Outline: Focus on Phonons in Clathrates
• Introduction:
Clathrate crystal structures; contrast to diamond structure
• Brief discussion of computational method
• Sn based clathrates (Types I & II)
Equations of state (Etot vs. V), Bandstructures (Ek)
Phonons (ωk), Raman spectra, Theory-Experiment comparisons
• Si, Ge, Sn based clathrates (Type II)
Phonons (ωk); Raman spectra, Theory-Experiment comparisons
• Si, Ge based clathrates (Type II). K. Biswas, PhD work
Guests (Impurities): Bands, Phonons, Theory-Experiment comparisons
• Si, Ge based clathrates (Type I). E. Nenghabi PhD work
“Alloys”: Bands, Phonons, Theory-Experiment comparisons
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 structure = Diamond Structure
• Each atom is tetrahedrally (4-fold) coordinated (4 near-neighbors) with
sp3 covalent bonding. Bond angles: Perfect, tetrahedral = 109.5º
Si, Ge are semiconductors, Sn (α-tin or gray tin) is a semimetal
• Another Sn structure (β-tin or white tin), a
body centered tetragonal lattice, 2 atoms
per unit cell, is metallic
• ALSO!! Si, Ge, Sn:
Form clathrate structures.
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
• The lattice contains hexagonal & pentagonal rings,
fused together with sp3 bonds to form large “cages”.
• The pure materials: Metastable, expanded
volume phases of Si, Ge, Sn
• Few pure elemental phases have been made. Most are
compounds with Group I & II atoms (Na, K, Cs, Ba).
• Potential applications: Thermoelectrics
• The lattices are open, cage-like structures, with
large “cages” of Si, Ge, or Sn atoms.
These are “Buckyball-like” cages of 20, 24, & 28 atoms.
• Two primary varieties have been studied:
Type I (X46) & Type II (X136)
X = Si, Ge, or Sn
The 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.”
• This talk: Group IV crystals with the same
crystal structure as clathrate-hydrates (ice).
Type I clathrate-hydrate crystal structure X8(H2O)46:
• 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:
Type II Clathrate
28 atom cage:
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. Lab synthesis. “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
• 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
 A 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:
Focus of some recent experiments.
Heat transport theory: The low frequency rattler
modes can scatter efficiently with the acoustic
modes of the host  Lowers the thermal conductivity
 A good thermoelectric!
• Clathrates of interest to experimenters:
Sn (mainly Type I). Si & Ge, (mainly Type II).
Most Recently, “Alloys” of Ge & Si (Type I ).
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 over many years.
• We’ve calculated equilibrium geometries, equations of state,
bandstructures, phonon (vibrational) spectra, mean square
atomic displacements, thermodynamic properties, ...
• 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: Etot vs. V (at T = 0)
• The total binding energy is minimized (in the LDA or GGA) by
optimizing the internal coordinates at a given volume Etot(V)
• 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.
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
• 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.
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.
Lattice Vibrations (Phonons)
• At optimized geometry:
Get the ground state energy: Ee(R1,R2,R3, …..RN)
• Harmonic Approximation: “Force constant” matrix:
Φ(i,i´)  (∂2Ee/∂Ui∂Ui´); Ui = atomic displacements from equilibrium.
Instead of directly computing derivatives, use the
• Finite displacement method: Compute Ee for many different (small;
harmonic approximation!) Ui; Forces  Ui. Divide forces by Ui gives Φ(i,i´) &
the dynamical matrix Dii´(q) for the lattice vibration calculation. Group theory
limits number & symmetry of the Ui required. (Materials of high symmetry!).
• Lattice dynamics in the harmonic approximation:  The classical
eigenvalue (normal mode) problem
det[Dii(q) - ω2δii´] = 0
Dynamical matrix Dii´(q) obtained from force constant matrix Φ in usual way.
First principles force constants!
NO FITS TO DATA!
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.
 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 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
• Experiments: Focused on rattling modes in Type II
clathrates (for possible 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)
Na16Cs8Si136
Na16Cs8Ge136
Na
Na
Cs
Na rattlers (20-atom cages)
~ 118 -121 cm-1
Cs rattlers (28-atom cages)
~ 65 - 67 cm-1
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.
• 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!
Na
Very small
frequencies!
