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J. Phys. Chem. B 2008, 112, 1482-1486
Determinants of Thermal Conductivity and Diffusivity in Nanostructural Semiconductors C. C. Yang,* J. Armellin, and S. Li School of Materials Science and Engineering, The UniVersity of New South Wales, NSW 2052, Australia ReceiVed: NoVember 4, 2007; In Final Form: NoVember 20, 2007
The origin of size effects in the thermal conductivity and diffusivity of nanostructural semiconductors was investigated through the establishment of a unified nanothermodynamic model. The contributions of sizedependent heat capacity and cohesive energy as well as the interface scattering effects were considered during the modeling. The results indicate the following: (1) both the thermal conductivity and diffusivity decrease with decreasing nanocrystal sizes (x) of Si and Si/SiGe nanowires, Si thin films and Si/Ge(SiGe) superlattices, and GaAs/AlAs superlattices when x > 20 nm; (2) the heat transport in semiconductor nanocrystals is determined largely by the increase of the surface (interface)/volume ratio; (3) the interface scattering effect predominates in the reduction of thermal conductivity and diffusivity while the intrinsic size effects on average phonon velocity and phonon mean free path are also critical; (4) the quantum size effect plays a crucial role in the enhancement of the thermal conductivity with a decreasing x ( 14 nm.17 An enhancement in κL with decreasing x is also reported when 4.4 < x < 14 nm.18 Different from this finding, κL was found to decrease with the decreasing of x when x > 7 nm whereas it increases with decreasing x when 3 < x < 7 nm.21 Publications such as these have demonstrated that the relationship between κL and x is non-monotonic and κL should have a minimum value at a critical x. A reasonable explanation as to the cause of this is eagerly awaited. Aside from the change of size, the change of dimensionality also results in different κL and D in the nanocrystals due to the various dimensionalities of confinement, and the different phonon dispersion relations that occur as a consequence.8,16 While two-dimensional thin film and superlattice structures show promising properties,12-25 one-dimensional nanowires may possess even more desirable characteristics and prove to be the source of even further improvements in device performance.10,11
10.1021/jp710588z CCC: $40.75 © 2008 American Chemical Society Published on Web 01/15/2008
Nanostructural Thermoelectric Semiconductors For this reason, the study of the dimensionality dependence of κL and D, as well as the differences in the heat-transport process between nanowires and thin films or superlattices, is beneficial to the understanding of the nature of heat transport in nanostructures and hence to industry. In complement to experimental efforts, several computer simulations27,28 and analytical models7,29-36 have also been developed to study the size dependence of κL in nanostructural materials. These theoretical methods have established a framework for the size dependence of κL. Recently, a theory with a phenomenological basis was proposed to describe the size and dimensionality dependencies of κL on the nanometer scale.37 By incorporation of the effect of interface scattering, it could reproduce the trends of observations. However, (i) the size dependence of D, which is directly measured in experiments, has not been considered; (ii) the variation of κL associated with the change of x in the range of x < 20 nm has not been explored; (iii) the size dependence of the heat capacity C has not been included in these approaches. Therefore, a general quantitative model needs to be developed to (1) reveal the intrinsic factor that dominates the variation of both κL and D at the nanometer scale and (2) determine κL and D of Si and GaAs nanocrystals with respect to size and dimensionality effects. Recently, we found that the size dependence of the cohesive energy Ec can be used to describe the size-dependent physicalchemical properties of nanocrystals.38 In this work, a unified model is established to investigate the size-, dimensionality-, and specular scattering parameter-dependent κL and D of semiconductor nanocrystals on the basis of the size-dependent C and Ec models as well as scattering mechanisms in nanocrystals. The accuracy of the developed model was verified through comparison with experimental data from the open literature. An expanded understanding of κL and D of semiconductor nanocrystals will advance the development of their applications in optoelectronic and thermoelectric devices. 