ARTICLE pubs.acs.org/JPCC
Mechanical Properties and Low Elastic Anisotropy of Semiconducting Group 14 Clathrate Frameworks Antti J. Karttunen,* Ville J. H€ark€onen, Mikko Linnolahti, and Tapani A. Pakkanen Department of Chemistry, University of Eastern Finland, P.O. Box 111, FI-80101 Joensuu, Finland
bS Supporting Information ABSTRACT: We have investigated the mechanical properties of semiconducting clathrate frameworks composed of group 14 elements carbon, silicon, germanium, and tin. The bulk moduli, Young’s moduli, and elastic anisotropies of 13 structurally different clathrate frameworks were determined using quantum chemical methods. The predicted elastic properties were compared to the properties of diamondlike, dense α-phases and experimentally known semiconducting group 14 clathrate structures. The predicted bulk and Young’s moduli of the studied carbon, silicon, germanium and tin frameworks suggest them to possess low compressibility and high stiffness, which are almost comparable to the diamondlike, dense α-phases of the elements. In particular, the studied microporous clathrate frameworks exhibit remarkably low elastic anisotropy, being clearly more isotropic than the denser α-phases.
’ INTRODUCTION The elastic properties of all known single crystals are anisotropic, that is, their elastic moduli depend on the orientation of the crystal.1,2 Since most materials exist anyway as polycrystalline aggregates,3 the anisotropy of the single-crystalline structures is not usually an issue for the design of practical applications. However, in some areas, such as the engineering of advanced semiconductor devices, the elastic anisotropy has to be taken into account during the device design,4 because the most commonly utilized semiconductor materials such as single-crystalline α-silicon and α-germanium do show elastic anisotropy at the atomic level.1 The most isotropic materials known to date are metals, such as α-tungsten.5 Furthermore, quasicrystalline materials have been predicted to reach even lower elastic anisotropies than conventional singly crystalline materials,6 and the quasicrystalline AlCuLi has actually been shown to be even more isotropic than α-tungsten.7 The group 14 elements play a key role in the semiconductor industry, and any new structural modifications resulting in electronic, optical, or elastic properties different from the bulk α-phases are of great interest. In particular, the modification of the elastic properties of the group 14 element based semiconductor materials toward lower elastic anisotropy could facilitate the engineering of new semiconductor devices. Semiconducting group 14 clathrates, which are three-dimensional microporous frameworks composed of fused, polyhedral atomic cages,8 10 are one promising example of structural modifications completely different from the bulk α-phases. Semiconducting clathrate frameworks composed of group 14 atoms are normally occupied by guest atoms (Figure 1), the most typical guest atoms in the cavities being alkali, earth-alkaline, and halogen atoms. Because of their structural and electronic characteristics, the guest-occupied semiconducting clathrates have high application potential as efficient thermoelectric materials.11 However, in addition to the r 2011 American Chemical Society
guest-occupied clathrate frameworks, almost empty, “guest-free” Si and Ge clathrates of the structure type II are known as well.12 14 In the case of carbon, the intercalation/deintercalation processes involved in the synthesis of the guest-free frameworks have been recently shown to require extreme conditions.15 In addition to being promising materials for optoelectronics or photovoltaics,13,16 the guest-free clathrate frameworks are also new allotropes of silicon and germanium. The structural characteristics of the microporous clathrate allotropes are very different from the diamondlike, dense α-phases, and their electronic and mechanical properties are also different from the α-phases. The majority of the experimental and theoretical work concerning the elastic properties of semiconducting group 14 clathrates has focused on the two most common structure types known as type I and II. For example, silicon clathrate framework II has been shown to possess very low compressibility.17 Moreover, the bulk moduli of the empty clathrate frameworks of type I, II, and IV have been predicted theoretically for all group 14 elements,18 and the elastic properties of hypothetical clathrate-I frameworks composed of carbon have been investigated in more detail both for empty19 and guest-occupied structures.20 The elastic properties of the guest-occupied Ba8Ga16Ge30 and Sr8Ga16Ge30 clathrates have been also determined experimentally.21,22 We have recently elucidated the structural characteristics and electronic properties of a large number of group 14 clathrate frameworks.23 Understanding the building principles of the semiconducting group 14 clathrate frameworks greatly facilitates further comprehensive studies on their properties, and here we investigate their elastic properties by using quantum chemical methods and a recently developed computational approach for Received: June 17, 2011 Revised: August 8, 2011 Published: September 07, 2011 19925
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different crystal directions were calculated from the elastic compliance constants using the expressions derived previously31 (see Supporting Information for details).
