臺灣博碩士論文加值系統

本研究藉由研究矽奈米線，尋找提高其熱電優值方法，以取代目前熱電材料。目前熱電材料如鉍鍗化合物(Bi2Te3)轉換效率ZT可至約1.0；若以矽奈米線(Silicon nanowire, SiNW)取代，除了材料取得便利，與目前各種矽基電子技術也有理想相容介面，運用在商品上潛力無窮。
熱電材料其效能受到材料本身電子傳導率(electrical conductivity)、席貝克係數(Seebeck coefficient)及熱傳導係數(thermal conductivity)等傳輸性質影響；而這些性質會隨著材料的能帶結構(band structure)、能隙(band gap)以及態密度分布(density of state, DOS)等材料性質改變而影響。而以上描述之材料性質會隨著改變材料尺寸大小及成長截面方向的不同而有所改變。是故本研究第一階段以密度泛函理論(density functional theory, DFT)，建立矽奈米線原子團簇(Silicon cluster)，以改變長度、截面積大小及成長截面方向，其中主要探討矽在常壓下鑽石結構以及高壓(7.5~10.5GPa)下beta-phase結構的差異性，配合週期性邊界條件模擬計算矽奈米線之材料性質；另外以密度泛函微擾理論(density functional perturbation theory, DFPT)計算聲子頻散關係(phonon dispersion relation)、聲子態密度(phonon density of state)並後處理聲子群速度(group velocity)以及定容比熱(specific heat)與聲子供應之熱傳導係數。
第二部分研究以波茲曼傳輸方程式(Boltzmann transport equation)計算影響熱電材料轉換效率之電子傳導率、熱傳導係數、席貝克係數、熱電功率因子(power factor)以及熱電優值ZT (figure of merit)。本論文計算結果發現若以目前矽材料中擁有最好熱電轉換效率的[110]鑽石結構矽奈米線為比較基準，高壓beta-phase結構矽奈米線轉換效率最高可增加約5100倍，應用上將有極為龐大潛力。

The purpose of this research is to use a First Principles study in an effort to enhance the Figure of Merit of silicon nanowires. Nowadays, most of the constituents in thermoelectric materials are rare earth elements; however, those elements are usually too expensive for commercial applications. Silicon is the second most abundant element in the Earth’s crust that enables a relatively cost effective extraction and a sound availability, thus its low price. Moreover, silicon is an ideal material for compatibility with electronic technology. On the basis of the above, we focus on silicon nanowires and investigate its thermoelectric properties under different pressure conditions.
The efficiency of thermoelectric materials will mainly be affected by the electrical conductivity, Seebeck coefficient, and thermal conductivity, etc. All these parameters will be inevitably influenced by changing the basic material properties which include; the band structure, band gap, density of state, to name a few. Theoretically, these material properties will also vary with different sizes, orientations and crystal structures (e.g., diamond structure for 1 atm. and beta-phase for 7.5~10.5GPa). Therefore, silicon cluster models in these different conditions were built and their transport properties employing a Density Functional Theory (DFT) approach were evaluated. Additionally, these silicon cluster models will be approximated to real nanowires for the sake of periodic crystal structures using periodic boundary conditions. This research is also concerned on the thermodynamic properties including; the phonon dispersion relation and phonon density of states, as well as specific heat. All these calculations are determined on the basis of the Density Functional Perturbation Theory (DFPT).
Furthermore, using the Boltzmann transport equation will include a holistic sum of the previously calculated properties. As a result, key parameters concerning the figure of merit of thermoelectric materials of interest will be obtained. Finally, a comparison between the electric and thermal properties of both diamond and beta-phase Si NW structures will be conducted place in order to determine the structure with superior efficiency.

摘要 I
英文摘要 II
致謝 IV
目錄 V
表目錄 VII
圖目錄 VIII
參數定義 X
第一章 緒論 1
1.1前言 1
1.2 研究動機與目的 5
1.3 矽基底熱電材料簡介 6
1.4 高壓矽材料簡介及相關文獻回顧 7
1.5 矽基底熱電材料文獻回顧 7
第二章 計算量子力學理論與波茲曼傳輸方程式 10
2.1 計算量子力學理論 10
2.1.1 密度泛函理論 10
2.1.1.1 Hohenberg-Kohn定理 11
2.1.1.2 Kohn-Sham系統 12
2.1.1.3交換相關泛函 14
2.1.1.4 基底函數集合 14
2.1.1.5 贗勢與超軟贗勢 15
2.1.4.6自洽場計算 17
2.2線性微擾近似法 19
2.2.1動力矩陣計算 20
2.2.2微擾自洽場計算 23
2.2.3離散傅立葉轉換 24
2.3 一維波茲曼傳輸方程式 27
2.3.1波茲曼傳輸方程式 27
2.3.2 電子傳導率 33
2.3.3 席貝克係數 33
2.3.4 熱傳導係數(電子提供) 34
第三章 系統模型建構與模擬方法 36
3.1 模擬計算流程 36
3.2 計算參數設定 38
第四章 計算與實驗結果討論 41
4.1 不同結構矽塊材電性比較 41
4.2 矽奈米線幾何結構最佳化 43
4.3 矽奈米線電學性質分析 46
4.4 矽奈米線熱學性質分析 51
4.5 矽奈米線熱電轉換及傳輸性質分析 58
第五章 結論與未來工作建議 69
5.1結論 69
5.2未來工作建議 70
參考文獻 71

[1] T. M. Tritt, “Holey and unholey semiconductors”, Science, Vol. 283, pp.804-805, 1999.
