Abstract
One of the most accurate approaches for calculating lattice thermal conductivity, \(\kappa _\ell\), is solving the Boltzmann transport equation starting from thirdorder anharmonic force constants. In addition to the underlying approximations of abinitio parameterization, two main challenges are associated with this path: high computational costs and lack of automation in the frameworks using this methodology, which affect the discovery rate of novel materials with adhoc properties. Here, the Automatic Anharmonic Phonon Library (AAPL) is presented. It efficiently computes interatomic force constants by making effective use of crystal symmetry analysis, it solves the Boltzmann transport equation to obtain \(\kappa _\ell\), and allows a fully integrated operation with minimum user intervention, a rational addition to the current highthroughput accelerated materials development framework AFLOW. An “experiment vs. theory” study of the approach is shown, comparing accuracy and speed with respect to other available packages, and for materials characterized by strong electron localization and correlation. Combining AAPL with the pseudohybrid functional ACBN0 is possible to improve accuracy without increasing computational requirements.
Introduction
Lattice thermal conductivity, \(\kappa _\ell\), is the key materials’ property for many technologies and applications, such as thermoelectric materials,^{1,2,3} heat sink materials,^{4} rewritable density scanningprobe phasechange memories,^{5} and thermal medical devices.^{6} Fast and robust predictions of this quantity remain a challenge:^{7} semiempirical models^{8,9,10} are computationally inexpensive but require some experimental data. Similarly, classical molecular dynamics combined with Green–Kubo relations^{11,12,13} is reasonably quick but requires the knowledge of specific force fields. On the contrary, frameworks based on the quasiharmonic Debye model, such as GIBBS^{14} or the AutomaticGibbsLibrary (AGL),^{15,16} are extremely efficient as prescreening techniques but they lack quantitative accuracy.
The quasiharmonic approximation (QHA), alone has also been used in different models to predict \(\kappa _\ell\).^{7,17} Although QHAbased models overall improve accuracy of \(\kappa _\ell\), they are far from the results obtained from calculating the anharmonic force constants and solving the associated Boltzmann transport equation (BTE).^{8,18} To the best of our knowledge, solving the BTE is the optimal method for systematically and accurately calculating thermal conductivity.^{19,20,21} This approach has been successfully applied to many systems during the last decade. It has been recently implemented in packages including Phono3py,^{22} PhonTS,^{23} ALAMODE,^{24} and ShengBTE,^{25} which compute \(\kappa _\ell\) by calculating the anharmonic force constants and solving the BTE. Nevertheless, there is a lack of a robust framework able to calculate \(\kappa _\ell\) with minimum intervention from the user, and therefore targeted to highthroughput automated and accelerated materials discovery.
Many challenges need to be tackled. I. The thirdorder interatomic force constants (IFCs) up to a certain distance cutoff are computationally expensive to obtain from first principles. Overall they represent the major concern for the method. Effective use of crystalline symmetry of the system must be employed to map, through appropriate tensorial transformations, dependent IFCs and therefore reduce the number of calculations. The task is performed by the internal AFLOW pointfactorspace group calculator.^{26} Recently, it has also been proposed to obtain the IFCs by inverting the results of many entangled calculations with the use of compressive sensing.