Materials Transactions, Vol. 48, No. 4 (2007) pp. 684 to 688
Special Issue on ACCMS Working Group Meeting on Clusters and Nanomaterials
#2007 Society of Nano Science and Technology
Electronic Structure and Thermoelectric Properties of Noble Metal Clathrates:
Ba8 M6 Ge40 (M = Cu, Ag, Au)
Koji Akai1 , Kenji Koga2 and Mitsuru Matsuura3
1
Media and Information Technology Center, Yamaguchi University, Ube 755-8611, Japan
Faculty of Science and Engineering, Tokyo University of Science Yamaguchi, Sanyo-Onoda 755-0884, Japan
3
Yamaguchi Learning Center, The University of The Air, Sanyo-Onoda 755-0884, Japan
2
Electronic structure and thermoelectric properties of noble metal clathrates Ba8 M6 Ge40 (M = Cu, Ag, Au) were calculated by using a first
principle based method to discuss prospect of high performance thermoelectric materials. The calculated band structures show that these
clathrate compounds are degenerate semiconductors with p type carrier. The band gaps become narrow by noble metal doping. The band gap
narrowing is caused by the valence band lifting due to anti-bonding nature between Ge and M, and conduction band widening due to frameworkguest atom coupling via Ba orbitals at 6d sites. The thermoelectric power of Ba8 M6 Ge40 increases with temperatures monotonically and the
magnitude is smaller than 100 mV/K under 1000 K. The carrier concentration of Ba8 M6 Ge40 is a little larger for the high performance
thermoelectric materials. The carrier concentration is controlled by varying the composition of Ba8 Aux Ge46x . The optimized carrier
concentration in Ba8 Aux Ge46x was obtained as 4:4 1026 m3 (p type) and 6:4 1026 m3 (n type), respectively.
[doi:10.2320/matertrans.48.684]
(Received December 17, 2006; Accepted February 4, 2007; Published March 25, 2007)
Keywords: thermoelectric properties, electronic structure, group-IV clathrates, noble metal doping
1.
Introduction
Group-IV element-based clathrates are candidates of high
performance thermoelectric materials. The performance of
thermoelectric materials are characterized by a dimensionless
figure of merit: Z T ¼ S2 T=ðl þ e Þ, where T is an
absolute temperature, S is a thermoelectric power, is an
electric conductivity, l and e are a lattice and a carrier parts
of thermal conductivity, respectively. The condition ZT > 1
is one of the criterions of high performance materials for
device applications. Recently Sramat et al. achieved the high
figure of merit ZT ¼ 1:35 at 900 K in the Ba8 Ga16 Ge30 single
crystal sample.1)
The germanium-based clathrate Ba8 Ga16 Ge30 has a type-I
crystal structure which is composed of two type Ge/Ga cages
containing Ba atoms: Ge20 and Ge24 . These cages are linked
each other by sharing face and a nano-cage network is
fabricated. As the capsulated Ba atom vibrates in the cage
unharmonically, which is called rattling, and interacts with
heat transfer phonons, clathrate compounds have the low
thermal conductivity like glasses.2,3) On the other hand, the
cage network is composed of sp3 hybridized orbitals.
Therefore the sp3 network conducts carriers like Ge crystals
with tetrahedral structure and large carrier mobility is
expected. In 1990s Slack suggested the concept of the high
performance thermoelectric material; phonon glass and
electron crystal (PGEC).4) Group-IV based clathrates are
expected as PGEC materials.
The binary clathrate Ba8 Ga46 is a metal. Because excess
electrons of Ba atoms are concentrated in insulating
conduction bands of the framework Ge46 . In generally as
metallic carrier density causes low thermoelectric power,
carrier density of semiconductor levels is demanded for the
high performance thermoelectric material. The ternary compound Ba8 Ga16 Ge30 is a semiconductor compensated carriers
due to Ga substitution into Ge atoms. Then Ga atoms position
in host sites randomly. The carrier mobility of clathrates is
smaller than that of tetradedral Ge, i.e. magnitude of the Hall
mobility of Ba8 Ga16 Ge30 is 1:4 103 m2 /Vs.5) Usually
carrier mobility of clathrate semiconductors is low relatively.