2. Methodology Based on a kinetic theory, κL of a bulk crystal is given by κL ) Cνsλ/3 where νs is the average phonon velocity and λ is the phonon mean free path (average distance traveled by phonons between anharmonic interactions with other phonons or scattering events with imperfections, electrons, and impurities).3,4,14,25,33 The thermal diffusivity is related to κL through the equation D ) κL/(FC) where F ) M/V is the density with M and V being the molar weight and volume, respectively.22,23 Note that the values of κL in experiments are calculated from this formula. Hence, D ) νsλ/(3F). As the crystal size has very limited influence on the atomic or molecular diameter h and the change of h is usually in the range of 0.1-2.5% even when x < 20 nm, the size effect on F can be neglected.39 Therefore, the size-dependent κL and D can be obtained on the basis of the size dependencies of C, νs, and λ. According to the approximation of an isotropic continuum, νs is proportional to the characteristic Debye temperature Θ, and νs(∞) ∝ Θ(∞) where ∞ denotes the bulk.37,40 A simple expression of the relationship between λ(∞) and bulk melting temperature Tm(∞) could be written as λ(∞) ) 20aTm(∞)/(γ2T) at a certain temperature where a is the lattice constant, γ is the Gruneisen constant, and T is the absolute temperature.41 Thus, λ(∞) ∝ Tm(∞).37 Since Θ2(∞) ∝ Tm(∞) ∝ Ec(∞),38 we have νs(∞) ∝ Ec1/2(∞) and λ(∞) ∝ Ec(∞). The relationship between Ec(∞) and κL(∞) or D(∞) are given as κL(∞) ∝ C(∞)Ec3/2(∞) and D(∞) ∝ Ec3/2(∞), respectively. It is assumed that this relationship can be extended to the nanoscale range as a first-order
J. Phys. Chem. B, Vol. 112, No. 5, 2008 1483 approximation.38 Thus, κL(d,x)/κL(∞) ) [C(d,x)/C(∞)][Ec(d,x)/ Ec(∞)]3/2 and D(d,x)/D(∞) ) [Ec(d,x)/Ec(∞)]3/2 where d is the dimensionality. Considering the similar size dependencies between the heat capacity and the entropy,42 the C(d,x)/C(∞) function can be expressed as follows,
C(d,x)/C(∞) ) 1 - 1/(x/x0 - 1)
(1)
where x0 ) 2(3 - d)h is a critical size where the dimensionality d ) 0 for nanoparticles, d ) 1 for nanowires, and d ) 2 for thin films or superlattices. Combining the Ec(x) function reported in the literature38 and the consideration of dimensionality effect, the Ec(d,x)/E(∞) function is obtained by
Ec(d,x) Ec(∞)
(
) 1-
) (
)
2Sb 1 1 exp 12x/x0 - 1 3R 12x/x0 - 1
(2)
where Sb ) Eb/Tb is the bulk solid-vapor transition entropy of a crystal with Eb and Tb being the bulk solid-vapor transition enthalpy and the solid-vapor transition temperature, respectively, and R is the ideal gas constant. The above models focus primarily on the intrinsic size effects of semiconductor nanocrystals on νs and λ. However, the phonon scattering mechanisms may affect the values of κL(d,x) and D(d,x). Recent advances in the thermal transport in nanostructures show that (1) the variation of κL(d,x) and D(d,x) are dominated by interface scattering in nanocrystals and (2) the phonon transport at the interfaces is partially specular and partially diffuse with the probability p and 1 - p (p is a specular scattering parameter).7,31-33,35 When p is equal to 0 or 1, the scattering is considered to be fully diffuse or fully specular, respectively. In the case of interfaces with partially specular and partially diffuse, 0 < p < 1. Moreover, the interface scattering shows an exponential suppression in nanocrystals as compared with that in the corresponding bulk crystals.31,35,37 As a result, a term p exp(-λ/x) should be incorporated to correct the expressions of κL(d,x) and D(d,x). Note that the size effect on λ has been considered in the above discussion and it is assumed to be a constant here. On the basis of eqs 1 and 2 as well as the consideration of the interface scattering effect, the size, dimensionality, and specular scattering parameter-dependent κL(p,d,x) and D(p,d,x) of semiconductor nanocrystals can be expressed respectively as
κL(p,d,x) κL(∞)
( )(
) p exp -
)[(
)
λ 1 1 11× x x/x0 - 1 12x/x0 - 1 3/2 2Sb 1 exp (3a) 3R 12x/x0 - 1
(
)]
and
( )[(
D(p,d,x) λ ) p exp x D(∞)
)
1 × 12x/x0 - 1 2Sb 1 exp 3R 12x/x0 - 1
1-
(
)]
3/2
(3b)
Note that both κL and D are size- and temperature-dependent parameters because C, νs, and λ are subject to change with temperature.4,25,43 In this work, we investigated the determinants of thermal conductivity and diffusivity in nanostructural semi-
1484 J. Phys. Chem. B, Vol. 112, No. 5, 2008
Figure 1. κL(p,d,x)/κL(∞) (solid lines calculated from eq 3a) of Si and Si/SiGe nanowires (NW), Si thin films, and Si/Ge(SiGe) superlattices (SL) as well as D(p,d,x)/D(∞) (dashed line calculated from eq 3b) of Si thin films. The symbols 2,10 1,11 b,13 [,14 O,15,16 +,17 3,18 ×,19 solid sideways triangle,20 and 421 are experimental data of κL(p,d,x)/ κL(∞) and the symbol ]12 is an experimental datum of D(p,d,x)/D(∞).