^{27} Further studies need to be carried out to address the scaling of the algorithm with respect to cutoffs and accuracy. II. For a rational software for accelerated materials development, all the geometric optimizations, symmetry analyses, supercell creation, preprocessing and postprocessing, and automatic error corrections to get the IFCs, in addition to the appropriate integration for the BTE must be performed by a single code. Here, we present Automatic Anharmonic Phonon Library (AAPL), which computes the IFCs and solves the BTE to predict \(\kappa _\ell\) as part of the AFLOW highthroughput framework,^{28,29,30,31,32,33,34,35,36,37,38,39} automatizing the entire process. The software is being finalized for an official opensource release during 2017, within the GNU GPL license. III. The accuracy of the method mostly depends on the accuracy of the computed forces, and therefore it will inherit the same limitations as the abinitio method used. For materials characterized by strong electron localization and correlation, accurate hybrid functionals for Density Functional Theory parameterizations might not even be feasible, as they would drastically increase computational costs, with respect to more basic LDA or GGA functionals. In that case, new strategies should be developed to contain computational demands. Here, an example is given: combining AAPL with the pseudohybrid functional Agapito Curtarolo Buongiorno Nardelli abinitio DFT functional (ACBN0) improves the accuracy without increasing computational requirements.^{40,41,42,43,44,45,46}
Results and discussion
The Automatic Anharmonic Phonon Library
The Boltzmann transport equation
The Boltzmann equation for phonons, originally formulated by Peierls in 1929, is an important approach for studying phonon transport.^{8} Its solution has posed a challenge for the last several decades. Callaway^{9} and Allen^{47} proposed models based on parameters that are fitted to experimental data. In 2003, Deinzer et al. used density functional perturbation theory (DFPT) to study the phonon linewidths of Si and Ge.^{48} Since then, many authors have used the solution of the BTE to calculate the lattice thermal conductivity of solids.^{19,20,21} The most used approach is the iterative solution of the BTE proposed by Omini et al. and successfully applied in the prediction of the \(\kappa _\ell\) tensor for different materials:^{49,50,51}
where superscripts α and β are two of the Cartesian direction indices and the subscript λ comprises both phonon branch index i and a wave vector q. The variables ω _{ λ } and v _{ λ } are the angular frequency and group velocity of the phonon mode λ, respectively, while f _{0}(ω _{ λ }) is the phonon distribution function according to Bose–Einstein statistics. All these quantities are obtained through the calculation of the IFCs by using a finitedifference supercell approach: forces vs. small displacement of inequivalent atoms. In this approach, a reference unit cell of volume Ω is used to create the supercell up to the cutoff distance. For the various summations, the Brillouin zone, BZ, is discretized into a Γcentered orthogonal regular grid of N ≡ N _{1} × N _{2} × N _{3} qpoints, where subscripts 1, 2, and 3 indicate the lattice vector indices.
The meanfree displacement F _{ λ } follows the definition of the Bose–Einstein phonon distribution, f _{ λ }, in the presence of a temperature gradient ∇T. For small perturbations, \(\nabla T \sim 0\), f _{ λ } can be expanded as \(f_{\lambda} \sim f_0\left( {\omega _{\lambda} } \right) + {g_{\lambda}}\), where g _{ λ } is the firstorder nonequilibrium contribution linear in ∇T:
$${g_{\lambda}} \equiv  {\bf{F}_{\lambda}} \cdot \nabla T\frac{{{\rm d}f_0}}{{{\rm d}T}}.$$Finally, the BTE can be expressed as a linear system of equations for F _{ λ }, as^{20,49,50,51,52,53}
with ξ _{ λλ′} = ω _{ λ }/ω _{ λ′}. The frequently used relaxation time approximation (RTA), corresponds to neglecting the Δ _{ λ } correction. For the full solution, F _{ λ } can be selfconsistently solved starting from the RTA guess, until convergence of \(\kappa _\ell\), Eq. (1). The other quantities present in these formulas, the relaxation time \(\tau _{\lambda} ^0\), and the threephonon scattering rates \({{\Gamma }}_{\lambda \lambda {\prime}\lambda {\prime}{\prime}}^ \pm\), will be illustrated in the next section.
Vibrational modes and group velocities
The vibrational modes are obtained by diagonalizing the dynamical matrix D(q)^{8,54,55,56}
where M(j) is the mass of j atom, e _{ λ } is the eigenvector for λ, R _{ l } is the position of lattice point l and \({\rm{\Phi }} ({i, \ j})_{\alpha \beta }\) are the secondorder force constants. D(q) is a Hermitian 3n _{a} × 3n _{a} matrix, where the factor “3” comes from the dimensionality of the problem, and n _{a} represents the number of atoms in the unit cell.
