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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
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

group iv crystals
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.



C Crystal Structures

  • Graphite & Diamond Structures
    • Diamond:Insulator or wide bandgap


    • 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 

  • 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
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.

clathrate building blocks
Clathrate Building Blocks

24 atom cage:

Type I Clathrate

Si46, Ge46, Sn46,


Simple Cubic

20 atom cage:

Type II Clathrate

Si136, Ge136, Sn136


Face Centered


28 atom cage:

clathrate lattices courtesy george s nolas u of south florida
Clathrate Lattices(Courtesy, George S. Nolas, U. of South Florida)

Type I Clathrate 

Si46, Ge46, Sn46,(C46?):

simple cubic



Type II Clathrate 

Si136, Ge136, Sn136 ,(C136?):

face centered cubic



group iv clathrates
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?

type i clathrate with guest rattlers
Type I Clathrate(with guest “rattlers”)

20 atom cage

with guest atom




24 atom cage

with guest atom



  • 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

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.
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.

  • 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.

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!
  • 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.
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.
equations of state for c solids birch murnhagan fits to lda e vs v curves
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”


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

c solids equation of state parameters birch murnhagan fits to lda e vs v curves
C Solids: Equation of State ParametersBirch-Murnhagan fits to LDA E vs. V curves


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”

ground state properties
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.


C46 C136

LDA gap Eg 3.67 eVLDA gap Eg 3.61 eV

Semiconductors(Hypothetical materials. Indirect band gaps)

The LDAUNDER-estimates bandgaps

lattice vibrations phonons
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!)


C46 C136


max  1269 cm-1 max  1257 cm-1

Flat optic bands!

Large unit cell  Small Brillouin Zone

reminiscent of “zone folding”

li containing c clathrates
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.

li containing c clathrates equation of state parameters birch murnhagan fits to lda e vs v curves
Li-Containing C ClathratesEquation of State ParametersBirch-Murnhagan fits to LDA E vs. V curves


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!)

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

Li8C46 orLi24C136 ?

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.

Transition pressures 54 - 62 GPa

(Transition from positive to negative H, where a reaction is favored)

summary conclusions
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 )

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!