Electronic cooling using thermoelectric devices

Thermoelectric coolers are proposed for many applications such as small refrigerators for beverage cooling and cooling of car seats, semiconductor lasers, compact imaging, and detecting sensors.¹ Thermoelectric devices integrated into electronic devices (packages) have also been proposed to cool the hotspots.^2,3 At smaller scales (few hundreds of micron, which is the typical size of hot spots of an IC chip), integrated thermionic coolers are proposed, wherein n-p legs are replaced by layered materials with tall barriers in the path of electrons to filter out the low-energy electrons and therefore to enhance the Peltier cooling.^4–6 

Typical commercial thermoelectric devices are made out of Bi₂Te₃/Sb₂Te₃ for room temperature applications and PbTe for high temperature applications, and they have a ZT of around 1. Heat pumps, coolers, and refrigerators are working based on similar principles. In most cases, the aim is to pump heat against its natural flux and from the cold side to the hot side. For example, pumping heat from the inside of the refrigerator to the outside ambient room, pumping heat from the inside of the household to the outside hot environment in summer and pumping heat backward from the cold winter air into the household. Thermoelectric coolers are proposed for some of these cooling/heating applications. It has been shown that the performance of the cooler/heater is an increasing function of its material figure of merit, $ZT=σS2T/κ$⁠, where in $σ$ is the electrical conductivity, S is the Seebeck coefficient, T is the operating temperature, and $κ$ is the thermal conducitity.⁷ 

Thermoelectric devices were also tested for electronic cooling and site specific cooling. Manno et al.⁸ recently provided a good review of different device configurations for hot spot cooling tested by different groups. For example, Chowdhury et al.⁹ modeled heat transferred in a thermoelectric Peltier cooler attached to a localized hot spot and experimentally measured the on demand cooling achievable. The temperature of the localized hot spot was 124.5 °C without the thermoelectric cooler. A temperature-drop of 7.6 °C due to passive cooling was observed when the thermoelectric cooler was attached to the heat spreader but was not powered. Then, the cooler was powered and an additional 7.3 °C of on demand cooling was observed. Considering the low thermal conductivity of bismuth telluride based thermoelectric coolers used in this experiment, 1.2 W/mK, one can conclude that the passive cooling part could be significantly enhanced by replacing the thermoelectric legs with a good thermal conductor such as copper with the thermal conductivity of more than 300 W/mK. However, it is very difficult to enhance the active part of the cooling, as bismuth telluride used in this experiment has one of the highest thermoelectric power factors observed so far at 300 K–400 K temperature range. Then, the question is if we can achieve better results with passive cooling, what is the point of using thermoelectric materials? The purpose of this paper is to re-evaluate thermoelectric coolers for electronic cooling applications. We show that for this specific application, ZT is not the relevant parameter, and higher ZT values do not result in pumping more heat. Instead, we need to combine active and passive cooling and design a thermoelectric material with large power factor as well as large thermal conductivity. This conclusion could be extended to any application in which the aim is to pump heat from hot to cold in the direction of natural heat flux.

This paper is organized in the following manner. We first write the equations for a standard thermoelectric cooler, which pumps heat from the cold side to the hot side, following the same notations as was originally used by Goldsmid.¹⁰ We then re-write the same equations, this time for active cooling in which one simply would like to enhance the passive cooling or the heat transfer from hot to cold. To distinguish, we refer only to the first case as a thermoelectric refrigerator and to the second case as a thermoelectric active cooler. We then show the limitations currently present for obtaining active cooling fluxes comparable to passive cooling fluxes. Finally, we propose proper materials for electronic cooling applications as well as device geometries to enhance the cooling performances.

Consider a single p-n thermoelectric module schematically shown in Fig. 1. We assume constant materials properties in each leg. $Sn/p,ρn/p,κn/p$ are the Seebeck coefficient, the electrical resistivity, and the thermal conductivity of the n/p legs, respectively. The p-n legs are connected electrically in series and thermally in parallel. Therefore, the electrical resistance, R, and the thermal conductance, K, of the device ignoring the metallic connections and the interfacial resistances, could be written as: $R=ρplpAp+ρnlnAn$ and $K=κpAplp+κnAnln$⁠, and finally, the Seebeck coefficient of the TE device is $α=(Sp−Sn)$⁠.

