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Theoretical Investigation of Carbon-Based Clathrate Materials

Theoretical Investigation of Carbon-Based Clathrate Materials. Charles W. Myles, Texas Tech U. Jianjun Dong, Auburn U. Otto F. Sankey, 1 Arizona State U. 5 th Motorola Workshop on Computational Materials and Electronics, Nov. 13-14, 2003. 1 Supported in part by the NSF. Group IV Crystals.

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Theoretical Investigation of Carbon-Based Clathrate Materials

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  1. Theoretical Investigation of Carbon-Based Clathrate Materials Charles W. Myles, Texas Tech U. Jianjun Dong, Auburn U. Otto F. Sankey,1 Arizona State U. 5th Motorola Workshop on Computational Materials and Electronics, Nov. 13-14, 2003 1Supported in part by the NSF

  2. Group IV Crystals • Si, Ge, Sn: Ground state crystal structure = Diamond Structure.Each atom tetrahedrally coordinated, sp3 bonding.Bond angles:Perfect, tetrahedral = 109.5º.Si, Ge: Semiconductors. Sn(-tin or gray tin): Semimetal. • Sn: (-tin or white tin) - body centered tetragonal lattice, 2 atoms per unit cell. Metallic.

  3. C Crystal Structures • Graphite & Diamond Structures • Diamond:Insulator or wide bandgap semiconductor. • Graphite:Planar structure:  sp2 bonding 2d metal (in plane) • Ground state(lowest energy configuration) is graphiteat zero temperature & atmospheric pressure. Graphite-diamond total energy difference is VERYsmall! • Other Carbon Crystal Structures “Buckyballs” (C60)  “Buckytubes” (nanotubes), other fullerenes 

  4. Clathrates • Crystalline Phases of Group IV elements: Si, Ge, Sn Not C yet, which motivates this work! “New” materials, but known (for Si) since 1965! J. Kasper, P. Hagenmuller, M. Pouchard, C. Cros, Science 150, 1713 (1965) • Like the diamond structure, all Group IV atoms are 4-fold coordinated insp3bonding configurations. Distorted tetrahedra  Distribution of Bond angles instead of perfect 109.5º • Pure materials: Metastable, high energy phases of Si, Ge, Sn. Few pure materials yet. Compounds with groups I & II atoms (Na, K, Cs, Ba). • Applications (Si, Ge, Sn): Thermoelectrics. • Open, cage-like structures. Large “cages” of group IV atoms. • Hexagonal & pentagonal rings, fused together to form “cages” of 20, 24, & 28 atoms

  5. Si46, Ge46, Sn46, (C46?): (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, (C136?): (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.

  6. 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:

  7. Clathrate Lattices(Courtesy, George S. Nolas, U. of South Florida) Type I Clathrate  Si46, Ge46, Sn46,(C46?): simple cubic [100] direction Type II Clathrate  Si136, Ge136, Sn136 ,(C136?): face centered cubic [100] direction

  8. 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, Ba). • Many experiments on Si, Ge, & Sn-based clathrates! • C Clathrateswith Li in the cages (so far hypothetical materials): Possible high pressure synthesis starting with Li intercalated graphite?

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

  10. Clathrates • Pure materials: Semiconductors. • Guest-containing materials: • Some are superconducting materials (Ba8Si46) from sp3bonded, Group IV atoms. • Guests weakly bonded in cages: Minimal effect on electronic transport • Host valence electrons taken up in sp3bonds  Guest valence electrons go to conduction band of host ( heavy n-type doping density). With no compensation, these aremetallicmaterials. • Guests vibrate with low frequency (“rattler”) modes Strong effect on vibrational properties Guest Modes  Rattler Modes

  11. Possible applications of Si, Ge, & Sn clathrates as thermoelectric materials. Good thermoelectrics have low thermal conductivity! • Guest Modes  Rattler Modes: Heat transport theory:Low frequency rattler modes can scatter efficiently with host acoustic modes  Lowers the thermal conductivity  Good thermoelectric • Experiments show:Some guest containing Ge & Sn clathrates have low thermal conductivities.

  12. Possible technological applications of(so far hypothetical) C Clathrate materials: 1. Very hard materials (Speculation: possibly harder than diamond) 2.Large bulk moduli materials 3. High Tc superconductors? 4.C materials with high n-type doping (Li & other alkali metals in the cages). Even if these turn out not to be true, it is of theoretical interest to investigate a possible new crystalline phase of carbon. Since Si, Ge & Sn all form clathrates, this phase should also be possible for C.

  13. Calculations • Computational package: VASP- Vienna Austria Simulation Package. First principles! Many electron effects:LocalDensityApproximation (LDA) Can include GGA corrections if needed. Exchange-correlation:Ceperley-Adler Functional Ultrasoft pseudopotentials, Planewave basis • Extensively tested on a wide variety of systems • We’ve used this previously to successfully describe properties of Si, Ge, & Sn-based clathrates. Good agreement with experiment for a number of properties.

  14. 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 offrequencies & symmetries of several observed Raman modes by comparison with theory.