Figure 2. κL(p,d,x)/κL(∞) (solid lines calculated from eq 3a) and D(p,d,x)/D(∞) (dashed lines calculated from eq 3b) of GaAs/AlAs superlattices. The symbols O,22 4,23 ×,24 and 325 are experimental data of κL(p,d,x)/κL(∞) and the symbols b22 and 223 are experimental data of D(p,d,x)/D(∞).
conductors at room temperature. Therefore, the temperature dependence of κL and D is not considered in our model. 3. Results and Discussion 3.1. Modeling. Figures 1 and 2 plot the following: (1) the results calculated from eq 3; (2) the experimental data of κL(p,d,x)/κL(∞) and D(p,d,x)/D(∞) for Si and Si/SiGe nanowires, Si thin films, Si/Ge(SiGe) superlattices, and GaAs/AlAs superlattices. The related parameters used in the modeling are as follows: for Si, λ ) 43 nm,14 h ) (x3/4)a ) 0.2352 nm with a ) 0.5431 nm (diamond crystal structure),44 and Sb ) Eb/Tb ) 144 J mol-1 K-1 with Eb ) 456 kJ mol-1 and Tb ) 3173 K;44 for GaAs, λ ) 5.8 nm,45 h ) (x3/4)a ) 0.2447 nm with a ) 0.565 nm (zinc blende crystal structure),46 and Sb ) 13R as a first-order approximation.38 It is discernible that the results of both κL(p,d,x)/κL(∞) and D(p,d,x)/D(∞) calculated from eq 3 generally decrease with decreasing p and x. This observation is in good agreement with experimental findings, which serves to demonstrate the accuracy of the developed model. 3.2. Specular Scattering Parameter, Dimensionality, and Size Effects on Thermal Conductivity and Diffusivity. As shown in Figures 1 and 2, both κL(p,d,x)/κL(∞) and D(p,d,x)/ D(∞) functions drastically decrease (even to the extent of 1-2 orders of magnitude reduction) with decreasing p or p exp(λ/x) due to the increase of surfaces or interfaces in nanocrystals. The larger value of p corresponds to the smaller interface
Yang et al. roughness, i.e., the smoother interface, thus, the more probability of specular scattering, vice versa.7,31-33,35 The parameter p depends greatly on fabrication methods of nanostructures. From eq 3, it can be seen that κL(p,d,x) f κL(∞) and D(p,d,x) f D(∞) when both x f ∞ and p f 1. Therefore, carefully fabricated structures with smoother surfaces will bring forth a larger κL(p,d,x) and D(p,d,x) under the same conditions, which is beneficial for the optoelectronic devices. Conversely, a rougher surface corresponds to the higher probability of diffusive scattering, resulting in the reduction of κL(p,d,x) and D(p,d,x). A rougher surface is hence beneficial toward the enhancement of the figure of merit of thermoelectric materials, which is the central issue in developing advanced materials for thermoelectric devices applications. In Figures 1 and 2, different values of p (a fitting parameter) have been applied to indicate the difference of the quality of surface (interfacial) structures that is mainly associated with experimental processes and fabrication techniques. It is found that the p in nanowires is smaller than that in films or superlattices due to the nature of their