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. 14-15, 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-99-86706
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
• 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)
Type I Clathrate: Si46 , Ge46 or Sn46
Type II Clathrate: Si136 , Ge136, Sn136
Clathrate Structures
24 atom cages
20 atom cages
28 atom cages
Type I ClathrateSi46, Ge46, Sn46
simple cubic
Type II ClathrateSi136, Ge136, Sn136
face centered cubic
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)
Type I Clathrate(with guest “rattlers”)
20 atom cage with guest atom
+
24 atom cage with guest atom
[100]direction
[010]direction
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)
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.
• Start with given interatomic distances & bond angles.• Supercell approximation • Interatomic forces act to relax lattice to equilibrium
configuration (distances, angles). – Schrdinger 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
Birch-Murnaghan Eqtn of StateSn 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
Bandstructures• At relaxed lattice configuration (“optimized
geometry”) use one electron Hamiltonian + LDA many electron corrections to solve Schrdinger Eq. for bandstructures Ek.
Sn46 & Sn136 BandstructuresC.W. Myles, J. Dong, O. Sankey, Phys. Rev. B 64, 165202 (2001).
The LDA UNDER-estimates bandgaps!
LDA gap Eg 0.86 eV
LDA gap Eg 0.46 eVSemiconductors of pure tin!!!!
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
Cs8Ga8Sn38 BandstructureC.W. Myles, J. Dong, O. Sankey, Phys. Rev. B 64, 165202 (2001).
LDA gap Eg 0.61 eV
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
Sn46 & Sn136 PhononsC. Myles, J. Dong, O. Sankey, C. Kendziora, G. Nolas,
Phys. Rev. B 65, 235208 (2002)
Flat optic bands!
Cs8Ga8Sn38 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: Due to Cs motion in large & small cages
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.
C. Myles, J. Dong, O. Sankey, C. Kendziora, G. Nolas, Phys. Rev. B 65, 235208 (2002)
Experimental & theoreticalrattler (& other) modes in very good agreement.
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.
Type II Clathrate PhononsWith “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!
Na16Cs8Si136 & Na16Cs8Ge136 Phonons J. Dong, A. Poddar, C. Myles, O. Sankey, unpublished
Cs rattler modes ~ 65 cm-1 Cs rattler modes ~ 21 cm-1
Cs24Sn136 Phonons C. Myles, J. Dong, O. Sankey, unpublished
Cs rattler modes: ~ 25-30 cm-1 (small cage)
~ 5 cm-1 (large cage) !
Cs in 8 large cages: Extremely anharmonic & “loosely” fitting. Very small frequencies: ~ 5 cm-1
Raman Spectra
• Again, estimate Raman scattering intensities using empirical (two parameter) bond charge model.
G. Nolas, C. Kendziora, J. Gryko, A. Poddar, J. Dong, C. Myles, O. Sankey J. Appl. Phys. (accepted).
• Experimental & theoreticalrattler (& other) modes in very good agreement.Also (not shown) detailed identification of frequencies & symmetries of several observed Raman modes by comparison with theory.
0 100 200 300 400 500 600
AA1g
T=300K=514nm
HV
VV
Si136
T2g
Eg
T2g
Eg
T2g
T2g
BA1g"Rattle"
T=300K=700nm
Cs8Na16Si136
HV 45oHV 0oVV 45oVV 0o
RamanIntensity(arb.units)
Raman Shift (cm-1)
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.
0 50 100 150 200 250 300 350
A1g
"Rattle"
HV
VV
T=300K=514 nm
Cs8Na
16Ge
136
Ram
an In
tens
ity (a
rb. u
nits
)
Raman Shift (cm-1)
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.
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!
Trends
• Trends in Cs “rattling” modes as host is changed from Si Ge SnNa16Cs8Si136 (~ 65 cm-1), Cs in large cages Na16Cs8Ge136 (~ 21 cm-1), Cs in large cages
Cs24Sn136 (~ 20-30 cm-1), Cs in large cages
(~ 5 cm-1), Cs in small cages
• In progress: Phonons in Na16Cs8Sn136