The nonanalytical contributions to the dynamical matrix are included by using the mixedspace formulation of Wang et al.^{57}
This contribution requires the calculation of the Born effective charge tensors, Z, and the highfrequency static dielectric tensor, \(\epsilon _\infty\), i.e. the contribution to the dielectric permittivity tensor from the electronic polarization.^{58} The mixedspace approach has been successfully applied in the calculation of phonon and thermal properties for a wide range of polar materials.^{59} Materials with high Z and low \(\epsilon _\infty\) are the cases in which the nonanalytical contributions are crucial for appropriate description of the phonon spectra as they cause the LO–TO splitting of the spectrum (between longitudinal and transverse optical phonon frequencies).^{58}
The group velocities, v _{ λ }, follow the Hellmann–Feynman theorem:
Scattering time
The total scattering time is a sum of terms representing different phenomena:
\(\tau _{\lambda} ^{{{\rm{iso}}}}\) indicates the isotopic or elastic scattering time and it is due to the isotopic disorder^{60,61}
where \(g(i) = \mathop {\sum}\nolimits_s {\kern 1pt} f_s(i)\left[ {1  M(i)^s{\rm{/}}\overline M (i)^s} \right]^2\) is the Pearson deviation coefficient of masses M(i)^{s} of isotopes s for the i atom, f _{ s } is the relative fraction of isotope s, and \(\overline M (i)^s\) is the average mass of the element.^{62}
\(\tau _{\lambda} ^{{{\rm{bnd}}}}\) is the time associated with scattering at the grain boundaries^{63,64}
where L is the average grain size. The effect of the boundaries on \(\kappa _\ell\) can also be calculated by restricting the summation to the modes with a mean free path, Λ = F _{ λ } · v _{ λ }/v _{ λ }, shorter than L:^{25}
\(\tau _{\lambda} ^{{{\rm{anh}}}}\) is the threephonon scattering time. It is the largest contribution to \(\tau _{\lambda} ^0\) for single crystals at mediumtemperature ranges and it is the most computationally expensive quantity to obtain:
Conservation of the quasimomentum requires that q″ = q ± q′ + Q in the summation ∑^{±}, for some reciprocal lattice vector Q such that q″ is in the same image of the Brillouin zone as q and q′. The threephonon scattering rates, \({{\Gamma }}_{\lambda \lambda {\prime}\lambda {\prime}{\prime}}^ \pm\), are computed as
and
The scattering matrix elements, \(V_{\lambda \lambda {\prime}\lambda {\prime}{\prime}}^ \pm\), are given by^{52,53}
where Φ(i, j, k)_{ αβγ } are the anharmonic force constants (introduced below) and \(e_{p{\prime}, \pm {\bf{q{\prime}}}}^\beta (j)\) is the element of the eigenvector of branch p′ at point ±q′ that corresponds to j atom in the βdirection. Note the indices {i ∈ uc (unit cell)} while {j, k ∈ sc (supercell)}. The conservation of energy, enforced by the Dirac distribution, can cause numerical instability during the calculations. Thus, we follow Li et al.^{25} and substitute δ with a normalized Gaussian distribution g: \(\delta ( \cdots ) \to g( \cdots )\) in Eqs. (12) and (13) with
where \(Q_\nu ^\alpha\) is the component in the Cartesian direction, α, of the reciprocalspace lattice vector Q _{ ν } and N _{ ν } is the number of points of the qpoints grid in the reciprocalspace direction ν. In principle, the parameter ζ could be taken equal to one. However, it can be adjusted to lower values to increase the speed of the calculations, without much effect on the overall accuracy of the integrations.
Interatomic force constants
The n ^{th}order IFCs, \({\rm{\Phi }}(i,j, \ldots )_{\alpha \beta \cdots }\) are tensorial quantities representing derivatives of the potential energy (V) with respect to the atomic displacements from equilibrium:
Labels i, j, k, … span atoms of the cell and indices α, β, γ, … are the Cartesian directions of the displacement. Secondorder harmonic IFC, Φ(i, j)_{ αβ }, calculations were already implemented in the original harmonic AutomaticPhononLibrary (APL),^{28} which obtains dispersion curves using three different approaches: direct force constant,^{65,66,67} linear response and projectoraugmented wave (PAW) potentials,^{68} and the frozen phonon methods.^{69,70}
Thirdorder IFCs, Φ(i, j, k)_{ αβγ } contain information about the anharmonicity of the lattice and they tend to rule phonon scattering in single crystals in the mediumtemperature ranges.^{71,72} Given the choice of a supercell size, the finite difference method to calculate the thirdorder IFCs leads to:
where {±h(i)^{α}}, {±h(j)^{β}} represent displacements of magnitude h of the i, j atoms in the Cartesian directions ±α, ±β and ψ(±h(i)^{α}, ±h(j)^{β}, k)_{ γ } are the γ components of the forces felt by the k atom in the distorted configurations caused by the i and j atoms.
The thirdorder IFCs’ calculation is computationally intensive: each Φ(i,j,k)_{ αβγ } requires four supercell calculations (Eq. (17)). Effective use of crystal symmetry can help the process.^{73} AAPL uses point, factor, and space group symmetry operations computed by the AFLOW symmetry engine^{26} to identify equivalence between single, pairs, and triplets of atoms (positions) and test equivalence between other field quantities, such as differential Φ or finite difference forces ψ (covariantly transforming). To avoid confusion, here indices as superscripts \(^{\left\{ {\alpha \beta \gamma \cdots } \right\}}\) or subscripts \(_{\left\{ {\alpha \beta \gamma \cdots } \right\}}\) are used to identify the character of the symmetry transformation to be applied.^{74}
The reduction of thirdorder IFC calculations is performed through the following steps:

1.