A current, I, is passed through the legs to pump heat from top to bottom or from cold to hot. In this geometry, the Peltier current, $αIT$⁠, is going against the natural heat flux $K∇T$⁠.¹¹ The heat rate extracted from the cold side (⁠$QC$⁠) can be written as¹² 

The heat generated as a result of Joule heating exists the two ends of the device with the same probability resulting in $−RI22$ term in Eq. (1).¹³ 

The work done to operate the cooler is

The coefficient of performance (COP) is defined as COP $=QCW$ and is a measure of the efficiency of these coolers. The COP of commercial home refrigerators and air-conditioners is about 2–4, and those of Stirling refrigerators are about 5.¹⁴ It is estimated that the thermoelectric coolers would require having ZT values close to 4 to compete with such coolers. Other estimates suggest requirements of even larger ZT values of 9.2.¹⁵ 

The performance of a TE refrigerator is a function of the applied current. Depending on the type of the application, one might be interested in the optimum COP or the optimum heat extracted by these coolers. The optimum current to maximize the heat extracted from the cold side, Q_C (Eq. (1)), is $I=αTCR$⁠. Therefore, the maximum heat extracted and the corresponding COP are

As can be seen, the COP and the maximum cooling are both increasing function of Z. It is also noted that the refrigerator only works for sufficiently small values of $ΔT<ZTc2/2$⁠. Beyond that, the refrigerator could not pump heat from the cold side to the hot side, and regardless of the applied current, the natural back heat flux would be larger than the Peltier cooling. Finally, the negative term in Eq. (3a) is proportional to the thermal conductance, while the first term is independent of it. As the thermal conductance increases, the negative term increases and Q_C decreases, and therefore, the refrigerator could not function when $K>PFDTc2/2ΔT$⁠. The optimum heat flux per leg (Eq. (3c)) divided by the temperature gradient is shown in Fig. 1(a), the area with the negative $Q′C_max$⁠, correspond to the range at which the refrigerator cannot function. This area clearly increases for larger values of $ΔT$ (compare blue and red negative contour lines).

Now, we switch to the geometry indicated in Fig. 1(b), an active thermoelectric cooler. Here, the natural conduction heat flux is in the same direction as the Peltier current, and the only competition is the Joule heating generated inside the TE legs. The heat extracted from the hot side could be written as

$KΔT$ is the passive cooling term. The additional $αITH−RI22$ term is the active cooling term. This term should be larger than zero for the active cooling to be effective. Therefore, the cooling range of operation is when $2αTHR>I$⁠. For any current larger than this, Joule heating will heat up the legs and will result in a net heat flux smaller than the natural conduction heat flux. The net work done by applying an electric current can be written as

The equivalent COP is defined as COP $=QHW.$

Looking at the expression for work, we see that there is one very interesting case where the work done is zero and COP becomes infinite. Such situation could be achieved when $I=αΔTR$⁠. This current is the short-circuiting current. It basically means that one can simply enhance the cooling performance by connecting the lower ends of the p and the n legs by a wire. Because of the existence of a temperature difference between the top and the bottom, the electrons and holes will start to circulate without an externally applied electric field. This natural circulation of current results in extra cooling. The heat extracted from the hot side when p and n's are short-circuited can be written as

Here, $TM$ is the average temperature $(TC+TH)/2$⁠. To maximize the cooling, one needs to use a device with large power factor (PF) and large thermal conductance. In cases where efficiency is not an issue and the only interest is to maximize heat extraction from the hot side, the optimum current is $I=αTHR$⁠. Under this optimum current, the maximum heat current and the corresponding COP could be written as

Leading to a maximum heat flux per leg:

The first term is the active cooling term, and the second term is the passive cooling term. Again, it can be seen that a better performance is achievable when both the thermal conductance and the power factor are large. Comparing Eqs. (3b) and (7b), one can easily see that a thermoelectric refrigerator performs better when the temperature difference is smaller and ZT is larger while a thermoelectric active cooler performs better under larger temperature differences and smaller ZT values.

It is possible to combine active and passive cooling to achieve a better overall cooling performance. The main challenge is the low power factor of conventional thermoelectric materials. Power-factor multiplied by temperature (⁠$PFT$⁠) has the same unit as the thermal conductivity. Best thermoelectric materials reported so far have power factors on the order of 5 W/mK. In comparison, thermal conductivity of copper is more than 300 W/mK. Perhaps, one can think of using copper, combining it with another metal with the opposite Seebeck coefficient to serve as p-n legs to have some improvements in the cooling. But the power factor of copper is very small and is about 1 W/mK at 500 K. Therefore, overall, there will be only a minor enhancement (∼0.3%), which is not worth the effort. On the other hand, if we use a common thermoelectric material such as Bi₂Te₃, the thermal conductivity and the power factor will be comparable, and therefore, active cooling achieved could double the cooling performance, but the overall cooling will be poor, because now both thermal conductivity and power factor are small and the sum will be less than the passive cooling achievable by using copper. In Table I, we have summarized some of the materials power factors (S²T/ρ) reported in the literature for different materials. Instead of including only good thermoelectric materials (high ZT), we focused on the reported materials with large power factors. As the base for comparison, we included the relevant values for copper.

