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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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
Outline
• Introduction to clathrates
Crystal structures. Contrast to diamond structure
• Brief discussion of computational method
• Sn clathrates (Types I & II)
–
–
–
–
Equations of state (Etot vs. volume)
Electronic bandstructures (Ek)
Vibrational (phonon) properties (ωk)
Raman spectra & comparison with experiment
• Si, Ge, & Sn clathrates (Type II)
– Vibrational (phonon) properties (ωk)
– Raman spectra & comparison with experiment
Group IV Elements


• Valence electron configuration: ns2 np2
[n = 2, C; n = 3, Si; n = 4, Ge; n = 5, Sn]
Group IV Crystals
• Si, Ge, Sn: Ground state crystalline
structure = Diamond Structure
– Each atom tetrahedrally (4-fold) coordinated (4
nearest-neighbors) with sp3 covalent bonding
– Bond angles: Perfect, tetrahedral = 109.5º
Si, Ge: Semiconductors
Sn: (α-tin or gray tin) - Semimetal
• Sn: (β-tin or white tin) - body centered
tetragonal lattice, 2 atoms per unit cell.
Metallic.
• Si, Ge, Sn: The clathrates.
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”.
• 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
• Why “clathrate”? The same crystal
structure as clathrate hydrates (ice).
• 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)
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:
28 atom cage:
Type II Clathrate
 Si136, Ge136, Sn136
(C136?)
Face Centered
Cubic
Clathrate Lattices
Type I Clathrate 
Si46, Ge46, Sn46
simple cubic
[100]
direction
Type II Clathrate 
Si136, Ge136, Sn136
face centered
[100]
cubic
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, 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
Type I Clathrate
20 atom cage
with guest atom

(with guest “rattlers”)
[100]
direction
+
24 atom cage
with guest atom

[010]
direction
• 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 sp3 bonds
– 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
• 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.
Calculations
• Computational package: VASP- Vienna
Austria Simulation Package. First principles!
Many electron effects:
Local Density Approximation (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.
• 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.
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.
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
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!
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:
Sn46 -- V: 12% larger, E: 41 meV higher, K0: 13% “softer”
Sn136 -- V: 14% larger, E: 38 meV higher, K0: 13% “softer”
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.
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
Diamond Structure Sn Bands
M.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)

Sn46 & Sn136 Bandstructures
C.W. Myles, J.J. Dong, O.F. Sankey, Phys. Rev. B 64, 165202 (2001)
The LDA UNDER-estimates bandgaps!
Sn46

LDA gap Eg  0.86 eV
Sn136

LDA gap Eg  0.46 eV
Semiconductors of pure tin!!!!
(Hypothetical materials! Indirect band gaps)
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
• Later: Si136,Ge136, Sn136: Semiconducting.
Na16Cs8Si136, Na16Cs8Ge136, Cs24Sn136: Metallic
• For EACH guest-containing clathrate, including those
with compensating atoms in the framework:
• THE ENTIRE LDA procedure 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
Cs8Ga8Sn38 & Cs8Zn4Sn42 Bands
C.W. Myles, J.J. Dong, O.F. Sankey, Phys. Rev. B 64, 165202 (2001).
The LDA UNDER-estimates bandgaps!
Cs8Ga8Sn38
Cs8Zn4Sn42