Inequivalent atoms, pairs, and triplets are identified using space group symmetries. The user chooses the neighborshell cutoff and only pairs/triplets completely contained are considered.

2.
The IFC tensors belonging to inequivalent triplets are analyzed. The symmetry operations mapping the representative inequivalent to the equivalent Φ are saved: Φ(i, j, k)_{ αβγ } → Φ′(i′, j′, k′)_{ α′β′γ′}.

3.
Each inequivalent tensor Φ(i, j, k)_{ αβγ } contains 3 × 3 × 3 = 27 coefficients. Every static abinitio calculation produces the vectorial force field for all the k atoms of the supercell (where each k, combined with the inequivalent pair (i, j), possibly generates (i, j, k) inequivalent triplets) starting from a combination of deformed positions for the i and j atoms belonging to inequivalent pairs. This requires the evaluation of 3 × 3 = 9 configurations. Following Eq. (17) four forces ψ(±h(i)^{α}, ±h(j)^{β}, k)_{ γ } are required for every entry Φ(i, j, k)_{ αβγ }. To conclude, a total of 36 static calculations are necessary to parameterize \({\rm{\Phi }}(i,j,\forall k \in {\rm{sc}})\).

4.
A large lookup table of all the necessary finite difference forces ψ(h(i)^{α}, h(j)^{β}, k) is prepared at the beginning of the process. Every ψ can be constituent of many inequivalent Φ(i, j, k)_{ αβγ }, and, within each Φ, be a term in several internal coefficients. To exploit redundancy, the force field generated by every static abinitio calculation is mapped through symmetry operations to recover as many possible other \(\psi \left( {h(i)^\alpha ,h(j)^\beta ,\forall k \in {\rm{sc}}} \right)_\gamma \to \psi \left( {h(i{\prime})^{\alpha {\prime}},h(j{\prime})^{\beta {\prime}},\forall k{\prime} \in {\rm{s}}c} \right)_{\gamma {\prime}}\). Calculated and symmetry reproduced ψ are then removed from the table, and the algorithm moves to the next one to characterize. The process is repeated until all the ψ are found. The process guarantees that only the minimum amount of calculations are performed, compatible with the model of Eq. (17).

5.
During the process, many equivalent entries of the tensors Φ(i, j, k)_{ αβγ } are generated by the static abinitio calculations. Because of unavoidable numerical noise, often equivalent entries have slightly different values, and the final value needs to be symmetrized somehow. This is performed during the resymmetrization necessary to address the “sum rules” conservation.
Sum rules and resymmetrization
Invariance with respect to any global rigid displacement translates into “sum rules” for anharmonic IFCs:
Due to finite size effects, the calculated IFCs are not perfectly symmetric and do not strictly satisfy Eq. (18), causing numerical instabilities. To tackle the issue, different strategies have been proposed such as Lagrange multipliers^{25} and orthogonalization procedures.^{75} We implement an iterative algorithm which corrects Φ(i, j, k) and fulfills the constraints.
Given a set of Φ(i, j, k) the error x of each sum rule at step \({\cal N}\) is defined as
Each iteration is composed of correction and resymmetrization of equivalent IFCs. Correction, \({\rm{\Phi }}(i,j,k)_{\alpha \beta \gamma }^{\cal N} \to {\rm{\Phi }}(i,j,k)_{\alpha \beta \gamma }^{{\cal N} + 1}\) is given by:
where the term \(x^{\cal N}\left {{\rm{\Phi }}^{\cal N}} \right{\rm{/}}{\sum} {\kern 1pt} \left {{\rm{\Phi }}^{\cal N}} \right\) corrects Φ based on the total error times the absolute contribution of Φ in the “sum rule”. The sum over the combination of indices {i′, j′, k′, α′, β′, γ′} giving IFCs equivalent to Φ(i, j, k)_{ αβγ } (there are n _{eq}) is meant to symmetrize the error across all the entries. The mixing fraction in the iterative process, μ, can be adjusted by the user to optimize convergence rate and robustness. Overall, with increasing number of neighbor shells, the user can effectively reduce this systematic error and achieve effective convergence of \(\kappa _\ell\).
