Rattling Atoms in Type I and Type II

Rattling Atoms in Type I and Type II Clathrate Materials Charles W. Myles, Texas Tech U. Jianjun Dong, Auburn U. Otto F. Sankey, 1 Arizona State U. March National APS Meeting Austin, TX, Tues. , March 4, 2003 1 Supported in part by NSF Grant NSF-DMR-99 -86706

• Si 46, Ge 46, Sn 46: ( Type I Clathrates) 20 atom (dodecahedron) “cages” & 24 atom (tetrakaidecahedron) cages, fused together through 5 atom rings. Crystal structure = simple cubic 46 atoms per cubic unit cell. • Si 136, Ge 136, Sn 136: ( Type II Clathrates) 20 atom (dodecahedron) “cages” & 28 atom (hexakaidecahedron) cages, fused together through 5 atom rings. Crystal structure = face centered cubic, 136 atoms per cubic unit cell.

Clathrates • Pure framework materials: Usually semiconductors. • Pure materials not easily fabricated. Normally have impurities (“guests”) encapsulated inside cages. Guests “Rattlers” • Guests: Group I atoms (Li, Na, K, Cs, Rb) or Group II atoms (Be, Mg, Ca, Sr, Ba) – Guests weakly bound in cages Minimal effect on electronic transport – Host valence electrons taken up in sp 3 bonds Guest valence electrons go to conduction band of host (heavy doping density). – Guests vibrate (“rattle”) with low frequency modes Strongly affect lattice vibrations (thermal conductivity)

Compensation • Guest-containing clathrates: Valence electrons from guests go to conduction band of host (heavy doping). Change material from semiconducting to metallic. • Sometimes compensate for this by replacing some host atoms in the framework by Group III atoms. Si 46, Ge 46, Sn 46 : Semiconducting Cs 8 Sn 46 : Metallic. Cs 8 Ga 8 Sn 38 : Semiconducting Si 136, Ge 136, Sn 136 : Semiconducting Na 16 Cs 8 Si 136, Na 16 Cs 8 Ge 136, Cs 24 Sn 136 : Metallic

Calculations • Computational package: VASP: Vienna Austria Simulation Package • First principles technique. – Many electron effects: Correlation: Local Density Approximation (LDA). Exchange-correlation energy: Ceperley-Adler Functional – Ultrasoft pseudopotentials. – Planewave basis • Extensively tested on a wide variety of systems • We’ve computed equations of state, bandstructures & vibrational phonon spectra.

• Start with given interatomic distances & bond angles. – Supercell approximation • Total binding energy minimized by optimizing internal coordinates at a given volume. – Interatomic forces to relax lattice to equilibrium configuration (distances, angles). – Schr dinger Eq. for interacting electrons, Newton’s 2 nd Law motion for atoms. • Repeat for several volumes until LDA minimum energy configuration is obtained. • Once equilibrium lattice geometry is obtained, all ground state properties can be obtained: – Vibrational dispersion relations: Our focus here! – Electronic bandstructures

Lattice Vibrational Spectra • Optimized LDA geometry: Calculate total ground state energy: Ee(R 1, R 2, R 3, …. . RN) • Harmonic Approx. : “Force constant” matrix: (i, i ) ( 2 Ee/ Ui Ui ), Ui = atomic displacements • Finite displacement method: Ee for many different (Small) Ui. Forces Ui. Dividing force by Ui gives (i, i ) & dynamical matrix Dii (q). Group theory limits number & symmetry of Ui required. • Positive & negative Ui for each symmetry: Cancels out 3 rd order anharmonicity (beyond harmonic approx. ) Once all unique (i, i ) are computed, do lattice dynamics. • Lattice dynamics in the harmonic approximation: det[Dii (q) - 2 ii ] = 0

Cs 8 Ga 8 Sn 38 Phonons C. Myles, J. Dong, O. Sankey, C. Kendziora, G. Nolas, Phys. Rev. B 65, 235208 (2002) Ga modes Cs guest “rattler” modes (~25 - 40 cm-1) “Rattler” modes: Cs motion in large & small cages

Raman Spectra Group theory determines Raman active modes. First principles frequencies, empirical intensities. C. Myles, J. Dong, O. Sankey, C. Kendziora, G. Nolas, Phys. Rev. B 65, 235208 (2002) Experimental & theoretical rattler (& other) modes in very good agreement!

• Reasonable agreement of theory and experiment for Raman spectrum. UNAMBIGUOUS IDENTIFICATION of low frequency (25 -40 cm-1) “rattling” modes of Cs guests in Cs 8 Ga 8 Sn 38 – Also: (not shown) Detailed identification of frequencies & symmetries of several experimentally observed Raman modes by comparison with theory.

Type II Clathrate Phonons With “rattling”atoms • Current experiments: Focus on rattling modes in Type II clathrates (thermoelectric applications). Theory: Given success with Cs 8 Ga 8 Sn 38: Look at phonons & rattling modes in Type II clathrates Search for trends in rattling modes as host changes from Si Ge Sn – Na 16 Cs 8 Si 136 : Have Raman data & predictions – Na 16 Cs 8 Ge 136 : Have Raman data & predictions – Cs 24 Sn 136: Have predictions, NEED DATA!