  15. C Clathrates: We’ve computed equilibrium geometries, 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). • Schrdinger Equation for interacting electrons, Newton’s 2nd Law of motion for atoms. Equations of State • Total binding energy is minimized by optimizing the internal coordinates at given volume. • Repeat for several volumes. Gives LDA binding energy vs. volume curve. Fit to empirical equation of state (4 parameter): “Birch-Murnaghan” equation of state.

  16. Equations of State for C SolidsBirch-Murnhagan fits to LDA E vs. V curves C Clathrates: (compared to diamond) expanded volume high energy phases “negative pressure” 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

  17. C Solids: Equation of State ParametersBirch-Murnhagan fits to LDA E vs. V curves Å C Clathrates:(compared to diamond): Expanded volume, high energy, “softer”C phases C46 -- V: 15% larger, E: 0.16 eV higher, K0: 16% “softer” C136 -- V: 16% larger, E: 0.13 eV higher, K0: 15% “softer”

  18. Ground State Properties • Equilibrium lattice geometry:  Cubic Lattice Constant a (& other internal coordinates) C46 a = 6.62 Å C136 a = 6.74 Å • Once equilibrium geometry is obtained, all ground state properties can be obtained (at min. energy volume) • Electronic bandstructures • Vibrational dispersion relations Bandstructures: At the equilibrium geometry configuration use the one electron Hamiltonian + LDA many electron corrections to solve the Schrdinger Equation for bandstructures Ek.

  19. Bandstructures C46 C136   LDA gap Eg 3.67 eVLDA gap Eg 3.61 eV Semiconductors(Hypothetical materials. Indirect band gaps) The LDAUNDER-estimates bandgaps

  20. Lattice Vibrations (Phonons) • At optimized LDA geometry, the total ground state energy: Ee(R1,R2,R3, …..RN) • Harmonic Approx.:“Force constant” matrix: (i,i)  (2Ee/Ui Ui) Ui= displacements from equilibrium. Instead of directly computing derivatives, we use the • Finite displacement method:Compute Eefor many different (small; harmonic approx.)Ui. Compute forces  Ui. Group theory limits number & symmetry of the Uirequired. Positive & negative Uifor each symmetry: Cancels out 3rd order anharmonicity (beyond harmonic approx.). Once all unique (i,i) are computed, do lattice dynamics in the harmonic approximation: det[Dii(q) - 2 ii] = 0 (NO FITTING) (Of course, for C clathrates, there is NO data to fit!)

  21. C46 C136 Phonons max  1269 cm-1 max  1257 cm-1 Flat optic bands! Large unit cell  Small Brillouin Zone reminiscent of “zone folding”

  22. Li-Containing C Clathrates • Guest-containing clathrates:Impurities in the cages. In general, guest valence electrons go to the conduction band of the host (heavy n-type doping). Change material from semiconducting tometallic. • For C Clathrates consider Li in large & small cages: Type I: Li8C46 Type II: Li24C136 1. Compute some basic properties(hypothetical material) 2. Investigate the possibility of high pressure synthesis of Li-containing C clathrates from Li intercalated graphite. Compute enthalpies to determine whether this is favored.

  23. Li-Containing C ClathratesEquation of State ParametersBirch-Murnhagan fits to LDA E vs. V curves Å Li-containing C Clathrates:(compared to pure phases): Expanded volume, “softer” phases Lattice Constants: C46 6.62 Å, Li8C46 6.68 Å C136 6.74 Å, Li24C136  6.87 Å  Li expands cage size(too large to fit easily inside!)

  24. High pressure synthesis of Li8C46&Li24C136 starting with Li intercalated graphite (LiC6): • Under high pressure, can LiC6 be converted to Li8C46 orLi24C136 ?

  25. Consider the reactions: LiC6 Li8C46 + Cgraphite LiC6 Li8C46 + Cdiamond 4LiC6 Li24C136 + Cgraphite 4LiC6 Li24C136 + Cdiamond • J.J. Dong’s calculations:Enthalpy change vs. pressure to determine whether these reactions are favorable.

  26. Transition pressures 54 - 62 GPa (Transition from positive to negative H, where a reaction is favored)

  27. Summary & Conclusions • Pure C Clathrates-- Predictions 1. Compared to diamond: Expanded volume, high energy, “softer”C phases 2. Equilibrium lattice geometry: Lattice Constants a =6.62 Å (C46 ), 6.74 Å (C136) 3. Bandstructures: Semiconductors Eg 3.67 eV(C46 )  3.61 eV (C136) 4. Phonon spectra: max  1269 cm-1 (C46 ) 1257 cm-1 (C136 )

  28. Li-Containing C Clathrates-- Predictions 1. Compared to pure clathrate phases Expanded volume, high energy, “softer”C phases 2. Equilibrium lattice geometry: Lattice Constants a =6.68 Å(Li8C46),6.87 Å (Li24C136 ) 3. Bandstructures: Metallic 4. Synthesis from Li intercalated graphite? Transition pressures(for H  0) 54 - 62 GPa  Li - containing C clathrates would likely be difficult to synthesize from Li interalated graphite!

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