Supporting Information: Capturing Anharmonicity in a Lattice Thermal Conductivity Model for High-throughput Predictions Samuel A. Miller,† Prashun Gorai,‡,¶ Brenden R. Ortiz,¶ Anuj Goyal,‡,¶ Duanfeng Gao,§ Scott A. Barnett,† Thomas O. Mason,† G. Jeffrey Snyder,† Qin Lv,§ Vladan Stevanovi´c,‡,¶ and Eric S. Toberer∗,‡,¶ †Northwestern University, Evanston, USA ‡National Renewable Energy Laboratory, Golden, USA ¶Colorado School of Mines, Golden, USA §University of Colorado Boulder, Boulder, CO 80309, USA E-mail:
[email protected]
Model Details Semi-empirical Model The thermal conductivity is determined within the semi-empirical model using ground state calculations and structural data from the Inorganic Crystal Structure Database (ICSD). 1 Assuming Umklapp scattering as the dominant scattering source and using the high temperature heat capacity limit, it can be shown that with a simplified Debye-Callaway model, 2 acoustic phonons can be approximated using: 3
κL,ac =
M vs3 (6π 2 )2/3 , 4π 2 T γ 2 V 2/3 n1/3 1
(1)
where M is the average atomic mass, vs the speed of sound, T is the temperature, γ is the Gr¨ uneisen parameter, V the volume per atom, and n is the number of atoms in the primitive cell. In most systems, acoustic phonon modes dominate the lattice thermal conductivity. However, omitting the optical modes produces a model which erroneously predicts that κL asymptotes to zero in materials with large n. To account for optical modes, we assume that their wavelength is related to the mean free path, 4 leading to:
κL,opt
1 3kb π 1/3 vs 1 − 2/3 , = 2 6 V 2/3 n
(2)
where kb is the Boltzmann constant. In the original model definition, the total lattice thermal conductivity is given as the sum of Equations 1 and 2, the Gr¨ uneisen parameter is absorbed into the parameter A1 assuming it is a constant, and A2 is the product of the constants in the optical portion.
Characterization All samples were checked for phase purity using X-ray diffraction. Using a Netzsch LFA 457, thermal diffusivity was measured with the data being fit using a Cowan plus Pulse Correction model. The total thermal conductivity of samples was calculated using:
κ = DρCp ,
(3)
where D is the diffusivity, ρ is the geometrical mass density, and Cp is the heat capacity from the Dulong-Petit approximation. An effective medium theory model 5 was used to correct for porosity in samples that were not fully dense by treating porosity as small spheres dispersed in a uniform matrix. The thermal conductivity of the matrix material is given by
κm =
κc (2κc + κa − 3κa ) , κc (2 − 3P ) + κa
2
(4)
where m, c, and a represent the matrix, composite, and air, respectively, and P is the porosity. Non-oxide samples were at least 95% dense and oxide samples were greater than 85% of theoretical density.
