# Application of Thermoelectric Generators for Low-Temperature-Gradient Energy Harvesting
## 2 Basic Principles of Thermoelectric Generators
### The Seebeck Coefficient  
The Seebeck coefficient \(S\) is defined from the voltage \(\Delta V\) produced due to a small temperature gradient (\(\Delta T=T_{hot}-T_{cold}\), between the two sides of the TEG) between the two semiconductor materials at the junction under open circuit conditions [21; 27; 28] as,  
\[S=-\Delta V/\Delta T=-(V_{hot}-V_{cold})/(T_{hot}-T_{cold}) \tag{1}\]  
The voltage difference \(\Delta V\) is due to the difference in the electrochemical potentials of the two semiconductor materials in contact [28]. Therefore, the Seebeck coefficient measures the magnitude of the thermoelectric voltage induced due to a temperature difference between the two materials. In general, the Seebeck coefficient depends on the molecular structure of the materials and on the absolute temperature. For most semiconductor materials, the voltage \(\Delta V\) generated due to the Seebeck effect is very low, on the order of a few tens or hundreds of \(\mu\)V/K. For example, for the commonly used Bi\({}_{2}\)Te\({}_{3}\) semiconductor, the Seebeck coefficient \(S\) is 160 \(\mu\)V/K for p-type material and \(-170\)\(\mu\)V/K for n-type material [29]. TEG materials must have high electrical conductivity to minimize the Joule effect, a large Seebeck coefficient for maximum conversion of heat to electrical energy, and low thermal conductivity to minimize thermal conduction through the material [21]. These properties are combined into a \(z\)-metric, the figure of merit, which quantifies the overall output of the TEG, and is defined as [30],  
\[z=S^{2}\rho^{-1}\kappa^{-1}\left[\text{K}^{-1}\right] \tag{2}\]  
where \(\rho\) is the electrical resistivity and \(\kappa\) the thermal conductivity. The Seebeck coefficient \(S\) is typically determined over a 5-10 K range, so the figure of merit \(z\) is valid only for a small temperature difference [31]. The dimensionless figure of merit \(z_{T}\) is commonly used to characterize TEG's behavior [30],  
\[z_{T}=S^{2}\rho^{-1}\kappa^{-1}T\left[-\right] \tag{3}\]  
Figure 1: (**a**) Schematics of a thermoelectric generator indicating the charge carriers, adapted from [25]. (**b**) Series connections of different thermoelectric couples, adapted from [26]. where \(T=(T_{hot}+T_{cold})/2\) is the average temperature. When assuming a one-dimensional steady state heat transfer process and no heat loss through the heat exchanger wall [32], uniform temperature distribution across each surface of the material [31], a rough estimation of the maximum thermo-electrical conversion efficiency of the TEG is determined by [30; 31; 33; 34; 35],  
\[\eta_{\text{TEG,max}}=\frac{T_{hot}-T_{cold}}{T_{hot}}\times\frac{\left(1+z_{T} \right)^{1/2}-1}{\left(1+z_{T}\right)^{1/2}+\frac{T_{cold}}{T_{hot}}} \tag{4}\]  
where the term \((T_{hot}-T_{cold})/\)\(T_{hot}\) is the Carnot efficiency. According to (4), the efficiency of a TEG depends on the hot and cold junction temperatures and on the thermoelectric properties of the material through the merit factor \(z_{T}\). However, (4) only predicts \(\eta_{max}\) accurately for small temperature differences \(\Delta T=T_{hot}-T_{cold}\), or for materials with \(z_{T}\) almost constant over a wide temperature range [36]. In practical scenarios, the efficiency of a TEG can be more complex when considering the thermal and electrical resistances, heat leakage, Thompson effect, and the temperature gradient in the heat sinks at the hot and cold ends of the TEG [37]. From (3) and (4), it is evident that the efficiency of TEGs depends on the properties of the internal semiconductor materials through the figure of merit and the temperature, so to improve the efficiency, materials with a high figure of merit are required. TEGs are mostly based on three semiconductor materials, i.e., Bi\({}_{2}\)Te\({}_{3}\), PbTe and SiGe. The election of the material depends on the heat source characteristics, cold sink and TEG design.