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

• 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.

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 sp3 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.

Si46, Ge46, Sn46 : Semiconducting

Cs8Sn46 : Metallic. Cs8Ga8Sn38 : Semiconducting

Si136,Ge136, Sn136 : Semiconducting

Na16Cs8Si136, Na16Cs8Ge136, Cs24Sn136 : 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). – Schrdinger 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

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: 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 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

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: Cs motion in large & small cages

Raman SpectraGroup 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 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 (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!

Phonons C. Myles, J. Dong, O. Sankey, submitted, Phys. Status Solidi B

Na rattlers (20-atom cages) ~ 118 -121 cm-1

Cs rattlers (28-atom cages) ~ 65 - 67 cm-1

Na16Cs8Si136 Na16Cs8Ge136

Na rattlers (20-atom cages) ~ 89 - 94 cm-1

Cs rattlers (28-atom cages) ~ 21 - 23 cm-1

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.

Si136, Na16Cs8Si136

Na16Cs8Ge136

Raman Spectra

• 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)

Cs24Sn136 Phonons C. Myles, J. Dong, O. Sankey, submitted, Phys. Status Solidi B

Cs rattler modes (20-atom cages) ~ 25 - 30 cm-1

Cs rattler modes (28-atom cages) ~ 5 - 7 cm-1

• Cs24Sn136: A

hypothetical material! Cs in large (28-atom) cages:Extremely anharmonic & “loose” fitting. Very small frequencies!

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!

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.

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.

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, Si28 cage:Δr 1.18 Å “oversized”

K 2.2 eV/(Å)2

KSi-Si 10 eV/(Å)2

Cs weakly bound• Ge28 cage:Δr 1.22 Å “oversized”

K 0.2 eV/(Å)2

KGe-Ge 10 eV/(Å)2

Cs very weakly bound• Largest, Sn28 cage: Δr 1.62 Å extremely “oversized”

K 0.02 eV/(Å)2, KSn-Sn 8 eV/(Å)2

Cs extremely weakly bound Largest alkali atom (Cs) in largest possible clathrate cage (Sn28)!

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.