interactions and rapidly decaying matrix elements, while coarse-grid calculations provide the interpolated inputs for transport and dynamics.
- Motivation: Ultra-dense k- and q-point grids are needed for converged transport and dynamics, making direct DFPT calculations computationally prohibitive.Perturbo therefore uses interpolation techniques to reduce the cost of evaluating electron–phonon matrix elements.
- Electronic structure: Wannier interpolation constructs localized orbitals from coarse-grid DFT Bloch states and represents the electronic Hamiltonian through rapidly decaying hopping integrals.The interpolated Hamiltonian yields band energies and velocities at arbitrary k-points.
- Phonon dispersion: DFPT dynamical matrices are Fourier transformed into real-space force constants, which can reconstruct phonon dispersions at arbitrary q-points when the force constants are short-ranged.Phonon frequencies and displacement eigenvectors are obtained by diagonalizing the reconstructed dynamical matrix.
- Electron–phonon interactions: Perturbo computes coarse-grid electron–phonon matrix elements from DFT Bloch states and DFPT perturbation potentials, then transforms them into a localized Wannier basis.The localized matrix elements behave like real-space couplings and decay rapidly when electron–phonon interactions are short-ranged.
- Electron–phonon interactions: The interpolation approach requires short-ranged electron–phonon interactions and rapidly decaying local-basis matrix elements.The current release provides Wannier interpolation routines, while atomic-orbital interpolation remains under development and testing.
2.3. Polar corrections for phonons and e-ph interactions
Polar materials violate the short-range assumptions behind standard interpolation because dipoles generate long-range phonon and electron–phonon interactions. Perturbo separates these long-range contributions analytically from short-range parts treated with Wannier interpolation, with a modified formulation for 2D materials.
- Motivation: Polar semiconductors and insulators have long-range dipole interactions that produce LO–TO splitting and invalidate the short-range interpolation assumption.The associated Fröhlich electron–phonon interaction diverges as 1/q when q → 0 in bulk materials.
- Polar correction: Wannier interpolation alone can fail at small q for polar electron–phonon matrix elements, so Perturbo splits them into long-range dipole and short-range parts.Only the short-range component is interpolated in real space.
- Polar correction: The long-range matrix elements are evaluated from the dipole-field potential using Born effective charges, atomic positions, the dielectric tensor, and the unit-cell volume.The reciprocal-space summation is evaluated with the Ewald method, and the Wannier gauge enables the smooth phase approximation.
- 2D extension: For 2D materials, the polar correction replaces the dielectric tensor with effective screening and the unit-cell volume with twice the unit-cell area.These substitutions extend the long-range electron–phonon formulation to two-dimensional systems.
3. Capabilities and workflow
Perturbo provides executables for converting QE DFT/DFPT results and for interpolating electronic, phonon, and electron–phonon quantities before transport and dynamics calculations. Its capabilities include phonon-limited transport, nonequilibrium carrier dynamics, broad pseudopotential and SOC support, and features described as unique among existing codes.
- Code organization: The package contains qe2pert.x for interfacing QE with Perturbo and perturbo.x for interpolating quantities and performing transport and dynamics calculations.qe2pert.x transforms coarse-grid DFT and DFPT results into the localized Wannier representation used by perturbo.x.
- Supported features: The current release supports norm-conserving and ultrasoft pseudopotentials, calculations with or without spin–orbit coupling, and Coulomb truncation for 2D materials.These options are exposed through the QE-to-Perturbo workflow.
- Capabilities: Perturbo supports band, phonon, and electron–phonon calculations on arbitrary grids or paths, scattering properties for electronic states, transport coefficients, and nonequilibrium carrier dynamics.Transport calculations include conductivity, mobility, and Seebeck coefficient using either RTA or an iterative BTE solution.
- Novelty: Several Perturbo features, including nonequilibrium dynamics, are described as unique and unavailable in other existing codes.Additional features are under development or testing for future releases.
- Workflow: The workflow uses coarse-grid DFT Bloch states, DFPT lattice properties and perturbation potentials, Wannier90 rotation matrices, and fine-grid interpolation.DFPT data can be generated on irreducible q-points and extended using space-group and time-reversal symmetries.
4. Technical aspects
Perturbo reduces the cost of electron–phonon calculations by combining symmetry-based perturbation-potential unfolding, Wannier interpolation, and tailored Brillouin-zone integration. Benchmarks show that its symmetry treatment matches direct DFPT results, whereas scalar wavefunction rotations can fail when SOC is included.
- Electron–phonon matrix elements: Perturbo computes coarse-grid electron–phonon matrix elements from DFT states and DFPT perturbation potentials, using lattice-periodic quantities and commensurate k- and q-point grids.The q-point grid is chosen so k + q maps onto a coarse-grid point plus a reciprocal lattice vector.
- Electron–phonon matrix elements: Self-consistent perturbation potentials are computed only in the irreducible Brillouin-zone wedge and unfolded to symmetry-equivalent q-points, while pseudopotential terms are computed directly.The unfolding uses space-group and, for non-magnetic systems, time-reversal symmetry.
- Electron–phonon matrix elements: Wavefunction rotations provide an alternative symmetry approach, but SOC calculations require symmetry operators to act on both spatial coordinates and spin space.Scalar treatment of spinor wavefunctions can produce significant electron–phonon matrix-element errors.
- Benchmarking: Perturbo’s matrix elements match direct DFPT benchmarks to high accuracy for silicon and monolayer MoS2, validating its perturbation-potential rotation approach.The benchmark includes SOC and all six symmetry-equivalent (k, q) pairs for the lowest acoustic mode.
