Introduction

Designing materials with specific properties is a long-term goal in materials science.1,2 High-throughput ab initio materials screening and design is a new and rapidly growing area in computational materials research.2,3 The application of high-throughput calculations has recently made formidable progress and led to novel insights in this field.3 The lattice thermal conductivity (κ) is a crucial physical property of crystalline materials for enormous practical implications, such as electronic cooling, thermoelectrics, phase change memories, and etc.1,4 Therefore, the fast evaluation of reliable κ for variety of materials plays a key role in identifying suitable materials for targeted applications.

Currently, first-principles based anharmonic lattice dynamics method coupled with the phonon Boltzmann transport equation (BTE) is one of the most featured methods to obtain the κ, which involves calculation of interatomic force constants (IFCs).5,6,7 Basically, the energy of a system can be expressed with Taylor expansion

$$E = E_0 + \frac{1}{{2!}}\mathop {\sum}\limits_{\begin{array}{*{20}{c}} {i,j} \cr {\alpha ,\beta } \end{array}} \Phi _{ij}^{\alpha \beta }u_i^\alpha u_j^\beta + \frac{1}{{3!}}\mathop {\sum}\limits_{\begin{array}{*{20}{c}} {i,j,k} \cr {\alpha ,\beta ,\gamma } \end{array}} {\Phi _{ijk}^{\alpha \beta \gamma }u_i^\alpha u_j^\beta u_k^\gamma + \cdots ,}$$
(1)

where \(u_i^\alpha\) is the displacement of atom i along α direction and Φ are the nth order IFCs.8 Based on the second order IFCs the dynamical matrix can be constructed and the phonon dispersion are obtained by diagonalizing the dynamical matrix. The anharmonic nature of the system is described using the third-order IFCs, while the contributions of the fourth and higher order terms are usually neglected.9 The anharmonic IFCs are evaluated based on the third-order derivatives of the total energy (E) with respect to the atomic displacements

$$\Phi _{ijk}^{\alpha \beta \gamma } = \frac{{\partial ^3E}}{{\partial u_i^\alpha \partial u_j^\beta \partial u_k^\gamma }}\;,$$
(2)

which is the response along the α direction on atom-i due to the displacement of atom-j (β-direction) and atom-k (γ-direction). From the anharmonic IFCs the scattering matrix can be constructed, based on which one can calculate the three-phonon scattering rates and then obtain the phonon lifetime.7,10 Finally, the κ can be obtained in the framework of BTE based on the single mode relaxation time approximation (RTA).10

For obtaining the anharmonic IFCs (\(\Phi _{ijk}^{\alpha \beta \gamma }\)) based on Eq. (2), the supercell-based finite displacement difference method is widely employed. The calculations are very time and resource consuming due to the huge number of computational runs using large supercell. Specially, j and k loop over the atoms in the whole supercell, while i only loops over one unit cell, which is due to the translational symmetry.10 Each element of \(\Phi _{ijk}^{\alpha \beta \gamma }\) requires four calculations with different supercell configurations. Thus, for a system of N1 × N2 × N3 supercells with n atoms per unit cell, the number of calculations is 4 × 9 × N1 × N2 × N3 × n2.10 It is a hard task to perform so huge number of computational runs even for a simple system, while for complex systems (large n) it is sometimes impossible to accomplish the calculations. In practice, a cutoff radius (rcutoff) is introduced to disregard the interatomic interactions outside a certain region (Inset of Fig. 1). Thus, the number of calculations can be largely reduced since atom indices of j and k only loop over atoms in the region defined by the rcutoff. However, severe problem arises for choosing an appropriate (usually large enough) rcutoff, in terms of getting satisfactory results. A not large enough rcutoff may give a relatively larger κ than the real value or even yield a diverged κ, in particular for low-dimensional systems, such as graphene and phosphorene.11,12,13 In principle, the rcutoff should exceed the range of physically relevant interactions to get accurate results.6,8 However, this statement is vague and is not useful for practical operation. Currently, in practice one has to perform the so-called convergence test of κ by gradually increasing the rcutoff. Such calculations are very time and resource consuming due to the aforementioned huge computational costs as rcutoff increases. In most cases, an empirical or arbitrary rcutoff is chosen, which could lead to unreliable results.12,14

