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Rattling Atoms in Type I and Type II Clathrate Materials

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.

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Rattling Atoms in Type I and Type II Clathrate Materials

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  1. 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 1Supported in part by NSF Grant NSF-DMR-99-86706

  2. Si46, Ge46, Sn46: (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. • Si136, Ge136, Sn136: (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.

  3. 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 insp3bonds  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)

  4. 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. Si46, Ge46, Sn46: Semiconducting Cs8Sn46 : Metallic. Cs8Ga8Sn38 : Semiconducting Si136,Ge136, Sn136 : Semiconducting Na16Cs8Si136, Na16Cs8Ge136, Cs24Sn136 : Metallic

  5. Calculations • Computational package: VASP: Vienna Austria Simulation Package • First principles technique. • Many electron effects: Correlation: LocalDensityApproximation (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.

  6. 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 2nd 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

  7. Lattice Vibrational Spectra • Optimized LDA geometry: Calculate total ground state energy: Ee(R1, R2, R3, …..RN) • Harmonic Approx.: “Force constant” matrix: (i,i)  (2Ee/Ui Ui),Ui= atomic displacements • Finite displacement method: Eefor many different (Small) Ui. Forces  Ui. Dividing force by Ui gives (i,i) & dynamical matrix Dii(q). Group theory limits number & symmetry of Uirequired. • Positive & negative Uifor each symmetry: Cancels out 3rd 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

  8. Cs8Ga8Sn38 PhononsC. Myles, J. Dong, O. Sankey, C. Kendziora, G. Nolas,Phys. Rev. B 65, 235208 (2002)  Ga modes  Cs guest“rattler”modes (~25 - 40cm-1) “Rattler” modes:Cs motion in large & small cages

  9. 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!

  10. Reasonable agreement of theory and experiment for Raman spectrum.  UNAMBIGUOUS IDENTIFICATION of low frequency (25-40 cm-1) “rattling” modes of Cs guests in Cs8Ga8Sn38 • Also:(not shown)Detailed identification of frequencies & symmetries of several experimentally observed Raman modes by comparison with theory.

  11. Type II Clathrate PhononsWith “rattling”atoms • Current experiments: Focus on rattling modes in Type II clathrates (thermoelectric applications).  Theory:Given success with Cs8Ga8Sn38: Look at phonons & rattling modes in Type II clathrates Search for trends in rattling modes as host changes from Si  Ge  Sn • Na16Cs8Si136 : Have Raman data & predictions • Na16Cs8Ge136 : Have Raman data & predictions • Cs24Sn136: Have predictions, NEED DATA!

  12. PhononsC. Myles, J. Dong, O. Sankey, submitted, Phys. Status Solidi B Na16Cs8Si136 Na16Cs8Ge136 Narattlers(20-atom cages) ~ 118 -121 cm-1 Csrattlers(28-atom cages) ~ 65 - 67 cm-1 Narattlers (20-atom cages) ~ 89 - 94 cm-1 Cs rattlers (28-atom cages) ~ 21 - 23 cm-1

  13. Si136, Na16Cs8Si136 Na16Cs8Ge136 Raman Spectra 1st 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.

  14. 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 Na16Cs8Si136(~ 65 - 67 cm-1) Na16Cs8Ge136 (~ 21 - 23 cm-1)

  15. Cs24Sn136 PhononsC. Myles, J. Dong, O. Sankey, submitted, Phys. Status Solidi B • Cs24Sn136:A • hypothetical • material! • Cs in large (28-atom) cages: • Extremely anharmonic & “loose” fitting. •  Very small • frequencies! Csrattler modes (20-atom cages) ~ 25 - 30 cm-1 Csrattler modes (28-atom cages) ~ 5 - 7 cm-1

  16. Predictions • Cs24Sn136: 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!

  17. Trend • Trend in “rattling” modes of Cs in large (28-atom) cages as host changes Si  Ge  Sn Na16Cs8Si136(~ 65 - 67 cm-1) Na16Cs8Ge136 (~ 21 - 23 cm-1) Cs24Sn136 (~ 5 - 7 cm-1) • Correlates with size of cages in comparison with “size” of Cs atom.

  18. 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- (rX + rCs) rcage  LDA-computed average Cs-X distance rX  (LDA-computed average X-X near- neighbor distance)  covalent radius of atom X rCsionic radius of Cs (1.69 Å) (rX + rCs)  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.

  19. 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

  20. Smallest, Si28cage: • Δr  1.18 Å  “oversized” • K  2.2 eV/(Å)2 • KSi-Si  10 eV/(Å)2 •  Cs weakly bound • Ge28cage: • Δr  1.22 Å  “oversized” • K  0.2 eV/(Å)2 • KGe-Ge  10 eV/(Å)2 •  Csvery weakly bound K vs. Δr • Largest, Sn28cage:Δr  1.62 Å  extremely “oversized” • K  0.02 eV/(Å)2, KSn-Sn  8 eV/(Å)2 •  Csextremely weakly bound • Largest alkali atom (Cs) in largest possible clathrate cage (Sn28)!

  21. Conclusions • LDA calculations of lattice vibrations • Type I clathrate: Cs8Ga8Sn38 • Good agreement with Raman data for Cs rattler modes & also host framework modes! • Type II clathrates: Na16Cs8Ge136, Na16Cs8Si136 • Good agreement with Raman data for Cs rattler modes & also host framework modes! • Type II clathrate: Cs24Sn136 (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.

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