Predicting accurate phonon spectra: An improved description of the

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Predicting accurate phonon spectra: An improved description of the lattice dynamics in thermoelectric clathrates based on the SCAN meta-GGA functional Holger Euchner, and Axel Gross Chem. Mater., Just Accepted Manuscript • DOI: 10.1021/acs.chemmater.9b00216 • Publication Date (Web): 18 Mar 2019 Downloaded from http://pubs.acs.org on March 21, 2019

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Chemistry of Materials

Predicting accurate phonon spectra: An improved description of the lattice dynamics in thermoelectric clathrates based on the SCAN meta–GGA functional Holger Euchner∗,† and Axel Groß†,‡ †Helmholtz Institut Ulm, Helmholtzstr. 11, D-89081 Ulm, Germany ‡Institute of Theoretical Chemistry, Ulm University, Albert-Einstein-Allee 11, D-89069 Ulm, Germany E-mail: [email protected]

Abstract Due to their thermoelectric properties, intermetallic clathrates are technologically highly interesting materials. Especially their strongly reduced lattice thermal conductivity has been largely investigated and strongly debated. While density functional theory based lattice dynamics calculations have helped clarifying the thermal transport in clathrates, significant discrepancies remain between experiment and theory, especially in Ge-based clathrates. In this work, we show that the recently released meta-GGA functional SCAN is able to overcome these issues and provides much improved agreement between experimental and theoretical phonon spectra, thus enabling quantitative predictions of clathrate phases with high potential for thermoelectric applications.

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Introduction Due to the increasing importance of the efficient use of energy, the search for thermoelectric materials has recently been an active field of reasearch. A low lattice thermal conductivity is one of the prerequisites for a good thermoelectric material, and consequently phonon engineering has been found to be a very efficient strategy for developing thermoelectric materials . 1–3 Intermetallic clathrates combine a low lattice thermal conductivity with a high electron mobility, thus making them interesting materials for thermoelectric applications . 4 One of the peculiarities of clathrates is their complex crystal structure, consisting of cage like cavities which form a covalently bound framework (typically with the group IV elemts Ge, Si or Sn being the main constituents). Usually, alkaline or alkaline earth metals are entrapped inside these cavities . 4 The crystal structure of such a type–I clathrate with the typical composition Ba8 Ge40 TM6 is depicted in Fig. 1. In spite of the fact that Si and Ge are the main constituents, the lattice thermal conductivity in Si- and Ge- based intermetallic clathrates is drastically decreased as compared to diamond structured Si or Ge. While the origin of the low lattice thermal conductivity in clathrates was debated essentially since its discovery, recent studies were able to identify the underlying microscopic mechanisms. Experimental and computational studies of the lattice dynamics of clathrates could show that a hybridization of acoustic modes with low lying optical rattling modes, results in a reduction of the acoustic bandwidth, concomitant with a reduction of the sound velocity . 5–8 The evidenced low energy optical phonons arise from the vibrations of the entrapped guest atoms which in clathrates or skutterudites, have been found to coherently couple with the host framework . 5,6,9 The hybridization of the acoustic modes with these low energy optical modes goes along with the simultaneous bending of the linear acoustic dispersion and corresponding changes in the spectral weight of these phonon modes, which has been shown to be essentially harmonic in nature . 5,6 These findings were further corroborated by an experimental study that showed the thermal transport being dominated by phonons with long mean free path, extending to at least 20 unit cells at 300 K. 10 2


