formulas4TE

热电领域内一些常用公式和经典文献中公式的总结

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        在热电研究中，我们常常需要反复地使用一些公式。 尽管目前 MicroSoft Office 提供了非常方便的内嵌公式编辑器，但是由于公式的形式较为复杂， 因此我们还是需要花费大量的时间来输入这些公式。这里，我们总结了一些高频使用的和经典文献里的公式， 不但能够方便截图使用，同时也能够非常容易地粘贴为 MicroSoft Office 的内嵌公式（参考后面的 附录：如何在 MicroSoft Office 中使用这里的公式 ），提高工作效率。这里强烈推荐使用后面的方法在 MicroSoft Office 中粘贴使用公式， 一方面可以对公式进行进一步的修改和调整，满足自己的需求， 同时也可以获得比网页截图更好的显示效果，比如上、下标显示的相对大小更加合理。 非常欢迎大家在讨论区（Issues）里勘误、建议以及补充！

• Basic Principles
• APS-SPB Model
• APS-SKB model
• Boltzmann Equation Description of Electron Transport
• Expressions for Multiband Conduction
• Equations in Papers
  • P001: Engineering Thermoelectric Model (ZTeng) (H.S. Kim et al., 2015)
  • P002: Restructure SPB (RSPB) Model (J. Zhu et al., 2021)
  • P003: Device Figure-of-Merit (ZTdev) (G.J. Snyder et al., 2017)
  • P004: Thermoelectric compatibility factor (CF) (G.J. Snyder et al., 2003)

• 附录：如何在 MicroSoft Office 中使用这里的公式
  • 方法一（适用于 Word 和 PPT）
  • 方法二（仅适用于 Word）

Basic Principles

$$ZT=\frac{\sigma S^2}{\kappa} \cdot T$$

$$PF=\sigma S^{2}$$

$$\sigma = ne\mu$$

$$ \mu = e \frac{\tau}{m^{\ast}} $$

$$\kappa = \kappa_{L} + \kappa_{e} + \kappa_{bip} $$

$$\kappa_{L} = \frac{1}{3} C_{v}vl$$

$$\kappa_{e} = L \sigma T$$

$$\eta = \frac{T_{h}-T_{c}}{T_{h}} \cdot \frac{\sqrt{1+ZT}-1}{\sqrt{1+ZT}+T_{c} / T_{h}}$$

$$s = \frac{\sqrt{1+ZT}-1}{S \cdot T}$$

$$ \beta = \frac{\mu_{0} (m^{\ast}/m_{e})^{3/2} T^{3/2}}{\kappa_{L}}T $$

APS-SPB Model

Single parabolic band (SPB) model with acoustic phonon scattering (APS) mechanism and deformation potential theory

$$ E = \frac{\hbar ^{2} k^{2}}{2 m^{\ast}} $$

$$ g(E) = \frac{(2 m_{d} ^{\ast})^{3/2}}{2 \pi^{2} \hbar^{3}} E^{1/2} = \frac{N_{v} (2 m_{b} ^{\ast})^{3/2}}{2 \pi^{2} \hbar^{3}} E^{1/2} $$

$$ v^{2}(E) = \frac{2E}{m_{b} ^{\ast}} $$

$$ \tau(E) = \frac{N_{v} \hbar C_{ii}}{\pi k_{B} T {\Xi ^{2}}} \frac{1}{g(E)} = \frac{2 \pi \hbar^{4} C_{ii}}{(2 m_{b} ^{\ast})^{3/2} k_{B} T {\Xi ^{2}}} E^{-1/2} $$

$$ F_{n}(\eta) = \int _{0}^{+\infty} {{\frac{x^{n}}{1+\exp (x-\eta)}}} dx $$

$$ x = \frac{E}{k_{B}T}, \eta = \frac{E_{F}}{k_{B}T} $$

$$ n = \frac{N_{v}( 2 m_{b} ^{\ast} k_{B} T )^{3/2}}{2\pi ^{2} \hbar ^{3}} F_{1/2}( \eta ) $$

$$ \mu = \frac{4 \pi e \hbar ^{4} C_{ii}}{m_{I} ^{\ast} (2 m_{b} ^{\ast} k_{B} T)^{3/2} \Xi ^{2}} \frac{F_{0}(\eta)}{3 F_{1/2}(\eta)} $$

$$ \sigma = \frac{2 N_{v} e^2 \hbar C_{ii}}{3 \pi m_{I} ^{\ast} \Xi ^{2} } F_{0}(\eta)$$

$$ S = \frac{k_{B}}{e} \left( \frac{2 F_{1}(\eta)}{F_{0}(\eta)} - \eta \right) $$

$$ L = \frac{k_{B}^{2}}{e^{2}} \left[ 3 \frac{F_{2}(\eta)}{F_{0}(\eta)} - 4 \left( \frac{F_{1}(\eta) }{F_{0}(\eta)} \right) ^{2} \right] $$

$$ r_{H} = \frac{n}{n_{H}} = \frac{\mu_{H}}{\mu} = \frac{3 F_{1/2}(\eta) F_{-1/2}(\eta)}{4 F_{0}^{2}(\eta)} $$

