Study of Rattling Atoms in Type I and Type II Clathrate Semiconductors

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