Calculation workflows
The calculations parameters and methods are described in the Methods section.
• Anharmonic scattering time \(\tau _{\lambda} ^{{{\rm{anh}}}}\):
• Elastic scattering time \(\tau _{\lambda} ^{{{\rm{iso}}}}\) (isotopic disorder) and grain boundaries scattering time \(\tau _{\lambda} ^{{{\rm{bnd}}}}\) (polycrystalline materials):
• Conductivity \(\kappa _\ell ^{\alpha \beta }\):
Analysis of results
Different statistical parameters are used to measure the qualitative and quantitative agreements of AAPL with respect to experimental values. The Pearson correlation coefficient r[{X}, {Y}] is a measure of the linear correlation between two variables, {X} and {Y}:
where \(\overline X\) and \(\overline Y\) are the averages of {X} and {Y}.
The Spearman rank correlation coefficient ρ[{X}, {Y}] is a measure of the monotonicity of the relationship between two variables. The values of the two variables {X} and {Y} are sorted in ascending order, and are assigned rank values {x} and {y} which are equal to their position in the sorted list. The correlation coefficient is then given by
ρ is useful for determining how well the values of one variable can predict the ranking of the other variable.
The rootmeansquare relative deviation (RMSrD), of the calculated κ vs. the experiment is also investigated. The RMSrD will measure the quantitative difference between AAPL and experimental results:
Lower values of RMSrD indicate better agreement.
Performance
Scaling
The calculation of the anharmonic IFCs is the most computationally expensive step in the method. First, we test the number of required calculations for some structural prototypes, such as diamond (spacegroup: Fd\(\overline 3\)m, #227; Pearson symbol: cF8; Strukturbericht designation: A4; AFLOW Prototype: A_cF8_227_a^{26} (http://aflow.org/CrystalDatabase/A_cF8_227_a.html.)), rocksalt (Fm\(\overline 3\)m, #225, cF8, B1, AB_cF8_225_a_b^{26} (http://aflow.org/CrystalDatabase/AB_cF8_225_a_b.html.)), fluorite (Fm\(\overline 3\)m, #225, cF12, C1, AB2_cF12_225_a_c^{26} (http://aflow.org/CrystalDatabase/AB2_cF12_225_a_c.html.)), wurtzite, (P6_{3}mc, #186, hP4, B4, AB_hP4_186_b_b^{26} (http://aflow.org/CrystalDatabase/AB_hP4_186_b_b.html.)), ZrO_{2} (P4/nmc, #137, tP6) and corundum (R\(\overline 3\)c, #167, hR10, D5_{1}, A2B3_hR10_167_c_e^{26} (http://aflow.org/CrystalDatabase/A2B3_hR10_167_c_e.html.)) for which there are abundant available experimental data. The number of calculations and how they scale are compared with respect to the chosen cutoff for the IFCs (see Fig. 1) for different available software (Phono3py and ShengBTE software packages). The number of required static calculations increases with the cell’s complexity, the total number of atoms, and the number of inequivalent positions in the primitive cell. AAPL reduces the number of required calculations compared to the other two codes for the six tested prototypes, indicating that the AAPL algorithm is efficient at handling symmetry equivalence. For example, in silicon and using the minimum shell cutoff, AAPL only needs 21 calculations, while ShengBTE requires 76. The advantage is preserved while increasing the range of the interactions. For example, Phono3py requires 616 static calculations for CaF_{2} with seventhneighbor shells, whereas AAPL needs less than onethird of this amount (176). Figure 1 summarizes the scaling results.
Validation with experiments
A data set of 30 compounds is used to validate our framework. The list of materials includes semiconductors and insulators that belong to different structural prototypes, such as diamond (A_cF8_227_a^{26}), rocksalt (AB_cF8_225_a_b^{26}), and fluorite (AB2_cF12_225_a_c^{26}). To maximize the heterogeneity of the data set, materials are selected containing as many different elements as possible from the sblocks, pblocks, and dblocks of the periodic table. The comparison of calculated vs. experimental values of \(\kappa _\ell\) is summarized in Table 1 and Fig. 2.
Different statistical quantities are used to measure qualitative and quantitative agreements between the AAPL and experimental results (Table 2). AAPL results strongly correlate with experimental findings, with relatively small RMSrD from experiment demonstrating the reliability and robustness of the framework. The algorithm should not be blamed for systematic errors in the abinitio characterization of the compounds (such as the ones containing Pb).
