Rattling Atoms in Type I and Type II
Clathrate Materials
Charles W. Myles, Texas Tech U.
Jianjun Dong, Auburn U.
Otto F. Sankey,1 Arizona State U.
March National APS Meeting
Austin, TX, Tues., March 4, 2003
1Supported
in part by NSF Grant NSF-DMR-99-86706
• 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.
Clathrates
• Pure framework materials: Usually semiconductors.
• Pure materials not easily fabricated. Normally
have impurities (“guests”) encapsulated inside
cages. Guests  “Rattlers”
• Guests: Group I atoms (Li, Na, K, Cs, Rb) or
Group II atoms (Be, Mg, Ca, Sr, Ba)
– Guests weakly bound in cages
 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 (“rattle”) with low frequency modes
 Strongly affect lattice vibrations (thermal conductivity)
Compensation
• Guest-containing clathrates: Valence electrons
from guests go to conduction band of host (heavy
doping). Change material from semiconducting to
metallic.
• Sometimes compensate for this by replacing some
host atoms in the framework by Group III atoms.
Si46, Ge46, Sn46 : Semiconducting
Cs8Sn46 : Metallic. Cs8Ga8Sn38 : Semiconducting
Si136,Ge136, Sn136 : Semiconducting
Na16Cs8Si136, Na16Cs8Ge136, Cs24Sn136 : Metallic
Calculations
• Computational package: VASP: Vienna Austria
Simulation Package
• First principles technique.
– Many electron effects: Correlation:
Local Density Approximation (LDA).
Exchange-correlation energy:
Ceperley-Adler Functional
– Ultrasoft pseudopotentials.
– Planewave basis
• Extensively tested on a wide variety of systems
• We’ve computed equations of state, bandstructures &
vibrational phonon spectra.
• Start with given interatomic distances & bond angles.
– Supercell approximation
• Total binding energy minimized by optimizing
internal coordinates at a given volume.
– Interatomic forces to relax lattice to equilibrium
configuration (distances, angles).
– Schrdinger Eq. for interacting electrons, Newton’s 2nd
Law motion for atoms.
• Repeat for several volumes until LDA minimum
energy configuration is obtained.
• Once equilibrium lattice geometry is obtained, all
ground state properties can be obtained:
– Vibrational dispersion relations: Our focus here!
– Electronic bandstructures
Lattice Vibrational Spectra
• Optimized LDA geometry: Calculate total
ground state energy: Ee(R1, R2, R3, …..RN)
• Harmonic Approx.: “Force constant” matrix:
(i,i)  (2Ee/Ui Ui), Ui = atomic
displacements
• Finite displacement method:  Ee for many
different (Small) Ui. Forces  Ui. Dividing force by Ui
gives (i,i) & dynamical matrix Dii(q). Group theory
limits number & symmetry of Ui required.
• Positive & negative Ui for each symmetry: Cancels out 3rd
order anharmonicity (beyond harmonic approx.) Once all
unique (i,i) are computed, do lattice dynamics.
• Lattice dynamics in the harmonic approximation:
det[Dii(q) - 2 ii] = 0
Cs8Ga8Sn38 Phonons
C. Myles, J. Dong, O. Sankey, C. Kendziora, G. Nolas,
Phys. Rev. B 65, 235208 (2002)
 Ga modes
 Cs guest
“rattler” modes
(~25 - 40 cm-1)
“Rattler” modes: Cs motion in large & small cages
Raman Spectra
Group theory
determines Raman
active modes. First
principles frequencies,
empirical intensities.
C. Myles, J. Dong, O.
Sankey, C. Kendziora,
G. Nolas, Phys. Rev.
B 65, 235208 (2002)
Experimental & theoretical
rattler (& other) modes in
very good agreement!
• Reasonable agreement of theory and
experiment for Raman spectrum.
 UNAMBIGUOUS IDENTIFICATION of
low frequency (25-40 cm-1) “rattling”
modes of Cs guests in Cs8Ga8Sn38
– Also: (not shown) Detailed identification of
frequencies & symmetries of several experimentally
observed Raman modes by comparison with theory.
Type II Clathrate Phonons
With “rattling”atoms
• Current experiments: Focus on rattling modes in
Type II clathrates (thermoelectric applications).
 Theory: Given success with Cs8Ga8Sn38:
Look at phonons & rattling modes in Type II
clathrates
 Search for trends in rattling modes as host
changes from Si  Ge  Sn
– Na16Cs8Si136 : Have Raman data & predictions
– Na16Cs8Ge136 : Have Raman data & predictions
– Cs24Sn136:
Have predictions, NEED DATA!
Phonons
C. Myles, J. Dong, O. Sankey, submitted, Phys. Status Solidi B
Na16Cs8Si136
Na16Cs8Ge136
Na rattlers (20-atom cages)
~ 118 -121 cm-1
Cs rattlers (28-atom cages)
~ 65 - 67 cm-1
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. Nolas, C. Kendziora, J. Gryko,
A. Poddar, J. Dong, C. Myles,
O. Sankey J. Appl. Phys. 92,
7225 (2002).
Experimental & theoretical
rattler (& other) modes 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 “rattling”
modes (of Cs in 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. Myles, J. Dong, O. Sankey, submitted, Phys. Status Solidi B
• Cs24Sn136: A
hypothetical
material!
Cs in large (28atom) cages:
Extremely
anharmonic &
“loose” fitting.
 Very small
frequencies!
Cs rattler modes (20-atom cages)
Cs rattler modes (28-atom cages)
~ 25 - 30 cm-1
~ 5 - 7 cm-1
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! Effective potential for Cs in 28-atom
cage is very anharmonic: Cs is very loosely bound
there. Calculations were done in the harmonic
approximation.  More accurate calculations
taking anharmonicity into account are needed.
 Potential thermoelectric applications.
NEED DATA!
Trend
• Trend in “rattling” modes of Cs in large
(28-atom) cages as host changes
Si  Ge  Sn
Na16Cs8Si136
(~ 65 - 67 cm-1)
Na16Cs8Ge136 (~ 21 - 23 cm-1)
Cs24Sn136
(~ 5 - 7 cm-1)
• Correlates with size of cages in comparison
with “size” of Cs atom.
Model for Trend
• 28-atom cage size in host framework 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   (LDA-computed average X-X nearneighbor distance)  covalent radius of atom X
rCs  ionic radius of Cs (1.69 Å)
(rX + rCs)  Cs-X distance if Cs were tight fitting in cage
 Δr  How “oversized” the cage is compared to
Cs “size”. Geometric measure of how loosely
fitting a Cs atom is inside a 28-atom cage.
Model
• Simple harmonic oscillator model for Cs,
with assumption that only Cs moves in its
oversized 28-atom cage.
• Equate LDA-computed rattler frequency to:
R = (K/M)½
K  Effective force constant for rattler mode
K  A measure of strength (weakness) of
guest atom-host atom interaction.
M  Mass of Cs
K vs. Δr
• Smallest, Si28 cage:
Δr  1.18 Å  “oversized”
K  2.2 eV/(Å)2
KSi-Si  10 eV/(Å)2
 Cs weakly bound
• Ge28 cage:
Δr  1.22 Å  “oversized”
K  0.2 eV/(Å)2
KGe-Ge  10 eV/(Å)2
 Cs 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
Largest alkali atom (Cs) in largest possible clathrate cage (Sn28)!
Conclusions
• LDA calculations of lattice vibrations
• 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 Cs guests
• Simple model for trend in Cs rattler modes (28atom cage) as host changes from Si to Ge to Sn.