J. Phys. Chem. C 2010, 114, 13947–13953
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Improved Thermoelectric Properties of La1-xSrxCoO3 Nanowires Yang Wang and Hong Jin Fan* DiVision of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological UniVersity, 21 Nanyang Link, 637371, Singapore ReceiVed: June 11, 2010; ReVised Manuscript ReceiVed: July 15, 2010
Nanotechnology has brought novel and controllable thermoelectric (TE) characteristics to conventional materials. Here we report a simple method to synthesize single-crystalline La1-xSrxCoO3 nanowire compacts with an improved TE efficiency. The figure of merit ZT in the nanowires can reach 0.19 at 300 K, which is nearly twice that in the bulk, and this room-temperature ZT is very high among oxide materials. The enhancement of TE efficiency can be attributed to both extrinsic and intrinsic factors: First, the unique nanowire structure together with numerous interfaces in the compact samples strongly suppresses the phonon thermal transport. Second, the small dimensions of nanowires give rise to an unexpected increase in the thermopower due to a reduction of the effective bandwidth and variation of the electron configuration. Our results indicate that the TE response of conventional oxides can be effectively improved by the form of nanowires, which can be synthesized much easier than single crystals and epitaxial thin films. 1. Introduction Environmentally friendly thermoelectric (TE) materials can convert a temperature difference into electricity via the Seebeck effect and vice versa via the Peltier effect. The efficiency of a TE material is determined by the dimensionless figure of merit
ZT )
S2T Fκ
(1)
where S is the thermopower, F is the resistivity, κ is the thermal conductivity, and T is the absolute temperature. For real applications, ZT > 1 is considered as a prerequisite condition. As the parameters of ZT are generally not independent, very few alloys exhibit ZT > 1 to date,1,2 and these TE alloys remain state of the art and have been widely used as commercial TE materials. Recently, the search of new TE materials using lowdimensional (nanostructured) systems has become an active topic because band structure and phonon engineering can be used to overcome the efficiency barriers imposed by the physics of conventional bulk materials.3-5 Compared with bulks, the density of electronic states (DOS) in a nanostructured system usually has sharp peaks and theoretically a large S.6,7 More importantly, the phonon dynamics and heat transport in a nanostructured system can be tailored. Nanostructures with one or more dimensions smaller than the mean free path of phonons but larger than that of electrons will noticeably reduce κ without affecting electrical transport.5 In other words, phonon transport will be strongly disturbed while the electronic structure remains bulk-like in nanostructured systems. On the basis of such a concept, efficient TE performance has been observed in several low-dimensional nanomaterials.3,8-10 Design of nanostructured materials (semiconductor alloys/composites) has been shown to be an effective way of suppressing κ and improving TE efficiency by both experiment and theory.11-20 * To whom correspondence should be addressed. E-mail: fanhj@ ntu.edu.sg.
Compared with the traditional TE alloys, metal oxides are more suitable for TE applications due to their structural and chemical stabilities, oxidation resistance, and low cost. The discovery of a large TE response in several oxides, such as NaCoO2,21 Ca3Co4O9,22 Sr1-xLaxTiO3,23 and La1-xSrxCoO3,24,25 has attracted new attention to oxide thermoelectrics. The origins of the unusual TE characteristics in these oxides are quite complex and not fully understood but may be ascribed to the high spin/orbital degeneracy as well as the large electron effective mass.23,26 In order to explore the origin of the large TE response and improve the TE performance in the oxide systems, a large number of investigations have been conducted on these oxide materials of various forms, including single crystals, ceramics, and thin films.27-32 Nevertheless, studies on the TE properties of nanostructured oxides are very scanty. In these narrow-band oxide systems, S and F usually increase or decrease simultaneously due to the strong electronic correlation. Therefore, suppressing phonon thermal transport by nanostructuring is a potential approach to effectively enhance ZT in oxides. In this study, we report the TE properties of single-crystalline La1-xSrxCoO3 nanowires. LaCoO3 has a nonmagnetic ground state and with the increase in temperature the Co3+ ions undergo 6 ) to intermediate spin (IS, a transition from low spin (LS, t2g 5 1 33-35 t2geg) occurring at ∼90 K. By divalent Sr2+ doping, the nonmagnetic ground state is suppressed and the LS-IS transition is shifted to lower temperatures, suggesting that the doping stabilizes the IS state.36 Furthermore, the Sr2+ doping results in a small F together with a relatively large S, making La1-xSrxCoO3 a potential candidate for p-type TE oxide.24,25,32 We ever investigated the TE properties of bulk La1-xSrxCoO3 ceramics and found that the optimal TE performance exists at x ) 0.1.37 Thus, herein we only present the results of La0.9Sr0.1CoO3 nanowires. 2. Experimental Section Synthesis of Materials. Single-crystalline La1-xSrxCoO3 nanowires were synthesized via the hydrothermal method. First, stoichiometric amounts of La(NO3)3, Sr(NO3)2, and Co(NO3)2
10.1021/jp105367r 2010 American Chemical Society Published on Web 07/28/2010
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Figure 1. (a, b, and c) SEM micrographs for as-prepared nanowires with diameters of 56, 83, and 107 nm, respectively; the insets are the magnifications for individual nanowires. (d) Diameter distribution histograms for the 56 (red), 83 (blue), and 107 nm (green) nanowires, respectively; the inset shows a SEM micrograph for the cold high-pressed 107 nm nanowires. (e) SAED pattern for the 107 nm nanowires; the inset is the TEM image. (f) HRTEM image of a portion of the 107 nm nanowires.