AAPL is also compared with approximate phenomenological frameworks, such as AFLOWAGL^{15} and AFLOWQHAAPL.^{7,86} Qualitatively, all the three frameworks have high linear correlation with experiments (Pearson, r); AAPL and QHAAPL are also very effective in rank ordering the compounds (Spearman, ρ). Quantitatively, AAPL has the lowest RMSrD value, followed by QHAAPL and AGL. Accuracy strongly correlates with computational costs (AAPL \(\gg\) QHAAPL > AGL), so that the users can choose which technique best fulfills their screening needs.
Singlecrystal and nanocrystalline silicon
Silicon is the perfect benchmark for testing the reliability of AAPL: extensive availability of experimental data for wellcharacterized samples^{87,78} and limited computational cost due to the diamond crystal structure with two atoms in the primitive cell and fcc lattice. Figure 3a depicts the calculated lattice thermal conductivity at different temperatures for singlecrystal and polycrystalline samples compared to singlecrystal experimental values from ref. 78. The calculations published by Carbogno et al. using the Green–Kubo formalism are also included.^{88} Boundary effects can be included in two ways: i. by calculating the complete \(\kappa _\ell ^{\alpha \beta }(L)\) (workflow (23)) for average grain sizes having different \(\tau _{\lambda} ^{{{\rm{bnd}}}}(L)\) (workflow (22)) or ii. by neglecting τ ^{bnd} from the total scattering time \(\tau _{\lambda} ^0\) (Eq. (7)) and restricting the summation to the modes with a mean free path shorter than L (Λ<L, Eq. (10)): \(\kappa _{\ell ,({{\Lambda }} < L)}^{\alpha \beta }\). Both approaches are implemented in AAPL. Comparison with experimental values for different polycrystalline Si samples (average grain size L = 64, 76, 80, 155, 550 and 20,000 nm) at 300 K are presented in Fig. 3b. Both approximations of grain boundary scattering effects show the same trend and are very close to the experimental results, corroborating the validity of our approaches.
Noncubic systems
The lattice thermal conductivity is addressed for three wellcharacterized systems.

i.
Tetragonal zirconia (tZrO_{2}) is used for energy and biomedical applications because of its combination of strength, fracture toughness, and ionic conductivity, as well as a low thermal conductivity coating material for protecting metals from excessive heat. Some properties strictly depend on the stabilization of the tetragonal phase as the monoclinic polymorph (mZrO_{2}) is the stable phase at room temperature.^{98} Stabilization is performed by alloying the system with aliovalent ions, such as yttrium.^{98} Carbogno et al. recently calculated \(\kappa _\ell\) for tZrO_{2} with a combination of abinitio molecular dynamics and Green–Kubo formalism.^{88} Figure 3c shows a comparison: AAPL values (PBE and LDA functionals) for tZrO_{2} and mZrO_{2}, Carbogno et al. for tZrO_{2}, and the experimental values for mZrO_{2} and yttriumstabilized tZrO_{2} (YSZ) tZrO_{2} (YSZ).^{89,90} Methods using molecular dynamics with ions responding “classically” to abinitio forces tend to overestimate the thermal conductivity below the Debye temperature. Classical dynamics follows the Dulong–Petit limit for the specific heat—Boltzmann distribution of phonon energies—and the quantum of transferred heat is unavoidably overestimated (\(\kappa _\ell\) increases with C _{V} ^{2}). On the contrary, forceconstants methods have phonons energies following the Bose–Einstein distribution by construction (Eq. (1)). They might neglect higherorder phonons’ renormalization effects, thus producing underestimated \(\kappa _\ell\).^{99} Above the Debye temperature (~590 K for ZrO_{2} ^{100,101}), the phonons mean free path reduces and the statistical distributions become classic. In that regime, the two methods will approach the correct experimental values from different directions.

ii.
Wurtzite zinc oxide (ZnO) is another wellcharacterized system. Olorunyolemi et al.^{91} and Barrado et al.^{92} reported \(\kappa _\ell\) of polycrystalline ZnO with an average particle size of 20 and 1000 nm, respectively. AAPL, performed with average grain size of 500 nm, is in very good agreement with the experimental results capturing the effects of the grain boundaries’ scattering (Fig. 3d).

iii.