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Beyond the crystal complexity, the decreased lattice thermal conductivity in clathrates can be attributed to two main factors. First, it was shown that Si–based clathrates exhibit an altered interaction potential due to charge transfer from encapsulated Ba atoms to the Si46 framework . 7 This then leads to a reduction of the sound velocity and a depression of the acoustic regime. Filling of the cages results in a further decrease of the sound velocity and moreover shifts the acoustic limit to lower energies. In addition, occupying the Ba(6d) site results in a significantly increased anharmonicity in the low energy limit . 7 Together with the increasing phase space, which is a consequence of the opening of further scattering channels due to flat Ba modes, this results in shorter phonon lifetimes, especially of the acoustic modes. Despite the qualitative agreement of lattice dynamics simulations with measured phonon dispersions and phonon densitiy of states , 5,6,10–12 Ge–based clathrates tend to show significant discrepancy between experimental and calculated slope of the acoustic dispersion (see Ba8 Ge40 Ni6 , Ba8 Ge40 Au6 , Ba8 Ge40 Zn6 ). In general, ab initio lattice dynamics studies of Ge– clathrates show clearly decreased acoustic and rattling mode frequencies, when compared to experimental data. As GGA calculations are known to overestimate the lattice parameter, somewhat decreased phonon frequencies are indeed to be expected. However, while for most crystalline materials the differences in the acoustic frequencies typically amount to a few percent, in Ge-based clathrates a huge discrepancy for the acoustic modes was observed (up to 40%). In previous studies this issue was solved by either introducing a scaling factor for the acoustic modes or by rescaling the lattice parameter . 5,6,13,14 Both solutions are not ideal as they point to some shortcomings of the calculations and, moreover, make true first principle predictions rather difficult. Interestingly, while this discrepancy is present for Gebased clathrates, in Si-based Ba8 Si46 phases this problem has not been observed to the same extent. In the following, we present results from lattice dynamics calculations obtained with the recently developed SCAN (strongly constrained and appropriately normed) functional

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and show that this indeed cures the observed mismatch for Ge–based clathrates. SCAN is a

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semi–local meta–GGA functional that is constructed in a way to fullfil all exact constraints that are known for this type of functional . 15 Due to this construction scheme SCAN allows for an improved predictive accuracy comparable to computationally much more expensive hybrid functionals. 16 Moreover, it has been shown that SCAN is able to describe complex bonding scenarios with different bonding characteristics – such as the loosely bound guest atoms in the covalent host matrix of intermetallic clathrates – in a much improved manner. 16 Consequently, the SCAN functional seems an ideal candidate for investigating the lattice dynamics in thermelectric clathrates.

Computational Methods In this work, we have used the periodic density functional theory (DFT) code VASP 17,18 for structure optimization as well as for the determination of the harmonic force constants necessary for the investigation of lattice dynamical properties. The projector augmented wave (PAW) method 19 was used to account for electron–ion interaction, while exchange and correlation in the investigated materials are accounted for by the general gradient approximation in the formulation of Perdew, Burke and Enzernhof (PBE)

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and the meta–GGA

SCAN. 15 All structures were optimized with respect to volume and atomic position, using an energy cutoff of 500 eV and a 5×5×5 Γ–centered k–point mesh, using a convergence criterion of residual forces of less than 0.1 meV/˚ A. Identical settings were then used for the determination of the harmonic forces, which were obtained via the finite displacement method. The harmonic force constants were extracted from the DFT calculated Helmann–Feynman forces by displacing selected atoms by 0.03 ˚ A out of their equilibrium along symmetry non– equivalent directions. The dynamical matrix was then constructed from the harmonic forces, and the phonopy code

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was used to calculate dispersion curves and phonon density of states

(DOS) of the different compounds. In addition to the calculations using the PBE and the SCAN functional, we have fur-

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thermore selected the Ba8 Ge40 Au6 compound to seek comparison with other exchange– correlation functionals. Apart from the local density approximation (LDA) we have also investigated the potential impact of van der Waals interactions. Since the importance of van der Waals interactions of the encaged Ba atoms with the surounding network was suggested to possibly be of importance, we have investigated two types of including it. The optPBE functional , 22 which is a van der Waals functional and the so–called Grimme–D3–correction , 23 which is a computationally simple extenstion that has been shown to improve the description of materials properties . 24 Finally, it has to be pointed out that the SCAN functional, being a meta–GGA, is computationally more expensive than a standard PBE calculation but still at comperably moderate cost (same order of magnitude).

Results In Fig.

2 the phonon dispersion curves of Ba8 Ge40 Au6 , Ba8 Ge40 Ni6 and Ba8 Si46 , as ob-

tained from calculations using the SCAN functional, are compared to experimentally available data for compounds with Ba7.81 Ge40.67 Au5.33 , Ba8 Ge40 Ni6 and Ba7.5 Si46 , Ba8 Ge42.1 Ni3.5 stoichiometries . 5,6,10 The agreement of the transverse acoustic branch is almost perfect in case of Ba8 Ge40 Au6 and Ba8 Si46 , while it is only slightly too low for Ba8 Ge40 Ni6 . The slight deviations in case of Ba8 Ge40 Ni6 may be due to the experimentally somewhat different stoichiometry corresponding to Ba8 Ge42.1 Ni3.5 . Similarly, the low–lying rattling modes fall in the correct frequency range for all three compounds. To further evaluate the quality of the results, we have calculated the dynamical structure factor S(Q, ω), which corresponds to the spectral weight of a given phonon mode and can be accessed by inelastic X-ray or neutron scattering (see colour code in Fig 2). In all three cases the experimental data points fall on the regions where high S(Q, ω) shows high intensity, indeed further corroborating the agreement of the dispersion curves. It should be noted that S(Q, ω) is a quite sensitive test, as in principle good dispersion curves are no guarantee for correct eigenstates, whereas S(Q, ω) is