Computational Details The density functional theory (DFT) calculations were performed with the plane-wave VASP code 6 using the generalized gradient approximation of Perdew-Burke-Ernzerhof (PBE) approach 7 within the projector augmented wave (PAW) formalism. 6 The calculations were performed within the Pylada framework 8 using our previously described approach for structure relaxations. 9 A suitable on-site correction in the form of Hubbard U in the rotationallyinvariant form 10 was applied for transition metals. For compounds containing transition metals, we performed a limited search for the magnetic ground state by enumerating over all possible magnetic orders on a primitive cell. Because the number of configurations scales as 2N with number of magnetic atoms (N ), we limited the total number of configurations to 32, including the ferromagnetic order. The exchange correlation functionals such as LDA 11 and GGA 7 that are routinely used in computational materials screening do not account for van der Waals (vdW) interactions, which are especially important in quasi-2D materials such as SnSe and SnS. It is well known that using LDA or GGA leads to overbinding and underbinding of the quasi-2D layers in these materials, respectively, and consequently, overestimation and underestimation of B, respectively. The underestimation in GGA is especially severe; in some cases, the value of B can be underestimated by an order of magnitude compared to experiments. For instance, B calculated with GGA for MoTe2 is ∼5 GPa while the experimental value is ∼40 GPa. As a consequence, calculated κL with GGA (0.3 Wm−1 K−1 ) is more than an order of magnitude smaller than the actual value (4 Wm−1 K−1 ). 12 To account for long-range vdW interactions, we employed the optB86 vdW-corrected exchange correlation functional, 13 which significantly improves predictions of κL . 14 Structural relaxations are performed with the 3
vdW-corrected functional using a plane-wave cutoff of 400 eV. As seen in Figure S1, it is necessary to use van der Waals corrected functionals to accurately compute the bulk modulus and thus lattice thermal conductivity of layered materials. 1
1 0
In I S n O In S e B i2 T e 3
2
0
1 0
M o d e le d
κL
(W /m K )
M o T e
1 0
-1
1 0
-1
1 0
E x p e r im e n ta l
0
1 0
κL
1
(W /m K )
Figure S1: Experimental lattice thermal conductivity shown against predicted κL with standard functionals (filled symbols) vs. van der Waals-corrected functionals (open symbols) using the original model from Yan et al. 15
Statistical Coefficients The factor difference between predicted and experimental values is of particular interest in evaluating how close predictions are to experimental values. We used a log scale such that for a given experimental value of 100, predictions of 50 or 200 were both a factor of 2 from the “true” value (likewise predictions of 0.5 and 2 for experimental measurement of 1 were both off by a factor of 2 as well). To assess whether the improved model was predicting κL in a linearly increasing relationship with experimental values, the Pearson and Spearman rank correlation coefficients were calculated. The Pearson correlation coefficient is a linear correlation between two variables and is defined as the covariance over the product of the two standard deviations. This was used to measure the degree of linear dependence between the variables and can range from 4
±1, inclusive, with 0 indicating no correlation, -1 total negative correlation, and 1 total positive correlation. The Spearman rank correlation can be used instead of the usual Pearson one to assess the relationship between the ordinal ranking of the predicted and experimental κL . This is determined by sorting the raw variables in ascending order and assigning a rank to each (using the average position when there is more than one variable with the same value) according to their position in the sorted list. The Spearman rank correlation is then calculated using the same covariance over standard deviation product definition, except with the rank rather than the raw values. Though the same range and interpretations apply, the Spearman rank correlation is used to determine how well the ranking order of one variable predicts that of the other. Both the Pearson and Spearman rank correlation coefficients were used to compare the present model with that of other high-throughput predictions. 16,17
Model Dataset New Measurements of Compounds The κL model was weighted toward common thermoelectric materials and binary compounds with simple structures and must be validated outside this set. Eight compounds were synthesized and their thermal conductivity was measured for the first time in bulk polycrystalline form in the present study. The synthesis procedure is described in detail below for each compound. • Ba3 In2 O6 : Stoichiometric BaCO3 and In2 O3 were ground in agate mortar and pestle. Powders were pressed into 1⁄2 inch diameter pellets using uniaxial press and steel die at 200 MPa. Pellets were fired for 18 hours at 1000°C, ball milled in a tungsten carbide vial for 1 hour, pressed again into pellet form, and fired at 1350°C for 36 hours. • Ba2 SnO4 : Stoichiometric BaCO3 and SnO2 were ground in agate mortar and pestle. Powders were pressed into 1⁄2 inch diameter pellets using uniaxial press and steel die at 5
200 MPa. Pellets were fired for 18 hours at 1200°C, ball milled in a tungsten carbide vial for 1 hour, pressed again into pellet form, and fired at 1425°C for 24 hours. • Cu3 TaTe4 : A stoichiometric mixture of Cu granules, Ta and Te chunks was ball milled in a stainless steel vial. The resulting powder was annealed at 600°C for 18 hours followed by hot pressing at 600°C with 3 hour densification followed by 18 hours stress free anneal. • Cu2 ZnSiTe4 : A stoichiometric mixture of Cu granule, Zn powder, Si and Te chunk was ball milled in a tungsten carbide vial. The resulting powder was annealed at 615°C for 72 hours followed by hot pressing at 554°C with 3 hour densification and 18 hour stress free anneal. A small amount (