Cs rattler modes (20-atom cages)
Cs rattler modes (28-atom cages)
Cs
Cs
~ 25 - 30 cm-1
~ 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.
Simple Model for the Trend
• 28-atom cage size in host lattice compared with Cs guest atom “size”.
For 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 near-neighbor distance)
rX  “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” 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  Effective force constant for the rattler mode
K  Measure of the strength (weakness!) of 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.
Koushik Biswas, TTU PhD (2007)
K. Biswas, C.W. Myles, Phys. Rev. B 74, 115113 (2006); 75, 245205 (2007);
J. Phys.: Condensed Matter 19, 466206 (2007)
C.W. Myles, K. Biswas, E. Nenghabi, Physica B 401-402, 695 (2007).
K. Biswas, C.W. Myles, M. Sanati, and G.S. Nolas, J. Appl. Phys. 104, 033535 (2008).
• Type II Si clathrates: “filled” cages: Na16Rb8Si136,
K16Rb8Si136, Cs8Ga8Si128, Rb8Ga8Si128. Plus Similar Type
II Ge clathrates
• Bandstructures, electronic densities of states, phonons,
vibrational densities of states, mean square atomic
displacements of rattlers, comparisons with experiment,
thermodynamic properties,…
2 Examples
1. Mean square atomic displacement parameters
2. Temperature dependence of heat capacity Cv
Mean Square Atomic Displacement Parameters (ADP) Uiso(T)
(measured in X-ray experiments)
K. Biswas, C.W. Myles, Phys. Rev. B 74, 115113 (2006); 75, 245205 (2007);
Na16Rb8Ge136
Na “Rattlers”
Rb “Rattlers”
Na16Cs8Ge136
Na “Rattlers”
Cs “Rattlers”
Phonon Contribution to Constant Volume Heat Capacity
CV(T) in Si136 & Ge136 K. Biswas, C.W. Myles, M. Sanati, and G.S. Nolas, J.
Appl. Phys. 104, 033535 (2008).
Theory
First-principles phonon modes &
DOS g(ω). Calculate the
Helmholtz Free Energy
(also other thermodynamic properties)
Fvib(T) =
kBT∫{(½)ħω + (kBT)ln[1 – exp(-ħω/kBT)]} g(ω)dω
Cv = -T(∂2Fvib/∂V2)V
Emmanuel Nenghabi, PhD Student (2004-2009)
E. Nenghabi and C.W. Myles, Phys. Rev. B 77, 205203 (2008);
Phys. Rev. B 78, 195202, (2008); J. Phys.: Condensed Matter, 20, 415214 (2008).
Type I Si-Ge clathrate “alloys”
M8N16SixGe30-x
M = Ba or Sr, N = Ga or In, 0 ≤ x ≤ 15
Of interest to experimenters: Thermoelectric applications
J. Martin, S. Erickson, G.S. Nolas, P. Alboni, T.M. Tritt, & J. Yang
J. Appl. Phys. 99, 044903 (2006)
• Bandstructures, electronic densities of states, phonons, vibrational
densities of states, mean square atomic displacements of rattlers,
thermodynamic properties. Effect of Si-Ge “alloying” on all of these.
Trends with composition x.
• Note: X-ray data shows that these are NOT random alloys, but
ordered materials.
Trends with x for Ba8Ga16SixGe30-x
Lattice Constant
Bulk Modulus
Phonon Dispersion Relations
Ba8Ga16SixGe30-x
Sr8Ga16SixGe30-x
These show: Upshift in the optic modes as x increases. Largest for the optic modes,
in which bond-stretching modes are important. Ba8Ga16SixGe30-x, highest optic
modes are 253, 334, 373 cm−1 for x = 0,5, 15. Sr8Ga16SixGe30-x these are 327,
350, 428 cm−1 for x = 0,5, 15.
Explanation: Ge substitution by Ga & Si strengthens bonds. Calculated lattice constants a
show that a in Ba compounds is larger than in the Sr materials because the Ba atomic mass is
larger than Sr’s. So, a larger strain effect occurs when Ba is in the cages than if Sr is in them.
Also: Because the Si atom is smaller than Ba, Sr, Ge, & Ga atoms, the lattice constant
a decreases as x increases. The nearest-neighbor bond distances in Ba8Ga16SixGe30x range from 2.53–2.63 Å. In Sr8Ga16SixGe30-x these range from 2.44–2.62 Å.
