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Theoretical Investigation of Carbon-Based Clathrate Materials

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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.

5th Motorola Workshop on Computational Materials and Electronics, Nov. 13-14, 2003

1Supported in part by the NSF

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.

- 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

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

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

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(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

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”)

20 atom cage

with guest atom

[100]

direction

+

24 atom cage

with guest atom

[010]

direction

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

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.

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.

- 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.

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).
- Schrdinger 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 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

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”

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 Schrdinger Equation for bandstructures Ek.

Bandstructures

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)

- 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!)

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 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!)

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

- 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!

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