Rhombohedral aluminum oxide, αAl_{2}O_{3}, which contains 30 atoms in its conventional unit cell, is chosen to test the framework’s robustness. To the best of our knowledge, this is the largest unit cell ever characterized for lattice thermal conductivity with abinitio means (FeSb3, containing 16 atoms was studied in ref. 102). AAPL results are again in excellent agreement with experimental values^{93} for a wide range of temperatures (Fig. 3d).
Extension to ACBN0 pseudohybrid functional
The accuracy of the results ultimately relies on the quality of the computed IFCs with abinitio. The use of hybrid functionals^{103} or advanced electronic structure methods, such as GW ^{104} to compute the IFCs is limited^{105,106} because of their computational costs. It was also reported that hybrid functionals can predict false lattice instability in some cases.^{107} Recently, the ACBN0 functional was introduced in order to facilitate the accurate characterization of electronic properties of correlated materials.^{40} ACBN0 is a pseudohybrid Hubbard density functional that introduces a new selfconsistent abinitio approach to compute U without the need for empirical parameters. ACBN0 can improve not only the description of the electronic structure, but also the prediction of the structural and the vibrational parameters of solids.^{45} One of the reasons for this is the better prediction of the Born charges, Z, and the dielectric tensor, \(\epsilon _\infty\), compared to LDA or GGA functionals.^{45} If the ACBN0 functional improves the vibrational parameters of solids, it can be assumed that the calculations of other temperaturedependent properties, such as \(\kappa _\ell\), may be improved too. As a testbed, calcium fluoride, CaF_{2}, is chosen, because of the ample available experimental data.^{81,95–97} CaF_{2} has been extensively used in optical devices due to its low refractive index, wide band gap, low dispersion, and large broadband radiation transmittance.^{108,100,110}
A package implementing a minimalist AFLOWPython approach to highthroughput abinitio calculations for the generation of tightbinding hamiltonians and the calculation with the ACBN0 functional (AFLOWπ)^{111} is used to obtain the ACBN0 electronic structure of CaF_{2}. A U _{eff} of 13.43 is obtained for F–p orbitals. This value is then used inside AAPL for the rest of the calculations. AAPL + ACBN0 almost perfectly predicts the experimental \(\kappa _\ell\) in contrast with AAPL + PBE which greatly underestimates \(\kappa _\ell\) over the entire temperature range (Fig. 3e). Phonon band structures have also been calculated with the harmonic library (APL) to explain the difference (Fig. 3f). ACBN0 reproduces the phonon dispersion better than the PBE functional. PBE, as a GGA functional, overestimates bond length and hence it tends to underestimate vibrational frequencies.^{112} On the contrary, ACBN0 describes the bond length more accurately, obtaining frequencies higher than PBE and closer to the experimental values.^{95} Major differences between ACBN0 and PBE results come from the optical bands, so the two main properties that are involved in the splitting of the optical band due to the nonanalytical contributions to the dynamical matrix are compared. While the Born charges are similar for ACBN0 and PBE (2.33 e and 2.34 e, respectively), there are significant differences in the highfrequency dielectric constant \(\epsilon _\infty\) (2.083 and 2.305, respectively). The value obtained using ACBN0 is closer to the experimental \(\epsilon _\infty\) (2.045)^{97} than that obtained using PBE.
Conclusions
In this article, we have presented the AAPL, which was developed to compute the thirdorder IFCs and solve the BTE within the highthroughput AFLOW framework. The software automatically predicts the lattice thermal conductivity of singlecrystals and polycrystalline materials by using a single input file and no further user intervention. The symmetry analysis has been optimized to further reduce the number of static calculations compared to other packages. The robustness and accuracy of the code have been tested with a set of 30 materials that belong to different space groups. APL has been combined with the ACBN0 pseudohybrid functional to predict the lattice thermal conductivity of CaF_{2}. Our results demonstrate that using ACBN0 can improve not only the electronic structure description of the material compared to the GGA functional, but also phonondependent properties, such as the thermal conductivity.
Methods
Geometry optimization
All structures are fully relaxed using the automated framework AFLOW^{28,29,30,31,32,33,34,35,36,37,38,39} and the VASP package.^{113} Optimizations are performed following the AFLOW standards.^{34} The PAW potentials^{114} are used and the exchange and correlation functionals parameterized by the generalized gradient approximation proposed by Perdew–Burke–Ernzerhof (PBE).^{115} All calculations use a highenergy cutoff, which is 40% larger than the maximum recommended cutoff among all component potentials, and a kpoint mesh of 8000 kpoints per reciprocal atom. Primitive cells are fully relaxed (lattice parameters and ionic positions) until the energy difference between two consecutive ionic steps is smaller than 10^{−4} eV and forces in each atom are below 10^{−3} eV/Å.