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contstructed from phonon eingenvectors. Thus, SCAN yields very satisfying agreement with the experimental dispersion curves. Now, the phonon dispersion curves of several type–I clathrates are depicted, showing a direct comparsion of results obtained by using SCAN (black) and PBE (red), respectively. While for Cu– and Zn–containing clathrates the differences are less pronounced, for Au–, Ag– and Ni–clathrates we find the TA mode to be tremendeously lower in the PBE calculation. In order to shed some light on the observed discrepancies, below we focus on the Ba8 Ge40 Au6 system. Apart from PBE and SCAN calculations, we have investigated the dispersion curves obtained with other approximations of the exchange–correlations interaction. Hence, we are showing dispersion curves for Ba8 Ge40 Au6 which were obtained by LDA, optPBE and PBED3 calculations to see how they perform with respect to the frequency rescaling. In Fig. 4 each panel shows the comparison of SCAN with one of the other functionals. The two dispersion curves that are obtained by accounting for van der Waals interaction are depicted in panels 4c) and 4d). We clearly see that both types of treating the van der Waals interaction do not significantly improve the dispersion curves. Consequently, we conclude that the van der Waals interactions in Ge–clathrates cannot be the origin of the differences with respect the to experimental data. Moreover, in panel 4a) we compare the SCAN functional with results from LDA calculations. As expected, the LDA results show slightly increased frequencies (also for the TA mode), which is a well–known finding and can be attributed to overbinding in the local density approximation. However, while the LDA calculations almost match the SCAN results for the LA modes, the TA modes are still significantly too soft. To ensure that the improvement of the acoustic dispersion is not on the expense of the high energy end of the vibrational spectrum, we have investigated the phonon DOS for the different functionals (see Fig. 4e). Interestingly, the SCAN functional clearly shows a shift of the low energy part of the spectrum towards higher energies (especially the acoustic and Ba–dominated optic modes are clearly improved), while the high energy part is less affected. In fact, the upper cutoff of the SCAN calculation almost exactly matches the PBE

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and is close to the LDA frequencies. This also clearly points to the shortcomings of applying a simple rescaling of the energy to e.g. the PBE results for a better matching of the low energy modes. Looking in detail into the optimized structures, we indeed see that, apart from LDA and SCAN, the lattice parameter is overestimated. However, the overestimation of the lattice parameter is not so significant that it would explain the full differences. Here it should be noted that the LDA calculation, despite having the lowest lattice parameter, still yields significantly softened TA modes. As a side note, it has to be pointed out that the lattice parameter obtained for the SCAN functional is very close to the experimental value for all investigated compounds (see Table 1 and 2). To further quantify the impact of the lattice parameter we have determined the dispersion curve for both PBE and SCAN, using the SCAN optimized lattice parameters. Indeed, the rescaling to the SCAN lattice parameter improves the PBE calculation, however, differences with respect to the TA branch remain, thus clearly pointing to the shortcomings of a rescaling of the lattice parameter (see Fig. 4b). Interestingly, the high–symmetry sites are not changed and thus only the Ge–sites slightly adapt their positions. Furthermore, the only significant change is associated with the Ge–Au bondlength which is about 0.025 ˚ A shorter when using the SCAN functional, however, still corresponds to a rather small modification. Thus, the observed differences do not stem from significant structural changes. While our findings clearly indicate a much improved predictability achieved by employing the SCAN functional, it remains to elucidate why the other functionals show these significant problems. Before addressing this issue we have a look at panel f) of Fig. 3. There, the comparison of SCAN and PBE for Ba8 Si46 is depicted. The fact that for the binary transition metal (TM) free Si clathrate, the agreement of both functionals is rather good – in fact the differences can essentially be attributed to the slightly varying lattice parameter – points to the TMs being at the origin of the observed differences. This makes us conclude that the origin of the discrepancies are related to the TM and consequently might be visible in the

