Texas Tech University
Clathrate Semiconductors:
Novel, Open Framework, Crystalline
Materials Based on Si, Ge, and Sn
Charles W. Myles and Dong Xue
Department of Physics, Texas Tech University
Charley.Myles@ttu.edu
http://www.phys.ttu.edu/~cmyles
APS/CNM Users Meeting
Argonne National Laboratory
Tuesday, May 12, 2015
• Introduction:
Outline
– Tutorial on the clathrate crystal structures. Contrast
with the diamond structure.
– Brief discussion of our computational method
• Selected Earlier Work:
– Sn based Clathrates (Types I & II): Bandstructures
(Ek) , Phonons (ωk), Raman spectra, TheoryExperiment comparisons
– Si, Ge based Clathrates (Type II): “Guests”
(Impurities): Bands, Phonons, Theory-Experiment
comparisons
– Si, Ge based Clathrates (Type I): “Alloys”: Bands,
Phonons, Theory-Experiment comparisons
Group IV Crystals
Si, Ge, Sn: Ground state lattice 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),
is a body centered tetragonal lattice,
2 atoms per unit cell. It is metallic.
ALSO!! Si, Ge, Sn
Form Clathrate Structures
Clathrates: An Introduction
• Clathrates are Crystalline Phases of the 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  A 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 are metastable, expanded
volume (in comparison with the diamond structure)
phases of Si, Ge, Sn
• Few pure elemental phases have been synthesized. 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.
• There are many crystal structure types, but only two
primary types have been studied:
Type I (X46) & Type II (X136)
X = Si, Ge, or Sn
Meaning of “Clathrate” ?
• Wikipedia: “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.
• This talk: Group IV Crystals with the same crystal
structures as clathrate-hydrates (ice).
.
Type I clathrate-hydrate crystal structure X8(H2O)46
H2O Molecules
Methane
Molecule: CH4
Cubic Unit Cell
Clathrate Cages
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
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 Lattices
Type I Clathrate 
Si46, Ge46, Sn46
simple cubic
[100]
direction
Type II Clathrate 
Si136, Ge136, Sn136
face centered cubic
[100]
direction
Group IV Clathrates
• Not found in nature. Synthesis is difficult!
– Outside the scope of this talk. I am a theorist!
• The Pure Group IV Clathrate Materials are
semiconductors.
– But, they are 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)
• Guest-Containing Clathrate Materials:
– The guests are weakly bonded in cages:
 They have minimal effects on
electronic transport
• The host valence electrons are taken up in the sp3 bonds
 Guest valence electrons go to the host conduction band.
•
•
•
•
( Effectively a heavy doping density)
Guests vibrate with low frequency (“rattler”) modes
 Strong effect on vibrational properties
Guest Modes  Rattler Modes
Possible applications as thermoelectric materials.
Good thermoelectrics need low thermal conductivity!
Guest Modes  Rattler Modes:
The focus of some recent experiments.
Heat Transport Theory: Low frequency rattler modes
can scatter efficiently with the acoustic modes of the host
 Lowering the thermal conductivity
 A good thermoelectric!
Compensation
• Guest-Containing Clathrates:
The 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, semiconductors are wanted!!
• 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.
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,
Typical Electronic Band Structures
Cs8Ga8Sn38 & Cs8Zn4Sn42 Bands
C. Myles, J. Dong, O. 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 equilibrium optimized geometry:
Get the ground state energy: Ee(R1,R2,R3, …..RN)
• The Harmonic Approximation:
– The “Force constant” matrix: Φ(i,i´)  (∂2Ee/∂Ui∂Ui´)
– Ui = atomic displacements from equilibrium.
– From Φ(i,i´) & obtain the dynamical matrix Dii´(q) used in
the lattice vibration calculation.
• Lattice dynamics in the harmonic approximation:
 The classical eigenvalue (normal mode) problem
det[Dii(q) - ω2δii´] = 0
• The dynamical matrix Dii´(q) is obtained from the force
constant matrix Φ in the usual way.
First principles force constants!
NO FITS TO DATA!
Typical Phonon Dispersion Curves
Cs8Ga8Sn38
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
C. Myles, J. Dong, O. Sankey, C.
Kendziora, G.S. Nolas,
Phys. Rev. B 65, 235208 (2002).
• Experimental & theoretical
rattler (& other!) modes
are in good agreement!
 UNAMBIGUOUS
IDENTIFICATION
of low (25-40 cm-1)
frequency rattler modes of
Cs guests.
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)
Type II Si & Ge Clathrates
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 clathrates with “filled” cages:
Na16Rb8Si136, K16Rb8Si136, Cs8Ga8Si128,
Rb8Ga8Si128,Na16Rb8Ge136, K16Rb8Ge136,
Cs8Ga8Ge128, Rb8Ga8Ge128.
2 Examples of Results
1. Mean square atomic displacement parameters
2. Temperature dependence of heat capacity Cv
Mean Square Atomic Displacement
Parameters (ADP) Uiso(T) (X-ray experiments)
K. Biswas, C.W. Myles, Phys. Rev. B 74, 115113 (2006); 75, 245205 (2007);
Na16Rb8Ge136
Na “Rattlers”
Rb “Rattlers”
Uiso(T) 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
Type I Si-Ge clathrate “alloys”
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).
