kALDo

What kALDo is

kALDo computes thermal conductivity from harmonic and anharmonic force constants. It solves the Boltzmann transport equation for crystals and applies the quasi-harmonic Green-Kubo method to disordered and amorphous systems. Both methods take the same inputs, so one code treats crystalline and glassy transport. It runs on GPUs through TensorFlow and accepts force constants from DFT and from interatomic potentials. On the map it serves phonon frequencies, lifetimes, mean free paths and the resulting conductivities.

WebsiteRepositoryOn the mapAll codesLicense: BSD-3-Clause

In more detail

Boltzmann transport in crystals

kALDo builds the dynamical matrix and the three-phonon scattering tensor from second and third order force constants on a supercell. It then solves the linearized BTE in the relaxation time approximation, by self-consistent iteration or by direct inversion of the scattering matrix. The iteration converges only when the off-diagonal part of the scattering matrix is smaller than the diagonal, so inversion is the reference when memory allows (Barbalinardo 2020). For diamond silicon with DFPT second-order constants from Quantum ESPRESSO, inversion with quantum statistics gives 147 W/m/K at 300 K, RTA about 5% less, the classical limit 123 W/m/K and Tersoff 305 (Barbalinardo 2020).

Quasi-harmonic Green-Kubo for disordered solids

QHGK writes the conductivity as a sum over pairs of modes with a generalized velocity and a generalized lifetime. It is a real-space method at the Gamma point and costs the same as BTE-RTA. In the Si116Ge20 type-II clathrate replicated 5x5x5 (17,000 atoms) it gives 31.0 W/m/K at 300 K, close to the BTE reference for the periodic alloy. Placing the Ge atoms at random over the 17,000-atom cell drops kappa to 1.7 W/m/K, a factor 200 below diamond silicon. Third-order splines interpolate the lifetimes for the large cell from a 3x3x3 cell of 3672 atoms (Barbalinardo 2020). Mode diffusivities show the loss builds up below 3 THz.

Numerics and scale

The expensive step is the projection of the third-order potential on phonon modes, which scales as (Nk 3N)^3. kALDo casts it as 2Nk(3N) tensor multiplications of size (Nk 3N)^2 for the BTE and (3N)^2 for QHGK, executed by NumPy and TensorFlow on multithreaded CPUs or GPUs. Memory is (3N)^2 numbers for QHGK, Nk(3N)^2 for RTA and Nk^2(3N)^2 for the full inversion. The energy-conserving delta is Gaussian, Lorentzian or triangular, and a frequency threshold stores the generalized velocity as a sparse tensor. Quasi-momentum conservation cuts the phonon state from (Nk 3N)^2 to 2Nk(3N)^2 entries (Barbalinardo 2020).

Interfaces and uses

Force constants come from any ASE calculator, from a native LAMMPS interface in the USER-PHONON package, or from hiPhive fits. The CNT study paired kALDo with GPUMD equilibrium MD and corrected finite-length BTE with the McKelvey-Shockley flux method, where the Matthiessen rule overestimates kappa for every length above 10 nm (Barbalinardo 2021). Zhang 2022 used kALDo for the QHGK reference in amorphous silicon (Zhang 2022). Liu 2025 ran kALDo on a neuroevolution potential to get eigenvectors for an AlN-SiC interface (Liu 2025). The code is on GitHub.

What it computes on the map

28 quantities, drawn by tier with the data flow between them. Dashed boxes are intermediate quantities. Select one for its formula and the other codes that compute it.

Each quantity with the interface the representation records.