Phonons C. Myles, J. Dong, O. Sankey, submitted, Phys. Status Solidi B Na 16 Cs 8 Si 136 Na rattlers (20 -atom cages) ~ 118 -121 cm-1 Cs rattlers (28 -atom cages) ~ 65 - 67 cm-1 Na 16 Cs 8 Ge 136 Na rattlers (20 -atom cages) ~ 89 - 94 cm-1 Cs rattlers (28 -atom cages) ~ 21 - 23 cm-1

Si 136, Na 16 Cs 8 Si 136 Na 16 Cs 8 Ge 136 Raman Spectra 1 st principles frequencies. G. Nolas, C. Kendziora, J. Gryko, A. Poddar, J. Dong, C. Myles, O. Sankey J. Appl. Phys. 92, 7225 (2002). Experimental & theoretical rattler (& other) modes in very good agreement! Not shown: Detailed identification of frequencies & symmetries of observed Raman modes by comparison with theory.

• Reasonable agreement of theory & experiment for Raman spectra, especially “rattling” modes (of Cs in large cages) in Type II Si & Ge clathrates. UNAMBIGUOUS IDENTIFICATION of low frequency “rattling” modes of Cs in Na 16 Cs 8 Si 136 (~ 65 - 67 cm-1) Na 16 Cs 8 Ge 136 (~ 21 - 23 cm-1)

Cs 24 Sn 136 Phonons C. Myles, J. Dong, O. Sankey, submitted, Phys. Status Solidi B • Cs 24 Sn 136: A hypothetical material! Cs in large (28 atom) cages: Extremely anharmonic & “loose” fitting. Very small frequencies! Cs rattler modes (20 -atom cages) Cs rattler modes (28 -atom cages) ~ 25 - 30 cm-1 ~ 5 - 7 cm-1

Predictions • Cs 24 Sn 136: Low frequency “rattling” modes of Cs guests in 20 atom cages (~25 -30 cm-1) & in 28 -atom cages (~ 5 - 7 cm-1, very small frequencies!) – Caution! Effective potential for Cs in 28 -atom cage is very anharmonic: Cs is very loosely bound there. Calculations were done in the harmonic approximation. More accurate calculations taking anharmonicity into account are needed. Potential thermoelectric applications. NEED DATA!

Trend • Trend in “rattling” modes of Cs in large (28 -atom) cages as host changes Si Ge Sn Na 16 Cs 8 Si 136 (~ 65 - 67 cm-1) Na 16 Cs 8 Ge 136 (~ 21 - 23 cm-1) Cs 24 Sn 136 (~ 5 - 7 cm-1) • Correlates with size of cages in comparison with “size” of Cs atom.

Model for Trend • 28 -atom cage size in host framework compared with Cs guest atom “size”. • For host atom X = Si, Ge, Sn, define: Δr rcage- (r. X + r. Cs) rcage LDA-computed average Cs-X distance r. X (LDA-computed average X-X nearneighbor distance) covalent radius of atom X r. Cs ionic radius of Cs (1. 69 Å) (r. X + r. Cs) Cs-X distance if Cs were tight fitting in cage Δr How “oversized” the cage is compared to Cs “size”. Geometric measure of how loosely fitting a Cs atom is inside a 28 -atom cage.

Model • Simple harmonic oscillator model for Cs, with assumption that only Cs moves in its oversized 28 -atom cage. • Equate LDA-computed rattler frequency to: R = (K/M)½ K Effective force constant for rattler mode K A measure of strength (weakness) of guest atom-host atom interaction. M Mass of Cs

K vs. Δr • Smallest, Si 28 cage: Δr 1. 18 Å “oversized” K 2. 2 e. V/(Å)2 KSi-Si 10 e. V/(Å)2 Cs weakly bound • Ge 28 cage: Δr 1. 22 Å “oversized” K 0. 2 e. V/(Å)2 KGe-Ge 10 e. V/(Å)2 Cs very weakly bound • Largest, Sn 28 cage: Δr 1. 62 Å extremely “oversized” K 0. 02 e. V/(Å)2, KSn-Sn 8 e. V/(Å)2 Cs extremely weakly bound Largest alkali atom (Cs) in largest possible clathrate cage (Sn 28)!

Conclusions • LDA calculations of lattice vibrations • Type I clathrate: Cs 8 Ga 8 Sn 38 – Good agreement with Raman data for Cs rattler modes & also host framework modes! • Type II clathrates: Na 16 Cs 8 Ge 136, Na 16 Cs 8 Si 136 – Good agreement with Raman data for Cs rattler modes & also host framework modes! • Type II clathrate: Cs 24 Sn 136 (A hypothetical material) – Prediction of extremely low frequency “rattling” modes of Cs guests • Simple model for trend in Cs rattler modes (28 atom cage) as host changes from Si to Ge to Sn.