APS-SKB model

Single Kane band (SKB) model with acoustic phonon scattering (APS) mechanism and deformation potential theory

$$ E \left( 1+\frac{E}{E_{g}} \right) = \frac{\hbar ^{2} k^{2}}{2 m^{\ast}} $$

$$ g(E) = \frac{N_{v} (2 m_{b} ^{\ast})^{3/2}}{2 \pi^{2} \hbar^{3}} E^{1/2} \left( 1+\frac{E}{E_{g}} \right) ^{1/2} \left( 1 + 2 \frac{E}{E_{g}} \right) $$

$$ v^{2}(E) = \frac{2E}{m_{b} ^{\ast}} \left( 1+\frac{E}{E_{g}} \right) \left( 1 + 2 \frac{E}{E_{g}} \right) ^{-2} $$

$$ \tau(E) = \frac{N_{v} \hbar C_{ii}}{\pi k_{B} T \Xi ^{2}} \frac{1}{g(E)} \frac{3(1+2E/E_{g})^{2}}{(1+2E/E_{g})^{2}+2} $$

$$ F^{n}_{m,k}(\eta, \alpha) = \int _{0}^{+\infty}{x^{n} (x+\alpha x^{2})^{m}[(1+2\alpha x)^{2} + 2]^{k/2} \left( -\frac{\partial f}{\partial x} \right)}dx $$

$$ x = \frac{E}{k_{B}T}, \eta = \frac{E_{F}}{k_{B}T}, \alpha = \frac{k_{B}T}{E_{g}} $$

$$ n = \frac{N_{v}( 2 m_{b} ^{\ast} k_{B} T )^{3/2}}{3\pi ^{2} \hbar ^{3}} {F^{0}_{3/2,0}(\eta, \alpha)} $$

$$ \mu = \frac{2 \pi e \hbar ^{4} C_{ii}}{m_{I} ^{\ast} (2 m_{b} ^{\ast} k_{B} T)^{3/2} \Xi ^{2}} \frac{3 F^{0} _{1,-2}(\eta, \alpha)}{F^{0} _{3/2,0}(\eta, \alpha)} $$

$$ \sigma = \frac{2 N_{v} e^{2} \hbar C_{ii}}{\pi m_{I} ^{\ast} \Xi ^{2} }{F^{0} _{1,-2}(\eta, \alpha)} $$

$$ S = \frac{k_{B}}{e} \left( \frac{F^{1} _{1,-2}(\eta, \alpha)}{F^{0} _{1,-2}(\eta, \alpha)} - \eta \right) $$

$$ L = \frac{k_{B}^{2}}{e^{2}} \left[ \frac{F^{2} _{1,-2}(\eta, \alpha)}{F^{0} _{1,-2}(\eta, \alpha)} - \left( \frac{F^{1} _{1,-2}(\eta, \alpha)}{F^{0} _{1,-2}(\eta, \alpha)} \right) ^{2} \right] $$

$$ r_{H} = \frac{n}{n_{H}} = \frac{\mu_{H}}{\mu} = \frac{{F^{0} _{3/2,0}(\eta, \alpha)}\cdot {F^{0} _{1/2,-4}(\eta, \alpha)}}{\left( {F^{0} _{1,-2}(\eta, \alpha)} \right) ^{2}} $$

Boltzmann Equation Description of Electron Transport

$$ f = \frac{1}{1+\exp \left( \frac{E - E_{F}}{k_{B} T} \right)} $$

$$ n = \int_{0}^{+\infty} {f \cdot g(E)} dE $$

$$ p = \int_{-\infty}^{0} {(1-f) \cdot g(E)} dE $$

$$ \sigma(E) = e^2 \tau(E)v^{2}(E)g(E) $$

$$ \sigma = \int_{-\infty}^{+\infty}{\sigma(E) \cdot \left( -\frac{\partial f}{\partial E} \right)} dE $$

$$ S = -\frac{1}{\sigma \cdot eT} \int_{-\infty}^{+\infty}{\sigma(E) \cdot (E-E_{F})\left( -\frac{\partial f}{\partial E} \right)} dE $$

$$ \kappa_{e} = \frac{1}{e^{2}T} \int_{-\infty}^{+\infty}{\sigma(E) \cdot (E-E_{F})^{2} \left( -\frac{\partial f}{\partial E} \right)} dE - \sigma S^{2} T $$