structures and complex of growth mechanism. In the case of the nanowires, this results in a rough surface (interface), thus intensifying interface scattering. For example, p in Si/SiGe nanowires11 (denoted as 1 in Figure 1) is only 0.1, giving rise to a very low κL(p,d,x). Note that some p values shown in Figures 1 and 2 are slightly greater than those in the reported data calculated from other theoretical models.37 We believe that the disparity is caused by the contribution of size effects on the heat capacity, which was not included in other methods. Moreover, κL(p,d,x) and D(p,d,x) are anisotropic18,20,32,33 on the nanoscale. Recent experimental data on Si/Ge superlattices20,32 show the following: (1) compared with κL(∞), the κL(p,d,x) of both in-plane (parallel to the interface) and cross-plane (perpendicular to the interface) decrease significantly with decreasing x; (2) the κL(p,d,x) of in-plane is 5-6 times higher than that of cross-plane. This is because the phonons may experience much stronger interface scattering in the cross-plane of the superlattices.20,32,33 Therefore, the anisotropic κL(p,d,x) or D(p,d,x) should be described with different p values. On the other hand, the different thermal transport processes between in-plane and crossplane would result in different contributions to κL(p,d,x) or D(p,d,x). For the same p and x, the values of κL(p,d,x) and D(p,d,x) as determined by eq 3 are smaller for nanowires compared with that of thin films or superlattices due to different values of d and thus different values of x0 in the equation. As a result, phonon dispersion in nanowires is more pronounced than that in thin films or superlattices with comparable values of p and x.16 Our model can also predict the dimensionality dependence of κL(p,d,x) and D(p,d,x) while nanocrystals with different dimensionality have different surface (interface)/volume ratios. Note that the interaction energy between the various layers in the superlattices of Si/Ge(SiGe) and GaAs/AlAs was not considered in our model. This would not affect the determination accuracy because of the following: (1) the structures and physical properties of Si and Ge or GaAs and AlAs are similar, resulting in a very small interaction energy; (2) the interaction energy is also size-dependent and it decreases with decreasing x in nanocrystals.47 Therefore, the interaction energy of superlattices should have a very limited influence on the determination accuracy and it can be excluded in our modeling. 3.3. Intrinsic Determinants of Size Effect on Thermal Conductivity and Diffusivity. From Figures 1 and 2, both κL(p,d,x)/κL(∞) and D(p,d,x)/D(∞) functions decrease with decreasing x when x > 20 nm. In consideration of the size and
Nanostructural Thermoelectric Semiconductors
Figure 3. Comparisons between eq 3a (solid lines) and the predictions from the relation κL(p,x)/κL(∞) ≈ p exp(-λ/x) (dashed lines) with p ) 1 for Si and GaAs nanowires.