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
Ba8Ga16SixGe30-x 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.
Recent Work: With Dong Xue (PhD Student)
Experimental collaborators G.S. Nolas, et al,
• Effect of Alkali Metal Filling on the
Properties of the Type-II Clathrate AxM136
(A = Na,K,Rb,Cs, M = Group IV atom, 0 ≤ x ≤24).
Motivation
• Extensive XRD data on
NaxSi136 by Nolas, Beekman, et al.
NaxSi136
• Note that, due to the effective heavy doping when the
cages are filled with Na, NaxSi136 is metallic.
Si136 lattice
• Highlighted Si20 & Si28
cages can be occupied by Na.
• As Na is added (in the cages), a
lattice expansion might be
expected. Such an expansion has
been observed for many different
guest atoms in many different
Type II and Type I clathrates.
• Also based on past observations, in this type of metallic
clathrate, significant charge transfer from guest to host is also
expected, which likely could contribute to a lattice contraction.
• Beekman & Nolas (U. of South Florida) &
collaborators performed powder X-ray diffraction
(XRD) experiments on
NaxSi136 (0 ≤ x ≤ 24).
• They measured the unit cell volume as a function of x.
Their results are VERY intriguing!
• They observed an initial lattice contraction as
Na is added (0 ≤ x ≤ 8), followed by an
eventual lattice expansion as more Na is
added (8 ≤ x ≤ 24).
Beekman & Nolas, et al. XRD Results
• Experimental normalized cage occupancies for Na in
the Si20 & the Si28 cages as a function of Na content x.
As x increases, the larger Si28 cages fill preferentially.
After all 8 Si28 cages in the unit cell are filled (x = 8), as
x increases further, the smaller Si20 cages begin to fill.
• These observations guided our structural optimizations.
• A model was chosen in which Na exclusively fills
the larger Si28 cages first as x is increased.
• The small Na size compared to the Si28 cage size
plus charge transfer between Na and Si causes the Si
to move towards the Na, leading to lattice
contraction until x = 8.
• As x increases above 8, the smaller Si20 cages begin
to fill. In this case, the size difference and charge
transfer effects cause the Si to move away from the
Na, leading to a lattice expansion.
Beekman & Nolas, et al. XRD Results
NaxSi136
• Experimental cubic unit
cell parameter as a
function of Na content x.
• Inset: LDA- Calculated
cubic unit cell parameter
as a function of x.
• In agreement with the data, our calculations clearly predict
that the incorporation of the small Na guests in the larger Si28
cages induces a contraction of the cell volume. Upon further
Na incorporation (x > 8), the smaller Si20 cages begin to fill,
resulting in a lattice expansion.
Another Interesting Plot
• Experimentally-derived
trends in the Si20 and Si28
cage volumes as a function
of the Na content, relative
to the x = 8 composition
and the analogous trend in
the unit cell volume.
• Clearly, the Si28 cages contract as they are filled, and the Si20
cages expand. These opposite effects, combined with
preferential occupation, help explain the minimum in the
lattice parameter near x = 8.
• These data indicate that the local guest-cage interaction drives
the behavior of the lattice parameter as a function of x.
Some Very Recent Results
• Motivated by the very interesting effects found in
NaxSi136, we have recently been exploring the general
problem of the effects of alkali metal filling on the
Properties of the Type-II Clathrates AxM136
(A = Na,K,Rb,Cs, M = Group IV atom, 0 ≤ x ≤24
• Question: Will the effect seen in NaxSi136, of an
initial lattice contraction as the Na atoms are
added (0 ≤ x ≤ 8), followed by an lattice
expansion as more Na atoms are added
(8 ≤ x ≤ 24) also hold for other alkali guests and
for alkali guests in the other Type II Clathrates
Ge136 and Sn136?
Lattice Constant vs. x: AxSi136
A = Na, K, Rb, Cs
Preliminary Results!
Lattice Constant vs. x: AxGe136
A = Na, K, Rb, Cs
Preliminary Results!
Lattice Constant vs. x: AxSn136
A = Na, K, Rb, Cs
Preliminary Results!
Lattice Constant vs. x: NaxM136
M = Si, Ge, Sn
Preliminary Results!
Mean Square Atomic Displacement Uiso(T)
Na24Si136, Cs8Ga8Si136, Cs8Ga8Sn136
Rb8Ga8Si136, Rb8Ga8Sn136,
Preliminary Results!
Comments & Conclusions
• Group IV clathrates are interesting “new” materials!
• For NaxSi136 XRD data shows that Na guests must go
into the small Si20 cages for x > 8, lattice contraction
occurs for
0 ≤ x ≤ 8, followed by subsequent lattice expansion
when x approaches 24 from 8.
• Our calculations for K, Rb, and Cs guests in Si136 show
that the dependence of the lattice constant on guest
content is qualitatively similar to that for NaxSi136.
• In contrast, our calculations predict that, for alkali guest
atoms in AxSn136, the lattice constant should be an
increasing function of x in the entire range 0 ≤ x ≤ 24.