# American Institute of Mathematical Sciences

2013, 2013(special): 737-746. doi: 10.3934/proc.2013.2013.737

## Modeling the thermal conductance of phononic crystal plates

 1 Rudolf Peierls Center for Theoretical Physics, University of Oxford, Oxford OX1 3NP, United Kingdom 2 Department of Electrical Engineering and Computer Science, University of Applied Sciences Zittau/Görlitz, D-02826 Görlitz, Germany

Received  September 2012 Published  November 2013

The paper presents a model to compute the phonon thermal conductance of phononic crystal plates. The goal is the optimization of the figure of merit for these materials, which is the primary criterion for the efficiency of a thermoelectric device. Values of about three or higher allow for the construction of thermoelectric generators based on the Seebeck effect, which are more efficient than conventional electrical generators. The paper introduces a numerical method to optimize the phonon thermal conductance of a given phononic material by varying the geometrical structure with respect to the width and thickness of a sample as well as pore size, shape, and mass density.
Citation: Stefanie Thiem, Jörg Lässig. Modeling the thermal conductance of phononic crystal plates. Conference Publications, 2013, 2013 (special) : 737-746. doi: 10.3934/proc.2013.2013.737
##### References:
 [1] D. Y. Chung, T. Hogan, J. Schindler, L. Iordarridis, P. Brazis, C. R. Kannewurf, B. Chen, C. Uher, and M.G. Kanatzidis, Complex bismuth chalcogenides as thermoelectrics, 16th International Conference on Thermoelectrics, 1 (1997), 459-462. [2] A. Grigorevskii, V. Grigorevskii, and S. Nikitov, Dispersion curves of bulk acoustic waves in an elastic body with a two-dimensional periodic structure of circular holes, Acoustical Physics, 54 (2008), 289-293. [3] T. C. Harman, P. J. Taylor, M. P. Walsh, and B. E. LaForge, Quantum dot superlattice thermoelectric materials and devices, Science, 297 (2002), 2229-2232. [4] A. I. Hochbaum, C. Renkun, R. D. Delgado, W. Liang, E. C. Garnett, M. Najarian, A. Majumdar, and P. Yang, Enhanced thermoelectric performance of rough silicon nanowires, Nature, 451 (2008), 163-167. [5] J.-C. Hsu and T.-T. Wu, Efficient formulation for band-structure calculations of two-dimensional phononic-crystal plates, Physical Review B, 74 (2006), 144303. [6] W. Kuang, Z. Hou and Y. Liu, The effects of shapes and symmetries of scatterers on the phononic band gap in 2D phononic crystals, Physics Letters A, 332 (2004), 481-490. [7] M. S. Kushwaha, P. Halevi, G. Martínez, I. Dobrzynski, and B. Djafari Rouhani, Theory of acoustic band structure of periodic elastic composites, Physical Review B, 49 (1994), 2313-2322. [8] N. Mingo, Calculation of Si nanowire thermal conductivity using complete phonon dispersion relations, Physical Review B, 68 (2003), 113308. [9] G. G. Samsonidze., R. Saito, A. Jorio, M. A. Pimenta, A. G. Souza Filho, A. Grüneis, G. Dresselhaus, and M. S. Dresselhaus, The concept of cutting lines in carbon nanotube science, Journal of Nanoscience and Nanotechnology, 3 (2003), 431-458. [10] G. A. Slack, New materials and performance limits for thermoelectric cooling, in ''CRC Handbook of Thermoelectrics'' (ed. D. M. Rowe), CRC Press, (1995), 407-440. [11] Y. Tanaka, Y. Tomoyasu, and S. Tamura, Band structure of acoustic waves in phononic lattices: Two-dimensional composites with large acoustic mismatch, Physical Review B, 62 (2000), 7387-7392. [12] J. Tang, H.-T. Wang, D. H. Lee, M. Fardy, Z. Huo, T. P. Russell, and P. Yang, Holey silicon as an efficient thermoelectric material, Nano Letters, 10 (2010), 4279-4283. [13] J. O. Vasseur, P. A. Deymier, B. Djafari Rouhani, Y. Pennec, and A.-C. Hladky Hennion, Absolute forbidden bands and waveguiding in two-dimensional phononic crystal plates, Physical Review B, 77 (2008), 085415. [14] R. Venkatasubramanian, E. Siivola, T. Colpitts, and B. O'Quinn, Thin-film thermoelectric devices with high room-temperature figures of merit, Nature, 413 (2001), 597-602.

