Improved Thermoelectric Cooling Based on the Thomson Effect G. Jeffrey Snyder 1* , Eric S. Toberer 2 , Raghav Khanna 1 , Wolfgang Seifert 3 1 Materials Science, California Institute of Technology, 1200 E. California Blvd. Pasadena, CA 91125, USA 2 Department of Physics, Colorado School of Mines, Golden CO 80401, USA and 3 Institute of Physics, University Halle-Wittenberg, D-06099 Halle, Germany (Dated: May 21, 2018) Traditional thermoelectric Peltier coolers exhibit a cooling limit which is primarily determined by the figure of merit, 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 such a device behaves very differently from a Peltier cooler, and 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 the development of Thomson coolers will ultimately lead to solid state cooling to cryogenic temperatures. PACS numbers: 84.60.Rb, 05.70.Ce, 72.20.Pa, 85.80.Fi 1. INTRODUCTION Peltier coolers are the most widely used solid state cooling devices, enabling a wide range of applications from thermal management of optoelectronics and infra- red detector arrays to polymerase chain reaction (PCR) instruments. Thermoelectric coolers have been tradition- ally understood by means of the Peltier effect, which de- scribes the reversible heat transported by an electric cur- rent. This effect is traditionally understood in terms of absorption or release of heat at the junction of two dis- similar materials. The conventional analysis of a Peltier cooler approximates the material properties as indepen- dent of temperature (Constant Property Model (CPM)). This results in a maximum cooling temperature differ- ence ΔT max for a CPM cooler, which dependent on the figure of merit ZT of the device [1, 2]. ΔT max = ZT 2 c 2 (1) For the best commercial materials this leads to a ΔT max of 65K (single stage) [3], which translates to a device ZT at 300K of 0.74. In the CPM the device ZT is equal to the material zT . Material zT depends on the Seebeck coefficient (α), temperature (T ), electrical resis- tivity (ρ), and thermal conductivity (κ), zT = α 2 T ρκ . In the CPM, the only way to increase ΔT max for a single stage is to increase zT , leading to the focus of much ther- moelectric research on improving zT . It is well known that even further cooling to lower temperatures can be achieved using multi-stage Peltier coolers [1, 2]. In prin- ciple, each stage can produce additional cooling to lower temperatures, regardless of the zT of the thermoelectric material in the stage. In practice, the thermal losses and complications of fabrication limit the performance of such devices. The 6-stage cooler of Marlow achieves aΔT max of 133 K; this doubling of ΔT max compared to a single stage cooler is achieved despite using materials with similar zT [3]. Alternatively, such ΔT max with a single-stage CPM cooler would require ZT to be 2.5. The transport properties across a single thermoelec- tric leg can be manipulated to improve cooling perfor- mance, although it has been less effective in reducing ΔT max than a multi-stage approach. One common strat- egy is to engineer a change in extrinsic dopant concen- tration across a thermoelectric element which can sig- nificantly alter α, ρ and even κ. For example, this has been demonstrated for thermoelectric generators in n- type PbTe doped with I [4]. Similar efforts have been done with cooling materials, as has been reviewed in ref [5]. The simplest explanation for an improvement is an in increase in the local zT at some temperatures by spa- tially adjusting the dopant composition within a material [6]. Early theoretical work by Sherman et al for TEC found that different ΔT max could be predicted from materials have the same or similar average zT but different temper- ature dependence of the individual properties α, ρ, κ [7]. This demonstrated that optimizing cooler performance is significantly more complex than simply maximizing zT . More recently, M¨ uller et al. [8–10] and Bian et al. [11, 12] used different numerical approaches to predict substan- tial gains in cooling to ΔT max from functionally grading where an average zT remains constant in an effort to determine the best approach to functionally grading. Different material classes optimized for different tem- arXiv:1111.5300v3 [cond-mat.mtrl-sci] 16 Jun 2012