Fig. 1
figure 1

The accelerated ratio with interactions taken into account up to different nearest neighbors for different systems of the proposed new approach. The ratio is defined as the accumulated number of cases to be run when doing the convergence test of κ with respect to rcutoff (counting all the nth nearest neighbors) over the number of cases to be run for the specific rcutoff. A typical integral calculation for one system can save (achieve) ~10 times the computational resources (speedup) by reconstructing the \(\Phi _{ijk}^{\alpha \beta \gamma }\) or performing calculations in increments. Inset: Sketch of cutoff in the calculations of third-order IFCs. The black circles represent atoms and the colored circles indicate the region with a certain cutoff radius (rcutoff), beyond which the interactions are neglected. The patterns of atomic displacement with a small cutoff radius (\(r_1^{{\mathrm{cutoff}}}\)) is a subset of that with a large cutoff radius (\(r_2^{{\mathrm{cutoff}}}\))

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In this paper, based on the analysis of the harmonic (second order) IFCs, we propose a concise strategy to efficiently accelerate the evaluation process of obtaining accurate and converged κ by solving the cutoff distance problem. The proposed strategy is efficient for directly determining when the κ converges with respect to rcutoff and how to fast get satisfactorily converged κ. With this strategy, we study the divergence problem of thermal conductivity of graphene, a long debate of two-dimensional heat conduction in literature. The feasibility of the method is also confirmed by other systems. In addition, the computing speed with our approach could be one order of magnitude faster compared to the traditional method in most situations, where the ~20th nearest neighbors are considered. The quantitative strategy proposed herein can be a good candidate for fast evaluating the reliable κ and thus provides useful tool for high-throughput materials screening and design with targeted thermal transport properties.

Results and discussion

Methodology

Our proposed strategy

The first step in our strategy is to roughly determine rcutoff based on the analysis of second order IFCs (\(\Phi _{ij}^{\alpha \beta }\)), which are the harmonic response of the force acting on atom i (α-direction) resulted from the displacement of atom j (β-direction). Based on the finite displacement difference method, the harmonic IFCs tensor can be obtained as

$$\Phi _{ij}^{\alpha \beta } = \frac{{\partial ^2E}}{{\partial u_i^\alpha \partial u_j^\beta }} = \left[ {\begin{array}{*{20}{c}} {\frac{{\partial ^2E}}{{\partial u_i^x\partial u_j^x}}} & {\frac{{\partial ^2E}}{{\partial u_i^x\partial u_j^y}}} & {\frac{{\partial ^2E}}{{\partial u_i^x\partial u_j^z}}} \cr {\frac{{\partial ^2E}}{{\partial u_i^y\partial u_j^x}}} & {\frac{{\partial ^2E}}{{\partial u_i^y\partial u_j^y}}} & {\frac{{\partial ^2E}}{{\partial u_i^y\partial u_j^z}}} \cr {\frac{{\partial ^2E}}{{\partial u_i^z\partial u_j^x}}} & {\frac{{\partial ^2E}}{{\partial u_i^z\partial u_j^y}}} & {\frac{{\partial ^2E}}{{\partial u_i^z\partial u_j^z}}} \end{array}} \right]\,.$$
(3)

To quantify the strength of interatomic interactions described by the harmonic IFCs, we define the root mean square (RMS) of the elements of the IFC tensor (Frobenius norm)15

$${{\mathrm{RMS}}}\left( {\Phi _{ij}} \right) = \left[ {\frac{1}{9}\mathop {\sum}\limits_{\alpha ,\beta } \left( {\Phi _{ij}^{\alpha \beta }} \right)^2} \right]^{\frac{1}{2}}\;.$$
(4)