show all references

##### References:
 [1] D. Y. Chung, T. Hogan, J. Schindler, L. Iordarridis, P. Brazis, C. R. Kannewurf, B. Chen, C. Uher, and M.G. Kanatzidis, Complex bismuth chalcogenides as thermoelectrics, 16th International Conference on Thermoelectrics, 1 (1997), 459-462. [2] A. Grigorevskii, V. Grigorevskii, and S. Nikitov, Dispersion curves of bulk acoustic waves in an elastic body with a two-dimensional periodic structure of circular holes, Acoustical Physics, 54 (2008), 289-293. [3] T. C. Harman, P. J. Taylor, M. P. Walsh, and B. E. LaForge, Quantum dot superlattice thermoelectric materials and devices, Science, 297 (2002), 2229-2232. [4] A. I. Hochbaum, C. Renkun, R. D. Delgado, W. Liang, E. C. Garnett, M. Najarian, A. Majumdar, and P. Yang, Enhanced thermoelectric performance of rough silicon nanowires, Nature, 451 (2008), 163-167. [5] J.-C. Hsu and T.-T. Wu, Efficient formulation for band-structure calculations of two-dimensional phononic-crystal plates, Physical Review B, 74 (2006), 144303. [6] W. Kuang, Z. Hou and Y. Liu, The effects of shapes and symmetries of scatterers on the phononic band gap in 2D phononic crystals, Physics Letters A, 332 (2004), 481-490. [7] M. S. Kushwaha, P. Halevi, G. Martínez, I. Dobrzynski, and B. Djafari Rouhani, Theory of acoustic band structure of periodic elastic composites, Physical Review B, 49 (1994), 2313-2322. [8] N. Mingo, Calculation of Si nanowire thermal conductivity using complete phonon dispersion relations, Physical Review B, 68 (2003), 113308. [9] G. G. Samsonidze., R. Saito, A. Jorio, M. A. Pimenta, A. G. Souza Filho, A. Grüneis, G. Dresselhaus, and M. S. Dresselhaus, The concept of cutting lines in carbon nanotube science, Journal of Nanoscience and Nanotechnology, 3 (2003), 431-458. [10] G. A. Slack, New materials and performance limits for thermoelectric cooling, in ''CRC Handbook of Thermoelectrics'' (ed. D. M. Rowe), CRC Press, (1995), 407-440. [11] Y. Tanaka, Y. Tomoyasu, and S. Tamura, Band structure of acoustic waves in phononic lattices: Two-dimensional composites with large acoustic mismatch, Physical Review B, 62 (2000), 7387-7392. [12] J. Tang, H.-T. Wang, D. H. Lee, M. Fardy, Z. Huo, T. P. Russell, and P. Yang, Holey silicon as an efficient thermoelectric material, Nano Letters, 10 (2010), 4279-4283. [13] J. O. Vasseur, P. A. Deymier, B. Djafari Rouhani, Y. Pennec, and A.-C. Hladky Hennion, Absolute forbidden bands and waveguiding in two-dimensional phononic crystal plates, Physical Review B, 77 (2008), 085415. [14] R. Venkatasubramanian, E. Siivola, T. Colpitts, and B. O'Quinn, Thin-film thermoelectric devices with high room-temperature figures of merit, Nature, 413 (2001), 597-602.
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