Phonon calculations
Phonon calculations are performed using the Automatic Phonon Library, APL, as implemented in AFLOW, and by using VASP to obtain the secondorder IFCs via the finitedisplacement approach.^{86} The magnitude of the displacement is 0.015 Å. Electronic selfconsistent field iterations for static calculations are stopped when the difference of energy between the last two steps is less than 10^{−5} meV. The threshold ensures a good convergence for the wavefunction and sufficiently accurate values for forces and harmonic constants. Nonanalytic contributions to the dynamical matrix are also included using the formulation developed by Wang et al.^{57} Frequencies and other related phonon properties are calculated on a 21 × 21 × 21 qpoint mesh in the Brillouin zone, which is a tradeoff between the computational cost, convergence of the phonon density of states, pDOS, and the derived thermodynamic properties. Integrations within the Brillouin zone are obtained by using the linear interpolation tetrahedron method available in AFLOW.
Lattice thermal conductivity
Anharmonic force constants are extracted from a 4 × 4 × 4 supercell using a cutoff that includes all 4thneighbor shells. Thermal conductivity is evaluated on a 21 × 21 × 21 qpoint mesh using ζ = 0.1 for the Gaussian smoothing, Eq. (15). The dense mesh ensures the convergence of the values obtained for \(\kappa _\ell\).^{25} The IFC calculations are iterated selfconsistently until all sum rules are lesser than 10^{−7} eV/Å^{3}.
Acronyms in the framework
The following acronysms are used in the article. AAPL: AutomaticAnharmonicPhononLibrary; AGL: AFLOWGibbsLibrary;^{15} APL: AutomaticPhononLibrary;^{28,35,16} QHA: quasiharmonic approximation;^{7} ACBN0: Agapito Curtarolo Buongiorno Nardelli abinitio DFT functional;^{40} AFLOWπ: A minimalist AFLOWPython approach to highthroughput abinitio calculations for the generation of tightbinding hamiltonians and the calculation with the ACBN0 functional.^{111}
Data availability
All the data and codes are freely available to the public as part of the AFLOW online repository and can be accessed through www.aflow.org.
Additional information
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Acknowledgements
The authors thank Drs. Natalio Mingo, David Hicks, Mike Mehl, Ohad Levy, Christian Carbogno, Matthias Scheffler, and Corey Oses for various technical discussions. We acknowledge support by the DOE (DEAC0205CH11231), specifically the Basic Energy Sciences program under Grant # EDCBEE. C.T., M.F., M.B.N., and S.C. acknowledge partial support by DODONR (N000141310635, N000141110136, and N000141512863). The AFLOW consortium acknowledges Duke University–Center for Materials Genomics—for computational support. S.C. acknowledges the Alexander von Humboldt Foundation for financial support (FritzHaberInstitut der MaxPlanckGesellschaft, 14195 BerlinDahlem, Germany).
Author information
Affiliations
Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC, 27708, USA
 Jose J. Plata
 , Pinku Nath
 , Demet Usanmaz
 & Cormac Toher
Institute of Materials Chemistry, TU Wien, A1060, Vienna, Austria
 Jesús Carrete
Department of Materials Science and Engineering, University of California, Berkeley, 210 Hearst Memorial Mining Building, Berkeley, USA
 Maarten de Jong
 & Mark Asta
SpaceX, 1 Rocket Road, Hawthorne, CA, 90250, USA
 Maarten de Jong
Department of Physics and Science of Advanced Materials Program, Central Michigan University, Mount Pleasant, MI, 48859, USA
 Marco Fornari
Department of Physics and Department of Chemistry, University of North Texas, Denton, TX, USA
 Marco Buongiorno Nardelli
Materials Science, Electrical Engineering, Physics and Chemistry, Duke University, Durham, NC, 27708, USA
 Stefano Curtarolo
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Contributions
J.J.P. developed and implemented the AAPL framework within AFLOW, with advice and assistance from J.C., C.T., P.N., M.d.J., M.A., and S.C. D.U., P.N. and J.J.P. ran the abinitio calculations to obtain the lattice thermal conductivity. M.F. and M.B.N. performed the ACBN0 calculations to obtain the selfconsistent U values for CaF_{2}. All authors contributed to the analysis of the results and the writing of the paper.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to Stefano Curtarolo.
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