[2] D. M. Rowe, CRC handbook of thermoelectrics, Boca Raton, CRC Press Inc. ,FL, 1995. (ISBN: 0849301467).
[3] P. A. Childs and C. C. C. Leung, “A one-dimensional solution of the boltzmann transport equation including electron–electron interactions”, J. Appl. Phys., Vol. 79, pp.222-227, 1996.
[4] L. H. Shi, D. L. Yao, G. Zhang, and B. W. Li, “Size dependent thermoelectric properties of silicon nanowires”, Appl. Phys. Lett., Vol. 95, pp. 063102-1-063102-3, 2009.
[5] P. Pichanusakorn, P. Bandaru, “Nanostructured thermoelectrics”, Materials Science and Engineering R, Vol. 67, pp. 19-63, 2010.
[6] S. Minomura, H. G. Drickamer, “Pressure induced phase transitions in silicon, germanium and some III-V compounds”, J. Phys. Chem. Solid, Vol. 23, pp. 451-456, 1962.
[7] J. C. Jamieson, “Crystal structures at high pressure of metallic modifications of silicon and germanium”, Science, Vol. 139, pp. 762-764, 1963.
[8] M. T. Yin, M. L. Cohen, “Microscopic theory of the phase transformation and lattice dynamics of Si”, Phys. Rev. letters, Vol. 45, No. 12, 1980.
[9] H. K. Poswal, N. Garg, S. M. Sharma, E. Busetto, S. K. Sikka, G. Gundiah, F. L. Deepak and C. N. Rao, “Pressure-induced structural phase transformations in silicon nanowires”, J. Nanosci. Nanotechnol., Vol. 5, pp. 729-732, 2005.
[10] V. V. Shchennikov Jr, S. V. Ovsyannikov, V. V. Shchenniko, N. A. Shaidarova, A. Misiuk, S. V. Smirnov and D. Yang, “Variations of high-pressure thermoelectric and mechanical properties of Si single crystals under doping with n and p–t pre-treatment”, Material Science and Engineering A, Vol. 462, pp. 347-350, 2007.
[11] P. B. Sorokin, P. V. Avramov, V. A. Demin, et al., “Metallic beta-phase silicon nanowires: structure and electronic properties”, Jounal of Experimental and Theoretical Physics Letters, Vol. 92, No. 5, 2010.
[12] H. J. Goldsmd, R. W. Douglas, “The use of semiconductors in thermoelectric refrigeration”, Br. J. Appl. Phys. Vol. 5, No. 11, 1954.
[13] M. C. Steele, F. D. Rosi, ”Thermal conductivity and thermoelectric power of germanium-silicon alloys”, J. Appl. Phys. Vol. 29, pp. 1517-1520, 1958.
[14] B. Abeles, D. S. Beers and G. D. Cody, et al., “Thermal conductivity of Ge-Si alloys at high temperatures”, Phys. Rev., Vol. 125, pp. 44–46, 1962.
[15] D. M. Rowe, C. M. Bhandari, Modern thermoelectrics, Prentice Hall, 1983. (ISBN-10: 0835945936).
[16] G. A. Slack, M. A. Hussain, ”The maximum possible conversion efficiency of silicon‐germanium thermoelectric generators”, J. Appl. Phys., Vol. 70, pp. 2694-2718, 1991.
[17] L. D. Hicks, M. S. Dresselhaus, ”Effect of quantum-well structures on the thermoelectric figure of merit”, Phys. Rev. B, Vol. 47, pp. 12727–12731, 1993.
[18] L. D. Hicks, M. S. Dresselhaus, ”Thermoelectric figure of merit of a one-dimensional conductor”, Phys. Rev. B, Vol. 47, pp. 16631–16634, 1993.
