Study of Rattling Atoms in Type I and
Type II Clathrate Semiconductors
Charles W. Myles,1 Texas Tech U.
Jianjun Dong, Auburn U.
Otto F. Sankey,2 Arizona State U.
4th Motorola Workshop on Computational
Materials and Electronics, Nov. 14-15, 2002
1Supported
in part by a Texas Tech U. Faculty
Development Leave. Thanks to ASU for hospitality!
2Supported in part by NSF Grant NSF-DMR-99-86706
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 diamond structure, all Group IV atoms are 4fold coordinated in sp3 bonding configurations.
– Metastable, high energy phases of Si, Ge, Sn
– Few pure elemental phases yet. Usually compounds
with groups I and II elements (Na, K, Cs, Ba).
– Applications: 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: ( Type I Clathrates)
– 20 atom (dodecahedron) “cages” and 24 atom
(tetrakaidecahedron) cages, fused together
through 5 atom pentagonal rings.
– Crystal structure = simple cubic, 46 atoms per
cubic unit cell.
• Si136, Ge136, Sn136: ( Type II Clathrates)
– 20 atom (dodecahedron) “cages” and 28 atom
(hexakaidecahedron) cages, fused together
through 5 atom pentagonal rings.
– Crystal structure = face centered cubic, 136
atoms per cubic unit cell (34 atoms/fcc unit cell)
Type I Clathrate: Si46 , Ge46 or Sn46
Type II Clathrate: Si136 , Ge136, Sn136
Clathrate Structures
24 atom cages
Type I Clathrate
Si46, Ge46, Sn46
simple cubic
20 atom cages
28 atom cages
Type II Clathrate
Si136, Ge136, Sn136
face centered
cubic
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 atoms (Li, Na, K, Cs, Rb)
or Group II atoms (Be, Mg, Ca, Sr, Ba)
Type I Clathrate
(with guest “rattlers”)
20 atom cage
with
guest atom
[100]
direction
+
24 atom cage
with
guest atom
[010]
direction
Clathrates
• Semiconductors or semimetals.
– Also superconducting materials made from sp3
bonded, Group IV atoms! (Ba8Si46)
• Guests weakly bound in cages:
– Host valence electrons taken up in sp3 bonds
– Guest valence electrons go to conduction band
of host (heavy doping density).
– Guests weakly bonded in cages  Minimal
effect on electronic transport
– Guests vibrate (“rattle”) with low frequency
modes  Strong effect on vibrational
properties (thermal conductivity)
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 &
phonon spectra.
• Start with given interatomic distances & bond angles.
• Supercell approximation
• Interatomic forces act to relax lattice to
equilibrium configuration (distances, angles).
– Schrdinger Eq. for interacting electrons, Newton’s 2nd
Law motion for atoms.
Equations of State
• Total binding energy minimized by optimizing
internal coordinates at given volume.
• Repeat for several volumes.
– Gives LDA binding energy vs. volume curve.
– Fit to empirical eqtn of state (4 parameter): “BirchMurnaghan” equation of state
Birch-Murnaghan Eqtn of State
Sn Clathrates = Metastable, expanded volume 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
Bandstructures
• At relaxed lattice configuration (“optimized
geometry”) use one electron Hamiltonian +
LDA many electron corrections to solve
Schrdinger Eq. for bandstructures Ek.
Sn46 & Sn136 Bandstructures
C.W. Myles, J. Dong, O. Sankey, Phys. Rev. B 64, 165202 (2001).
The LDA UNDER-estimates bandgaps!
LDA gap Eg  0.86 eV
LDA gap Eg  0.46 eV
Semiconductors of pure tin!!!!
Compensation
• Guest-containing clathrates: Valence electrons
from guests go to conduction band of host (heavy
doping). Change material from semiconducting to
metallic.
• Compensate for this by replacing some host atoms
in the framework by Group III atoms.
–
–
–
–
–
–
Sn46 : Semiconducting
Cs8Sn46 : Metallic
Cs8Ga8Sn38 : Semiconducting
Cs8Zn4Sn42 : Semiconducting
Sn136 : Semiconducting
Cs24Sn136 : Metallic
Cs8Ga8Sn38 Bandstructure
C.W. Myles, J. Dong, O. Sankey, Phys. Rev. B 64, 165202 (2001).