Therefore increase of the carrier mobility in clathrate
semiconductors is significant to achieve high thermoelectric
performance. We have attention to noble metal doping in
clathrate compounds. It is known that doped noble metal
atoms position at the 6c sites in the type-I clathrate structure,
which is shown in Fig. 1.6,7) It is expected that the ordering of
substituting atoms brings improvement of carrier mobility. In
addition to the ordering effects the noble metal doping
reduces the number of substituting atoms par a unit cell. In
the previous work it was shown that noble metal atoms acts as
acceptors with three holes par an atom.8,9)
Fig. 1 Crystal structure of type-I clathrate Ba8 M6 Ge40 . Large spheres are
Ba atoms, small and white spheres are Ge atoms, small and black spheres
are noble metal atoms M. Ba atoms position at 2a sites and 6d sites,
respectively. Ge atoms position at 16i sites and 24k sites. M atoms position
at 6c sites. The cubic frame with dotted lines denotes a conventional unit
cell.
Electronic Structure and Thermoelectric Properties of Noble Metal Clathrates: Ba8 M6 Ge40 (M = Cu, Ag, Au)
Table 1 Lattice parameters of Ba8 M6 Ge40 . The lattice parameter x is
related to the position of 16i: (x; x; x), lattice parameters y; z are position
parameters of 24k: (0; y; z) in the Pm-3n symmetry.
a(nm)
x
y
z
Ba8 Cu6 Ge40
1.082
0.183
0.317
0.199
Ba8 Ag6 Ge40
1.102
0.183
0.307
0.115
Ba8 Au6 Ge40
1.099
0.183
0.309
0.117
In the present study we discuss about the electronic
structure of noble metal clathrates Ba8 M6 Ge40 (M = Cu, Ag,
Au) and its thermoelectric properties. In the second section
the details of the ab-initio electronic structure calculation of
Ba8 M6 Ge40 and the brief theory of transport coefficients are
introduced. In the third section the results of electronic
structure calculation of Ba8 M6 Ge40 are discussed form the
point of view of band structure and bond picture. In the fourth
section the thermoelectric properties are discussed.
2.
Computational Details
Electronic structure of Ba8 M6 Ge40 is calculated by the Full
Potential Linearized Augmented Plane Wave (FLAPW)
method with the Generalized Gradient Approximation
(GGA).10) The exchange- correlation energy parameterized
by Perdew-Burke-Ernzerhof is used.11) The cut-off energies
of plane waves are 13.2 Ry. The maximum magnitude of the
reciprocal lattice vector G a=2 is 14, which is used in the
Fourier expansion of the charge density. Here a is the lattice
constant. The radii of atomic spheres are given by following;
Ba: 3.0a.u., Ge: 2.2a.u., Cu: 2.2a.u., Ag: 2.4a.u., Au: 2.4a.u.
In the self-consistent field calculation, the Brillouin sampling
of 20 k points is used in the irreducible wedge. In the density
of state (DOS) calculation the sampling of 84 k points are
used.
The lattice parameters were calculated in the previous
work, the results are shown in Table 1.8) Type-I clathrate
structure has the highly crystal symmetry: the space group
Pm-3n, and the five kinds of Wycoff positions: 2a(Ba),
6d(Ba), 6c(M), 16i(Ge), 24k(Ge) as shown in Fig. 1.
By using the linearized Boltzmann equation with the
relaxation time approximation, the thermoelectric power S
and the electric conductivity are given as:
Z
e2
@ f ð"Þ
¼
ð"Þvð"Þ2 ð"Þ;
ð1Þ
d" @"
3
Z
e
@ f ð"Þ
d" ð"Þvð"Þ2 ð"Þð" Þ;
ð2Þ
S¼
3
@"
where f ð"Þ is a Fermi distribution function as a function of
energy ", ð"Þ is a density of states of electronic states, vð"Þ is
a velocity of an electron, ð"; TÞ is a carrier relaxation time, is a chemical potential. In this study the energy dependence
of the relaxation time is neglected for simplicity. Then the
thermoelectric power does not depend on the relaxation time,
because the relaxation time in the numerator of (2) cancels
out that in of the denominator. The DOS ð"Þ and the
velocity vð"Þ are calculated by using the calculated band
structure.
3.