Shorter bonds strengthen the structures, resulting in larger force constants.
Vibrational State Densities (VDOS)
• The VDOS increases at the bottom of the
optic band just above the acoustic modes.
Eigenvector analysis shows that these
additional modes are from the Sr & Ba
guest atoms.
• The VDOS is higher for x = 5, than for x
= 0 & higher for x =15 than for x = 5.
This is due to the smaller Sr mass than for
Ba atom in Ba8Ga16SixGe30-x.
• These optic modes compress the acoustic
bandwidth. For x = 0,5,15, the tops of the
acoustic bands in Ba8Ga16SixGe30-x at 33,
36, 30 cm-1. In Ba8Ga16SixGe30-x, these
are at 40, 42, and 33 cm-1for x = 0, 5, 15.
• These acoustic bandwidths are reduced by
~16%–40%, depending on the value of x,
in comparison to that of pristine Ge46.
Mean Square Atomic Displacement Parameters (ADP)
Uiso ~ (kBT)/φ
φ = calculated force constant
for Ba, Sr vibrations.
x=5
• Results for the Ba, Sr in 20 atom cages
& in 24 atom cages are both shown.
• Uiso values for Sr are larger than for Ba.
In qualitative agreement with experiments
by Bentien et al. in Ba8Ga16Ge30,
Ba8Ga16Si30, Ba8In16Ge30,
Sr8Ga16Ge30.
• Because of the Sr small atom in
comparison to Ba, the Sr atoms are more
off-centered in the cages than Ba, which
leads to a larger ADP.
x = 15
Thermal Properties: Cv, S, F for Ba8Ga16SixGe30-x
Cv
S
F
Heat Capacity, Cv Entropy S, & Helmholtz Free Energy F
• Of course, because of their low frequencies of vibration, the Ba
guest atoms contribute little to these properties.
• As can be seen, the dependence on the Si composition x is very
small for each of these properties.
• Similar calculations for Sr8Ga16SixGe30-x for these properties
shows that the Ba-containing materials are thermodynamically
more stable than the Sr-containing materials.
Brief Tribute to
Emmanuel N.
Nenghabi
Emmanuel N. Nenghabi
Born: October 11, 1972
Cameroon (West Coast of Africa)
Died: January 27, 2009
Lubbock, Texas
Physicist
PhD Physics Student
Teacher
Researcher
Collaborator
Friend!!
• Emmanuel Nenghabi, my PhD student, passed away
unexpectedly following an illness Tues., Jan. 27, 2009.
• At Texas Tech, he was an excellent graduate student, earning
mostly A's in the courses. He also was a very good teacher
who was well liked and very popular with the
undergraduates in the Physics Lab courses he taught. Student
evaluations of his teaching uniformly indicate that he was a
teacher with rigorous standards, who also treated the
students fairly.
• He was a highly intelligent, very hard working, very
ambitious and productive PhD student, who was well on his
way to becoming an excellent researcher in computational
materials theory. It was my privilege to have been his
research supervisor for about the last 3 years of his life. He
was extremely close to finishing his PhD. We had planned
his defense to be late in the Spring Semester of 2009.
• Emmanuel was self-motivated and wrote papers without me asking him
to. Sometimes, he even had to push me to get papers submitted to the
publisher. At the time of his death, together, we have published four
papers in high quality, peer-reviewed physics journals based on his work.
At least two more papers will be submitted soon. He also gave three
presentations of our work at Texas Section APS meetings. For one of
these, he won a presentation award. He also made one presentation to the
National APS meeting in New Orleans in March, 2008. He had planned
to give another presentation at the National APS meeting in Pittsburgh in
March, 2009. I gave it in his place.
• In addition to the obvious loss to my research program, I also miss
Emmanuel a lot personally. I was privileged to be his friend, as well as
his mentor. He was a wonderful friend, not just to me, but also to most
people in the Department of Physics. He was well-liked by everyone who
knew him and, in turn, he liked everyone. He was always cheerful and
was always willing to help others. He also often volunteered for tasks
when few others would come forward.
• As his PhD supervisor, I miss him a lot. As his friend, I am very sad that I
cannot have even one more conversation with him.