Rattling Atoms in Group IV Clathrate Materials

Charles W. MylesProfessor, Department of Physics

Texas Tech University

Charley.Myles@ttu.edu

http://www.phys.ttu.edu/~cmyles

Colloquium, Auburn U., Friday, April 4, 2003

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Collaborators

• Otto F. Sankey: Arizona State University

• J.J. Dong: Auburn University– Was Otto Sankey’s post-doc at Arizona State

• George S. Nolas: University of South Florida– Materials synthesis & electrical characterization

• Chris Kendziora: Naval Research Labs– Experimentalist: Raman spectroscopy

• Jan Gryko: Jacksonville State U. (Alabama)– Experimentalist: Materials synthesis

Outline• Introduction to clathrates

Crystal structures. Contrast to diamond structure

• Brief discussion of computational method• Sn clathrates (Type I)

– Equations of state (Etot vs. volume)

– Electronic bandstructures (Ek)

– Vibrational (phonon) properties (k)

– Raman spectra & comparison with experiment

• Si, Ge, & Sn clathrates (Type II)

– Vibrational (phonon) properties (k)

– Raman spectra & comparison with experiment

Group IV Elements

• Valence electron configuration: ns2 np2

[n=2, C; n=3, Si; n=4, Ge; n=5, Sn]

Group IV Crystals• Si, Ge, Sn: Ground state crystalline structure = Diamond

Structure.

– Each atom tetrahedrally (4-fold) coordinated (4 nearest-neighbors) with sp3 covalent bonding

– Bond angles: Perfect, tetrahedral = 109.5º– Si, Ge: Semiconductors.– Sn: (-tin or gray tin) - Semimetal

Carbon Crystals• C: Graphite & Diamond Structures

– Diamond Insulator or wide bandgap

semiconductor

– Graphite Planar structure

sp2 bonding

2d metal (in plane) – Ground state (lowest energy configuration) is graphite at zero

temperature & atmospheric pressure. Graphite-diamond total energy difference is VERY small!

Other Group IV Crystal Structures(Higher Energy)

• C: “Buckyballs” (C60)

“Buckytubes” (nanotubes),

other fullerenes

• Sn: (-tin or white tin) - body centered tetragonal lattice, 2 atoms per unit cell. Metallic.

• Si, Ge, Sn: The clathrates.

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.

• Bond angles: Distorted tetrahedra Distribution of angles instead of perfect tetrahedral 109.5º

• Lattice contains hexagonal & pentagonal rings, fused together with sp3 bonds to form large “cages”.

• Pure materials: Metastable, expanded volume phases of Si, Ge, Sn

• Few pure elemental phases yet. Compounds with Group I & II atoms (Na, K, Cs, Ba).

• Possible application: Thermoelectrics.• Open, cage-like structures, with large “cages” of Si, Ge, or

Sn atoms. “Buckyball -like” cages of 20, 24, & 28 atoms.

• Two varieties: Type I (X46) & Type II (X136)

X = Si, Ge, or Sn

• Why “clathrate”? Same crystal structure as clathrate hydrates (ice).

• 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.

• 24 Atom Cage:

• 20 Atom Cage:

• 28 Atom Cage:

Clathrate Building Blocks

Clathrate Structures

24 atom cages

Type I ClathrateSi46, Ge46, Sn46

simple cubic

Type II ClathrateSi136, Ge136, Sn136

face centered cubic

20 atom cages

28 atom cages

Clathrate Lattices

Type I Clathrate Si46, Ge46, Sn46

simple cubic

Type II Clathrate Si136, Ge136, Sn136

face centered cubic

[100]direction

[100]direction

Group IV 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 (alkali) atoms (Li, Na, K, Cs, Rb) or Group II

(alkaline earth) atoms (Be, Mg, Ca, Sr, Ba)

• Synthesis: NaxSi46 (A theorists view!)– Start with Zintl phase NaSi compound.

– Ionic compound containing Na+ and (Si4)-4 ions

– Heat to thermally decompose. Some Na vacuum.– Si atoms reform into clathrate framework around Na.

– Cages contain Na guests

Type I Clathrate(with guest “rattlers”)

20 atom cage with guest atom

+

24 atom cagewith guest atom

[100]direction

[010]direction

Clathrates • Pure materials: Semiconductors.

• Guest-containing materials:– Some are superconducting materials (Ba8Si46) from sp3

bonded, Group IV atoms!

– Guests weakly bonded 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 with low frequency (“rattler”) modes Strong effect on vibrational properties

Guest Modes Rattler Modes

• Possible use as thermoelectric materials.Good thermoelectrics should have

low thermal conductivity!

• Guest Modes Rattler Modes: A focus of experiments.

Heat transport theory: Low frequency rattler

modes can scatter efficiently with acoustic

modes of host Lowers thermal conductivity

Good thermoelectric!

