利用熱力學方法探討半導體材料的熱電效應

Abstract

半導體材料的熱電效應可以藉由熱力學方法中 : 一系統存在電場的情況下，而詳加研究。併入Clausis-Mossotti方程式的這個新的方法被提出來計算固態材料的介電常數，然後便可以計算出一個決定材料熱電轉換效率的重要因子 : 熱電係數。 一般來說，在Clausis-Mossotti方程式中的極化率包含三個部份 : 電子極化率、原子極化率與方向性極化率，然而對於半導體材料的主要貢獻來自於電子極化率。從極化率得到的介電常數因為擁有粒子聚集的重要巨觀特性，因此，從電子極化率衍生而來的介電常數便可以將此巨觀特性描述為一個固態性質。Ab initio量子化學的理論被運用在計算隨著電場變化的電子極化率之上。 在目前的工作中，三種半導體的熱電材料被考慮 : 矽化鎂、二矽化鐵和矽化鍺。第一步，APT 和Mulliken電荷經由密度泛涵理論的方法在一變化的電場下而計算出來，變化的電場如下 : -0.01、-0.0075、-0.005、-0.0025、0.00、0.0025、0.005、0.0075和0.01原子單位。四種密度泛涵理論的方法被選用，如下 : B3LYP、BLYP、M05和M052X，再加上一系列的基底函數，如下 : Pople形式的基底函數，例如6-311G、6-311G(d)…等等 ﹔effective core potential形式的基底函數，例如CEP-31G、CEP-121G和LANLDZ。第二步，APT和Mulliken電荷被使用來計算在電場中的偶極矩，然後電子極化率可經由偶極矩對電場求一階導數而計算出來。最後，介電常數便可以從Clausis-Mossotti方程式與電子極化率而求得。經與實驗測量比較之下，對於矽化鎂、二矽化鐵和矽化鍺而言，在B3LYP方法之下所模擬計算出來的介電常數顯示最正確的結果。介電常數從目前的計算方法及他們相對應的實驗結果，分別如下 : 對於矽化鎂而言，εr = 11.86和13.3 ﹔對於矽化鐵而言，εr = 27.806和27.6 ﹔對於矽化鍺而言，εr = 13.571和13.95。 熱電係數可以從熱力學方法中的chemical potential計算求得，這種方法比能帶結構理論還簡單的多。Helmholtz自由能在不同溫度下被計算，而且基於溫度為一變量之下，可以得到某些分析性的函數。因此，熱電係數可經由Helmholtz自由能對溫度求一階導數而計算出來。然而，這個熱電係數還要除以先前計算的介電常數，最後才能真正表示固態材料的熱電係數。對於熱電係數而言，目前計算矽化鎂、二矽化鐵和矽化鍺的結果與實驗測量的結果非常相近。經由目前計算的熱電係數，分別如下 : 矽化鎂在溫度(300, 800)K下，Se = (284,334) μV/K ﹔ 二矽化鐵在溫度(300,900)K下，Se = (118.8,140.4) μV/K ﹔矽化鍺在溫度(300, 900)K下，Se = (196.3,220.9)。與之相對應的實驗結果，分別如下 : 矽化鎂在溫度(300, 800)K下，Se = (180,280) μV/K﹔二矽化鐵在溫度(300, 900)K下，Se = (190, 170) μV/K﹔矽化鍺在溫度(300, 900)K下，Se = (345,325) 。 總之，對於計算熱電係數而言，目前計算的方法明顯優於傳統能帶結構理論的方法。

Thermoelectric effect of semiconducting materials is studied by the thermodynamic method for a system in the presence of an electric field. The new method incorporating with the Clausis-Mossotti equation is proposed to calculate dielectric constant for solid-state materials, and then to compute the Seebeck coefficient that is key factor to determine thermoelectric conversion efficiency of the materials. The polarizability in the Clausis-Mossotti equation in general includes three parts; electronic polariability, atomic polariability and orientation polariability. The dominant contribution for semiconducting materials comes from electronic polariability. A dielectric constant derived from the polarizability is an important bulk property of a collection of particles. Therefore, the dielectric constant derived from electronic polariability can describe bulk as a solid. Ab initio quantum chemistry theory is utilized to compute electronic polariability directly with varying strength of electric field. In the present work, three semiconductor thermoelectric materials are considered; Mg2Si, FeSi2 and SiGe. In the first step, APT and Mulliken charges are computed with density functional theory (DFT) method at various electric fields; -0.01, -0.0075, -0.005, -0.0025, 0.00, 0.0025, 0.005, 0.0075 and 0.01 in atomic unit. Four kinds of DFT functionals are chosen: B3LYP, BLYP, M05 and M05-2X, plus a bunch of basis sets; Pople style basis sets including of 6-311G, 6-311G(d)…etc; effective core potential including of CEP-31G, CEP-121G and LANL2DZ. In the second step, APT and Mulliken charges are used to calculate dipole moments at given electric field above and then derivatives of dipole moments with respect to electric field lead to the electronic polarizability. In the final step, the dielectric constant is evaluated from the Clausis-Mossotti equation through the electronic polarizability. In comparison with experimental measurements, simulated dielectric constants with B3LYP method show the most accurate results for Mg2Si, FeSi2 and SiGe. The dielectric constants from the present calculations and their corresponding experiment results are ε¬r = 11.86 and 13.3 for Mg2Si, εr = 27.806 and 27.6 for FeSi2, and εr = 13.571 and 13.95 for SiGe, respectively. The Seebeck coefficient is calculated from the thermodynamic method with chemical potential. This method is much simpler than energy band structure theory. The Helmholtz free energies are computed at various temperatures, and then are fitted into the certain analytical function with respect temperatures as a variable. Thus, the Seebeck coefficient can be evaluated from partial derivative of Helmholtz free energy with respect to temperature. This Seebeck coefficient that must be divided by the dielectric constant evaluated previously can finally be considered as the Seebeck coefficient for a solid-state material. The present results show good agreements with experimental measurements for the Seebeck coefficients of Mg2Si, FeSi2 and SiGe. The Seebeck coefficients from the present calculations are Se = (284, 334)μV/K at the temperature (300, 800)K for Mg2Si, Se = (118.8, 140.4)μV/K at (300,900)K for FeSi2, and Se = (196.3,220.9)μV/K at (300,900)K for SiGe. Their corresponding experiment results are Se = (180, 280)μV/K at (300, 800)K for Mg2Si, Se = (190, 170)μV/K at (300,900)K for FeSi2, and Se = (345,325)μV/K at (300,900)K for SiGe. In conclusion, the present method surprisingly woks better than conventional energy band structure theory for calculating the Seebeck coefficient.