Then, the RMS(Φ ij ) with respect to the distance between atom i and atom j can be extracted and loop over all the atoms to get insight into how the interaction strength changes with distance increasing. Based on this parameter one can directly determine how large the rcutoff should be chosen to evaluate the anharmonic IFCs by effectively including the possibly present strong interaction strength as revealed by the large RMS(Φ ij ) value. Since the harmonic IFCs are quite easy to obtain, it is convenient to implement the strategy. With the determined rcutoff one can get accurate, converged, and reliable κ, which can be confirmed by the convergence test of κ vs. rcutoff.

Our second step is to speed up the convergence test of κ vs. rcutoff based on the fact that, when using the finite displacement method to evaluate the anharmonic IFCs (Eq. 2), the atomic configuration for a small rcutoff is theoretically a subset of that for a large rcutoff. Thus, the calculations of \(\Phi _{ijk}^{\alpha \beta \gamma }\) can be largely accelerated by only performing calculations incrementally based on previous calculations for a small rcutoff. Or equivalently, from the calculations with a certain rcutoff, one can immediately obtain the convergence result of κ vs. rcutoff without re-calculating the \(\Phi _{ijk}^{\alpha \beta \gamma }\) at smaller rcutoff. Figure 1 depicts the accelerated ratio between our new approach (one time evaluation for a certain rcutoff) and the traditional method (accumulative evaluation for all cutoff distance less than rcutoff). A slop of 1/2 is generally found despite the different systems tested. Thus, for a system with 20th nearest neighbors considered, the computational resource is reduced by one order of magnitude, compared to the traditional method of gradually increasing the rcutoff. The efficiency is more remarkable for complex systems with long-range interactions, such as phosphorene and bulk SnSe.13,16,17 More information on the number of cases of third-order IFCs can be found in the Supplemental Material.

In short, based on the analysis of harmonic IFCs, the large enough rcutoff can be directly determined with very little computational cost. Then, the convergence test of κ vs. rcutoff can be immediately obtained in a largely accelerated way by fast reconstructing the anharmonic IFCs for different rcutoff, which finally confirms the chosen rcutoff is appropriate to get accurate, converged, and reliable κ. Compared to the traditional method, the computing speed with our approach could be one order of magnitude faster in most situations, where up to ~20th nearest neighbors should be considered. With the accelerated procedure to evaluate the κ, the computational cost to do with high-throughput screening could be effectively reduced although the reduction is not too much. Thus, the quantitative strategy proposed herein for fast evaluating the reliable κ is anticipated to be beneficial for high-throughput materials screening and design with targeted thermal transport properties.

Computational details

All the first-principles calculations are performed based on the density functional theory using the projector augmented wave method18 as implemented in the Vienna ab initio simulation package.19 The Perdew–Burke–Ernzerhof (PBE) of generalized gradient approximation (GGA) revised for solids (PBEsol)20 is chosen as the exchange-correlation functional to describe interatomic interactions in graphene. While for phosphorene, the PBE of GGA is chosen as the exchange-correlation functional and van der Waals interactions are taken into account at the vdW–DF level with optB88 used as exchange functional.21 The kinetic energy cutoff of wave functions is set as 1000 and 700 eV for graphene and phosphorene, respectively. The Monkhorst–Pack22 k-mesh of 15 × × 15 × 1 and 15 × 11 × 1 are used to sample the Brillouin zone (BZ) for graphene and phosphorene, respectively, with energy convergence threshold set as 10−8 eV. A large vacuum spacing of 20 Å along the out-of-plane direction is used to hinder the interactions arising from the employed periodic boundary conditions. The cell is fully optimized and all atoms are allowed to relax until the maximal Hellmann–Feynman force acting on each atom is no larger than 10−8 eV/Å.