were dissolved in deionized water, and the pH of the mixed solution was adjusted to 13 by adding KOH. After ultrasonic stirring, the solution was transferred into a Teflon autoclave for hydrothermal treatment for 48 h at 200, 230, and 250 °C, respectively, to get nanowires with different diameters. After the autoclave was cooled down to room temperature, the soobtained solid products were washed with deionized water and ethanol and subsequently dried overnight. The bulk ceramic samples were synthesized by a conventional solid-state reaction as described in ref 37. Structure and Morphology Characterization. Phase identification and the crystal structure were determined by X-ray powder diffraction (XRD) using a XRD diffractometer (D8 Advanced) with Cu KR radiation. The data were collected from 10° up to 140° in 2θ with 0.02° step and a counting time of 30 s per step. The morphology was observed with scanning electron microscopy (SEM, JEOL JSM-6700F) and highresolution transmission electron microscopy (HRTEM, JEOL 2100F). By means of a JEOL 2100F, selected area electron diffraction (SAED) patterns of the samples were also obtained. Physical Properties Measurements. The nanowires were cold high pressed into compact pellets for physical properties measurements. First, the as-synthesized nanowire powders were placed in a 6 mm diameter hole drilled in the center of a gasket made of a dense, strong cardboard and loaded between two tungsten-carbide anvils. Subsequently, the powders in a gasket were cold pressed into pellets under a high pressure of 3 GPa with a special steel die for several minutes. After release of pressure, a thin, compact, pellet-shaped sample with metallic shine was obtained. Finally, the samples were annealed at 373 K to form hard bulks. The temperature dependences of resistivity, thermopower, thermal conductivity, specific heat, and Hall coefficient were all measured using the physical property measurement system (Quantum Design PPMS). Resistivity and thermopower measurements were performed by a standard four-
Figure 2. XRD pattern and the Rietveld refinement result for the 56 nm nanowires. Experimental data are shown as red dots; global fitting profile and difference curve are shown as solid lines; the calculated reflection positions are indicated by vertical lines. Insets: (a) sketch of the crystalline structure for LaCoO3; (b) sketch of the electron configurations for IS Co3+ ion with and without JT splitting, corresponding to the cases of bulk and nanowires, respectively.
