Clathrate Semiconductors: Novel Crystalline Phases of the Group IV Elements

1. 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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6. 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

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

8. Group IV Elements  • Valence electron configuration: ns2 np2 [n = 2, C; n = 3, Si; n = 4, Ge; n = 5, Sn] 

9. Group IV Crystals • Si, Ge, Sn:Ground state crystalline structure = Diamond Structure • Each atom tetrahedrally (4-fold) coordinated (4 nearest-neighbors) withsp3 covalent bonding • Bond angles:Perfect, tetrahedral = 109.5º Si, Ge: Semiconductors Sn: (α-tin or gray tin) - Semimetal

10. Sn:(β-tin or white tin) - body centered tetragonal lattice, 2 atoms per unit cell. Metallic. • Si, Ge, Sn:The clathrates.

11. 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 insp3 bonding configurations. • Bond angles:Distorted tetrahedra Distribution of angles instead of the perfect tetrahedral 109.5º • Lattice contains hexagonal & pentagonal rings, fused together with sp3bonds to form large “cages”.

12. 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

13. Why “clathrate”? The same crystal structure as clathrate hydrates (ice).

14. Si46, Ge46, Sn46: (Type IClathrates) 20atom (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 IIClathrates) 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.

15. 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:

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

17. 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 a Zintl phase NaSicompound. • An ionic compound containing Na+ and (Si4)-4 ions • Heat to thermally decompose. Some Na vacuum. Siatoms reform into a clathrate framework around Na. • Cages contain Naguests

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

19. 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 sp3bonds • 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

20. 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.

21. Calculations • Computational package: VASP- Vienna Austria Simulation Package. First principles! Many electron effects: LocalDensityApproximation (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.

22. 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.

23. 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.

24. 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

25. Equations of State for Sn SolidsBirch-Murnhagan fits to LDAE 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!

26. Equation of State ParametersBirch-Murnhagan fits to LDAE 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”

27. 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.

28. 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

29. Diamond Structure Sn BandsM.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) 

30. Sn46 & Sn136 BandstructuresC.W. Myles, J.J. Dong, O.F. Sankey, Phys. Rev. B64, 165202 (2001) TheLDAUNDER-estimates bandgaps! Sn46 Sn136   LDA gap Eg 0.86 eVLDAgap Eg 0.46 eV Semiconductorsof pure tin!!!! (Hypothetical materials! Indirect band gaps)

31. 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!! • COMPENSATEfor 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

32. For EACH guest-containing clathrate, including those with compensating atoms in the framework: • THE ENTIRELDAprocedure 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

33. Cs8Ga8Sn38 & Cs8Zn4Sn42 BandsC.W. Myles, J.J. Dong, O.F. Sankey, Phys. Rev. B64, 165202 (2001). TheLDAUNDER-estimates bandgaps! Cs8Ga8Sn38 Cs8Zn4Sn42   LDA gap Eg 0.61 eVLDA gap Eg 0.57 eV Semiconductors (Materials which have been synthesized. Indirect band gaps)

34. 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 Eefor many different(small; harmonic approximation!)Ui Compute forces  Ui. • Dividing forces by Ui gives Φ(i,i´) & thus thedynamical matrixDii´(q) used in the lattice vibration calculation.

35. Phonons • Group theory limits the number & symmetry of the Uirequired. (These materials have high symmetry!). • Use positive & negative Uifor 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 Φin the usual way. First principles force constants!NO FITS TO DATA!

36. 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

37. Diamond Structure Sn PhononsW. Weber, Phys. Rev. B15, 4789 (1977). 3 Acoustic branches 3 Optic branches

38. Sn46 & Sn136 PhononsC.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”

39. Guest-Containing Clathrates asThermoelectrics • Guest atoms: Weakly bound to the clathrate lattice. • Lattice Framework: Fullysp3 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.

40. 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.

41. Cs8Ga8Sn38 PhononsC.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 - 40cm-1) “Rattler” modes:Cs motion in the large & small cages

42. 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 Frequenciesare calculated from first principles as described. • Estimate the Raman scattering intensities using an empirical (two parameter) bond polarization model.

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

44. Type II Clathrate PhononsWith “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!

45. PhononsC.W. Myles, J.J. Dong, O.F. Sankey, Phys. Status Solidi B 239, 26 (2003) •  Na • Cs Na16Cs8Si136 Na16Cs8Ge136  Na  Cs  Na  Cs Narattlers(20-atom cages)Narattlers(20-atom cages) ~ 118 -121 cm-1 ~ 89 - 94 cm-1 Csrattlers(28-atom cages)Cs rattlers(28-atom cages) ~ 65 - 67 cm-1 ~ 21 - 23 cm-1

46. 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.

47. There is reasonable agreement of theory & experiment for Raman spectra, especially for the “rattling” modes of Csin the large cages in Type IISi & Geclathrates.  UNAMBIGUOUS IDENTIFICATION of low frequency “rattling” modes of Csin Na16Cs8Si136(~ 65 - 67 cm-1) Na16Cs8Ge136(~ 21 - 23 cm-1)

48. Cs24Sn136 PhononsC.W. Myles, J.J. Dong, O.F. Sankey, Phys. Status Solidi B 239, 26 (2003) • Cs24Sn136: • A hypothetical • material! • Csin the large (28-atom) cages is • extremely • anharmonic & • “loose” fitting! •  Very small • frequencies! Csrattlermodes (20-atom cages) ~ 25 - 30 cm-1 Csrattlermodes (28-atom cages) ~ 5 - 7 cm-1 Thermoelectric applications?

49. 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 forCs 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!

50. Trend • The trendin the Cs“rattling” modes in the large(28-atom)cagesas 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 Csatom.