LDA gap Eg  0.61 eV
LDA gap Eg  0.57 eV
Semiconductors
(Materials which have been synthesized. Indirect band gaps)
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 Ee for
many different (small; harmonic approximation!) Ui
Compute forces  Ui.
• Dividing forces by Ui gives Φ(i,i´) & thus the dynamical
matrix Dii´(q) used in the lattice vibration calculation.
Phonons
• Group theory limits the number & symmetry of the Ui required.
(These materials have high symmetry!).
• Use positive & negative Ui for 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 Φ in the usual way. First principles
force constants! NO FITS TO DATA!
• 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
Diamond Structure Sn Phonons
W. Weber, Phys. Rev. B 15, 4789 (1977).
3 Acoustic branches
3 Optic branches
Sn46 & Sn136 Phonons
C.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”
Guest-Containing Clathrates as
Thermoelectrics
• Guest atoms: Weakly bound to the clathrate lattice.
• Lattice Framework: Fully sp3 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.
• 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.
Cs8Ga8Sn38 Phonons
C.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 - 40 cm-1)
“Rattler” modes: Cs motion in the large & small cages
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 Frequencies are calculated from
first principles as described.
• Estimate the Raman scattering intensities
using an empirical (two parameter) bond
polarization model.
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 of frequencies
& symmetries of several
observed Raman modes by
comparison with theory.
Type II Clathrate Phonons
With “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!
Phonons
C.W. Myles, J.J. Dong, O.F. Sankey, Phys. Status Solidi B 239, 26 (2003)
 Na
Na16Cs8Si136
Na16Cs8Ge136
 Na
 Cs
Na rattlers (20-atom cages)
~ 118 -121 cm-1
Cs rattlers (28-atom cages)
~ 65 - 67 cm-1
Cs
 Na
 Cs
Na rattlers (20-atom cages)
~ 89 - 94 cm-1
Cs rattlers (28-atom cages)
~ 21 - 23 cm-1
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.
• There is reasonable agreement of theory &
experiment for Raman spectra, especially
for the “rattling” modes of Cs in the large
cages in Type II Si & Ge clathrates.
 UNAMBIGUOUS IDENTIFICATION of
low frequency “rattling” modes of Cs in
Na16Cs8Si136 (~ 65 - 67 cm-1)
Na16Cs8Ge136 (~ 21 - 23 cm-1)
Cs24Sn136 Phonons
C.W. Myles, J.J. Dong, O.F. Sankey, Phys. Status Solidi B 239, 26 (2003)
• Cs24Sn136:
A hypothetical
material!
Cs in the large
(28-atom) cages is
extremely
anharmonic &
“loose” fitting!
 Very small
frequencies!
Cs rattler modes (20-atom cages) ~ 25 - 30 cm-1
Cs rattler modes (28-atom cages) ~ 5 - 7 cm-1
Thermoelectric applications?
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 for Cs 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!
Trend
• The trend in the Cs “rattling” modes in the
large (28-atom) cages as 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 Cs atom.
A Simple Model for the Trend
• The 28-atom cage size in the host lattice compared with a Cs
guest atom “size”. For a host atom X = Si, Ge, Sn, define:
Δr  rcage - (rX + rCs)
rcage  LDA-computed average Cs-X distance
rX  (½)(LDA-computed average X-X nearneighbor distance)  The “covalent radius” of atom X
rCs  Ionic radius of Cs (1.69 Å)
(rX + rCs)  Cs-X distance if Cs were tight fitting in a cage
 Δr  How “oversized” the cage is compared to
Cs “size”. A geometric measure of how loosely
fitting a Cs atom is inside a 28-atom cage.
• Couple this geometric model with a
simple harmonic oscillator model for Cs in
the cage. Assume that only Cs moves in its
oversized 28-atom cage.
• Equate the LDA-computed rattler frequency to:
ωR = (K/M)½
(M  Mass of Cs)  This gives:
K  An effective force constant for the rattler mode
K  A measure of the strength (weakness!) of the
guest atom-host atom interaction.
K vs. Δr
• Smallest, Si28 cage:
Δr  1.18 Å  “oversized”
K  2.2 eV/(Å)2
KSi-Si  10 eV/(Å)2
 Cs is weakly bound!
• Ge28 cage:
Δr  1.22 Å  “oversized”
K  0.2 eV/(Å)2
KGe-Ge  10 eV/(Å)2
 Cs is very weakly bound!
• Largest, Sn28 cage: Δr  1.62 Å  EXTREMELY “oversized”
K  0.02 eV/(Å)2, KSn-Sn  8 eV/(Å)2
 Cs extremely weakly bound (almost “unbound”!)
Largest alkali atom (Cs) in the largest possible clathrate cage (Sn28)!
Conclusions: Phonons
• Type I clathrate: Cs8Ga8Sn38
– Good agreement with Raman data for Cs rattler modes &
also host framework modes!
• Type II clathrates: Na16Cs8Ge136, Na16Cs8Si136
– Good agreement with Raman data for Cs rattler modes &
also host framework modes!
• Type II clathrate: Cs24Sn136 (A hypothetical material!)
– Prediction of extremely low frequency “rattling” modes of
the Cs guests
– Possibly extremely low thermal conductivity?
• A simple model for the trend in the Cs rattler modes
(28-atom cage) as the host changes from Si to Ge to Sn.
Comments & Conclusions
• Group IV clathrates are interesting “new” materials!
• Experimental measurements (G. Nolas, et al.) show
guest-containing Ge & Sn materials have very low
thermal conductivities. Mainly Type I materials.
• Molecular dynamics simulations on Sr6Ge46
[J.J. Dong, O.F. Sankey, C.W. Myles, Phys. Rev. Lett. 86,
2361 (2001)] confirm this. Thermoelectric properties is
another talk!
• On-going & future work:
– Thermodynamic properties (J.J. Dong)
– Thermal conductivity calculations
– Carbon clathrates (not made in the lab yet). Should be very
“hard” materials
Carbon Crystals
• C: Graphite & Diamond Structures
Diamond 
Insulator or wide bandgap
semiconductor
Graphite 
Planar structure
sp2 bonding
 2d metal (in plane)
– Ground state (lowest energy configuration) is graphite
at zero temperature & atmospheric pressure. Graphitediamond total energy difference is VERY small!
Other Group IV Crystal Structures
(Higher Energy)
• C: “Buckyballs” (C60) 
“Buckytubes” (nanotubes),
other fullerenes

Clathrate Structures
24 atom cages
20 atom cages
28 atom cages
Type I Clathrate
Si46, Ge46, Sn46
simple cubic
Type II Clathrate
Si136, Ge136, Sn136
face centered
cubic
Clathrate Building Blocks
• 24 atom cage: 
• 20 atom cage: 
• 28 atom cage: 