dimensionality dependencies of the heat capacity function C(d,x), a more accurate model has been established to predict the κL(p,d,x)/κL(∞) of nanostructural semiconductors. Moreover, it has also been found that the reduction trends for κL(p,d,x)/ κL(∞) and D(p,d,x)/D(∞) are different due to C(d,x) effect, resulting in different expressions for κL(p,d,x)/κL(∞) and D(p,d,x)/ D(∞) in eq 3. In most of the theoretical models, C was treated as a size-independent constant.37 This could be expected to result in a certain deviation from the experimental data. Therefore, the above finding is crucial for reflecting the actual values of κL in experiments as κL ) DFC. In general, p exp(-λ/x) ≈ p(1 - λ/x), 1 - 1/(x/x0 - 1) ≈ 1 - x0/x, [1 - 1/(12x/x0 - 1)]exp{-[2Sb/(3R)]/[1/(12x/x0 - 1)]} ≈ 1 - [Sb/(18R) + 1/12]x0/x as a first-order approximation where x is sufficiently large. In this case, the above three formulas obey the thermodynamic rules of low dimensional materials in which the alternation of a size-dependent quantity is associated with the surface (interface)/volume ratio, or 1/x.38 This suggests that the increased surface (interface)/volume ratio with decreasing values of x is the dominant factor in the determination of the heat transport in semiconductor nanocrystals. Hence, the surface or interface engineering plays an important role in developing high-performance optoelectronic and thermoelectric nanomaterials. Moreover, it is clearly seen that p exp(-λ/x) is far smaller than the latter two terms for Si due to its bulk λ value (43 nm). But this difference is smaller for GaAs compared with that of Si. This is because GaAs has a smaller λ value (only 5.8 nm). In Figure 3, we compare the results calculated from eq 3a with the predictions from the relation κL(p,x)/κL(∞) ≈ p exp(-λ/x) with p ) 1 for Si and GaAs nanowires. It is discernible that the above two calculated results are almost equivalent for Si even when x < 20 nm, while for GaAs, they start to separate when x < 100 nm. Therefore, for semiconductor nanocrystals with larger λ, κL(p,x)/κL(∞) ≈ D(p,x)/D(∞) ≈ p exp(-λ/x) is a good approximation. It demonstrates that the interface scattering effect makes the most significant contribution to the reduction of κL and D. However, for semiconductor nanocrystals with smaller λ, the intrinsic size effects on λ and νs could not be neglected and should be taken into account in determining κL(p,d,x) and D(p,d,x) in addition to the main contribution of interface scattering. Based on the above discussion, the following is found: (1) interface scattering is the dominant phonon scattering mechanism in the determination of the heat transport in these semiconductor nanocrystals; (2) the intrinsic size effects on λ and νs also contribute to the thermal-physical properties of nanostructural semiconductors. As shown in Figure 1, the Si/Ge superlattices show complex thermal behavior when the lattice periodic thickness x < 20
J. Phys. Chem. B, Vol. 112, No. 5, 2008 1485 nm. The lower values of κL(p,d,x) in Si/Ge superlattices when x > 14 nm was attributed to the presence of a high density of dislocations and stacking faults in the samples, which act as the centers of phonon scattering.17 However, for GaAs/AlAs superlattices the lattice mismatch is very small, and thus an increase in defect concentration for large-period superlattices is unlikely to occur. As a result, a monotonic relationship between κL(p,d,x)/κL(∞) and x has been found in Figure 2 for GaAs/AlAs superlattices. Moreover, the quantum size effect plays a very crucial role in the enhancements in κL(p,d,x) with decreasing x when x < 20 nm.18,21 In thin superlattice samples, the wave feature of phonon scattering comes into play while it breaks down for thicker superlattices. In contrast to