Figure 1. The general structural motif of the group 14 semiconducting clathrates. The clathrate I structure illustrated here is composed of fused 20-membered (blue) and 24-membered (red) atomic cages. The cages are filled with guest atoms as illustrated by the transparent cages in the front right corner. Unit cell edges are drawn in black.
facile ab initio calculation of second-order elastic constants of solid-state materials.24 We determine the bulk moduli, Young’s moduli, and elastic anisotropies of 13 different clathrate frameworks and compare the predicted elastic properties to the properties of diamondlike, dense α-phases and experimentally known group 14 clathrate structures.
’ COMPUTATIONAL DETAILS The mechanical properties of the semiconducting clathrate frameworks were investigated using the PBE0 hybrid density functional25,26 and localized atomic basis sets composed of Gaussian-type functions. All calculations were performed using the CRYSTAL09 software package.27,28 In periodic calculations, the choice of the Gaussian-type localized atomic basis set requires careful consideration. Basis sets originally developed for molecular calculations contain diffuse basis functions to model the tails of wave function, but in periodic calculations, where the whole space is filled with basis functions, such diffuse functions are usually unnecessary and can result in numerical instability and severe degradation of performance.29 The splitvalence + polarization (SVP) level basis sets applied here were taken from our previous study, where they were shown produce results that were in good agreement with state-of-the-art projector augmented wave (PAW) DFT calculations.23 The fully optimized clathrate framework geometries and the shrinking factors (SHRINK) used for generating a Monkhorst-Pack-type30 grid of k-points in the reciprocal space were also taken from the previous study (the unit cell coordinates and k-point grids of the structures are available in the Supporting Information of ref 23). For the evaluation of the Coulomb and exchange integrals (TOLINTEG), tight tolerance factors of 8, 8, 8, 8, and 16 were used. The elastic constants of the structures were evaluated using the ELASTCON scheme implemented in CRYSTAL09.24 Default convergence thresholds for the ELASTCON scheme and an extra large integration grid (XLGRID) for the density-functional part were applied in the calculations. The Young’s moduli in
’ RESULTS AND DISCUSSION The studied group 14 semiconducting clathrate frameworks are illustrated in the Figure 2. The structures are composed of polyhedral building blocks and can be classified into basic frameworks and extended frameworks based on larger icosahedral building blocks, that is, Ih-clathrates. The structural principles of the clathrate frameworks have been investigated in detail previously.23 The previous study showed the clathrate II framework to be the energetically most favorable for all elements, in agreement with experimental results. Hexagonal polytypes of clathrate II (clathrates V and II-4H) were also found to be practically as stable as clathrate II. For Si, Ge, and Sn, the other frameworks were also found to be energetically rather close to the framework II, except for the frameworks VI and VII, which are less stable due to the presence of a large number of strained, fourmembered rings. The bulk moduli of the semiconducting Group 14 clathrate frameworks shown in Figure 3 clearly illustrate two general trends (for tabulated data, see Supporting Information). As can be expected from the properties of the elements, the absolute values of the bulk moduli decrease noticeably when moving from carbon to tin. Furthermore, the bulk moduli are relatively similar for all clathrate frameworks, especially when considering the density-normalized bulk moduli, which have been normalized with respect to the density of the diamondlike α-phase. The cubic VI and VII frameworks that have the smallest bulk moduli are also the most strained frameworks in terms of total energies.23 In addition to the data shown in Figure 3, we also investigated the periodic trends within the calculated bulk moduli by plotting them with respect to