Colloquium, University of North Texas Clathrate Semiconductors: Novel Crystalline Phases of the Group IV Elements Charles W. Myles Professor, Department of Physics Texas Tech University Charley.Myles@ttu.edu http://www.phys.ttu.edu/~cmyles Colloquium, University of North Texas Tuesday, April 19, 2005

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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: Univ. of South Florida Materials synthesis & electrical characterization Chris Kendziora: Naval Research Labs Experimentalist: Raman spectroscopy Jan Gryko: Jacksonville State Experimentalist: Materials synthesis

Outline Introduction to clathrates Crystal structures. Contrast to diamond structure Brief discussion of computational method Sn clathrates (Types I & II) Equations of state (Etot vs. volume) Electronic bandstructures (Ek) Vibrational (phonon) properties (ωk) Raman spectra & comparison with experiment Si, Ge, & Sn clathrates (Type II)

  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: (α-tin or gray tin) - Semimetal

Si, Ge, Sn: The clathrates. Sn: (β-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 the diamond structure, all Group IV atoms are 4-fold coordinated in sp3 bonding configurations. Bond angles: Distorted tetrahedra  Distribution of angles instead of the 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”? The 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) & 28 atom (hexakaidecahedron) Face Centered Cubic, 136 atoms per cubic unit cell.

Clathrate Building Blocks 24 atom cage: Type I Clathrate Si46, Ge46, Sn46, (C46?) Simple Cubic  20 atom cage: Type II Clathrate Si136, Ge136, Sn136 (C136?) Face Centered Cubic  28 atom cage:

Clathrate Lattices Type I Clathrate  Si46, Ge46, Sn46 simple cubic [100] direction Type II Clathrate  Si136, Ge136, Sn136 face centered cubic [100] direction

Group IV Clathrates Synthesis: NaxSi46 (A theorists view!) 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 a Zintl phase NaSi compound. An ionic compound containing Na+ and (Si4)-4 ions Heat to thermally decompose. Some Na  vacuum. Si atoms reform into a clathrate framework around Na. Cages contain Na guests

Type I Clathrate (with guest “rattlers”) 20 atom cage with guest atom  [100] direction + 24 atom cage with guest atom  [010] direction

Guest Modes  Rattler Modes Pure materials: Semiconductors. Guest-containing materials: Some are superconducting materials (Ba8Si46) from sp3 bonded, Group IV atoms! Guests are weakly bonded in cages:  A 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

Guest Modes  Rattler Modes: Possible use as thermoelectric materials. Good thermoelectrics should have low thermal conductivity! Guest Modes  Rattler Modes: A focus of recent experiments. Heat transport theory: The low frequency rattler modes can scatter efficiently with the acoustic modes of the host  Lowers thermal conductivity  A good thermoelectric! Among materials of experimental interest are tin (Sn) clathrates. Mainly Type I. Also, Si & Ge, Type II.

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 a lattice geometry from experiment or guessed (interatomic distances & bond angles). Use the supercell approximation Interatomic forces act to relax the lattice to an equilibrium configuration (distances, angles). Schrödinger Eqtn. for interacting electrons. Newton’s 2nd Law for atomic motion.

Equations of State The total binding energy is minimized in the LDA by optimizing the internal coordinates at a given volume. Repeat the calculation for several volumes. Gives the minimum energy configuration.  An LDA binding energy vs. volume curve. To save computational effort, fit this to an empirical equation of state (4 parameters): the “Birch-Murnaghan” equation of state.

Birch-Murnaghan Eqtn of State Fit the LDA total binding energy vs. volume curve to E(V) = E0 + (9/8)K0V0[(V0/V)⅔ - 1]2 {1 + (½)(4 - K´)[1 - (V0/V)⅔]} 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 Solids Birch-Murnhagan fits to LDA E vs 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 α-Sn: Sn46: V: 12% larger E: 41 meV higher Sn136: 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 α-Sn: Sn46 -- V: 12% larger, E: 41 meV higher, K0: 13% “softer” Sn136 -- V: 14% larger, E: 38 meV higher, K0: 13% “softer”

Ground State Properties Once the equilibrium lattice geometry is obtained, all ground state properties can be obtained at the minimum energy volume. Electronic bandstructures Vibrational dispersion relations Bandstructures At the relaxed lattice configuration, (“optimized geometry”) use the one electron Hamiltonian + LDA many electron corrections to solve the Schrödinger Equation for bandstructures Ek.