685
Electronic Structure
Band structure and DOS of Ba8 M6 Ge40 (M = Cu, Ag, Au)
are shown in Fig. 2. The broken lines denote the Fermi energy, i.e. the highest occupied states. The band structure and
DOS of Ba8 Ga16 Ge30 are shown for comparison with noble
metal doping cases. The Fermi level is in the valence band,
because two excess holes bring in the Ge framework par a unit
cell. Then Ba8 M6 Ge40 becomes a degenerate semiconductor
with p type carriers or a metal. The overall band structure
of noble-metal-doping clathrates is similar to that of Ba8 Ga16 Ge30 . The characteristic difference is seen in the valence
band, i.e., the sharp DOS peaks due to d orbitals of noble metal atoms are near 3 eV in Ba8 Cu6 Ge40 , 4 eV in Ba8 Ag6 Ge40 and Ba8 Au6 Ge40 . The d orbital character for electronic
states near the Fermi level is not dominant. In spite of this a
small peak or a shoulder structure appears at the edge of the
valence band. This state is characterized by p orbital of Ge
and d orbital of noble metals. Band gap between the unoccupied conduction band and the partially filled valence band become narrow by doping noble metals. The narrowing effect is
the largest in the Cu case and is the smallest in the Au case.
Figure 3 shows contour plot of electronic charge density of
valence electrons on Ba8 Au6 Ge40 . The (100) plane is a cross
section with a 6-members ring in the unit cell. Ba(2a) is at
center of the square and Ba(6d) is on sides of the square. The
(110) plane is sharing the vertical center line through center
of Ba(2a) with the (100) plane. Ge atoms on the (110) plane
are at 16i sites. The nearest neighbor bond of Ge(16i)Ge(16i) is seen in Fig. 3(b). The next nearest neighbor atom
of Ge(16i) is Ge(24k). Two Ge(16i)-Ge(16i) pairs are
coupled via Ge(24k). This bond network is shown as the
belt of charge density in Fig. 3(b). The large density area
above the Ba(2a) atom is correspond to the middle region
between Ge(24k) atoms in Fig. 3(a). From Fig. 3(a) and (b)
the three-dimensional charge network among host atoms is
illustrated. In addition to the whole structure the atomic or the
molecular charge structures are seen. Charge distribution for
Ge atoms is anisotropic which reflects to the form of bonding
sp3 orbitals. The charge distribution of Au is isotropic, as the
d shell of Au is almost closed. The charge density of a
Ge(24k)-Ge(24k) bond is similar to that of a Ge(24k)-Au(6c)
bond. It is shown that the Ge-Au bond is covalent-like one,
that is, the sd orbital of Au and sp3 -orbital are overlapping.
The charge density at the mid point between Ge(24k) and
Au(6c) is larger than that between 16i sites of Ge. Thus the
charge transfer between Ge(24k) and Au(6c) is large, but the
energy gain due to the covalent-like coupling is small as the
charge distribution of Au atoms is isotropic. The charge
density profile clearly shows that host atoms including noble
metal atoms make a charge network. On the other hand
Ba atoms at 2a sites and 6d sites are separated form the
framework. As almost 6s and 5d states of Ba(2a) and Ba(6d)
are empty, the charge density of the valence electrons with
atomic character of Ba is brought by the effect of hybridization among host atoms and guest atoms. The charge
density of Ba(2a) is larger than that of Ba(6d). As Ge20 cages
encapsulating Ba(2a) are smaller than Ge24 cages encapsulating Ba(6a), electrons on Ba(2a) atoms are confined more
strongly and hybridizes with orbitals of Ge20 cages.
686
K. Akai, K. Koga and M. Matsuura
Fig. 2 Band structure and total DOS of Ba8 M6 Ge40 and Ba8 Ga16 Ge30 : a) Ba8 Cu6 Ge40 , b) Ba8 Ag6 Ge40 , c) Ba8 Au6 Ge40 , d) Ba8 Ga16 Ge30 .
Broken lines denote the Fermi level, which is the origin of energy axis. The unit of DOS is states/eVunit cell.
Fig. 3 Charge density of valence electron in Ba8 Au6 Ge40 , a) the (100)
plane which is a cross section of a unit cell though Ba atoms at 2a and 6d,
b) the (110) plane in which the center of Ba atom is at 2a site. In the (100)
plane the Au-Ge-Ge-Au network is a part of six-members rings.
Figure 4 shows the charge density of valence bands and
conduction bands near the band gap, (a) and (b) are charge
densities of the highest 3 valence bands and the lowest
conduction band in Ba8 Au6 Ge40 , (c) and (d) are that in
Ba8 Cu6 Ge40 , respectively. The calculated charge profiles of
germanium clathrates are similar to that in silicon clathrates.8) As seen in silicon clathrates the bond feature is
different between the valence band and the conduction band.