this, however, the classical particle nature of phonon scattering dominates the heat transport in nanocrystals with larger value of x.30,32 The minimum value of κL(p,d,x) occurs at the crossover between the wave and particle-interference types of heat transport. In Figure 1, the experimental data of κL(p,d,x) for Si/Si0.9-0.95Ge0.1-0.05 nanowires11 (denoted as 1) and Si/Si0.7Ge0.3 superlattices19 (denoted as ×) are also plotted for comparison with the results calculated from eq 3. It can be seen that the calculated κL(p,d,x) decreases with decreasing x, which is consistent with the experimental data when x > 10 nm. Therefore, we could infer that the interface scattering is dominant in this case while the alloy scattering effect is negligible. In contrast, the experimental results of Si0.84Ge0.16/Si0.76Ge0.24 superlattice samples show that the thermal conductivity is size-independent and all experimental data collapse to the corresponding alloy value due to the dominance of alloy scattering.19 As a result, the determinant of the size effect on thermal conductivity in Si/ alloy nanowires and superlattices should also be interface scattering rather than the alloy scattering effect that predominates in the alloy/alloy superlattices. It should be noted however that our model suffers from a limitation in its description of the κL(p,d,x) in Si/Ge(SiGe) superlattices on the deep nanometer scale, and also of the κL(p,d,x) of alloy/alloy superlattices since the developed model is based upon the assumption of a continuous medium and crystal defects or alloy scattering effects have not been accounted for. The limitation accounts for why our model predictions demonstrate a certain extent of disparity from the experimental data when x < 10 nm as shown in Figure 1. Further theoretical efforts will be directed toward the issues of the quantum confinement effect and complicated interface structures of superlattices. 4. Conclusions A quantitative thermodynamic model was developed to investigate the intrinsic determinants of thermal conductivity and diffusivity in nanostructural semiconductors on the basis of size dependencies of heat capacity and cohesive energy as well as the interface scattering effect. The following was found: (1) both the thermal conductivity and diffusivity decrease with decreasing specular scattering and nanocrystal size x when x > 20 nm while the size dependencies of the thermal conductivity and diffusivity are different; (2) the heat transport in semiconductor nanocrystals is dominated by the surface (interface)/volume ratio; (3) the interface scattering effect is causative of the most significant contributions to the reduction of thermal conductivity and diffusivity in nanostructural semiconductors with larger phonon mean free paths; (4) in the case of semiconductor nanocrystals with smaller phonon mean free paths, the intrinsic size effects on the average phonon velocity
1486 J. Phys. Chem. B, Vol. 112, No. 5, 2008 and phonon mean free path become also substantial; (5) the enhancements in the thermal conductivity with decreasing x when x < 20 nm is a consequence of the quantum confinement effect. It demonstrates that the surface or interface engineering is an essential approach for the development of highperformance optoelectronic and thermoelectric nanomaterials. Acknowledgment. This project is financially supported by the Australia Research Council Discovery Program (Grant No. DP0666412). References and Notes (1) Gudiksen, M. S.; Lauhon, L. J.; Wang, J. F.; Smith, D. C.; Lieber, C. M. Nature 2002, 415, 617. (2) Venkatasubramanian, R.; Siivola, E.; Colpitts, T.; O’Quinn, B. Nature 2001, 413, 597. (3) Goodson, K. E.; Ju, Y. S. Annu. ReV. Mater. Sci. 1999, 29, 261. (4) Tritt, T. M.; Subramanian, M. A. MRS Bull. 2006, 31, 188. (5) Meng, J. F.; Chandra Shekar, N. V.; Chung, D.