For all the calculation of IFCs in graphene and phosphorene, 5 × 5 × 1 supercell is constructed and the Monkhorst–Pack k-mesh of 2 × 2 × 1 is used to sample the BZ. The space group symmetry properties are employed for reducing the computational cost and the numerical noise of the IFCs.23 The thickness for calculating κ is chosen as the structure thickness plus largest van der Waals diameter, which are 3.4 and 5.36 Å for graphene and phosphorene, respectively. The κ is obtained by solving the linearized phonon BTE using an iterative procedure as implemented in the ShengBTE package.8,10

Validation and applications

Now we take three systems (graphene, phosphorene, and bulk SnSe) as representative examples to demonstrate how the proposed strategy efficiently determines appropriate rcutoff and obtains converged and accurate κ.

Graphene

Graphene, a 2D carbon sheet with planar honeycomb structure, is one of the most fascinating and extensively studied materials in recent years due to its extraordinary mechanical, electronic, and thermal properties.24 Benefiting from its extremely high κ, thermal transport in graphene receives exceptional attention in both experimental and theoretical studies.11 Moreover, graphene also plays a benchmark role in the studies of thermal transport in broad 2D materials.25 Despite the intensive studies on the thermal transport of graphene, it has been a long time debate on whether the κ of graphene converges or not with respect to sample length.11,26,27 In the presence of extrinsic scattering mechanisms such as scattering from defects or boundary, or coupling to the substrate, the κ indeed converges as reported in lots of literature.25,28 However, for samples with infinite size the intrinsic κ of graphene is reported to logarithmically diverge with length, which was thought to be universal for all 2D materials.11,26,29 Such viewpoint was supported by numerous studies, such as those of silicene and phosphorene.12,30,31 The possible solution to the divergence problem was proposed in terms of higher-order phonon anharmonicity, which has not been conclusively proven yet.28 Recently, it was reported that the κ of graphene converges based on the exact solution of phonon BTE.32

Here, based on the harmonic IFCs analysis (Eq. 4), we demonstrate that there exist strong long-range interactions at the distance of 6.19 Å (Fig. 2a). If we choose the rcutoff smaller than 6.19 Å such as 5.94 Å (corresponding to 8th nearest neighbors), the divergence of κ is reproduced (Fig. 2b). All other results with smaller rcutoff also show diverged κ (Supplemental Material). However, when the rcutoff is larger than 6.19 Å such as 6.35 Å (corresponding to 9th nearest neighbors), the κ converges. The converged κ of graphene is 3267 W/mK, which is in good agreement with previous reports and experimental measurements.11,25,28,32 The inclusion of the long-range interactions effectively suppresses the lifetime of acoustic phonon modes near the Γ point of the BZ (corresponding to low frequency phonon modes). For the cases with small rcutoff, due to the insufficient involvement of the long-range interactions, the lifetime of acoustic phonon modes approaching the Γ point blows up, resulting in the diverged κ.33 Detailed information on the phonon lifetime can be found in the Supplemental Material. It is worth noting that, the κ of strained graphene (up to 10strain) also converges with the large rcutoff based on our calculations, in contrast to the diverged κ as reported previously.28,33,34,35

Fig. 2
figure 2

The study case of graphene. a The RMS(Φ ij ) vs. cutoff distance for graphene, revealing a strong interaction at the distance of 6.19 Å. Inset: absolute change of charge density due to the displacement (0.02 Å along z direction) of the central carbon atom within 5 × 5 × 1 supercell. The plot is top view with atoms marked on site and the unit is 10−5 eÅ−3. b Comparison of the (Q-grid) size effect on κ between different rcutoff. The κ converges when rcutoff 6.19 Å

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Phosphorene

Phosphorene, an elemental 2D semiconductor with high carrier mobility36,37,38 and intrinsic direct band gap,39 calls for fundamental understanding of thermal transport properties for its rapidly growing applications in nano-electronics/opto-electronics and thermoelectrics. The κ of phosphorene has been investigated theoretically by independent groups using various methods. However, the results obtained differ with each other unacceptably by even one order of magnitude.12,13,14,40,41,42,43,44,45,46 It is also claimed for phosphorene that there exists a size-dependent κ along the zigzag direction, which is similar to graphene.12