probe method, whereas the Hall coefficient was measured using a five-probe method with Cu wire serving as leads. The thermal conductivity and specific heat were measured using a steadystate technique in a closed cycle refrigerator with high vacuum (10-5 Torr). 3. Results and Discussion Morphology and Crystalline Structure. Representative SEM and TEM images as well as the SAED pattern are shown in Figure 1. It is observed that the as-prepared samples consist exclusively of nanowires. The XRD patterns of the nanowires (Figure 2) exhibit consistent diffraction peaks with those of the La0.9Sr0.1CoO3 ceramic counterparts37 and the JCPDS card, indicating the single-phase nature of these nanowires. The
Thermoelectric Properties of La1-xSrxCoO3 Nanowires TABLE 1: Room-Temperature Lattice Parameters for the La0.9Sr0.1CoO3 Bulk and Nanowires bulk 107 nm 83 nm 56 nm
a (Å)
b (Å)
c (Å)
R (deg)
V (Å3)
5.3716(4) 5.3729(6) 5.3772(7) 5.3868(8)
5.4343(4) 5.4357(5) 5.4413(7) 5.4516(7)
7.6469(5) 7.6481(6) 7.6540(6) 7.6638(7)
90.8124(8) 90.8037(8) 90.7704(9) 90.7217(9)
223.19(2) 223.34(3) 223.92(3) 225.03(3)
energy-dispersive spectrum (EDS) measurements, performed by JEOL JSM-6700F, confirmed that the Sr content in both nanowires and ceramic bulk is about 0.1. As the hydrothermal treatment temperature rises, the diameter of the obtained nanowires increases gradually. As shown in Figure 1a, 1b, and 1c, the typical diameters of the synthesized nanowires are 56, 83, and 107 nm at 200, 230, and 250 °C, respectively. The size distribution is reasonably small (see Figure 1d). The diameter of these nanowires is much smaller than the grain size of ∼3-5 µm for ceramics prepared by solid-state reaction.37 The SEAD and HRTEM patterns (e.g., see Figure 1e and 1f) recorded from different positions along the same nanowires are identical, indicating these nanowires are of high-quality single crystallinity. The inset of Figure 1d shows the typical top view of the compact samples. One can see that the nanowires retain their morphology and size after being compressed at a pressure of 3 GPa, and the pellet is quite dense without obvious pores. The density of the compact pellets is ∼7.0 g/cm3, close to the theoretical density of 7.2 g/cm3. Hence, the numerous interfaces between nanowires will play an important role in transport behavior, whereas the effect of porosity is negligible.
J. Phys. Chem. C, Vol. 114, No. 32, 2010 13949 The structure parameters were determined by Rietveld refinement based on the XRD data using a profile analysis program (Fullprof). It should be emphasized that while the LaCoO3 system has been long interpreted to have a rhombohedral R3jc symmetry,38,39 this space group is incompatible with cooperative JT distortion as IS Co3+ is strongly JT active. Recent high-resolution XRD studies revealed that LaCoO3 has a monoclinic symmetry (I2/a space group) that is compatible with JT distortion.34,40 Herein our Rietveld refinement of all samples was also carried out by considering a monoclinic structure described by the I2/a space group with a ≈ b ≈ 2ap and c ≈ 2ap (where ap is the lattice parameter of an ideal cubic-perovskite structure), and good agreement is obtained between the observed and calculated XRD patterns (see Figure 2). The refined quality factors are all below 10%, ensuring the reliability of the refinement. The SAED pattern in Figure 1e can also be indexed to a monoclinic structure. As listed in Table 1, the lattice parameters a, b, and c in the nanowires are all larger than those of bulk and the values of a, b, and c all increase gradually with the reduction of wire diameter. This behavior indicates expansion of the unit cell in the nanowires, which could be attributed to the surface effect in nanostructures.41,42 In nanostructures such as nanowires or nanoparticles, the ratio of the surface to the volume gradually increases with decreasing wire/particle size. The enhanced surface states can release surface strain and result in an increased number of broken bonds (e.g., Co3+-O and Co4+-O bonds in the case of La1-xSrxCoO3 system) that give rise to more unpaired electronic orbitals at the surface.42 These variations will cause
Figure 3. Rom-temperature structure parameters for the bulk and nanowires as a function of unit-cell volume V: (a) Co-O bond length dCo-O, (b) Co-O-Co bond angle θCo-O-Co, (c) degree of JT distortion D%, (d) distortion parameters φ and Q. The variations of these structure parameters clearly indicate that with the unit cell expanding (i.e., nanowires narrowing down), both global and local distortions decrease.
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Figure 4. Temperature dependences of (a) thermopower S and (b) resistivity F. The inset of b shows the difference of S between the nanowires and the bulk. Note that S-Sbulk is nearly temperature independent for the 107 nm nanowires, while there is a wide hump around 150 K for the 56 and 83 nm nanowires that can be attributed to a phonon drag effect. The solid lines are a guide for the eyes.