The charge density for the valence band mainly belongs to
six-members rings, Ge4 Au2 , and these rings are connected
via Ba(2a). In Fig. 4(a) the bonding feature of the valence
band between Ge and Au is shown. The dominant atomic
states of Au are d states, then the coupling between d states of
Fig. 4 Charge density for selected bands in the (100) plane, a) the highest
three valence bands of Ba8 Au6 Ge40 , b) the lowest conduction band of
Ba8 Au6 Ge40 , c) the highest three valence bands of Ba8 Cu6 Ge40 , d) the
lowest conduction band of Ba8 Cu6 Ge40 . Ba atoms at the center of squares
are at 2a sites.
Electronic Structure and Thermoelectric Properties of Noble Metal Clathrates: Ba8 M6 Ge40 (M = Cu, Ag, Au)
Au and sp states of Ge has both of features: anti-bonding and
bonding. Due to this anti-bonding coupling the energy of
valence bands make large. Indeed the valence bands shown in
Fig. 2 are flat and isolated from lower bands, and the DOS
has a shoulder structure at the edge of valence bands, which is
not seen in the DOS of Ba8 Ga16 Ge30 . The conduction band is
composed of the Ge(24k) and Ba(6d) network and d orbitals
of noble metal atoms, but noble metal atoms are isolated from
the network.
In Fig. 4(c) the charge density of the highest 3 valence
bands in Ba8 Cu6 Ge40 is shown. This profile is similar to that
of Ba8 Au6 Ge40 in Fig. 4(a), but the detail distribution is
different. The Cu has 3d orbitals that are localized compared
with 4d and 5d orbitals, because there are no inner d shells. In
spite of this, the charge density in the middle region between
Ge and Cu is higher than that between Ge and Au. This higher
charge density brings higher lifting of the flat valence bands.
The distance between Ge-Cu is 0.243 nm, 0.255 nm for GeAu. To contrary this the distance between Ge and Ba(2a) in
Ba8 Cu6 Ge40 , 0.367 nm, is larger than that in Ba8 Au6 Ge40 ,
0.363 nm. On the other hand the distance between Ge and
Ba(6a) in Ba8 Cu6 Ge40 is shorter than that in Ba8 Au6 Ge40 .
This is reasonable from the point of view of ion radii of noble
metal atoms. From Figs. 4b and d, it is seen that the charge
density in the middle region between Ge and Ba(6d) in
Ba8 Cu6 Ge40 is higher than that in Ba8 Au6 Ge40 . This difference of charge density causes difference of the bandwidth
and the band dispersion shown in Fig. 2, i.e., the bandwidth
of the lowest conduction band in Ba8 Cu6 Ge40 is larger than
that in Ba8 Au6 Ge40 . The band gap of Ba8 Cu6 Ge40 is larger
than that of Ba8 Au6 Ge40 , which is caused by the Band lifting
of the valence band and widening of the conduction band due
to enhancement of Ge-Cu and Ge-Ba(6d) couplings.
4.
687
Fig. 5 Temperature dependence of thermoelectric power. The solid line is
Ba8 Au6 Ge40 and the dotted line is Ba8 Cu6 Ge40 .
Thermoelectric properties
The temperature dependence of thermoelectric power of
Ba8 Au6 Ge40 and Ba8 Cu6 Ge40 is shown in Fig. 5. The
temperature dependence of these clathrates is similar. It is
like degenerate semiconductors and magnitude of S is smaller
than 100 mV/K. Then the Fermi DOS is given as F ¼ 14:6
states/eV (Ba8 Au6 Ge40 ), 16.2 states/eV (Ba8 Ag6 Ge40 ) and
15.1 states/eV (Ba8 Cu6 Ge40 ), respectively. For high performance thermoelectric materials the carrier concentration
in Ba8 M6 Ge40 is a little larger than the desirable one. The
carrier concentration can be controlled by varying the amount
of noble metal atoms substituted for Ge atoms. In
Ba8 Mx Ge46x the number of carrier per a unit cell is given
by Nc ¼ 16 3x. When Nc is zero, the clatrates Ba8 Mx Ge46x is an intrinsic semiconductor. When Nc > 0, it is a n
type semiconductors and when Nc < 0, it is a p type semiconductor. The Fig. 6 shows the carrier concentration
dependence of power factor P (¼ S2 ) in Ba8 Aux Ge40x .
