1. Introduction to Group IV
Clathrate Materials

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

3. Group IV Crystals Si, Ge, Sn: Ground state crystal structure:
= Diamond Structure E
Each atom is tetrahedrally (4-fold) coordinated (with 4 nearest-neighbors) with sp3 covalent bonding
Bond angles: Perfect, tetrahedral = 109.5º
Si, Ge are semiconductors
Sn: (α-tin or gray tin) is a semimetal

4. Another crystalline phase of Sn Si, Ge, Sn: The clathrates.
(β-tin or white tin)
This phase has a body centered tetragonal lattice, with 2 atoms per unit cell. It is metallic.
There is one more crystalline phase of the Column IV elemental solids!!
Si, Ge, Sn: The clathrates.

5. 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 in sp3 bonding configurations.
Bond angles: Distorted tetrahedra  Distribution of angles instead of the perfect tetrahedral 109.5º
Lattice contains hexagonal & pentagonal rings, fused together with sp3 bonds to form large “cages”.

6. X = Si, Ge, or Sn Two common varieties: Type I (X46) & Type II (X136)
Pure materials: Metastable, expanded volume phases of Si, Ge, Sn
Few pure elemental phases. Compounds with Group I & II atoms (Na, K, Cs, Ba).
Potential applications: Thermoelectrics
Open, cage-like structures, with large “cages” of Si, Ge, or Sn atoms.
“Buckyball-like” cages of 20, 24, & 28 atoms.
Two common varieties:
Type I (X46) & Type II (X136)
X = Si, Ge, or Sn

7. Meaning of “Clathrate” ?
From Wikipedia, the free encyclopedia:
“A clathrate or clathrate compound or cage compound is a
chemical substance consisting of a lattice of one type of molecule trapping and containing a second type of molecule. The word comes from the Latin clathratus meaning furnished with a lattice.”
“For example, a clathrate-hydrate involves a special type of gas
hydrate consisting of water molecules enclosing a trapped gas.
A clathrate thus is a material which is a weak composite, with
molecules of suitable size captured in spaces which are left by
the other compounds. They are also called host-guest complexes,
inclusion compounds, and adducts.”

8. Type I clathrate-hydrate crystal structure X8(H2O)46:
Here: Group IV crystals with the same crystal structure as clathrate-hydrates (ice).
Type I clathrate-hydrate crystal structure X8(H2O)46:

9. Si46, Ge46, Sn46: ( 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: ( Type II Clathrates)
28 atom (hexakaidecahedron) cages,
Face Centered Cubic, 136 atoms per cubic unit cell.

10. 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:

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

12. Group IV Clathrates Guests  “Rattlers”
Not found in nature. Lab synthesis. An “art” more than a science!.
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 NaSi compound.
An ionic compound containing Na+ and (Si4)-4 ions
Heat to thermally decompose. Some Na  vacuum.
Si atoms reform into a clathrate framework around Na.
Cages contain Na guests

13. Guest Modes  Rattler Modes
Pure materials: Semiconductors.
Guest-containing materials:
Some are superconducting materials (Ba8Si46) from sp3 bonded, Group IV atoms!
Guests are weakly bonded in the cages:
 A minimal effect on electronic transport
The host valence electrons taken up in sp3 bonds
Guest valence electrons go to the host conduction band
( heavy doping density)
Guests vibrate with low frequency (“rattler”) modes
 A strong effect on vibrational properties
Guest Modes  Rattler Modes

14.  Lowers the thermal conductivity
Possible use as thermoelectric materials.
Good thermoelectrics should have low thermal conductivity!
Guest Modes  Rattler Modes:
The focus of many recent experiments.
Heat transport theory: The low frequency “rattler”
modes can scatter efficiently with the acoustic modes of the host
 Lowers the thermal conductivity
 Good thermoelectrics!
Clathrates of recent interest: Sn (mainly Type I). Si & Ge, (mainly Type II). “Alloys” of Ge & Si (Type I ).

15. Ultrasoft pseudopotentials; Planewave basis
Calculations
Computational package: VASP- Vienna Austria Simulation Package. “First principles”!
Many electron / Exchange-correlation effects
Local Density Approximation (LDA)
with Ceperley-Alder Functional OR
Generalized Gradient Approximation (GGA)
with Perdew-Wang Functional
Ultrasoft pseudopotentials; Planewave basis
Extensively tested on a wide variety of systems
We’ve computed equilibrium geometries, equations of state, bandstructures, phonon (vibrational) spectra, ...

16. Equations of State Schrödinger Equation for the interacting electrons
Start with a lattice geometry from experiment or guessed (interatomic distances & bond angles).
Use the supercell approximation (periodic boundary conditions)
Interatomic forces act to relax the lattice to an equilibrium configuration (distances, angles).
Schrödinger Equation for the interacting electrons
Newton’s 2nd Law for the atomic motion (quantum mechanical forces!)
Equations of State
The total binding energy is minimized (in the LDA or GGA) by optimizing the internal coordinates at a given volume.
Repeat the calculation for several volumes.
Gives the minimum energy configuration.
 An LDA or GGA 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.

17. Birch-Murnaghan Eqtn of State
Fit the total binding energy vs. volume curve to
E(V) = E0 + (9/8)K0V0[(V0/V)⅔ - 1] {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

18. Equations of State for Sn Solids Birch-Murnhagan fits to LDA E vs
Equations of State for Sn Solids Birch-Murnhagan fits to LDA E 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!

19. Equation of State Parameters Birch-Murnhagan fits to LDA E vs. V curves
Sn Clathrates
Expanded volume, high energy, “soft” Sn phases
Compared to α-Sn
Sn V, 12% larger. E, 41 meV higher. K0, 13% “softer”
Sn V, 14% larger. E, 38 meV higher. K0, 13% “softer”

20. Ground State Properties
Once the equilibrium lattice geometry is obtained, all ground state properties are obtained at the minimum energy volume.
Electronic bandstructures
Vibrational (phonon) dispersion relations
Bandstructures
(will discuss these after the electronic bands chapter)
At the relaxed lattice configuration, (“optimized geometry”) use the one electron Hamiltonian + LDA or GGA many electron corrections to solve the Schrödinger Equation for bandstructures Ek.

21. 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!!
COMPENSATE for 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.
Si136, Ge136, Sn136: Semiconducting.
Na16Cs8Si136, Na16Cs8Ge136, Cs24Sn136: Metallic.

22. ASSUME an ordered structure
For EACH guest-containing clathrate, including those with compensating atoms in the framework:
ENTIRE LDA or GGA procedure is repeated:
Total energy vs. volume curve
 Equation of State
Birch-Murnhagan Eqtn fit to results.
At the minimum energy volume, compute the bandstructures & the lattice vibrations.
For the compensated materials:
ASSUME an ordered structure