LDA gap Eg  0.61 eV
Lattice Vibrational Spectra
• At optimized LDA geometry, calculate total
ground state energy: Ee(R1, R2, R3, …..RN)
• Harmonic Approx.: “Force constant” matrix:
(i,i)  (2Ee/Ui Ui)
Ui = displacements from equilibrium
• Derivatives  Ee for many different Ui. (Small Ui;
harmonic approximation)
• 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
Sn46 & Sn136 Phonons
C. Myles, J. Dong, O. Sankey, C. Kendziora, G. Nolas,
Phys. Rev. B 65, 235208 (2002)
Flat optic bands!
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: Due to Cs motion in large & small cages
Raman Spectra
• Do group theory necessary to determine
Raman active modes (frequencies
calculated from first principles as
described).
• Estimate Raman scattering intensities using
empirical (two parameter) bond charge
model.
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.
Conclusion
• 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 (for thermoelectric applications).
 Theory: Given our success with Cs8Ga8Sn38:
Look at phonons & rattling modes in Type II clathrates
 Search for trends in rattling modes as host is
changed from Si  Ge  Sn
– Na16Cs8Si136 : Have Raman data & predictions
– Na16Cs8Ge136 : Have Raman data & predictions
– Cs24Sn136: Have predictions, NEED DATA!
– Na16Cs8Sn136  Calculations are in progress!
Na16Cs8Si136 & Na16Cs8Ge136 Phonons
J. Dong, A. Poddar, C. Myles, O. Sankey, unpublished
Cs rattler modes ~ 65 cm-1
Cs rattler modes ~ 21 cm-1
Cs24Sn136 Phonons
C. Myles, J. Dong, O. Sankey, unpublished
Cs rattler modes:
 ~ 25-30 cm-1 (small cage)
 ~ 5 cm-1 (large cage) !
Cs in 8 large cages: Extremely anharmonic &
“loosely” fitting.  Very small frequencies: ~ 5 cm-1
Raman Spectra
• Again, estimate Raman scattering
intensities using empirical (two parameter)
bond charge model.
G. Nolas, C. Kendziora,
J. Gryko, A. Poddar,
J. Dong, C. Myles,
O. Sankey J. Appl. Phys.
(accepted).
• Experimental & theoretical
rattler (& other) modes in
very good agreement.
Also (not shown) detailed
identification of frequencies
& symmetries of several
observed Raman modes by
comparison with theory.
T=300K
=514nm
Si136
A
A1g
Ra
m
an
Int
en
sit
y
(ar B
b.
uni "Rattle"
ts)
VV
HV
Cs8Na16Si136
A1g
T2g
T2g
T2g
T=300K
=700nm
o
T2g
Eg
0
100
200
300
Eg
400
-1
Raman Shift (cm)
VV 0
o
VV 45
o
HV 0
o
HV 45
500
600
Raman Intensity (arb. units)
Cs8Na16Ge136
T=300K
=514 nm
"Rattle"
A1g
VV
HV
0
50
100
150
200
250
300
350
-1
Raman Shift (cm )
G. Nolas, C. Kendziora, J. Gryko, A. Poddar, J. Dong,
C. Myles, & O. Sankey, J. Appl. Phys. (accepted)
• Experimental & theoretical rattler (& other) modes in very
good agreement.
Conclusions
• Reasonable agreement of theory and 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 cm-1), Na16Cs8Ge136 (~ 21 cm-1)
• Also: (not shown) Detailed identification of
frequencies & symmetries of several experimentally
observed Raman modes by comparison with theory.
Prediction
• Cs24Sn136: Prediction of low frequency
“rattling” modes of Cs guests in small
(~20-30 cm-1) & large (~ 5 cm-1) cages (a very
small frequency!)
 Potential thermoelectric applications.
NEED DATA!
Trends
• Trends in Cs “rattling” modes as host is
changed from Si  Ge  Sn
Na16Cs8Si136 (~ 65 cm-1), Cs in large cages
Na16Cs8Ge136 (~ 21 cm-1), Cs in large cages
Cs24Sn136 (~ 20-30 cm-1), Cs in large cages
(~ 5 cm-1), Cs in small cages
• In progress: Phonons in Na16Cs8Sn136