an increased lattice constant compared to the bulk in order to attain stability.41,42 Such expansion of the unit-cell with the reduction of size in nanomaterials has been observed in several metal oxides with perovskite structure.42-44 The monoclinic angle R decreases with a reduction of diameter, which indicates that the global structure distortion of this system decreases gradually as the nanowires narrow down, viz. a higher symmetry (orthorhombic or cubic) is approached. Due to the close correlation of lattice structures with electric transport properties, in the following we exam the lattice structural distortions of the nanowires. The variations of Co-O bond length (dCo-O) and Co-O-Co bond angle (θCo-O-Co) as a function of unit-cell volume are presented in Figure 3a and 3b, respectively. The two different bond angles, θCo-O(1)-Co and θCo-O(2)-Co, both increase toward 180° as the unit cell expands (i.e., diameter decreases), which is consistent with the evolution of R, also implying a reduction of global structure distortion. In the I2/a space group there are three unequal Co-O bond lengths: one Co-O(1) and two Co-O(2). Therefore, the JT distortion that lifts the degeneracy of eg orbitals can be triggered. From the refinement results, all samples show a clear difference in the bond lengths, indicating that these samples are JT distorted. With the expansion of the unit cell, the Co-O(1) bond of medium length exhibits only a weak variation; the shorter Co-O(2)2 increases, while the longer Co-O(2)1 decreases gradually (note the average Co-O bond length djCo-O still increases with the expansion of the unit cell). This behavior suggests that the local CoO6 octahedral distortion (i.e., JT distortion), the same as the global distortion, is also suppressed as the nanowires narrow down.
To further validate the structure distortions in the samples, the variations of JT distortion and CoO6 octahedral distortion are shown in Figure 3c and 3d. The degree of JT distortion D% was calculated by dividing the difference between the two Co-O(2) bond lengths by their average. Another two measures of the local structural distortions, Q and φ, are described in the Supporting Information. The decreases in D%, Q, and φ clearly indicate that the local distortion lessens with the expansion of the unit cell. In summary, the variations of these structural parameters (dCo-O, θCo-O-Co, R, D%, Q, and φ) unambiguously reveal that the global and local structure distortions are both suppressed in the nanowires. The expansion of the unit cell is the reason for the reduced structure distortions. It has been reported that the structure distortions gradually decrease as the the unit cell expands in RCoO3 (R ) rare earth) bulks and LaCoO3 epitaxial films due to the release of lattice strain.43-45 In the following, we show that these variations of structure distortions strongly influence the TE properties in the nanowires. Electrical Transport Properties. Temperature dependences of resistivity F and thermopower S are presented in Figure 4. These samples have similar F-T and S-T dependences, but the values of F and S rise in the nanowires. Such changes result from both intrinsic and extrinsic factors as discussed below. The structure variations induce an increase in the bond angle θCo-O-Co together with elongation of the bond length dCo-O as mentioned above, which will lead to a change in the effective bandwidth. Within the tight-binding approximation, the effective bandwidth W is represented by46
W∝
cos[(π - θ¯ Co-O-Co)/2] 7/2 d¯Co-O
(2)
j Co-O-Co denote the average Co-O bond length where djCo-O and θ and Co-O-Co bond angle, respectively. The calculated relative bandwidth Wr () W/WLa0.9Sr0.1CoO3 bulk) decreases monotonously with the expansion of the unit cell (see Table 2), indicating that the bandwidth narrows in the nanowires. The variation of W is also evidenced by specific heat measurements, because the decrease in W means an enhancement of electron correlation, viz. the increase in the DOS of the conduction band as well as the electronic effective mass m*. From the specific heat results, the electron specific heat coefficient γ increases gradually with decreasing nanowires diameter (see Table 2), confirming the decrease in W and enhancement of electron correlation. The narrowing of the effective bandwidth will lead to localization of carriers. From the Hall coefficient measurements it is confirmed that the carrier concentration n decreases as W narrows (Table 2). On the other hand, the considerable interfaces in the nanowires will strongly scatter carriers, which is evidenced by the observable reduction of mobility µ (Table 2). Therefore, the nanowires have lower n and much smaller µ than the bulk, and consequently, compared with the bulk, these nanowires exhibit an increase in F. As the thermopower of materials is hardly affected by the interfaces or grain boundaries, the change in S in the nanowires
TABLE 2: Room-Temperature Carrier Concentration n and Mobility µ, Relative Bandwidth Wr, Electronic Specific Heat Coefficient γ, and TE Parameters at 300 K for the La0.9Sr0.1CoO3 Bulk and Nanowires bulk 107 nm 83 nm 56 nm
n (1020/cm3)
µ (cm2/(V s))
Wr
γ (mJ/K2 mol)
S (µV/K)
F (mΩcm)
κ (W/mK)
ZT
5.61(7) 5.22(7) 4.89(6) 4.66(7)
0.313(5) 0.092(3) 0.062(2) 0.039(2)
1 0.99851 0.99362 0.98936
3.81(3) 3.90(3) 4.02(4) 4.09(4)
159.7 ( 1.1 178.1 ( 1.5 213.2 ( 1.3 236.2 ( 1.6
3.58(6) 13.1(2) 20.6(3) 33.8(5)
2.20(9) 0.51(3) 0.35(2) 0.29(2)
0.10 0.14 0.19 0.17
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Figure 5. Temperature dependence of thermal conductivity κ. The inset presents the room-temperature κ, κph, and κcar for the bulk and nanowires. κcar is calculated from the Wiedemann-Franz law κcar ) L0T/F, where L0 is the Lorentz constant that L0 ) π2kB2/3e2 ) 2.45 × 10-8 W Ω K-2. The result indicates that κcar is less than 10% of the total κ.