-Y.; Kanatzidis, M.; Badding, J. V. J. Appl. Phys. 2003, 94, 4485. (6) Toprak, M. S.; Stiewe, C.; Platzek, D.; Williams, S.; Bertini, L.; Mu¨ller, E.; Gatti, C.; Zhang, Y.; Rowe, M.; Muhammed, M. AdV. Funct. Mater. 2004, 14, 1189. (7) Chen, G. Phys. ReV. B 1998, 57, 14958. (8) Cahill, D. G.; Ford, W. K.; Goodson, K. E.; Mahan, G. D.; Majumdar, A.; Maris, H. J.; Merlin, R.; Phillpot, S. R. J. Appl. Phys. 2003, 93, 793. (9) Gleiter, H. Acta Mater. 2000, 48, 1. (10) Li, D. Y.; Wu, Y. Y.; Kim, P.; Shi, L.; Yang, P. D.; Majumdar, A. Appl. Phys. Lett. 2003, 83, 2934. (11) Li, D. Y.; Wu, Y. Y.; Fan, R.; Yang, P. D.; Majumdar, A. Appl. Phys. Lett. 2003, 83, 3186. (12) Yu, X. Y.; Zhang, L.; Chen, G. ReV. Sci. Instrum. 1996, 67, 2312. (13) Asheghi, M.; Leung, Y. K.; Wong, S. S.; Goodson, K. E. Appl. Phys. Lett. 1997, 71, 1798. (14) Ju, Y. S.; Goodson, K. E. Appl. Phys. Lett. 1999, 74, 3005. (15) Liu, W.; Asheghi, M. Appl. Phys. Lett. 2004, 84, 3819. (16) Liu, W.; Asheghi, M. J. Heat Transfer 2006, 128, 75. (17) Lee, S.-M.; Cahill, D. G.; Venkatasubramanian, R. Appl. Phys. Lett. 1997, 70, 2957. (18) Borca-Tasciuc, T.; Liu, W. L.; Liu, J. L.; Zeng, T. F.; Song, D. W.; Moore, C. D.; Chen, G.; Wang, K. L.; Goorsky, M. S.; Radetic, T.;
Yang et al. Gronsky, R.; Koga, T.; Dresselhaus, M. S. Superlatt. Microstruct. 2000, 28, 199. (19) Huxtable, S. T.; Abramson, A. R.; Tien, C. L.; Majumdar, A.; LaBounty, C.; Fan, X. F.; Zeng, G. H.; Bowers, J. E.; Shakouri, A.; Croke, E. T. Appl. Phys. Lett. 2002, 80, 1737. (20) Yang, B.; Liu, W. L.; Liu, J. L.; Wang, K. L.; Chen, G. Appl. Phys. Lett. 2002, 81, 3588. (21) Chakraborty, S.; Kleint, C. A.; Heinrich, A.; Schneider, C. M.; Schumann, J.; Falke, M.; Teichert, S. Appl. Phys. Lett. 2003, 83, 4184. (22) Yao, T. Appl. Phys. Lett. 1987, 51, 1798. (23) Yu, X. Y.; Chen, G.; Verma, A.; Smith, J. S. Appl. Phys. Lett. 1995, 67, 3554. (24) Capinski, W. S.; Maris, H. J. Physica B 1996, 219 & 220, 699. (25) Capinski, W. S.; Maris, H. J.; Ruf, T.; Cardona, M.; Ploog, K.; Katzer, D. S. Phys. ReV. B 1999, 59, 8105. (26) Sun, C. Q. Prog. Solid State Chem. 2007, 35, 1. (27) Chen, Y. F.; Li, D. Y.; Lukes, J. R.; Majumdar, A. J. Heat Transfer 2005, 127, 1129. (28) Wang, Z. G.; Gao, F.; Crocombette, J.-P.; Zu, X. T.; Yang, L.; Weber, W. J. Phys. ReV. B 2007, 75, 153303. (29) Balandin, A.; Wang, K. L. Phys. ReV. B 1998, 58, 1544. (30) Simkin, M. V.; Mahan, G. D. Phys. ReV. Lett. 2000, 84, 927. (31) Zou, J.; Balandin, A. J. Appl. Phys. 2001, 89, 2932. (32) Yang, B.; Chen, G. Phys. ReV. B 2003, 67, 195311. (33) Yang, R. G.; Chen, G. Phys. ReV. B 2004, 69, 195316. (34) Mingo, N.; Broido, D. A. Phys. ReV. Lett. 2004, 93, 246106. (35) Ziambaras, E.; Hyldgaard, P. J. Appl. Phys. 2006, 99, 054303. (36) Au Yeung, T. C.; Gu, M. X.; Sun, C. Q.; Chen, G. C. K.; Wong, D. W. K.; Nosik, V. Phys. ReV. B 2006, 74, 155317. (37) Liang, L. H.; Li, B. W. Phys. ReV. B 2006, 73, 153303. (38) Yang, C. C.; Li, S. Phys. ReV. B 2007, 75, 165413. (39) Yang, C. C.; Xiao, M. X.; Li, W.; Jiang, Q. Solid State Commun. 2006, 139, 148. (40) Regel’, A. R.; Glazov, V. M. Semiconductors 1995, 29, 405. (41) Soyez, G.; Eastman, J. A.; Thompson, L. J.; Bai, G.-R.; Baldo, P. M.; McCormick, A. W.; DiMelfi, R. J.; Elmustafa, A. A.; Tambwe, M. F.; Stone, D. S. Appl. Phys. Lett. 2000, 77, 1155. (42) Zhang, Z.; Zhao, M.; Jiang, Q. Physica B 2001, 293, 232. (43) Gu, M. X.; Sun, C. Q.; Chen, Z.; Au Yeung, T. C.; Li, S.; Tan, C. M.; Nosik, V. Phys. ReV. B 2007, 75, 125403. (44) http://www.webelements.com/, Web Elements Periodic Table. (45) Sze, S. M. Physics of Semiconductor DeVices, 2nd ed.; Wiley, New York 1981; p 850. (46) Li, J. B.; Wang, L. W. Phys. ReV. B 2005, 72, 125325. (47) Liang, L. H.; Liu, D.; Jiang, Q. Nanotechnology 2003, 14, 438.