Bandstructures NB= # of valence bands Ne = # valence electrons per atom NA= # atoms per cell  NB = Ne  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 Bands M. L Cohen & J 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 (α-Sn): A semimetal (Eg = 0) 

Sn46 & Sn136 Bandstructures C.W. Myles, J.J. Dong, O.F. Sankey, Phys. Rev. B 64, 165202 (2001) The LDA UNDER-estimates bandgaps! Sn46 Sn136   LDA gap Eg  0.86 eV LDA gap Eg  0.46 eV Semiconductors of pure tin!!!! (Hypothetical materials! Indirect band gaps)

Compensation Guest-containing clathrates: Valence electrons from the guests go to the conduction band of the host (heavy doping!), changing the material from semiconducting to metallic. For thermoelectric applications, we want semiconductors!! COMPENSATE for this by replacing some host atoms in the framework by Group III or Group II atoms (charge compensates). Gets a semiconductor back! Sn46: Semiconducting. Cs8Sn46: Metallic. Cs8Ga8Sn38 & Cs8Zn4Sn42: Semiconducting Later: Si136,Ge136, Sn136: Semiconducting. Na16Cs8Si136, Na16Cs8Ge136, Cs24Sn136: Metallic

ASSUME an ordered structure For EACH guest-containing clathrate, including those with compensating atoms in the framework: THE ENTIRE LDA procedure is repeated: LDA total energy vs. volume curve  Equation of State Birch-Murnhagan Eqtn fit to LDA results. At the minimum energy volume, compute the bandstructures & the lattice vibrations. For the compensated materials: ASSUME an ordered structure

Cs8Ga8Sn38 & Cs8Zn4Sn42 Bands C. W. Myles, J. J. Dong, O. F Cs8Ga8Sn38 & Cs8Zn4Sn42 Bands C.W. Myles, J.J. Dong, O.F. Sankey, Phys. Rev. B 64, 165202 (2001). The LDA UNDER-estimates bandgaps! Cs8Ga8Sn38 Cs8Zn4Sn42   LDA gap Eg  0.61 eV LDA gap Eg  0.57 eV Semiconductors (Materials which have been synthesized. Indirect band gaps)

Lattice Vibrations (Phonons) At the optimized LDA geometry: Calculate the total ground state energy: Ee(R1,R2,R3, …..RN) Harmonic Approx.: “Force constant” matrix: Φ(i,i´)  (∂2Ee/∂Ui∂Ui´) Ui = atomic displacements from equilibrium. Instead of directly computing derivatives, we use the Finite displacement method: Compute Ee for many different (small; harmonic approximation!) Ui Compute forces  Ui. Dividing forces by Ui gives Φ(i,i´) & thus the dynamical matrix Dii´(q) used in the lattice vibration calculation.

Phonons det[Dii(q) - ω2δii´] = 0 force constants! NO FITS TO DATA! Group theory limits the number & symmetry of the Ui required. (These materials have high symmetry!). Use positive & negative Ui for each symmetry: Cancels out 3rd order anharmonicity (beyond the harmonic approximation). Once all Φ(i,i´) have been computed, do lattice dynamics! Lattice dynamics in the harmonic approximation:  The classical eigenvalue (normal mode) problem det[Dii(q) - ω2δii´] = 0 The dynamical matrix Dii´(q) obtained from the force constant matrix Φ in the usual way. First principles force constants! NO FITS TO DATA!

NA = # of atoms per unit cell  NB = 3  NA Eigenvalues: Squares of the vibrational frequencies ω2(q) (“phonon dispersion relations) NB = # of branches (modes) in ω(q) NA = # of atoms per unit cell  NB = 3  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 Phonons W. Weber, Phys. Rev. B 15, 4789 (1977). 3 Acoustic branches 3 Optic branches

Sn46 & Sn136 Phonons C. W. Myles, J. Dong, O. F. Sankey, C Sn46 & Sn136 Phonons C.W. Myles, J. Dong, O.F. Sankey, C. Kendziora, G.S. Nolas, Phys. Rev. B 65, 235208 (2002) Sn46 Sn136 Flat optic bands! Large unit cell  Small Brillouin Zone reminiscent of “zone folding”

Guest-Containing Clathrates as Thermoelectrics Guest atoms: Weakly bound to the clathrate lattice. Lattice Framework: Fully sp3 tetrahedrally bonded.  The guest atom electrons don’t participate in the bonding or affect electronic transport very strongly. The guests have low energy (“rattling”) phonon modes (guest atoms vibrating in the cages with small force constants). We will see this explicitly later.  These STRONGLY affect the vibrational properties & thus the phonon-phonon scattering & thermal conductivity.