originates from an intrinsic variation. The decrease in carrier concentration and enhancement of electron correlation can partially account for the enhanced thermopower in the nanowires because S ∝ γ/n in a simple Drude picture for such strongly correlated system.47 However, here the changes in n and γ cannot quantitively explain the pronounced increase in S (the room-temperature S increases from 159.7 µV/K for the bulk to 236.2 µV/K for the 56 nm nanowires). Considering the important role of the electron configuration in S in the cobaltites,26 the effects of spin/orbital states on S should also be taken into account. S of doped cobaltites is expected to be determined by Heikes formula, namely
S)-
[
kB g3 x ln e g4 1 - x
(
)]
(3)
where g3 (g4) denotes the number of possible configurations of the Co3+ (Co4+) ions.26,48 For the nanowires, the increase in djCo-O will reduce the crystal-field splitting ∆CF according to ∆CF ∝ 1/dj5Co-O,49 which implies that the IS state of Co3+ is more stable. Therefore, the lessening of JT distortion in the nanowires suggested that the JT splitting is essentially suppressed (see the sketch of the inset of Figure 2).50 Accordingly, the total degeneracy g3 ) 9 for the bulk with JT active Co3+, but it becomes 18 in the case of the absence of JT splitting. Using eq 3 one can calculate that the change in degeneracy will bring an increment of S that ∆S ≈ 59.6 µV/K. A detailed discussion on electron configuration and S are presented in the Supporting Information. When taking the degeneracy into account, the remarkable enhancement of S in the nanowires is well elucidated. This also indicates the crucial role of the electron configuration in determining S in the cobalt oxides. Thermal Conductivity and TE Characteristics. As presented in Figure 5, the thermal conductivity κ drops sharply with decreasing the nanowire diameter. Generally, κ can be expressed by the sum of phonon component κph, carrier component κcar, and magnetic spin wave component κm as κ ) κph + κcar + κm. The La0.9Sr0.1CoO3 system exhibits spin glass behavior at low temperature,33 so the specific heat C-T curves do not show any peaks (see Supporting Information). The kinetic expressions of the phonon and magnetic spin wave components are κph ) CphVph2τph/3 and κm ) CmVm2τm/3, where Cph (Cm), Vph (Vm), and τph (τm) are specific heat, mean velocity, and lifetime
of the phonon (spin wave), respectively. Using Cohn et al.’s method,51 the estimated κm is estimated to be near zero for the present system. As calculated from Wiedemann-Franz law, the carrier thermal conductivity κcar is also very small (see the inset of Figure 5). Therefore, the phonon thermal conductivity κph is dominant in κ in this system, and the variation of κ results mainly from the change in κph. The κ reduction can be attributed to reasons as discussed below. First, the noticeable interfaces in the nanowires strongly scatter phonons, similar to the reported nanostructured alloys.52,53 For example, Chen and Ren et al. found a very low κ in alloyed nanopowder compacts fabricated by hot pressure and attributed the suppression of κ to numerous grain boundaries.52-54 We can therefore conclude that the boundaries between nanowires play a similar role on κ in our nanowire compacts. The second factor is the unique κ reduction existing in nanowires,9,10,55,56 where the phonon mean free path is limited by nanowire boundary scattering as opposed to intrinsic Umklapp scattering.9 By fitting the low-temperature specific heat data using C ) γT + βT3, one can get the phonon specific heat coefficient β. Then the Debye temperature θD can be calculated to be 380 and 304 K for the bulk and 56 nm nanowires, respectively, according to
θD )
(
12π4Rp 5β
)
1/3
(4)
where p is the number of atoms per unit cell. The mean sound velocity V is described by
V)
( )
kBθD 1 p 6π2N
1/3
(5)
