熱電效應在1821年就已經被發現，但因熱電優值Z極低所以該領域並沒有特別發展，幸運的是，在1954年時發現半導體材料有較高的熱電優值[1]，引起多數人的注意，熱電材料的研究才開始蓬勃發展。
為了找出較高的熱電優值，必須要找到極低熱導係數的材料以及電導係數較高的材料，但根據Wiedemann-Franz law[2]發現電子熱導係數與電導係數是成正相關，而導熱係數主要是由電子以及聲子所貢獻，所以降低聲子熱導係數成為增加Z值的可行方法，1993年，推論出奈米元件的熱電優值比所知的巨觀材料高[3]，也有人推論出一維奈米元件熱電優值上升主要是來自於聲子熱傳導係數的下降[4]，所以目前都朝向找尋是否有適合的奈米材料提供較高的熱電優值。
因應元件尺度的縮小，電子的傳輸行為也與以前所認知的有所不同，所以需要新的方法去計算熱電相關的係數，而NEGF的方法是研究奈米元件的一個方法，因NEGF在理論上只要知道元件的Hamiltonian即可得到熱電的相關係數所以可以解決任意形狀元件的問題。
而這篇文章主要是重複一篇2009年的文章[5]，以及2003年關於量子傳輸的書[6]，從半古典的理論出發推論熱電的相關係數，並試著NEGF方法算出一維tingt binding 長鏈模型的number of modes並與半古典模型做比較。

Thermoelectric effect was discovered in 1821, but the related field didn’t develop be- cause of the small value of figure of merit ”Z”. Fortunately, in 1954 someone found that the figure of merit of semiconductor was larger than before[1]. So most of people start to pay attention to this field. The research of thermoelectric material began to flourish.
In order to find the higher figure of merit we need to find the material which have poor thermal conductance and good conductance. Disappointingly, according to Wiedemann- Franz law[2], the electron thermal conductance is proportional to the conductance. We know that electrons and phonons are contribute to the thermal conductance, so the way of increasing the figure of merit is to reduce the phonon thermal conductance. In 1993 L. D. Hicks and M. S. Dresselhaus found that the figure of merit of nano-scale meterial was higher than macroscopic material[3]. N. Mingo also discovered that the figure of merit of one-dimension nano-scale devices was increased because of reducing the phonon thermal conductance[4].Now people want to find suitable nano-scale mate- rial to provide good figure of merit.
Because of micro-scale view, the transport property of electron is different. So we need a new method to calculate the thermoelectric coefficients. The non-equilibrium Green’s function method(NEGF) provide a way to study nano-scale device. Ideally, if we know the Hamiltonian of material we can calculate the thermoelectric coefficients so we can solve the problem of any shape of device.
This report will be repeated the result of an article in 2009[5] and a book named Quan- tum transport:Atom to Transistor[6].We will calculate thermoelectric coefficients from semi-classical model[5], and compare the number of modes of 1-dimensional-chain tight binding model using NEGF method with semi-classical model.

[1] H J Goldsmid and R W Douglas. The use of semiconductors in thermoelectric refrigeration. J. Appl. Phys., 5(386), Nov. 1954.
[2] Charles Kittel. Introduction to Solid State Physics. John Wiley and Sons, Inc, 8 edition, 2004.
[3] L. D. Hicks and M. S. Dresselhaus. E↵ect of quantum-well structures on the thermoelectric figure of merit. Phys. Rev. B, 47:12727–12731, May 1993.
[4] N.Mingo. Thermoelectric figure of merit and maximum power factor in iii–v semi- conductor nanowires. App. Phys. Lett., 84:2652, 2004.
[5] Raseong Kim, Supriyo Datta, and Mark S. Lundstrom. Influence of dimensionality on thermoelectric device performance. J. Appl. Phys., 105:034506, 2009.
[6] Supriyo Datta. Quantum Transport Atom to Transistor. CAMBRIDGE UNIVER- SITY PRESS, 2005.
[7] Juan Carlos Cuevas and Elke Scheer. Molecular Electronics:An Introduction to Theory and Experiment. World Scientific, 2010.
[8] D.M. Rowe, editor. THERMOELECTRICS HANDBOOK MACRO TO NANO. CRC Press, 2006.
[9] K. chao and M.Larsson. Physics of Zero- and One-Dimensional Nanoscopic Sys- tems, volume 156. Springer, 2007.
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