• Among materials of experimental interest are tin (Sn) clathrates. Mainly Type I. Much of my work.

• Also, Si and Ge, Type II. Most recent work.

Calculations• Computational package: VASP- Vienna Austria

Simulation Package. First principles!

Many electron effects:

Local Density Approximation (LDA).

Exchange-correlation:

Ceperley-Adler Functional

Ultrasoft pseudopotentials

Planewave basis

• Extensively tested on a wide variety of systems • We’ve computed equilibrium geometries, equations of state,

bandstructures & phonon spectra.

• Start with lattice geometry from expt or guessed (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 for atomic motion.

Equations of State• Total binding energy minimized in the LDA by

optimizing internal coordinates at a given volume.

• Repeat calculation for several volumes.– Gives minimum energy configuration.

LDA binding energy vs. volume curve.

– To save computational effort, fit this to empirical equation of state (four parameters): “Birch-Murnaghan” equation of state.

Birch-Murnaghan Eqtn of StateFit LDA total binding energy vs. volume curve to

E(V) = E0 + (9/8)K0V0[(V0/V) -1]2

{1 + (4-K)[1- (V0/V)]}

4 Parameters:

E0 Minimum binding energy

V0 Volume at minimum energy

K0 Equilibrium bulk modulus

K dK0/dP Pressure derivative of K0

Equations of State for Sn SolidsBirch-Murnhagan fits to LDA E vs. V curves

Sn Clathrates:expanded volume, high energy,metastable Sn phases

Compared to -Sn:Sn46

V: 12% larger E: 41 meV higherSn136

V: 14% larger E: 38 meV higher

Clathrates: “Negative pressure” phases!

Equation of State ParametersBirch-Murnhagan fits to LDA E vs. V curves

Sn Clathrates:

Expanded volume, high energy, “soft” Sn phases

Compared to -Sn: Sn46 -- V: 12% larger, E: 41 meV higher, K0: 13% “softer” Sn136 -- V: 14% larger, E: 38 meV higher, K0: 13% “softer”

• Once equilibrium lattice geometry is obtained, all ground state properties can be obtained (at minimum energy volume)– Electronic bandstructures– Vibrational dispersion relations

Bandstructures

• At relaxed lattice configuration (“optimized geometry”) use one electron Hamiltonian + LDA many electron corrections to solve Schrdinger Eq. for bandstructures Ek.

Ground State Properties

Bandstructures• NB= # of valence bands

Ne = # valence electrons / atom

NA= # atoms per cell

NB = Ne x NA

• Diamond Structure & Clathrates: Ne = 4

Diamond: NA = 2 NB = 8

Clathrates:

X46: NA = 46 NB = 184

X136: NA = 136 NB = 544

Diamond Structure Sn BandsM.L Cohen & J. Chelikowsky, Electronic Structure and Optical

Properties of Semiconductors, (Springer) Solid State Science, 75 (1989).

Diamond Structure Sn (-Sn):

A semimetal (Eg = 0)*

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 eV

Semiconductors of pure tin!!!! (Hypothetical materials. Indirect band gaps)

Sn46 Sn136

Compensation• Guest-containing clathrates: Valence electrons

from guests go to conduction band of host (heavy doping). Change material from semiconducting to metallic. For thermoelectric applications, want semiconductors!!

• COMPENSATE for this by replacing some host atoms in the framework by Group III or Group II atoms (charge compensates). Gets semiconductor back!

• Sn46: Semiconducting. Cs8Sn46: Metallic. Cs8Ga8Sn38 & Cs8Zn4Sn42: Semiconducting

• Later: Si136,Ge136, Sn136: Semiconducting. Na16Cs8Si136, Na16Cs8Ge136, Cs24Sn136: Metallic

• For EACH guest-containing clathrate, including those with compensating atoms in framework:

• ENTIRE LDA procedure is repeated:

– LDA total energy vs. volume curve

Equation of State– Birch-Murnhagan Eqtn fit to LDA results.

– At minimum energy volume, compute bandstructures & lattice vibrations.

– Compensated materials:

ASSUME an ordered structure.

Cs8Ga8Sn38 & Cs8Zn4Sn42 BandsC.W. Myles, J. Dong, O. Sankey, Phys. Rev. B 64, 165202 (2001).

The LDA UNDER-estimates bandgaps!

LDA gap Eg 0.61 eV LDA gap Eg 0.57 eV

Semiconductors (Materials which have been synthesized. Indirect band gaps)

Cs8Ga8Sn38 Cs8Zn4Sn42

Lattice Vibrations (Phonons)• 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 = atomic displacements from equilibrium.

Instead of directly computing derivatives, we use

• Finite displacement method: Compute Ee for many different (small; harmonic approx.) Ui

Compute forces Ui.