Based on the proposed strategy, we show clearly in Fig. 3 that the rcutoff required for obtaining converged κ of phosphorene is consistent with the interactions revealed by the harmonic IFCs analysis. From Fig. 3a, we can see that interactions up to ~6 Å are still fairly strong and rcutoff should be larger than 6 Å to get satisfactorily converged results. Based on the κ vs. rcutoff as shown in Fig. 3b, the appropriate rcutoff should be 7.85 Å, which includes the interactions up to the 20th nearest neighbors. Such choice of rcutoff yields converged κ of phosphorene.13 In previous study which reports the diverged κ of phosphorene, the interactions are truncated only up to 4.4 Å, i.e., up to the 7th nearest neighbors.12 The reason for the size-dependent κ of phosphorene in the case of small rcutoff primarily lies in the quickly blowing up of the lifetime for phonons approaching the Γ point of the BZ, which is similar to the case of graphene. When the rcutoff is larger than 6 Å, the lifetime will be suppressed by the additional long-range interactions and then the κ of phosphorene converges. Detailed information can be found in the Supplemental Material. Note that the RMS(Φ ij ) value in graphene is relatively larger compared to phosphorene, which is due to the relatively stronger σ bond between carbon atoms (C–C bond length: 1.420 Å) compare to phosphorus atoms (P–P bond length: 2.235 Å).

Fig. 3
figure 3

The study case of phosphorene. a The RMS(Φ ij ) vs. distance for phosphorene, revealing strong interactions for the cutoff distance of ~6 Å. Inset: absolute change of charge density due to the displacement (0.02 Å along z direction) of the central phosphorus atom within 5 × 5 × 1 supercell. The plot is top view with atoms marked on site and the unit is 10−5 eÅ−3. The charge density perturbation in phosphorene shows long-range interactions along the [110] direction. b Comparison of the convergence test of κ vs. rcutoff with third-order IFCs obtained from reconstruction and direct calculations

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Moreover, Fig. 3b shows the results of κ vs. rcutoff based on \(\Phi _{ijk}^{\alpha \beta \gamma }\) from both reconstruction and direct calculations. Overall, the reconstructed results agree very well with those from direct calculations. The slight discrepancy can be attributed to the instability and inconsistency of the numerical precision in the first-principles calculations, where the wave function and charge density are slightly different for different calculations even with the same input files and parameters. The difference is due to the random initial guess and the iteration procedure of the self consistent field.19

SnSe and silicon

In addition to 2D systems, the proposed strategy can also be applied to bulk systems, for which much more computational resources are usually consumed. Taking bulk SnSe as an example, which is currently the record material for thermoelectric performance due to its ultralow κ.47 As revealed by the harmonic IFCs analysis (Fig. S11 in Supplemental Material), there exist strong interactions at the distance of 6.15 Å in SnSe. Such strong interactions correspond to the sharp drop of κ when the rcutoff is larger than ~6 Å (Fig. S1 in ref. 16 and Fig. 4a in ref. 17). The long-range interactions in SnSe is claimed to be due to the giant phonon anharmonicity caused by the unstable electronic structure, with orbital interactions leading to a ferroelectric-like lattice instability.17 Based on our proposed strategy, one can directly determine the appropriate rcutoff. Then, the convergence test of κ vs. rcutoff can be immediately obtained by fast reconstructing anharmonic IFCs, which confirms the accurate κ at the given rcutoff. The κ of SnSe obtained theoretically agrees very well with experimental reports.16,47 Besides SnSe, we would like to provide more discussions on the limitation of our proposed strategy with another example of the widely studied silicon. Lots of works have been conducted to study the thermal transport in silicon due to its extensive applications in modern electronic industry technologies.10,48,49 It has been widely acknowledged that for silicon including up to 4th nearest neighbors is enough to get converged κ.10 No long-range interaction is reported in silicon. Consequently, our proposed strategy might have limited performance for silicon.