Expressions for Multiband Conduction

$$ n = \sum_{i}{n_{i}} $$

$$ \sigma = \sum_{i} {\sigma_{i}} $$

$$ S = \frac{\sum_{i}{\sigma_{i} S_{i}}}{\sum_{i}{\sigma_{i}}} $$

$$ R_{H} = \frac{\sum_{i}{\sigma_{i}^{2} R_{H,i}}}{\left( \sum_{i}{\sigma_{i}} \right) ^{2}} $$

$$ \kappa_{e} = \left[ {\sum_{i}{L_{i} \sigma_{i}} + \sum_{i}{\sigma_{i} S_{i}^{2}} - \frac{\left( \sum_{i}{\sigma_{i} S_{i}} \right) ^{2}}{\sum_{i}{\sigma_{i}}} } \right] \cdot T $$

Equations in Papers

P001: Engineering Thermoelectric Model (ZTeng) (H.S. Kim et al., 2015)

$$ \eta_{max} = \eta _{c} \frac{\sqrt{1 + (ZT) _{eng} \alpha _{1} \eta _{c}^{-1}} - 1}{\alpha _{0} \sqrt{1 + (ZT) _{eng} \alpha _{1} \eta _c^{-1}} + \alpha _{2}} $$

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• H.S. Kim, W. Liu, G. Chen, C. Chu, Z. Ren, Relationship between thermoelectric figure of merit and energy conversion efficiency, Proceedings of the National Academy of Sciences 112 (27) (2015) 8205-8210. https://doi.org/10.1073/pnas.1510231112

P002: Restructure SPB (RSPB) Model (J. Zhu et al., 2021)

$$ S_{r} = \ln \left( 1.075 + \frac{e^{2}}{n_{r}} \right) $$

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• J. Zhu, X. Zhang, M. Guo, J. Li, J. Hu, S. Cai, W. Cai, Y. Zhang, J. Sui, Restructured single parabolic band model for quick analysis in thermoelectricity, npj computational materials 7 (1) (2021) 1-8. https://doi.org/10.1038/s41524-021-00587-5

• GitHub rSPB respository. https://github.com/JianboHIT/rSPB

P003: Device Figure-of-Merit (ZTdev) (G.J. Snyder et al., 2017)

$$ (ZT)_{dev} = \left[ \frac{T_{h} - T_{c}(1-\eta)}{T_{h}(1-\eta) - T_{c}} \right] ^{2} - 1 $$

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• G.J. Snyder, A.H. Snyder, Figure of merit zt of a thermoelectric device defined from materials properties, Energy Environ. Sci. 10 (11) (2017) 2280-2283. https://doi.org/10.1039/C7EE02007D.

P004: Thermoelectric compatibility factor (CF) (G.J. Snyder et al., 2003)

$$ s = \frac{\sqrt{1+ZT}-1}{ST} $$

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• G.J. Snyder, T.S. Ursell, Thermoelectric efficiency and compatibility, Phys. Rev. Lett. 91 (14830114) (2003). https://doi.org/10.1103/PhysRevLett.91.148301.

• W. Seifert, K. Zabrocki, G.J. Snyder, E. Müller, The compatibility approach in the classical theory of thermoelectricity seen from the perspective of variational calculus, physica status solidi (a) 207 (3) (2010) 760-765. https://doi.org/10.1002/pssa.200925460.

• W. Seifert, V. Pluschke, C. Goupil, K. Zabrocki, E. Müller, G.J. Snyder, Maximum performance in self-compatible thermoelectric elements, J. Mater. Res. 26 (15) (2011) 1933-1939. https://doi.org/10.1557/jmr.2011.139.

附录：如何在 MicroSoft Office 中使用这里的公式

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• 方法一（适用于 Word 和 PPT）

1. 在网页中的公式上右键，点击 Copy to Clipboard > TeX Commands；

2. 在 Word 或者 PPT 中，点击 插入 > 方程，或者使用快捷键 Alt + =；

3. 在 公式 选项卡中点击 {}LaTeX ，切换为 LaTeX 输入模式；

4. 粘贴内容（快捷键 Ctrl + V），然后按回车（Enter）键；

5. 完成。

• 方法二（仅适用于 Word）

1. 在网页中的公式上右键，点击 Copy to Clipboard > MathML Code；

2. 在 Word 中直接粘贴（快捷键 Ctrl + V）；

3. 完成。