[19] Y. S. Touloukian, Thermal conductivity: metallic elements and alloys, thermophysical properties of matter Vol. 1339, Springer, 1971, (ISBN-10:0306670216).
[20] L. Weber and E. Gmelin, ”Transport properties of silicon”, Appl. Phys. A, Vol. 53, pp. 136–140, 1991.
[21] A. Majumdar, P. D. Yang, A. I. Hochbaum, R. D. Delgado, W. Liang, C. Garnett, R. Chen and M. Najarian, “Enhanced thermoelectric performance of rough silicon nanowires”, Nature Vol. 451, 2008,
[22] J. T. Kheli, J. K. Yu, W. A. Goddard III, J. R. Heath, A. I. Boukai and Y. Bunimovich, “Silicon nanowires as efficient thermoelectric materials”, Nature, Vol. 451, pp.168-171, 2008.
[23] G. S. Nolas, J. Sharp, H. J. Goldsmid, Thermoelectrics-basic principles and new materials developments, Springer, 2001, (ISBN:354041245X).
[24] I. N. Levine, Quantum chemistry 6th ed., Prentice Hall, 2008, (ISBN: 0132358506).
[25] 邱創斌(洪哲文指導), “量子力學與分子動力分析酵素生物燃料電池性能影響因子”, 國立清華大學動力機械系博士論文, 1/2010.
[26] D. Vanderbilt, “Soft self-consistent pseudopotentials in a generalized eigenvalue formalism”, Physical Review B, Vol.41, pp.7892-7895, 1990.
[27] 蔡岳璁(洪哲文指導), “計算量子力學於CO在直接甲醇燃料電池觸媒之毒化研究”, 國立清華大學動力機械系碩士論文, 6/2006.
[28] M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, Ö. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian 09, Revision A.1, Gaussian, Inc., Wallingford CT, 2009.
[29] S. Baroni, A. D. Corso, S. Gironcoli, P. Giannozzi, C. Cavazzoni, G. Ballabio, S. Scandolo, G. Chiarotti, P. Focher, A. Pasquarello, K. Laasonen, A. Trave, R. Car, N. Marzari, A. Kokalj, PWscf, Trieste, Italy, 2002.
[30] L. Reggiani, Hot electron transport in semiconductors, topics in Physics, Springer, 1985, (ISBN-10: 0387133216).
[31] http://www.ioffe.ru/SVA/NSM//Semicond/Si/electric.html
[32] R. D. Meo, A. D. Corso and P. Giannozzi, et al., “Calculation of phonon dispersion on the grid using quantum espresso”, Trieste, Italy, 2008.
[33] J. Zou and A. Balandin, “Phonon heat conduction in a semiconductor nanowire”, Journal of Applied Physics, Vol. 89, No.5, 2001.
[34] Y. Zhang, J. X. Cao and Y. Xiao, et al., “Phonon spectrum and specific heat of silicon nanowires”, Journal of Applied Physics, Vol. 102, 104303, 2007.
[35] B. D. Kong, S. Paul and M. B. N. William and K. W. Kim, “First-principles analysis of lattice thermal conductivity in monolayer and bilayer graphene”, Phys. Rev. B, Vol. 80, 033406, 2009.
[36] T. L. Chan, C. V. Ciobanu and F. C. Chuang, et al., “Magic structure of h-passivated <110> silicon nanowires”, Nano Letters, Vol. 6, No.2, pp. 277-281, 2006.
[37] http://en.wikipedia.org/wiki/BFGS_method
[38] R. W. Godby, M. Schluter, ”Accurate exchange-correlation potential for silicon and its discontinuity on addition of an electron”, Physical Review Letter, Vol. 56, pp. 2415-2418, 1986.
[39] N. W. Ashcroft, N. D. Mermin, Solid state physics, Holt, Rinehart and Winston, 1976, (ISBN:0030493463).
[40] T. Vo, A. J. Williamson, et al, “First principles simulations of the structural and electronic properties of silicon nanowires”, Physical Review B, Vol.74, 045116, 2006.
[41] H. Peelaers, B. Partoens and F. M. Peeters, “Phonon band structure of si nanowires: a stability analysis”, Nano Letters, Vol. 9, No.1, pp. 107-111, 2009.
[42] M. Menon, E. Richter, and K. R. Subbaswamy, “Structural and vibrational properties of si clathrates in a generalized tight-binding molecular-dynamics scheme”, Physical Review B, Vol. 56, No. 19, pp. 12 290-12 295, 1997.
[43] K. J. Suthar, J. Patten and L. Dong, “Estimation of temperature distribution in silicon during micro laser assisted machining”, ASME 2008 International Manufacturing Science and Engineering Conference Vol. 2, pp. 301-309, 2008.