- Benchmarking: In the SOC benchmark, EPW agrees with DFPT only for the irreducible q-point, while symmetry-generated values show significant errors that can propagate to fine-grid interpolation.The paper attributes the discrepancy to rotating wavefunctions as scalars rather than spinors; without SOC, both codes agree with DFPT.
- Wannier interpolation: Wannier interpolation uses least-distance vectors to construct Wigner–Seitz supercells separately for electronic bands, phonons, and electron–phonon matrix elements.The BvK supercell must be large enough that longer-distance hopping terms are negligible.
- Brillouin-zone sampling and integration: Perturbo uses different Brillouin-zone integration strategies by task, including the tetrahedron method for transport integrations over k.Scattering rates and transport distribution functions require fine-grid Brillouin-zone integrations, while iterative BTE calculations retrieve stored matrix elements for scattering processes.
5. Examples
Perturbo is demonstrated across polar bulk, two-dimensional, and organic materials, covering electron–phonon interpolation, scattering analysis, transport, and ultrafast carrier dynamics. The examples show agreement with direct calculations or experiment in several settings while exposing material- and method-dependent behavior.
- Examples and workflow: Perturbo is demonstrated for GaAs, monolayer MoS2, and naphthalene using DFT, DFPT, Wannier interpolation, and electron–phonon matrix elements.The examples span polar bulk, 2D spin–orbit-coupled, and organic molecular materials.
- Interpolation of the e-ph matrix elements: Interpolated electron–phonon matrix elements for GaAs and MoS2 closely agree with direct DFPT calculations when polar corrections are included.The comparison is made through deformation potentials along high-symmetry paths.
- Scattering rates and electron mean free paths: At 300 K, naphthalene transport is dominated by states within about 100 meV of the valence band maximum, with scattering resolved across inter- and intra-molecular modes.The analysis includes acoustic modes associated with inter-molecular vibrations and optical modes associated with intra-molecular vibrations.
- Charge transport: In GaAs, the iterative transport approach gives higher electron mobility than the relaxation-time approximation, while the relaxation-time results agree better with experiment.The discrepancy between the approaches increases with temperature, and the reported difference is linked to band-structure and two-phonon-scattering corrections.
- Charge transport: At 300 K, GaAs has a computed Seebeck coefficient of about 130 µV/K at nc = 10^18 cm^-3, compared with an experimental value of about 150 µV/K.The coefficient increases with temperature and decreasing carrier concentration; only the diffusive contribution is computed because phonon drag is neglected.
- Charge transport: At room temperature, monolayer MoS2 has computed electron and hole mobilities of about 168 cm2/V s and 20 cm2/V s, respectively, with close RTA and ITA results.The electron mobility is compared with an experimental value of 150 cm2/V s.
- Ultrafast dynamics: In silicon, hot carriers approach energies near the band edge within 500 fs, while reaching the 300 K equilibrium Fermi–Dirac distribution takes several picoseconds.Hole relaxation is slightly faster than electron relaxation, and carrier concentration is conserved during the simulation.
6. Parallelization and performance
Perturbo uses hybrid MPI plus OpenMP parallelization to support computationally demanding transport and ultrafast dynamics calculations on HPC systems. The approach improves performance and scaling for larger systems, with near-linear scaling demonstrated up to 1,024 CPU cores.
- Parallelization strategy: Hybrid MPI plus OpenMP parallelization combines distributed-memory MPI across nodes with shared-memory OpenMP within nodes.This design targets the memory and computational demands of large transport and ultrafast dynamics calculations.
- Parallelization strategy: For larger systems with several to tens of atoms per unit cell, hybrid MPI plus OpenMP improves performance and CPU-core scaling over pure MPI.Compared with pure MPI, it reduces communication and memory consumption while improving load balance.
- Implementation: The strategy is applied to coarse-grid electron-phonon matrix elements and nonequilibrium carrier dynamics, the most time-consuming tasks of qe2pert.x and perturbo.x.MPI distributes q_c- or fine-grid k-points, while OpenMP handles local k-points or scattering pairs with limited communication.
- Performance tests: Carrier-dynamics tests on silicon used 1 fs time steps for 50 ps, totaling 50,000 steps on NERSC Cori Haswell nodes.The benchmark evaluated Perturbo’s parallelization under a long nonequilibrium dynamics workload.
- Performance results: Both qe2pert.x and perturbo.x scale close to the ideal linear limit up to 1,024 CPU cores.At 2,048 cores, qe2pert.x scaling becomes subideal because insufficient per-process workload makes communication and I/O bottlenecks; serial HDF5 I/O costs about 14% of wall time.
7. Conclusions and outlook
Perturbo provides first-principles tools for charge transport and ultrafast carrier dynamics in bulk and 2D materials. The current release combines electron-phonon calculations, Wannier interpolation, supported corrections, and hybrid HPC parallelization, while several additional capabilities remain planned.
- Conclusions: Perturbo computes charge-transport properties and simulates nonequilibrium excited-electron dynamics in bulk and 2D materials.Its core tasks include electron-phonon scattering rates, mean free paths, conductivity, mobility, and the Seebeck coefficient.
- Conclusions: The software uses DFT and DFPT results from Quantum ESPRESSO, with Wannier interpolation and symmetry reducing computational cost.Spin-orbit coupling and polar corrections for bulk and 2D materials are supported and tested.
- Performance: Perturbo demonstrates promising scaling on massively parallel HPC architectures through hybrid MPI plus OpenMP parallelization.The conclusion attributes this scaling to the effective implementation of the hybrid parallelization strategy.
- Outlook: Future releases are planned to add spin dynamics, large-polaron transport, coupled electron-phonon dynamics, atomic-orbital interpolation, and interfaces to additional codes.These capabilities are described as features under development or planned for later releases.