the average interatomic distance in each clathrate framework. The advantage of using the interatomic distance as the parameter is that the clathrate frameworks and α-phases for all studied group 14 elements can be included in the same plot. The results shown in Figure 4 demonstrate clearly how the bulk modulus decreases exponentially with the increasing group 14 element interatomic distance. The calculated bulk moduli demonstrate that the empty group clathrate frameworks with the largest bulk moduli show almost similar resistance to uniform compression as the dense diamondlike phases. The very low compressibility of the silicon clathrateII framework has been demonstrated experimentally, and the bulk modulus calculated here (86 GPa) is in agreement with the experimental value (90 ( 5 GPa).17 The bulk moduli obtained here for the frameworks I, II, and IV that are most often discussed in the literature are in agreement with the previous theoretical studies,18 notably, earlier theoretical predictions on guest-occupied Si clathrate-I compounds such as I8@Si46 have suggested that suitable guest atoms should make the bulk modulus of the clathrate framework similar to the α-silicon.20 In the case of the empty Ge clathrate-I framework studied here, the calculated bulk modulus (64 GPa) is actually very similar to the experimentally determined bulk moduli for the Ba8Ga16Ge30 (65 GPa) and Sr8Ga16Ge30 (64 GPa) clathrates that include guest atoms and framework heteroatoms.32 Similarly, the calculated bulk modulus of the empty silicon clathrate-I framework (86 GPa) is close to the reported experimental value of the filled K8Si46 clathrate (86 ( 5 GPa).18 However, the bulk 19926
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Figure 2. Studied group 14 clathrate frameworks and their building blocks. For each clathrate structure, the label shows the space group and the composition in terms of the polyhedral cages. Notation for the polyhedral cages: [51262] = A cage with twelve five-membered rings and two sixmembered rings.
moduli of the clathrates can also be strongly affected by the guest atoms, as shown by the notably lower bulk modulus of 61.4(7) GPa obtained for the partially filled Rb6.15Si46 clathrate I structure.33 Interestingly, Toulemonde et al. have very recently measured the bulk modulus of the type IX clathrate Ba24S100 to be only 65 GPa, clearly smaller than the empty Si-frameworks investigated here.34 As discussed by Toulemonde et al., the clathrate framework IX includes distorted M24f20 cages with four vacancies, leading to a number of three-coordinated group 14 atom positions. This lack of atoms in the covalently bonded group 14 atom framework is expected to have a significant effect on the mechanical properties of the structure.
The calculated Young’s moduli of the semiconducting group 14 clathrate frameworks in the basic crystal directions are shown in Figure 5 (for tabulated data, see Supporting Information). The difference between the Young’s moduli obtained for carbon and the heavier group 14 congeners is larger than in the case of bulk moduli, highlighting the structural stiffness of the carbon sp3 frameworks. For all studied group 14 elements, the diamondlike α-phase has the largest Young’s moduli, although some clathrate frameworks composed of Si, Ge, and Sn are actually stiffer than the α-phase in its [100] crystal direction. For all studied group 14 elements, the extended clathrate frameworks based on the large icosahedral building blocks (Ih clathrates23) show the largest 19927
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Figure 4. Bulk moduli of the semiconducting group 14 clathrate frameworks plotted as a function of the average bond distance (note the logarithmic y axis). R2 for the fit is 0.985.
Figure 3. Bulk moduli of the semiconducting group 14 clathrate frameworks. The density-normalized values were obtained by scaling the bulk moduli for porous clatrate phases with the factor Fdiamond/ Fclathrate, where the Fdiamond and Fclathrate are the densities of the diamondlike α-phase and the clathrate phase, respectively. The scale of the Y axis is different for carbon in comparison to the other elements.
