README-en.md

formulas4TE

Summary some formulas and equations for thermoelectricity(TE)

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    We often need to use some formulas repeatedly in thermoelectric research. Although, an efficient built-in equation editor is embedded in MicroSoft Office, it still takes some time to edit these numerous complex repetitive formulas. Here, some frequently used and reported formulas are collected, which not only facilitates the use of screenshots, but also can be easily copied to MicroSoft Office as an embedded equation (see Appendix: How to use equations in MicroSoft Office ), improving work efficiency. Any bug reports or suggestions are very welcome, which can be discussed at Issues page.

• 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)

• Appendix: How to use equations in MicroSoft Office
  • Method 1 (Both MicroSoft Word and PPT support)
  • Method 2 (Only Word supports)

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.

Appendix: How to use equations in MicroSoft Office

• Method 1 (Both MicroSoft Word and PPT support)

1. Right click on the equation, select Copy to Clipboard > TeX Commands.

2. Select Insert > Equation or press Alt + = in Word or PPT.

3. Click {}LaTeX to enable LaTeX mode in the Equation ribbon.

4. Paste (Ctrl + V) and press Enter.

5. Done.

• Method 2 (Only Word supports)

1. Right click on the equation, select Copy to Clipboard > MathML Code.

2. Paste (Ctrl + V) in Word.

3. Done.