 Lowers the thermal conductivity  A good thermoelectric! Good thermoelectrics should have low thermal conductivity! Guest Modes  Rattler Modes: A focus of experiments! Heat transport theory says: The low frequency rattler modes can scatter efficiently with the acoustic modes of the host  Lowers the thermal conductivity  A 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. W. Myles, J. J. Dong, O. F. Sankey, C Cs8Ga8Sn38 Phonons C.W. Myles, J.J. Dong, O.F. Sankey, C. Kendziora, G.S. Nolas, Phys. Rev. B 65, 235208 (2002)  Ga modes Compare to Sn46 results.  Cs guest “rattler” modes (~25 - 40 cm-1) “Rattler” modes: Cs motion in the large & small cages

Raman Spectra Do the group theory necessary to determine the Raman active modes. Raman spectroscopy probes only the modes at zone center (q = 0). Vibrational Frequencies are calculated from first principles as described. Estimate the Raman scattering intensities using an empirical (two parameter) bond polarization model.

theoretical rattler (& other!) modes in good agreement!  UNAMBIGUOUS C.W. Myles, J.J. Dong, O.F. Sankey, C. Kendziora, G.S. 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 the 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 Recent experiments: Focused 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 the rattling modes as the 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. W. Myles, J. J. Dong, O. F. Sankey, Phys Phonons C.W. Myles, J.J. Dong, O.F. Sankey, Phys. Status Solidi B 239, 26 (2003)  Na Cs Na16Cs8Si136 Na16Cs8Ge136  Na  Cs  Na  Cs 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

Si136, Na16Cs8Si136 Na16Cs8Ge136 Raman Spectra 1st principles frequencies. G.S Nolas, C. Kendziora, J. Gryko, A. Poddar, J.J. Dong, C.W. Myles, O.F. Sankey J. Appl. Phys. 92, 7225 (2002). Experimental & theoretical rattler (& other) modes are in very good agreement! Not shown: Detailed identification of frequencies & symmetries of observed Raman modes by comparison with theory.

There is reasonable agreement of theory & experiment for Raman spectra, especially for the “rattling” modes of Cs in the 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. W. Myles, J. J. Dong, O. F. Sankey, Phys Cs24Sn136 Phonons C.W. Myles, J.J. Dong, O.F. Sankey, Phys. Status Solidi B 239, 26 (2003) Cs24Sn136: A hypothetical material! Cs in the large (28-atom) cages is extremely anharmonic & “loose” fitting!  Very small frequencies! Cs rattler modes (20-atom cages) ~ 25 - 30 cm-1 Cs rattler modes (28-atom cages) ~ 5 - 7 cm-1 Thermoelectric applications?

 Potential 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! The effective potential for Cs in the 28 atom cage is very anharmonic. Cs is very loosely bound there. The calculations were done in the harmonic approximation.  More accurate calculations taking anharmonicity into account are needed!  Potential thermoelectric applications! DATA IS NEEDED!

Trend Si  Ge  Sn Na16Cs8Si136 (~ 65 - 67 cm-1) The trend in the Cs “rattling” modes in the large (28-atom) cages as the host changes Si  Ge  Sn Na16Cs8Si136 (~ 65 - 67 cm-1) Na16Cs8Ge136 (~ 21 - 23 cm-1) Cs24Sn136 (~ 5 - 7 cm-1) Correlates the with size of the cages in comparison with the “size” of a Cs atom.

A Simple Model for the Trend The 28-atom cage size in the host lattice compared with a Cs guest atom “size”. For a host atom X = Si, Ge, Sn, define: Δr  rcage - (rX + rCs) rcage  LDA-computed average Cs-X distance rX  (½)(LDA-computed average X-X near- neighbor distance)  The “covalent radius” of atom X rCs  Ionic radius of Cs (1.69 Å) (rX + rCs)  Cs-X distance if Cs were tight fitting in a cage  Δr  How “oversized” the cage is compared to Cs “size”. A geometric measure of how loosely fitting a Cs atom is inside a 28-atom cage.

(M  Mass of Cs)  This gives: Couple this geometric model with a simple harmonic oscillator model for Cs in the cage. Assume that only Cs moves in its oversized 28-atom cage. Equate the LDA-computed rattler frequency to: ωR = (K/M)½ (M  Mass of Cs)  This gives: K  An effective force constant for the rattler mode K  A measure of the strength (weakness!) of the 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 is weakly bound! Ge28 cage: Δr  1.22 Å  “oversized” K  0.2 eV/(Å)2 KGe-Ge  10 eV/(Å)2  Cs is 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 the 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 Type II clathrate: Cs24Sn136 (A hypothetical material!) Prediction of extremely low frequency “rattling” modes of the Cs guests Possibly extremely low thermal conductivity? A simple model for the trend in the Cs rattler modes (28-atom cage) as the host changes from Si to Ge to Sn.

Comments & Conclusions Group IV clathrates are interesting “new” materials! Experimental measurements (G. Nolas, et al.) show guest-containing Ge & Sn materials have very low thermal conductivities. Mainly Type I materials. Molecular dynamics simulations on Sr6Ge46 [J.J. Dong, O.F. Sankey, C.W. 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 the lab yet). Should be very “hard” materials

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 

Clathrate Structures Type I Clathrate Si46, Ge46, Sn46 simple cubic 24 atom cages Type I Clathrate Si46, Ge46, Sn46 simple cubic 20 atom cages Type II Clathrate Si136, Ge136, Sn136 face centered cubic 28 atom cages

Clathrate Building Blocks 24 atom cage:  20 atom cage:  28 atom cage: 