where N is the atom density per unit volume. Thus, one can calculate that V56 nm/Vbulk ≈ 0.8. Then on the basis of κ ) CVlph/ 3, the ratio of the mean phonon free path is estimated to be lph56 nm/lphbulk ≈ 0.17. The pronounced reductions of the mean phonon free path and sound velocity result in strong suppression of κ in the nanowires. Finally, the nanosized wires may cause more efficient phonon scatterings, as the dominant phonon wavelength at low temperatures is on the order of several nanometers.57 Consequently, at low temperatures, the longwavelength phonon modes, which contribute strongly to thermal transport in bulk, are efficiently scattered in the nanowires, and thus, the “shoulder” around 70 K in the κ-T curve of the bulk disappears. The wide hump around 150 K in S-T curves in the 56 and 83 nm nanowires (see the inset of Figure 4b) can be attributed to the phonon drag effect. Although the phonon drag peaks usually exist at very low temperatures (lower than 100 K),58 they can shift to higher temperatures in materials with small dimensions as the observation in Si nanowires.10 In nanowires, the contribution from phonon drag to thermopower (Sph) is ∝1/ µTκ,10 but should be ∝T3 as T f 0 because the phonon free path reaches the sample size and the specific heat tends to zero at low temperature.59 Accordingly, the phonon drag can give rise to a wide hump in the middle temperature range in the nanowires due to the decreases in µ and κ. The room-temperature ZT values are listed in Table 2 and shown in Figure 6. ZT is enhanced in these nanowires mainly due to the increase in S along with strong suppression of κ. The highest ZT (∼0.19) is obtained from the 83 nm nanowires, which is nearly twice that of the bulk. This room-temperature ZT value is very high among oxide materials. It is noted that as
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Figure 6. Plot of the ZT value versus nanowire diameter, as a comparison to the bulk.
the nanowires narrow down, ZT does not increase monotonously. This indicates that ZT may be maximized at an optimal size. Overall, our results suggest that the TE performance of oxide materials can be effectively improved by nanoengineering nanomorphology. 4. Conclusions La0.9Sr0.1CoO3 nanowires are found to have a ZT value up to 0.19 at room temperature, nearly double the bulk value. This ZT enhancement can be attributed to the unexpected increase in the thermopower and strong suppression of thermal conductivity, both of which are nanowire diameter dependent. On the basis of these results, it is concluded that (i) the nanowire samples exhibit unique transport properties which are caused by a combination of both intrinsic and extrinsic factors, (ii) the global/local structure distortions have a close correlation to the transport and TE properties in nanostructures of these strongly correlated oxide systems, and (iii) the size dependence of ZT is not monotonous; it is speculated that there exists an optimal size in nanomaterials for TE efficiency where ZT could be maximized. Acknowledgment. This work was supported by the start-up funding from Nanyang Technological University to F.H.J. The authors express their thanks to Dr. T. Sun, Prof. J. Ma, and Prof. H. H. Hng of the School of Materials Science and Engineering, Nanyang Technological University for HRTEM measurements and helpful discussions. Supporting Information Available: Crystalline structure, definition of the parameters Q and φ, detailed discussion on electron configuration and calculation of S, and plot of specific heat versus temperature. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Bell, L. E. Science 2008, 321, 1457–1461. (2) Snyder, G. J.; Toberer, E. S. Nat. Mater. 2008, 7, 105–114. (3) Venkatasubramanian, R.; Siivola, E.; Colpitts, T.; O’Quinn, B. Nature 2001, 413, 597–602. (4) Lin, Y. M.; Rabin, O.; Cronin, S. B.; Ying, J. Y.; Dresselhaus, M. S. Appl. Phys. Lett. 2002, 81, 2403–2405. (5) Majumdar, A. Science 2004, 303, 777–778. (6) Hicks, L. D.; Dresselhaus, M. S. Phys. ReV. B 1993, 47, 16631– 16634. (7) Humphrey, T. E.; Linke, H. Phys. ReV. Lett. 2005, 94, 096601. (8) Harman, T. C.; Taylor, P. J.; Walsh, M. P.; LaForge, B. E. Science 2002, 297, 2229–2232. (9) Hochbaum, A. I.; Chen, R. K.; Delgado, R. D.; Liang, W. J.; Garnett, E. C.; Najarian, M.; Majumdar, A.; Yang, P. D. Nature 2008, 451, 163–167.
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