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electronic structure. Therefore, we have derived the electronic density of states (DOS) for Ba8 Ge40 Au6 and Ba8 Si46 , again using both the SCAN and the PBE functional, as depicted in Fig. 5. Surprisingly, for both cases we find an electronic DOS that looks qualitatively very similar when SCAN and PBE are compared, however, the states are shifted towards lower energies for the SCAN calculation. Yet, this shift can be essentially accounted for by the differences in lattice parameter, meaning that the DOS is not significantly different. This means that neither crystal nor electronic structure differ significantly when SCAN and PBE are compared. Instead, the differences in the phonon dispersion seem to be a direct consequence of the exchange correlation energy and changes in the resulting energy landscape. Table 1: Experimental and calculated lattice parameter (in ˚ A) for Ba8 Ge40 Au6 Exp PBE-D3 10.80 10.90

optPBE LDA PBE SCAN 11.09 10.72 10.97 10.78

Table 2: Comparison of experimental and calculated (PBE and SCAN) lattice parameter for the other clathrate phases studied.

Exp PBE SCAN

Ba8 Ge40 Ni6 10.68 25 10.78 10.62

Ba8 Ge40 Zn6 10.75 26 10.92 10.75

Ba8 Ge40 Cu6 10.69 27 10.82 10.65

Ba8 Ge40 Ag6 10.84 28 11.01 10.83

Ba8 Si46 10.33 5 10.40 10.33

Discussion To summarize, we have shown that the recently released SCAN functional is able to reproduce experimentally observed dispersion curves of clathrates with much improved accuracy. Thus the so far applied empirical techniques to match experiment and theory – either rescaling phonon frequencies or fixing the lattice parameter to the experimental value – can be replaced by a truly first principles approach. This is especially important when computational approaches are used to predict material properties. Consequently, we hope that using 8


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the SCAN functional may allow for an improved search for materials with low lattice thermal conductivity.

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Figure 1: Structure of type–I clathrates. Structure of the type–I clathrate Ba8 Ge40 TM6 , with face sharing dodecahedral BaGe20 and tetrakaidecahedral BaGe20 TM4 cages as building units. Ge atoms are depicted in blue, Ba atoms in red and TM atoms in green.

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a)

b)

c)

Figure 2: Experimental and theoretical phonon dispersion curves. Comparison of results obtained with the SCAN functional with experimentally available data. Panels ac show the dispersion curves (black solid lines) for Ba8 Ge40 Au6 , Ba8 Ge40 Ni6 and Ba8 Si46 , connecting the high symmetry points Γ–X–M–Γ–R. In addition, the calculated dynamical structure is depicted as color coding. Experimental data (diamonds and squares) are extracted from . 5,6,10 Experimental data and S(Q, ω) are determined around the (6,0,0) Bragg reflection. The vertical dashed lines indicate the positions of the high symmetry points.

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a)

b)

c)

d)

e)

f)

Figure 3: Calculated phonon dispersion curves. Comparison of results obtained with the SCAN (black) and the PBE (red) functional, respectively. All panels show the phonon dispersions along the same pathway, connecting the high symmetry points Γ–X–M–Γ–R. The top panels a-c show the dispersion curves of Ba8 Ge40 Ni6 (left), Ba8 Ge40 Zn6 (middle) and Ba8 Ge40 Cu6 (right), respectively. The bottom panels d-e depict the dispersion for Ba8 Ge40 Ag6 , Ba8 Ge40 Au6 Ba8 Si46 (right). The vertical dashed linesindicate the positions of the high symmetry points.

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a)

b)

c)

d)

e)

Figure 4: Phonon dispersion in Ba8 Ge40 Au6 . All panels show the phonon dispersion for different exchange–correlation functionals calculated along the Γ–X–M–Γ–R pathway. The top panels show a comparison of a) SCAN with LDA and b) SCAN with PBE for the same lattice parameter. Panels c-d depict the comparison of SCAN to functionals including van der Waals interactions, namely c) vdW–D3 and d) optPBE. In e) the phono DOS in Ba8 Ge40 Au6 for the different functionals is shown. 16


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a)

b)

Figure 5: Electronic DOS. DOS as obtained using the SCAN (red) and the PBE (blue) exchange-correlation functional for a Ba8 Ge40 Au6 and b Ba8 Si46 , respectively. The areas filled in pink and cyan in panel a indicate the transition metal d–states, the dashed vertical lines mark the Fermi energy. 17


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Predicting accurate phonon spectra: An improved description of the

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