Rattling Atoms in Group IV Clathrate Materials Charles W. Myles Professor, Department of Physics Texas Tech University 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 (E tot vs. volume) –Electronic bandstructures (E k ) –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: ns 2 np 2 [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 sp 3 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 sp 2 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” (C 60 )  “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 sp 3 bonding configurations. Bond angles: Distorted tetrahedra  Distribution of angles instead of perfect tetrahedral 109.5º Lattice contains hexagonal & pentagonal rings, fused together with sp 3 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 (X 46 ) & Type II (X 136 ) X = Si, Ge, or Sn

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

Si 46, Ge 46, Sn 46 : (  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. Si 136, Ge 136, Sn 136 : (  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 Clathrate Si 46, Ge 46, Sn 46 simple cubic Type II Clathrate Si 136, Ge 136, Sn 136 face centered cubic 20 atom cages 28 atom cages

Clathrate Lattices Type I Clathrate  Si 46, Ge 46, Sn 46 simple cubic Type II Clathrate  Si 136, Ge 136, Sn 136 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: Na x Si 46 (A theorists view!) –Start with Zintl phase NaSi compound. –Ionic compound containing Na + and (Si 4 ) -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 cage with guest atom  [100] direction [010] direction

Clathrates Pure materials: Semiconductors. Guest-containing materials: –Some are superconducting materials (Ba 8 Si 46 ) from sp 3 bonded, Group IV atoms! –Guests weakly bonded in cages:  Minimal effect on electronic transport –Host valence electrons taken up in sp 3 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). –Schr  dinger Eq. for interacting electrons. Newton’s 2 nd 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 State Fit LDA total binding energy vs. volume curve to E(V) = E 0 + (9/8)K 0 V 0 [(V 0 /V)  -1] 2  {1 +  (4-K  )[1- (V 0 /V)  ]} 4 Parameters: E 0  Minimum binding energy V 0  Volume at minimum energy K 0  Equilibrium bulk modulus K   dK 0 /dP  Pressure derivative of K 0

Equations of State for Sn Solids Birch-Murnhagan fits to LDA E vs. V curves Sn Clathrates: expanded volume, high energy, metastable Sn phases Compared to  -Sn: Sn 46 V: 12% larger E: 41 meV higher Sn 136 V: 14% larger E: 38 meV higher  Clathrates: “Negative pressure” phases!

Equation of State Parameters Birch-Murnhagan fits to LDA E vs. V curves Sn Clathrates: Expanded volume, high energy, “soft” Sn phases Compared to  -Sn: Sn V: 12% larger, E: 41 meV higher, K 0 : 13% “softer” Sn V: 14% larger, E: 38 meV higher, K 0 : 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 Schr  dinger Eq. for bandstructures E k. Ground State Properties

Bandstructures N B = # of valence bands N e = # valence electrons / atom N A = # atoms per cell  N B = N e x N A Diamond Structure & Clathrates: N e = 4 Diamond: N A = 2  N B = 8 Clathrates: X 46 : N A = 46  N B = 184 X 136 : N A = 136  N B = 544

Diamond Structure Sn Bands M.L Cohen & J. Chelikowsky, Electronic Structure and Optical Properties of Semiconductors, (Springer) Solid State Science, 75 (1989). Diamond Structure Sn (  -Sn): A semimetal (E g = 0) *

Sn 46 & Sn 136 Bandstructures C.W. Myles, J. Dong, O. Sankey, Phys. Rev. B 64, (2001). The LDA UNDER-estimates bandgaps! LDA gap E g  0.86 eV LDA gap E g  0.46 eV Semiconductors of pure tin !!!! (Hypothetical materials. Indirect band gaps) Sn 46 Sn 136  

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! Sn 46 : Semiconducting. Cs 8 Sn 46 : Metallic. Cs 8 Ga 8 Sn 38 & Cs 8 Zn 4 Sn 42 : Semiconducting Later: Si 136,Ge 136, Sn 136 : Semiconducting. Na 16 Cs 8 Si 136, Na 16 Cs 8 Ge 136, Cs 24 Sn 136 : 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.

Cs 8 Ga 8 Sn 38 & Cs 8 Zn 4 Sn 42 Bands C.W. Myles, J. Dong, O. Sankey, Phys. Rev. B 64, (2001). The LDA UNDER-estimates bandgaps! LDA gap E g  0.61 eV LDA gap E g  0.57 eV Semiconductors (Materials which have been synthesized. Indirect band gaps) Cs 8 Ga 8 Sn 38 Cs 8 Zn 4 Sn 42  

Lattice Vibrations (Phonons) At optimized LDA geometry: Calculate total ground state energy: E e (R 1,R 2,R 3, …..R N ) Harmonic Approx.: “Force constant” matrix:  (i,i  )  (  2 E e /  U i  U i ) U i = atomic displacements from equilibrium. Instead of directly computing derivatives, we use Finite displacement method: Compute E e for many different (small; harmonic approx.) U i Compute forces  U i. Dividing forces by U i gives  (i,i  ) & thus dynamical matrix D ii (q).

