1. Effects of Si on the Vibrational and Thermal Properties of the Clathrates A 8 Ga 16 Si x Ge 30-x (A = Ba, Sr) For more details: See Emmanuel N. Nenghabi and Charles W. Myles, Phys. Rev, B 78, 195202 (2008) Charles W. Myles and Emmanuel N. Nenghabi * *Deceased Paper D28:0011: March APS Meeting, Pittsburgh, PA, March 16, 2009

2. What are Clathrates? Crystalline phases based on Group IV elements Group IV atoms are 4-fold coordinated in sp 3 bonding configurations, but with distorted bond angles.  A distribution of bond angles. Lattices have hexagonal & pentagonal rings, fused together with sp 3 bonds to form large, open “cages” of Group IV atoms. Cages of 20, 24 & 28 atoms. Meta-stable, high energy phases of group IV elements. Applications: Thermoelectric materials & devices. Not found naturally. Must be lab synthesized.

3. Type I: Formula: X 8 E 46 (simple cubic lattice) Type II: Formula: X 8 Y 16 E 136 (face centred cubic lattice) X,Y = alkali metal or alkaline earth atoms, E = group IV atom Clathrate Types “Building Blocks” 24 atom cages  28 atom cages  dodecahedra (D) hexakaidecahedra ( H ) 20 atom cages  tetrakaidecahedra (T) Type I: cage ratio: 6 D’s to 2 T’s E 46 sc lattice Type II: cage ratio 16 T’s to 8 H’s E 136 fcc lattice

4. Why Ba 8 Ga 16 Si x Ge 30-x & Sr 8 Ga 16 Si x Ge 30-x ? Some of these have been lab synthesized & have also been found to have promising thermoelectric properties J. Martin, S. Erickson, G.S. Nolas, P. Alboni, T.M. Tritt, & J. Yang J. Appl. Phys. 99, 044903 (2006) First Principles Calculations VASP (Vienna ab-initio Simulation Package) Many e - effects: Generalized Gradient Approximation (GGA). Exchange Correlation: the Perdew-Wang Functional Vanderbilt ultrasoft Pseudopotentials Plane Wave Basis Set

5. Ba 8 Ga 16 Si x Ge 30-x Sr 8 Ga 16 Si x Ge 30-x These show: An up-shift in the optic modes as x increases. Largest for the optic modes, in which bond-stretching modes are important. In Ba 8 Ga 16 Si x Ge 30-x : The highest optic modes are 253, 334, 373 cm −1 for x = 0,5, 15. In Sr 8 Ga 16 Si x Ge 30-x : These are 327, 350, 428 cm −1 for x = 0,5, 15. Phonon Dispersion Relations

6. The phonon modes show: An up-shift in the optic modes as x increases. Explanation: Ge substitution by Ga & Si strengthens bonds. Calculated lattice constants a show that a in Ba compounds is larger than in the Sr materials because the Ba atomic mass is larger than Sr’s. So, a larger strain effect occurs when Ba is in the cages than if Sr is in them. Also: Because the Si atom is “smaller” than Ba, Sr, Ge, & Ga atoms, the lattice constant a decreases as x increases. The nearest-neighbor bond distances in Ba 8 Ga 16 Si x Ge 30-x range from 2.53–2.63 Å. In Sr 8 Ga 16 Si x Ge 30-x these range from 2.44– 2.62 Å. Shorter bonds strengthen the structures, resulting in larger force constants.

7. Vibrational State Densities (VDOS) The VDOS increases at the bottom of the optic band, just above the acoustic modes. – Eigenvector analysis shows that these additional modes are from the Sr & Ba guests. The VDOS is higher for x = 5 than for x = 0 & higher for x = 15 than for x = 5. The optic modes compress the acoustic bandwidth. For x = 0,5,15, the tops of the acoustic bands in Ba 8 Ga 16 Si x Ge 30-x are at 33, 36, 30 cm - 1. In Sr 8 Ga 16 Si x Ge 30-x, these are at 40, 42, 33 cm -1 for x = 0, 5, 15. The acoustic bandwidths are reduced (in comparison to that of pristine Ge 46 ) by ~16%–40%, depending on the value of x.

8. U iso ~ (k B T)/φ φ = Calculated force constant for Ba, Sr vibrations. Results for the Ba, Sr in 20 atom cages & in 24 atom cages are both shown. U iso values for Sr are larger than for Ba. In qualitative agreement with experiments by Bentien et al. in Ba 8 Ga 16 Ge 30, Ba 8 Ga 16 Si 30, Ba 8 In 16 Ge 30, Sr 8 Ga 16 Ge 30. Because of the Sr small atom in comparison to Ba, the Sr atoms are more off-centered in the cages than Ba, which leads to a larger ADP. Mean Square Atomic Displacement Parameters (ADP) x = 5 x = 15

9. Heat Capacity, C v, Entropy S, Helmholtz Free Energy F Of course, because of their low frequencies of vibration, the Ba guests contribute little to these properties. As can be seen, the dependence on the Si composition x is also very small for each of these properties. Similar calculations for Sr 8 Ga 16 Si x Ge 30-x for these properties show that the Ba-containing materials are thermodynamically more stable than the Sr-containing materials. Cv Cv S F Thermal Properties: C v, S, F for Ba 8 Ga 16 Si x Ge 30-x

10. Conclusions We hope that our predicted vibrational and thermal properties for the clathrate alloys Ba 8 Ga 16 Si x Ge 30-x, Sr 8 Ga 16 Si x Ge 30-x will lead to investigations of the thermoelectric properties of these interesting materials. We also hope that these investigations will provide information about which of these materials will be useful in the search for better thermoelectric materials.