Fig. 4
figure 4

The orbital projected electronic band structures and density of states (DOS) of a graphene and b phosphorene. The p z orbital in graphene is totally independent of other orbitals (s, p x , and p y ), forming the delocalized π-bond. As for phosphorene, the s orbital is mainly confined 9 eV below the valence band maximum (VBM), showing weak hybridization with p x /p y /p z -orbitals. Thus the p electrons are highly delocalized owing to the weak sp hybridization, forming the resonant bonding

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Orbitally driven long-range interactions

We further analyze the origin of the long-range interactions in graphene and phosphorene based on the orbital-projected electronic structures. For graphene, the p z orbital is totally independent of other orbitals of s and p x /p y (Fig. 4a), which form the σ-bond between carbon atoms in the form of sp2 hybridization. The π-bond is solely contributed by the p z orbital, which is shared among carbon atoms in the honeycomb lattice. The electrons forming the σ-bond are localized while the electrons forming the π-bond are highly delocalized. The long-range interactions in graphene, as revealed by the harmonic IFCs analysis (Fig. 2a), is due to the delocalized π-bond. The interactions can also be revealed by the perturbation of charge density distribution induced by the displacement of one specific atom as shown in the inset of Fig. 2a. Thus, only when we choose the rcutoff larger than 6.19 Å (approximately two times the diameter of the hexagon ring) to include the long-range interactions, can the κ of graphene converge (Fig. 2b).

As for phosphorene, the p electrons forming the resonant bond are highly delocalized owing to the weak sp hybridization, as shown in Fig. 4b. The long-range interactions are due to the delocalized resonant bond, which lead to giant phonon anharmonicity associated with the soft transverse optical phonon modes and the low κ.13,50 Such long-range interactions are revealed by the harmonic IFCs analysis (Fig. 3a), which are also evidently shown by the long-range charge density perturbation induced by atomic displacement (inset of Fig. 3a). The perturbation of charge density distribution is particularly extended along the [110] direction, corresponding to the collinear bonding direction of the resonant bond in the hinge-like structure of phosphorene.13

In summary, based on the analysis of harmonic IFCs, we propose a concise strategy to efficiently accelerate the evaluation process of obtaining accurate κ in the framework of phonon BTE. The RMS parameter of the IFCs, which describes the strength of interatomic interactions, is extracted with respect to the interatomic distance to get insight into how the interaction strength changes with distance. Two steps are involved in our strategy. Firstly, the rough rcutoff needed to get satisfactory κ is directly determined based on the RMS parameter. Secondly, the convergence test of κ vs. rcutoff is largely accelerated by fast reconstructing the anharmonic IFCs. Based on the strategy, one can efficiently determine when the κ converges with respect to the cutoff distance and how to get satisfactorily converged thermal conductivity. We successfully apply the strategy to address the long-term divergence problem of thermal transport in graphene and phosphorene. The κ of both systems eventually converges based on our calculations when the orbitally driven long-range interactions are included with a large enough rcutoff. The proposed strategy can also be applied to bulk systems such as SnSe, for which much more computational resources are usually consumed. In addition, the computing speed with our approach could be one order of magnitude faster compared to the traditional method in most situations, where the ~20th nearest neighbors are considered. With the accelerated procedure to evaluate the κ, the computational cost to do with high-throughput screening could be effectively reduced although the reduction is not too much. The quantitative strategy proposed herein would be of great significance for further studies of phonon transport regarding the convergence/divergence problems of κ, and the fast evaluation of the reliable κ is anticipated to be beneficial for other fields such as high-throughput materials screening and design.

Data availability

The data that support the findings of this study and the code for the strategy proposed in this study are available from the corresponding author upon reasonable request. The request can also be sent to the first author, Guangzhao Qin: qin.phys@gmail.com