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Theoretical Investigation of Carbon-Based Clathrate MaterialsPowerPoint Presentation

Theoretical Investigation of Carbon-Based Clathrate Materials

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Theoretical Investigation of Carbon-Based Clathrate Materials

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Theoretical Investigation of Carbon-Based Clathrate Materials

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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

- 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.

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!

- Diamond:Insulator or wide bandgap
- 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.

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:

Type I Clathrate

Si46, Ge46, Sn46,(C46?):

simple cubic

[100]

direction

Type II Clathrate

Si136, Ge136, Sn136 ,(C136?):

face centered cubic

[100]

direction

- 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?

20 atom cage

with guest atom

[100]

direction

+

24 atom cage

with guest atom

[010]

direction

- 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!
- 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

- Schrdinger Equation for interacting electrons, Newton’s 2nd Law of motion for atoms.
- 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.

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

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”

- 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.

C46 C136

LDA gap Eg 3.67 eVLDA gap Eg 3.61 eV

Semiconductors(Hypothetical materials. Indirect band gaps)

The LDAUNDER-estimates bandgaps

- 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”

- 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.

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)

- 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!