Improved thermoelectric cooling based on the Thomson effect

Traditional thermoelectric Peltier coolers exhibit a cooling limit which is primarily determined by the figure of merit, $\mathit{zT}$. Rather than a fundamental thermodynamic limit, this bound can be traced to the difficulty of maintaining thermoelectric compatibility. Self-compatibility locally maximizes the cooler's coefficient of performance for a given $zT$ and can be achieved by adjusting the relative ratio of the thermoelectric transport properties that make up $zT$. In this study, we investigate the theoretical performance of thermoelectric coolers that maintain self-compatibility across the device. We find that such a device behaves very differently from a Peltier cooler, and we term self-compatible coolers “Thomson coolers” when the Fourier heat divergence is dominated by the Thomson, as opposed to the Joule, term. A Thomson cooler requires an exponentially rising Seebeck coefficient with increasing temperature, while traditional Peltier coolers, such as those used commercially, have comparatively minimal change in Seebeck coefficient with temperature. When reasonable material property bounds are placed on the thermoelectric leg, the Thomson cooler is predicted to achieve approximately twice the maximum temperature drop of a traditional Peltier cooler with equivalent figure of merit ($zT$). We anticipate that the development of Thomson coolers will ultimately lead to solid-state cooling to cryogenic temperatures.

Figure 1

The local reduced coefficient of performance ${\phi }_r$ is optimized at a specific reduced current density, termed $s$. If $u\ne s$, the ${\phi }_r$ is less than that predicted by the material $zT$. Here, $z=0.002$ K${}^{-1}$, $\alpha =200\mu \text{V}{\text{K}}^{-1}$.

Figure 2

(a) The CPM Peltier cooler and $u=s$ Thomson cooler are compared using the same constant $z=0.002$ K${}^{-1}$. The overall device $\phi$ of a CPM cooler crosses zero at a finite temperature, indicating $\Delta T_{\max }$ is reached, while $\phi$ remains positive for all temperatures for the $u=s$ cooler. (b) The local performance of a CPM cooler (${\phi }_r$) is significantly compromised at both the hot and cold ends. In contrast, ${\phi }_{r,\max }$ is achieved at all temperatures when $u=s$. In both panels, the performance calculated for an actual Bi${}_2$Te${}_3$ Peltier cooler leg is similar to the CPM.

Figure 3

Both the CPM and Bi${}_2$Te${}_3$ coolers have $u=s$ at one point along the leg. By definition, the $u=s$ is self-compatibile along the entire leg. The different slope signs of $u$ for CPM and $u=s$ reveals that these coolers are fundamentally distinct. The curves were generated for an optimized cooler at $\Delta T_{\max }$ with $z=0.002$ K${}^{-1}$, $T_h=300$ K.

Figure 4

The maximum temperature drop $\Delta T_{\max }$ of a $u=s$ Thomson cooler exceeds that of a Peltier cooler with the same $z$. Large band gap ($E_g$) thermoelectric materials are necessary at the hot junction to improve the performance ($T_h=300$ K).

Figure 5

(a) The Seebeck coefficient of the $u=s$ Thomson cooler varies exponentially, while it is by definition constant for the CPM cooler. As a degenerately doped semiconductor, the Seebeck coefficient of commercial Bi${}_2$Te${}_3$ increases gradually with temperature. (b) The curvature of $T\left(x\right)$ for the CPM Peltier cooler temperature profile is opposite that of the $u=s$ cooler because of the different sign of the Fourier heat divergence. Again, similar behavior is found between the CPM and commercial Bi${}_2$Te${}_3$ cooler.