• BareDynamicalMatrixHarmonicWithQ(q)._dynmat_fourier
• DielectricTensoratoms.info['dielectric']dimensionless
• DynamicalMatrixHarmonicWithQ(q).dynmat
• EigenvectorsPhonons.eigenvectors
• ForceConstants[order=2]ForceConstants.second.value
• ForceConstants[order=3]ForceConstants.third.valueeV_per_A3
• FrequencyPhonons.frequencylinear_THz
• GroupVelocityPhonons.velocityangstrom_linear_THz
• HeatCapacityPhonons.heat_capacityJ_per_K
• HelmholtzFreeEnergyPhonons.free_energyeV
• IsotopeAbundancesPhonons(g_factor=...)dimensionless
• Linewidth[channel=anharmonic_3ph]Phonons.bandwidthangular_THz
• Linewidth[channel=boundary]Conductivity(length=...).bandwidth + 2|v_α|/L_α (inline)angular_THz
• Linewidth[channel=isotope]Phonons.isotopic_bandwidthangular_THz
• Linewidth[channel=total]Phonons.bandwidthangular_THz
• MeanFreeDisplacement[bte_solver=direct_inverse]Conductivity(method='inverse').mean_free_path
• MeanFreeDisplacement[bte_solver=rta]Conductivity(method='rta').mean_free_path
• ModalDiffusivityConductivity(method='qhgk').diffusivitymm2_per_s
• MolarHeatCapacityN_A * np.sum(Phonons.heat_capacity) / n_q_points
• ParticipationRatioPhonons.participation_ratiodimensionless
• PhaseSpace3PhononPhonons.phase_space
• PhononDOSplotter.plot_dos(phonons, bandwidth, n_points)
• PotentialForceConstants(calculator=<ASE calculator>)
• TemperaturePhonons(temperature=...)
• ThermalConductivity[bte_solver=direct_inverse]Conductivity(method='inverse').conductivityW_per_m_per_K
• ThermalConductivity[bte_solver=rta]Conductivity(method='rta').conductivityW_per_m_per_K
• ThermalConductivity[transport_model=qhgk]Conductivity(method='qhgk').conductivityW_per_m_per_K
• VolumetricHeatCapacitynp.sum(Phonons.heat_capacity) / (V_cell * n_q_points)

Releases on record

Versions on record for this code, with the license read from the release itself.

References

The code

The papers that describe the code; cite them when you use it.

1. G. Barbalinardo, Z. Chen, N. W. Lundgren et al., Efficient anharmonic lattice dynamics calculations of thermal transport in crystalline and disordered solids, Journal of Applied Physics, 2020. doi.org/10.1063/5.0020443

Methods and inputs it builds on

The methods the code implements and the inputs it relies on, as published.

1. P. B. Allen, J. L. Feldman, Thermal conductivity of disordered harmonic solids, Physical Review B, 1993. doi.org/10.1103/PhysRevB.48.12581
2. L. Isaeva, G. Barbalinardo, D. Donadio et al., Modeling heat transport in crystals and glasses from a unified lattice-dynamical approach, Nature Communications, 2019. doi.org/10.1038/s41467-019-11572-4

Applications

Highly cited studies that cite the code, with a few picked by hand.

1. N. W. Lundgren, G. Barbalinardo, D. Donadio, Mode localization and suppressed heat transport in amorphous alloys, Physical Review B, 2021. doi.org/10.1103/PhysRevB.103.024204
2. G. Barbalinardo, Z. Chen, H. Dong et al., Ultrahigh Convergent Thermal Conductivity of Carbon Nanotubes from Comprehensive Atomistic Modeling, Physical Review Letters, 2021. doi.org/10.1103/PhysRevLett.127.025902
3. Z. Zhang, Y. Guo, M. Bescond et al., How coherence is governing diffuson heat transfer in amorphous solids, npj Computational Materials, 2022. doi.org/10.1038/s41524-022-00776-w
4. A. Fiorentino, P. Pegolo, S. Baroni, Hydrodynamic finite-size scaling of the thermal conductivity in glasses, npj Computational Materials, 2023. doi.org/10.1038/s41524-023-01116-2
5. F. Liu, R. Mao, Z. Liu et al., Probing phonon transport dynamics across an interface by electron microscopy, Nature, 2025. doi.org/10.1038/s41586-025-09108-6

How to cite

G. Barbalinardo, Z. Chen, N. W. Lundgren, D. Donadio, Efficient anharmonic lattice dynamics calculations of thermal transport in crystalline and disordered solids, J. Appl. Phys. 128, 135104 (2020)

doi.org/10.1063/5.0020443