1. Study of Rattling Atoms in Type I and Type II Clathrate Semiconductors
Charles W. Myles,1 Texas Tech U.
Jianjun Dong, Auburn U.
Otto F. Sankey,2 Arizona State U.
4th Motorola Workshop on Computational Materials and Electronics, Nov , 2002
1Supported in part by a Texas Tech U. Faculty
Development Leave. Thanks to ASU for hospitality!
2Supported in part by NSF Grant NSF-DMR

2. Clathrates
Crystalline Phases of Group IV elements: Si, Ge, Sn (not C yet!) “New” materials, but known (for Si) since 1965!
J. Kasper, P. Hagenmuller, M. Pouchard, C. Cros, Science 150, 1713 (1965)
As in diamond structure, all Group IV atoms are 4-fold coordinated in sp3 bonding configurations.
Metastable, high energy phases of Si, Ge, Sn
Few pure elemental phases yet. Usually compounds with groups I and II elements (Na, K, Cs, Ba).
Applications: Thermoelectrics.
Open, cage-like structures. Large “cages” of group IV atoms.
Hexagonal & pentagonal rings, fused together to form “cages” of 20, 24, & 28 atoms

3. Si46, Ge46, Sn46: ( Type I Clathrates)
20 atom (dodecahedron) “cages” and 24 atom (tetrakaidecahedron) cages, fused together through 5 atom pentagonal rings.
Crystal structure = simple cubic, 46 atoms per cubic unit cell.
Si136, Ge136, Sn136: ( Type II Clathrates)
20 atom (dodecahedron) “cages” and 28 atom (hexakaidecahedron) cages, fused together through 5 atom pentagonal rings.
Crystal structure = face centered cubic, 136 atoms per cubic unit cell (34 atoms/fcc unit cell)

4. Type I Clathrate: Si46 , Ge46 or Sn46

5. Type II Clathrate: Si136 , Ge136, Sn136

6. Clathrate Structures 24 atom cages 20 atom cages 28 atom cages
Type I Clathrate
Si46, Ge46, Sn46
simple cubic
20 atom cages
Type II Clathrate
Si136, Ge136, Sn136
face centered
cubic
28 atom cages

7. Clathrates Not found in nature. Synthesized in the lab.
Not normally in pure form, but with impurities (“guests”) encapsulated inside the cages. Guests  “Rattlers”
Guests: Group I atoms (Li, Na, K, Cs, Rb) or Group II atoms (Be, Mg, Ca, Sr, Ba)

8. Type I Clathrate (with guest “rattlers”)
20 atom cage
with
guest atom
[100]
direction +
24 atom cage
with
guest atom
[010]
direction

9. Clathrates Semiconductors or semimetals.
Also superconducting materials made from sp3 bonded, Group IV atoms! (Ba8Si46)
Guests weakly bound in cages:
Host valence electrons taken up in sp3 bonds
Guest valence electrons go to conduction band of host (heavy doping density).
Guests weakly bonded in cages  Minimal effect on electronic transport
Guests vibrate (“rattle”) with low frequency modes  Strong effect on vibrational properties (thermal conductivity)

10. 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 & phonon spectra.

11. Supercell approximation
Start with given interatomic distances & bond angles.
Supercell approximation
Interatomic forces act to relax lattice to equilibrium configuration (distances, angles).
Schrdinger Eq. for interacting electrons, Newton’s 2nd Law motion for atoms.
Equations of State
Total binding energy minimized by optimizing internal coordinates at given volume.
Repeat for several volumes.
Gives LDA binding energy vs. volume curve.
Fit to empirical eqtn of state (4 parameter): “Birch-Murnaghan” equation of state

12. Birch-Murnaghan Eqtn of State Sn Clathrates = Metastable, expanded volume phases
E(V) = E0 + (9/8)K V0[(V0/V) -1]2{1 + ½(4-K)[1- (V0/V)]}
E0  Minimum binding energy, V0  Volume at minimum energy
K  Equilibrium bulk modulus; K  dK/dP

13. Bandstructures
At relaxed lattice configuration (“optimized geometry”) use one electron Hamiltonian + LDA many electron corrections to solve Schrdinger Eq. for bandstructures Ek.

14. Sn46 & Sn136 Bandstructures C. W. Myles, J. Dong, O. Sankey, Phys. Rev
Sn46 & Sn136 Bandstructures C.W. Myles, J. Dong, O. Sankey, Phys. Rev. B 64, (2001).
The LDA UNDER-estimates bandgaps!
LDA gap Eg  0.86 eV
LDA gap Eg  0.46 eV
Semiconductors of pure tin!!!!