The power factor appears in the numerator of the figure of
merit Z. In this calculation the ridged band approximation is
employed, i.e., the calculated band structure of Ba8 Au6 Ge40
is used. The relaxation time to estimate the electric conductivity is put at 1 1014 s, which is reasonable magnitude
in Ge-base clathrates.12) In the overall region with increasing
temperatures the power factor becomes large until about
Fig. 6 Carrier concentration dependence of power factor on Ba8 Au6 Ge40 .
The negative sign denotes hole concentration and the positive sign denotes
electron concentration. The origin of the n axis indicates that the valence
bands are filled perfectly.
800 K. The magnitude of the maximum P for the p type case
is larger than that for the n type case, because the DOS of
valence band is larger than that of conduction bands. In the p
type case the power factor has the maximum at 4:4 1026
m3 and 800 K. In the n type case it has the maximum at
6:4 1026 m3 and 1000 K. At the condition values of
thermoelectric power are 172 mV/K (p type) and 131 mV/K
(n type), respectively. The results are similar to that of
Ba8 Ga16 Ge30 1) and it indicates that noble-metal-doping
clathrates have a highly potential as high performance
thermoelectric materials.
5.
Conclusion
We calculated the electronic structure of noble metal
doped clathrates and charge density for electronic states near
the band gap to discuss the effect of the noble metal doping.
688
K. Akai, K. Koga and M. Matsuura
Noble metal doping causes band gap narrowing due to the
valence band lifting and the bandwidth widening of the
conduction band. The band lifting of the valence bands is
caused by the difference between shapes of p orbital and of d
orbital of noble metal atoms, i.e., d-orbitals on noble metal
atoms bring the anti-bonding nature in 6-members rings. The
bandwidth widening is caused by increasing of coupling
between Ge24 cage and Ba(6d). The gap narrowing effect
becomes large with decreasing ionic radii of noble metal
atoms, as the ionic radius is related to rate of cage sizes
between Ge20 and Ge24 . By shrinking of 6-members ring due
to decreasing ionic radii of noble metal, the coupling between
the guest atom at 2a sites and the host framework become
weak and the coupling between the guest atom at 6d sites and
the host framework become strong.
By the calculation of thermoelectric properties of noble
metal doped clathrates, it was shown that noble metal doped
clathrates Ba8 Mx Ge46x are candidates the high performance
thermoelectric materials, but in fully M-ordering Ba8 M6 Ge40
(M = Cu, Ag, Au) the carrier concentration is large, i.e., the
order of the carrier concentration is 1027 m3 . The optimized
concentration is about 0:5 1027 m3 in both of the hole
doping case and the electron doping case. Then the doping
factor x of the noble atoms is corresponded to about 5.1
(n-type) and 5.6 (p-type), respectively.
REFERENCES
1) A. Saramat, G. Svenson, A. E. C. Palmqvist, C. Stiewe, E. Mueller,
D. M. Rowe, J. D. Bryan and G. D. Stucky: J. Appl. Phys. 99 (2006)
023708.
2) J. L. Cohn, G. S. Nolas, V. Fessatides, T. H. Metcalf and G.A. Slack:
Phys. Rev. Lett. 82 (1999) 779.
3) J. Dong, O. F. Sankey and C. W. Myles: Phys. Rev. Lett. 86 (2001)
2361.
4) G. A. Slack: CRC Hnadbook of Thermoelectrics, ed. Rowe, (CRC
Press, 1995), 407.
5) V. L. Kuznetsov, L. A. Kuznetsova, A. E. Kaliazin and D. M. Rowe:
J. Appl. Phys. 87 (2000) 245113.
6) G. Cordir and P. Woll: J. Less-Common Met. 169 (1991) 291.
7) R. F. W. Herrmann. K. Tanigaki, T. Kawaguchi, S. Kuroshima and
O. Zhou: Phys. Rev. B60 (1999) 13245.
8) K. Akai, K. Koga, K. Oshiro and M. Matsuura: Trans. MRSJ 29 (2004)
3647.
9) K. Akai, G. Zhao, K. Koga, K. Oshiro and M. Matsuura: Proc. of 24th
Int. Conf. on Thermoelectrics, (2005) 215.
10) P. Blaha, K. Schwarz, G. Madsen, D. Kvasnica and J. Luitz: program
package WIEN2k, Technical University of Vienna (2001).
11) J. P. Perdew, S. Burke and M. Ernzerhof: Phys. Rev. Lett. 77 (1996)
3865.
12) N. P. Blake, S. Latturuner, J. D. Bryan, G. D. Stucky and H. Metiu:
J. Chem. Phys. 115 (2001) 8060.