Young’s moduli. The present Ih clathrates are structurally related to the previously studied icosahedral diamondoids35 and polyicosahedral diamond nanowires, the latter of which have actually been shown to possess higher Young’s moduli than conventional diamond nanowires.36 The Young’s moduli of the clathrate frameworks have not been discussed in the literature nearly as often as their bulk moduli. The polycrystalline Young’s moduli for the guest-occupied Ba8Ga16Ge30 and Sr8Ga16Ge30 clathrates have been determined to be 110 and 91 GPa, respectively.32 Hence, the Young’s moduli of the Ba- and Sr-occupied species
differ much more from each other than their bulk moduli. For comparison, the polycrystalline Young’s moduli calculated here for the guest-free germanium clathrate-I is 107 GPa. Clear differences in other kind of properties of Ba- and Sr-occupied clathrates have also been discovered previously. In the case of silicon, studies on the superconductivity of (Ba1 xSrx)8Si46 structures showed the superconducting transition temperature Tc to decrease as the function of Sr content.37 In this case, the observations could be explained by the lower contribution of the Sr 4d electrons to the density of states at the Fermi energy in comparison to Ba 5d electrons. Similarly, the differences in the Ba/Sr guest framework interactions are also expected to play a role in the observed variation in the Young’s moduli of Ba8Ga16Ge30 and Sr8Ga16Ge30. Detailed investigation of the direction dependence of the calculated Young’s moduli reveals an interesting difference between the clathrate frameworks and the diamondlike α-phase. While Young’s moduli in different crystal directions can vary up to 30% for the α-phases, several clathrate frameworks possess practically equivalent Young’s moduli in all studied crystal directions. For example, the cubic clathrate frameworks II and II-100 show very little anisotropy in the studied [100], [110], and [111] directions for all four studied elements. The very low elastic anisotropy of the clathrate frameworks suggests that they could turn out to be elastically isotropic single-crystalline semiconductor materials. Because of the low direction dependence of the calculated Young’s moduli, we investigated the elastic anisotropy of the clathrates frameworks in more detail. Several different measures for the elastic anisotropy of single crystals have been proposed in the literature, the most recent one being the universal anisotropy index (AU).1 AU is a single-valued anisotropy measure calculated using the bulk and shear moduli of both Voigt and Reuss (for isotropic materials, AU = 0).3 Notably, AU is applicable to all kinds of crystal systems from triclinic to cubic, enabling convenient comparisons between the cubic, hexagonal, and tetragonal clathrate frameworks studied here. The calculated AU values for the group 14 clathrate frameworks are listed in Table 1. For several frameworks, the calculated AU values were found to be zero within the accuracy of the present computational approach, suggesting them to exhibit remarkably low elastic anisotropy for a semiconducting material. Numerically more 19928
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Table 1. Universal Anisotropy Index AU for the Studied Group 14 Clathrate Frameworks structure
C
Si
Ge
Sn
α (diamond)
0.057
0.238
0.309
0.478
I II
0.005 0.000
0.012 0.001
0.004 0.000
0.002 0.000
VI
0.047
0.056
0.084
0.089
VII
0.022
0.071
0.196
0.110
VIII
0.012
0.011
0.013
0.011
I-100
0.001
0.000
0.001
0.003
II-100
0.001
0.000
0.000
0.000
cubic clathrates
hexagonal clathrates IV
0.002
0.007
0.003
0.003
V H
0.000 0.004
0.000 0.006
0.001 0.004
0.001 0.004
II-4H
0.001
0.000
0.000
0.000
IV-100
0.001
0.000
0.002
0.005
0.002
0.001
tetragonal clathrates III
Figure 5. Young’s moduli of the semiconducting group 14 clathrate frameworks in different crystal directions. For the hexagonal frameworks, the [100] and [110] crystal directions are equivalent due to the properties of the hexagonal crystal system. The scale of the Y axis is different for carbon in comparison to the other elements.