15. Compensation
Guest-containing clathrates: Valence electrons from guests go to conduction band of host (heavy doping). Change material from semiconducting to metallic.
Compensate for this by replacing some host atoms in the framework by Group III atoms.
Sn46 : Semiconducting
Cs8Sn46 : Metallic
Cs8Ga8Sn38 : Semiconducting
Cs8Zn4Sn42 : Semiconducting
Sn136 : Semiconducting
Cs24Sn136 : Metallic

16. Cs8Ga8Sn38 Bandstructure C. W. Myles, J. Dong, O. Sankey, Phys. Rev
Cs8Ga8Sn38 Bandstructure C.W. Myles, J. Dong, O. Sankey, Phys. Rev. B 64, (2001).
LDA gap Eg  0.61 eV

17. Lattice Vibrational Spectra
At optimized LDA geometry, calculate total ground state energy: Ee(R1, R2, R3, …..RN)
Harmonic Approx.: “Force constant” matrix: (i,i)  (2Ee/Ui Ui)
Ui = displacements from equilibrium
Derivatives  Ee for many different Ui. (Small Ui; harmonic approximation)
Group theory limits number & symmetry of Ui required.
Positive & negative Ui for 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

18. Sn46 & Sn136 Phonons C. Myles, J. Dong, O. Sankey, C. Kendziora, G
Sn46 & Sn136 Phonons C. Myles, J. Dong, O. Sankey, C. Kendziora, G. Nolas, Phys. Rev. B 65, (2002)
Flat optic bands!

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

20. Raman Spectra
Do group theory necessary to determine Raman active modes (frequencies calculated from first principles as described).
Estimate Raman scattering intensities using empirical (two parameter) bond charge model.

21. C. Myles, J. Dong, O.
Sankey, C. Kendziora,
G. Nolas, Phys. Rev. B 65,
(2002)
Experimental & theoretical
rattler (& other) modes in
very good agreement.

22. Conclusion
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.

23. Type II Clathrate Phonons With “rattling”atoms
Current experiments: Focus on rattling modes in Type II clathrates (for thermoelectric applications).
 Theory: Given our success with Cs8Ga8Sn38:
Look at phonons & rattling modes in Type II clathrates
 Search for trends in rattling modes as host is changed from Si  Ge  Sn
Na16Cs8Si136 : Have Raman data & predictions
Na16Cs8Ge136 : Have Raman data & predictions
Cs24Sn136: Have predictions, NEED DATA!
Na16Cs8Sn136  Calculations are in progress!

24. Na16Cs8Si136 & Na16Cs8Ge136 Phonons J. Dong, A. Poddar, C. Myles, O
Na16Cs8Si136 & Na16Cs8Ge136 Phonons J. Dong, A. Poddar, C. Myles, O. Sankey, unpublished
Cs rattler modes ~ 21 cm-1
Cs rattler modes ~ 65 cm-1

25. Cs24Sn136 Phonons C. Myles, J. Dong, O. Sankey, unpublished
Cs rattler modes:
 ~ cm-1 (small cage)
 ~ 5 cm-1 (large cage) !
Cs in 8 large cages: Extremely anharmonic &
“loosely” fitting.  Very small frequencies: ~ 5 cm-1

26. Raman Spectra
Again, estimate Raman scattering intensities using empirical (two parameter) bond charge model.

27. G. Nolas, C. Kendziora,
J. Gryko, A. Poddar,
J. Dong, C. Myles,
O. Sankey J. Appl. Phys.
(accepted).
Experimental & theoretical
rattler (& other) modes in
very good agreement.
Also (not shown) detailed
identification of frequencies
& symmetries of several
observed Raman modes by
comparison with theory.

28. G. Nolas, C. Kendziora, J. Gryko, A. Poddar, J. Dong,
C. Myles, & O. Sankey, J. Appl. Phys. (accepted)
Experimental & theoretical rattler (& other) modes in very good agreement.

29. Conclusions
Reasonable agreement of theory and 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 cm-1), Na16Cs8Ge136 (~ 21 cm-1)
Also: (not shown) Detailed identification of frequencies & symmetries of several experimentally observed Raman modes by comparison with theory.

30. Prediction
Cs24Sn136: Prediction of low frequency “rattling” modes of Cs guests in small
(~20-30 cm-1) & large (~ 5 cm-1) cages (a very small frequency!)
 Potential thermoelectric applications.
NEED DATA!

31. Trends
Trends in Cs “rattling” modes as host is changed from Si  Ge  Sn
Na16Cs8Si136 (~ 65 cm-1), Cs in large cages Na16Cs8Ge136 (~ 21 cm-1), Cs in large cages
Cs24Sn136 (~ cm-1), Cs in large cages
(~ 5 cm-1), Cs in small cages
In progress: Phonons in Na16Cs8Sn136