Perhaps, best materials for combining passive and active cooling are metallic alloys such as Pd-Ag and Cu-Ni, wherein two metals from neighboring groups in transition metals with large d or f density of states and with similar electronic structures are chosen to preserve the large conductivity while enhancing the Seebeck coefficient. These metallic alloys are not investigated in depth in the thermoelectric community because of their large thermal conductivity. Pd-Ag is very expensive but if one can find a cheaper combination of materials (e.g., Cu-Ni) with large power factors, it is possible to replace copper sinks. To do so, we should push for power factor values close to 100 W/mK. Note that in Eq. (7c), there is a factor of $TH2ΔT$ in the active cooling term. Therefore, while the passive cooling term is always proportional to the thermal conductivity, the active cooling term is linearly increasing with the operating temperature. To show the improvement in Table I, we have added one extra column labeled as active cooling term, which is the same as the first term in the parenthesis of Eq. (7c) (⁠$PFTmTH2ΔT$⁠) and is calculated for $ΔT=50K$⁠.

The active cooling term increases with smaller temperature differences. For example, if only 10 K cooling is desired then active cooling terms listed in the table will increase by a factor of 5. Therefore, active cooling might be good to start the process, and once a small cooling is achieved, one can turn the cooler off and rely only on the passive cooling.

The total achievable cooling listed for different materials combinations are all less than that of the copper (350 W/mK at 500 K). But the values are not completely off. Therefore, there is a hope to find materials competitive with copper and even better. As we mentioned, metallic alloys have not been fully investigated because of their high thermal conductivity. Another set of candidates is carbon-based materials. These materials own large lattice thermal conductivities. Metallic carbon materials could also have large electronic thermal conductivities and possibly large power factors. Extremely large thermal conductivities as well as power factors are reported recently for p- and n-doped carbon nanotubes. Jiang et al.²⁰ theoretically calculated large power factors of 176 W/mK and 76 W/mK for p and n type carbon nanotubes at 400 K with lattice thermal conductivity of 701 W/mK. The resulting ZT is small and less that 0.25 but nevertheless both power factor and thermal conductivities are much larger than any other material listed above. The referenced paper presents a theoretical calculation and might overestimate the performance. However, it demonstrates that it is possible to have materials with large power factors and thermal conductivities to compete with the passive cooling possible using copper.

Perhaps, the best way of using thermoelectric materials for electronic cooling applications is to use them in the fin geometry. Fins are used very often to accelerate the cooling processes. Having thermoelectrics as fins, the cooling could be accelerated significantly. Thermoelectrics are traditionally cut as bar shaped legs, and therefore, they could be easily used as fins attached to a copper base. The geometry is similar to a regular Peltier cooler except for the fact that the insulation layers between p-n legs are replaced by convective heat flux to quickly remove the heat. This is schematically shown in Fig. 2. We can write the energy balance in a single fin (Fig. 2(b)). The resulting heat flux at the hot side for one leg can be written as¹³ 

The optimum current that maximizes the heat flux is

And therefore, the corresponding optimum heat flux is

Figure 2 shows the enhancement in the cooling performance as a result of using thermoelectrics in the fin geometry. Note that both passive and active cooling terms increase as $ωL$ increases. Interestingly, in this geometry, a larger thermal conductivity increases the passive cooling, while for the same $hp/A$ ratio, a larger thermal conductivity, $κ$⁠, results in a smaller $ω2=hpκA$ values and therefore lowers the active cooling term. Note that without the convective heat transfer, active cooling term is independent of $κ$⁠. Figure 2 demonstrates that small power factors of less than 50 W/mK will not affect the cooling processes significantly even in the presence of large convective heat transfer. Similar to what we discussed before, the active cooling term will be enhanced when the factor of $TH2ΔT$ increases. Furthermore, at large power factors, the effect of convective heat transfer is more significant and can enhance the active cooling term in a similar manner to the enhancement of the passive cooling term.

In summary, we have shown that for electronic cooling and any application in which the aim is to enhance the passive cooling and to pump heat from the hot side to the cold side, traditionally used high-Z materials are not useful. Instead, materials with larger power factors and large thermal conductivities should be used. By simply connecting p and n legs with a wire (short-circuiting), one could improve the cooling process without performing any external work. Optimizing the electric current passed through p-n legs could further enhance the cooling processes. Finally, adding the legs as fins attached to a copper sink, the cooling performance could be significantly enhanced. In all cases, the thermoelectric community needs to explore a new set of materials such as metallic alloys and metallic carbon based materials to increase the thermoelectric power factor and the thermal conductivity simultaneously.

The author would like to acknowledge K. Esfarjani and Y. Jaluria for their helpful feedback on the manuscript. This work was supported by the Air Force Young Investigator Award, Grant Number FA9550-14-1-0316.