accurate calculations would be required to conclude the materials to be truly isotropic, since the AU values of tungsten5 and quasicrystalline AlCuLi7 are still several orders of magnitude smaller than the smallest AU values here. In any case, the elastic properties of the present group 14 clathrate frameworks are several orders of magnitude less anisotropic in comparison to the dense, diamondlike α-phases. The most isotropic phases II, II-100, V, and II-4H showing very small elastic anisotropy for all four studied elements are all structurally related, the V and II-4H phases being hexagonal polytypes of the II phase.23
0.002
0.006
Almost all semiconducting clathrates known to date include guest atoms and heteroatoms within the frameworks, which are expected to change the elastic properties in comparison to the empty frameworks studied here. For comparison, we also calculated the AU values for the experimentally known Ba8Ga16Ge30 and Sr8Ga16Ge30 clathrate-I structures based on the experimentally determined elastic constants.32 The AU values of the Ba8Ga16Ge30 (AU = 0.091) and Sr8Ga16Ge30 (AU = 0.073) clathrates are much larger than for the empty Ge clathrate-I framework (AU = 0.004) but still clearly smaller in comparison to the dense α-Ge (AU = 0.309). The elastic anisotropy of the Ba8Ga16Ge30 and Sr8Ga16Ge30 clathrate-I structures was also discussed in the original paper in terms of cubic anisotropy factor A, and the increased isotropy with respect to α-Ge was suggested to be due to their complex crystal structure where the directionality of atomic bonding is averaged over for many different types of atomic bonds.32 Similarly, Blase et al. have previously discussed the low elastic anisotropy of the carbon clathrate frameworks I and II, resulting from the effect of the many different orientations of the sp3 bonds in clathrates in comparison to diamond having only four different orientations.19,38 The results obtained here suggest that the high number of bonds with different orientations results in a very low elastic anisotropy for the various other clathrate frameworks, as well. The present evaluation of the elastic properties of large number of clathrate frameworks for four group 14 elements also shows the low elastic anisotropy to be present regardless of the elemental constitution, which is notable due to the rarity of elastically isotropic semiconductor materials. For many practical applications, the question of elastic isotropy or anisotropy of single-crystalline materials is actually insignificant, since the target materials are in any case polycrystalline. However, in the modern semiconductor industry, built on α-silicon-based technologies, the elastic anisotropy has to be taken into account during the device design.4 Hence, from the point of view of engineering of advanced semiconductor devices, semiconductor materials with very low elastic anisotropies could 19929
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’ CONCLUSIONS We have systematically investigated the elastic properties of semiconducting group 14 clathrate frameworks by using quantum chemical methods. The predicted bulk and Young’s moduli of the studied carbon, silicon, germanium and tin frameworks suggest them to possess low compressibility and high stiffness, which are almost comparable to the diamondlike, dense α-phases of the elements. In particular, the studied microporous clathrate frameworks exhibit remarkably low elastic anisotropy, being clearly more isotropic than the denser α-phases. The elastic properties predicted here are in line with the elastic properties determined for the experimentally known guest-occupied semiconducting clathrates. Detailed theoretical and experimental studies on the elastic properties of various guest-occupied group 14 clathrate frameworks are expected to shed more light on their possible applications. In particular, experimental preparation of further guest-free clathrate frameworks could result in semiconductor materials with very low elastic anisotropy and attractive characteristics for the engineering of semiconductor devices. ’ ASSOCIATED CONTENT
bS
Supporting Information. Expressions for the Young’s moduli in different crystal directions and supplementary data tables. This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*E-mail: antti.j.karttunen@iki.fi.
’ ACKNOWLEDGMENT We gratefully acknowledge financial support from the Academy of Finland (A.J.K., Grant No. 138560/2010), Finnish Funding Agency for Technology and Innovation, and European Union/European Regional Development Fund (Grant No. 70026/08). ’ REFERENCES (1) Ranganathan, S. I.; Ostoja-Starzewski, M. Phys. Rev. Lett. 2008, 101, 055504. (2) Chung, D. H.; Buessem, W. R. J. Appl. Phys. 1967, 38, 2010–2012. (3) Hill, R. Proc. Phys. Soc. London, Sect. A 1952, 65, 349–354. (4) Hopcroft, M. A.; Nix, W. D.; Kenny, T. W. J. Microelectromech. 2010, 19, 229–238. (5) Featherston, F. H.; Neighbours, J. R. Phys. Rev. 1963, 130, 1324–1333. (6) Bak, P. Phys. Rev. B 1985, 32, 5764–5772. (7) Spoor, P. S.; Maynard, J. D.; Kortan, A. R. Phys. Rev. Lett. 1995, 75, 3462–3465. (8) Kovnir, K. A.; Shevelkov, A. V. Russ. Chem. Rev. 2004, 73, 923–938. (9) Beekman, M.; Nolas, G. S. J. Mater. Chem. 2008, 18, 842–851. (10) Christensen, M.; Johnsen, S.; Iversen, B. B. Dalton Trans. 2010, 39, 978–992. (11) Sootsman, J. R.; Chung, D. Y.; Kanatzidis, M. G. Angew. Chem., Int. Ed. 2009, 48, 8616–8639.
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