• Dividing forces by Ui gives (i,i) & thus dynamical matrix Dii(q).

• Group theory limits number & symmetry of Ui required. (Materials have high symmetry).

• Positive & negative Ui for each symmetry: Cancels out 3rd order anharmonicity (beyond harmonic approximation).

• Once have all unique (i,i), do lattice dynamics.• Lattice dynamics in the harmonic approximation:

classical eigenvalue (normal mode) problem

det[Dii(q) - 2 ii] = 0

Dynamical matrix Dii(q) obtained from force

constant matrix in usual way. First principles force constants! NO FITS TO DATA!

Phonons

• Eigenvalues: Squares of vibrational frequencies 2(q) (phonon dispersion relations)

NB = # of branches (modes) in (q)

NA = # of atoms / unit cell

NB = 3 x NA

• Diamond Structure: NA = 2 NB = 6

Clathrates:

X46: NA = 46 NB = 138

X136: NA = 136 NB = 408

• 3 Acoustic branches, NB - 3 Optic branches

Diamond Structure Sn PhononsW. Weber, Phys. Rev. B 15, 4789 (1977).

3 Acoustic branches 3 Optic branches

Sn46 & Sn136 PhononsC.W. Myles, J. Dong, O. Sankey, C. Kendziora, G. Nolas,

Phys. Rev. B 65, 235208 (2002)

Flat optic bands! Large unit cell Small Brillouin Zone reminiscent of “zone folding”

Sn46 Sn136

Guest-Containing Clathrates as Thermoelectrics

• Guest atoms: Weakly bound to clathrate framework.

• Framework: Fully sp3 tetrahedrally bonded.

Guest atom e- don’t participate in bonding or affect electronic transport very strongly.

• Guests have low energy (“rattling”) phonon modes (guest atoms vibrating in cages, small force constants). Will see this explicitly later in talk.

These strongly affect vibrational properties & thus phonon-phonon scattering & thermal conductivity.

• Good thermoelectrics should have low thermal conductivity.

• Guest Modes Rattler Modes:

A focus of experiments

Heat transport theory:

Low frequency rattler modes can scatter efficiently with acoustic modes of the host Lowers the thermal conductivity

Good thermoelectric!

Many experiments (e.g., Raman scattering) have focussed on the rattler modes of the guests. Our calculations have also done so.

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

Compare to Sn46 results.

Raman Spectra• Do group theory necessary to determine

Raman active modes.– Raman spectroscopy probes only modes at

zone center (q = 0).

• Frequencies calculated from first principles as described.

• Estimate Raman scattering intensities using an empirical (two parameter) bond polarization model.

C.W. Myles, J. Dong, O. Sankey, C. Kendziora, G. Nolas, Phys. Rev. B 65, 235208 (2002).• Experimental & theoretical rattler (& other!) modes in good agreement! UNAMBIGUOUS IDENTIFICATION of low (25-40 cm-1) frequency rattler modes of Cs guests. Not shown: Detailed identification of frequencies & symmetries of several 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!

– Note: These materials are metallic!

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

Na rattlers (20-atom cages) Na rattlers (20-atom cages) ~ 118 -121 cm-1 ~ 89 - 94 cm-1

Cs rattlers (28-atom cages) Cs rattlers (28-atom cages) ~ 65 - 67 cm-1 ~ 21 - 23 cm-1

Na16Cs8Si136 Na16Cs8Ge136

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:

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

Thermoelectric applications?

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.

Simple 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.

• Couple this geometric model with:

Simple harmonic oscillator model for Cs. Assumption that only Cs moves in its oversized 28-atom cage.

• Equate LDA-computed rattler frequency to:

R = (K/M)½

(M Mass of Cs) Gives: K Effective force constant for rattler mode

K Measure of strength (weakness!) of guest

atom-host atom interaction.

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 (almost “unbound”!)Largest alkali atom (Cs) in largest possible clathrate cage (Sn28)!

Conclusions: Phonons • 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 (Hypothetical material!)

– Prediction of extremely low frequency “rattling” modes of Cs guests

– Possibly extremely low thermal conductivity?• Simple model for trend in Cs rattler modes (28-atom cage)

as host changes from Si to Ge to Sn.

Comments & Conclusions• Clathrates are interesting “new” materials!• Experimental measurements (G. Nolas, et al.) show guest-

containing materials have very low thermal conductivities. Mainly Type I materials.

• Molecular dynamics simulations on Sr6Ge46

[J. Dong, O. Sankey, C. Myles, Phys. Rev. Lett. 86, 2361 (2001)] confirm this. Thermoelectric properties is another talk!

• On-going & future work:– Thermodynamic properties (J.J. Dong)

– Thermal conductivity calculations

– Carbon clathrates (not made in lab yet). Should be very “hard” materials