Group theory limits number & symmetry of U i required. (Materials have high symmetry). Positive & negative U i for each symmetry: Cancels out 3 rd 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[D ii (q) -  2  ii ] = 0 Dynamical matrix D ii (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) N B = # of branches (modes) in  (q) N A = # of atoms / unit cell  N B = 3 x N A Diamond Structure: N A = 2  N B = 6 Clathrates: X 46 : N A = 46  N B = 138 X 136 : N A = 136  N B = Acoustic branches, N B - 3 Optic branches

Diamond Structure Sn Phonons W. Weber, Phys. Rev. B 15, 4789 (1977). 3 Acoustic branches 3 Optic branches

Sn 46 & Sn 136 Phonons C.W. Myles, J. Dong, O. Sankey, C. Kendziora, G. Nolas, Phys. Rev. B 65, (2002) Flat optic bands! Large unit cell  Small Brillouin Zone reminiscent of “zone folding” Sn 46 Sn 136

Guest-Containing Clathrates as Thermoelectrics Guest atoms: Weakly bound to clathrate framework. Framework: Fully sp 3 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.

Cs 8 Ga 8 Sn 38 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: Cs motion in large & small cages Compare to Sn 46 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, (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 Phonons With “rattling”atoms Current experiments: Focus on rattling modes in Type II clathrates (thermoelectric applications).  Theory: Given success with Cs 8 Ga 8 Sn 38 : Look at phonons & rattling modes in Type II clathrates  Search for trends in rattling modes as host changes from Si  Ge  Sn –Na 16 Cs 8 Si 136 : Have Raman data & predictions –Na 16 Cs 8 Ge 136 : Have Raman data & predictions –Cs 24 Sn 136 : 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) ~ cm -1 ~ cm -1 Cs rattlers (28-atom cages) Cs rattlers (28-atom cages ) ~ cm -1 ~ cm -1 Na 16 Cs 8 Si 136 Na 16 Cs 8 Ge 136

1 st 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. Si 136, Na 16 Cs 8 Si 136 Na 16 Cs 8 Ge 136 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 Na 16 Cs 8 Si 136 (~ cm -1 ) Na 16 Cs 8 Ge 136 (~ cm -1 )

Cs 24 Sn 136 Phonons C. Myles, J. Dong, O. Sankey, submitted, Phys. Status Solidi B Cs rattler modes (20-atom cages) ~ cm -1 Cs rattler modes (28-atom cages) ~ cm -1 Cs 24 Sn 136 : Hypothetical material! Cs in large (28- atom) cages: Extremely anharmonic & “loose” fitting.  Very small frequencies! Thermoelectric applications?

Predictions Cs 24 Sn 136 : Low frequency “rattling” modes of Cs guests in 20 atom cages (~25-30 cm -1 ) & in 28-atom cages (~ 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 Na 16 Cs 8 Si 136 (~ cm -1 ) Na 16 Cs 8 Ge 136 (~ cm -1 ) Cs 24 Sn 136 (~ 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  r cage - (r X + r Cs ) r cage  LDA-computed average Cs-X distance r X   (LDA-computed average X-X near- neighbor distance)  Covalent radius of atom X r Cs  Ionic radius of Cs (1.69 Å) (r X + r Cs )  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 Si 28 cage: Δr  1.18 Å  “oversized” K  2.2 eV/(Å) 2 K Si-Si  10 eV/(Å) 2  Cs weakly bound Ge 28 cage: Δr  1.22 Å  “oversized” K  0.2 eV/(Å) 2 K Ge-Ge  10 eV/(Å) 2  Cs very weakly bound Largest, Sn 28 cage: Δr  1.62 Å  EXTREMELY “oversized” K  0.02 eV/(Å) 2, K Sn-Sn  8 eV/(Å) 2  Cs extremely weakly bound (almost “unbound”!) Largest alkali atom (Cs) in largest possible clathrate cage (Sn 28 )!

Conclusions: Phonons Type I clathrate: Cs 8 Ga 8 Sn 38 –Good agreement with Raman data for Cs rattler modes & also host framework modes! Type II clathrates: Na 16 Cs 8 Ge 136, Na 16 Cs 8 Si 136 –Good agreement with Raman data for Cs rattler modes & also host framework modes! Type II clathrate: Cs 24 Sn 136 (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 Sr 6 Ge 46 [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