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III. RESULTS A. Anisotropic dispersion curves The definitions in the 2D reciprocal space of the independent variables k and φ to obtain the throughput λ are explained in Fig. 3(a). The 2D BZ is elongated by a 20-fold factor in the reciprocal direction k z with respect to that k x . Indeed, the reduced length of the supercell slab in x is given by n x = 20. The dispersion-curve diagrams for φ = 0° (direction k x with k z = 0) and φ = 90° (direction k z with k x = 0) are displayed in Figs. 3(b) and 3(c), respectively. (a) (b) φ = 0 o K z =π/a k k = (0,0) K(φ) φ K x =π/d x =π/(n x a) 7-9 October 2009, Leuven, Belgium Different colors (blue, red, green, magenta and black from the lowest to the highest frequencies) represent different frequency ranges for the dispersion curves. As shown in Fig. 3(b), the dispersion curves for φ = 0° are very flat leading to low radial group velocities u m (k, 0) in the direction k x . In contrast, for φ = 90°, the radial group velocities u m (k, π / 2) are usually much higher than those obtained for φ = 0°. Indeed, as depicted in Fig. 3(c), the slopes of the dispersion curves for φ = 90° are usually much larger than those for φ = 0° [Fig. 3(b)]. As a consequence, we can already expect from the evolution of the dispersion curves a significant exaltation of the throughput thermal conductivity λ in the direction z (β = 90°) with respect to that x (β = 0°), as discussed in the following. B. Anisotropic thermal conductivity For the example nanodevice presented in Figs. 1 and 2, a low λ β varying from only 0.7 to 1.0 W/m/K is computed when β is increased from 0° to 24.4°, respectively, as shown in Fig. 4. Since 1 W/m/K corresponds (approximately) to the extreme low limit of the thermal conductivity of bulk amorphous Si [17,18], the hybrid nanomaterial can have a insulating behavior when β ≤ 25°. Second, when β is increased from 26.6 to 90°, the material operation regime becomes more dissipative since λ β grows, with a sigmoid curve, from 1.1 to 2.9 W/m/K, respectively. (c) φ = 90 o Fig. 3 (colors). Phonon band diagrams with the definition of the polar reciprocal coordinates k and φ (a) and dispersion curves obtained by lattice dynamics for φ = 0° (b) and φ = 90° (c). Fig. 4. Computed λ β vs. β curve with a sigmoid shape: The hybrid nanomaterial is that in Figs. 1 and 2 at room temperature T = 300 K. The circles, interpolated by the dotted line, are values for fcc crystallographic directions with integer Miller indices. ©EDA Publishing/THERMINIC 2009 207 ISBN: 978-2-35500-010-2
The regime transition at λ β = 1.7 W/m/K between the thermal insulating and dissipative behaviors is obtained when β = 45°. This value is related to the close-packed directions in a membrane plane (for the fcc group). A significant 4.1 exaltation factor of λ β is found out when β is increased from 0° (x direction) to 90° (z direction). IV. CONCLUSION A suspended thermal nanomaterial made up of a Si membrane with stretched SA Ge QDs forming phonon waveguides is proposed. A hybrid behavior, which can be either insulating or dissipative is shown for thermal transport in our nanomaterial. As a consequence, a wide range of applications can be possibly covered by the membranous nanomaterial from thermoelectrics to heat sinking. Phonon wave-guiding is analyzed as a function of an angle β in a membrane plane. This deflection angle is defined from the in-plane direction [100], showing a significant QD constriction, to that [001] of the QD stretching. As observed from the dispersion curves computed by lattice dynamics, the throughput thermal conductivity λ is significantly increased in the direction [001] with respect to that [100]. Numerical results show that (i) the suspended nanomaterial has a thermalinsulating behavior for moderate β-values while (ii) heat dissipation is much more significant when β is increased up to 90°. A significant exaltation factor of 4 to 5 folds is obtained for the throughput λ between these operation regimes for an example molecular-scale device. 7-9 October 2009, Leuven, Belgium [8] J.-N. Gillet, Y. Chalopin, and S. Volz, ASME J. Heat Transfer 131, 043206 (2009). [9] J.-N. Gillet, and S. Volz, J. Electron. Mater., in press. [10] J.-N. Gillet, Outstanding Scientific Paper Award, in Proc. 28 th International Conference on Thermoelectrics (ITC 2009), H. Bottner, Ed., Freiburg, Germany, 26-30 July 2009. [11] W. Kim, J. Zide, A. Gossard, D. Klenov, S. Stemmer, A. Shakouri, and A. Majumdar, Phys. Rev. Lett. 96, 045901 (2006). [12] T. M. Tritt, H. Bottner, and L. Chen, MRS Bulletin 33, 366-368 (2008). [13] M. T. Dove, Introduction to Lattice Dynamics, Cambridge Topics in Mineral Physics and Chemistry, No 4 (Cambridge Univ. Press, Cambridge, UK, 1993). [14] Z. Jian, Z. Kaiming, and X. Xide, Phys. Rev. B 41, 12915-12918 (1990). [15] Y. Chalopin, J.-N. Gillet, and S. Volz, Phys. Rev. B 77, 233309 (2008). [16] C. J. Glassbrenner, and G. A. Slack, Phys. Rev. 134, A1058-A1069 (1964). [17] D. G. Cahill, S. K. Watson, and R. O. Pohl, Phys. Rev. B 46, 6131-6140 (1992). [18] C. Chiritescu, D. G. Cahill, N. Nguyen, D. Johnson, A. Bodapati, P. Keblinski, and P. Zschack, Science 315, 351- 353 (2007). ACKNOWLEDGMENT The authors thank the European Consortium NANOPACK (www.nanopack.org) for its financial contribution. REFERENCES [1] R. Venkatasubramanian, Ed., Nanoscale Heat Transport - From Fundamentals to Devices, (Mater. Res. Soc. Symp. Proc. Volume 1172E, Warrendale, PA, 2009). [2] P. W. K. Rothemund, Nature (London) 440, 297-302 (2006). [3] V. Maurice, G. Despert, S. Zanna, M.-P. Bacos, and P. Marcus, Nat. Mater. 3, 687-691 (2004). [4] A. Condon, Nat. Rev. Genet. 7, 565-575 (2006). [5] C. R. Martin, and P. Kohli, Nat. Rev. Drug Discov. 2, 29- 37 (2002). [6] A. I. Yakimov, A. V. Dvurechenskii, and A. I. Nikiforov, J. Nanoelectron. Optoelectron. 1, 119-175 (2006). [7] S. Kiravittaya, H. Heidemeyer, and O. G. Schmidt, Appl. Phys. Lett. 86, 263113 (2005). ©EDA Publishing/THERMINIC 2009 208 ISBN: 978-2-35500-010-2
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International Workshop on THERMal I
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