Inhomogeneous Thermal Conductivity Enhances Thermoelectric Cooling

We theoretically investigate the enhancement of thermoelectric cooling performance in thermoelectric devices made of materials with inhomogeneous thermal conductivity, beyond the usual practice of enhancing thermoelectric figure of merit ZT. The dissipation of Joule heat in such thermoelectric devices is asymmetric which can give rise to better thermoelectric cooling performance. Although the thermoelectric figure of merit and the coefficient-of-performance are only slightly enhanced, both the maximum cooling power and the maximum cooling temperature difference can be enhanced significantly. This finding can be used to increase the heat absorption at the cold end. The asymmetric dissipation of Joule heat also leads to thermal rectification.

There has been great interests in thermoelectric (TE) devices that can directly convert electricity into thermal energy for cooling or heating and can harvest solar and waste heat into electric power [1,2]. The energy conversion efficiency of TE devices is determined by the figure of merit of TE materials [3,4] ) /( 2 ρλ α T ZT = , where α is the Seebeck coefficient, T is the absolute temperature, ρ is the electrical resistivity, and λ is the thermal conductivity which consists of electronic thermal conductivity and lattice thermal conductivity. High ZT materials are desirable for high efficiency TE devices. Even though TE devices have many advantages such as reliability and scalability, the commercial available materials with ZT~1 limits widespread applications of thermoelectrics. Great efforts in enhancing ZT have been made in past decades [5,6,7]. The performance of a TE cooler is evaluated with these three parameters: i). the maximum cooling power max ) ( c q that describes the maximum rate at which heat can be absorbed from the cold end, ii) the maximum cooling temperature difference max ) ( T ∆ which can be reached when the maximum cooling power falls to zero, max ) ( c q =0; and iii) the maximum coefficient-of-performance (COP) max φ which is the energy conversion efficiency. There have been many efforts in enhancing the performance of TE coolers through high ZT materials, system engineering [1], and even transient cooling [8,9,10]. In this work, we study the performance of TE devices made of materials with inhomogeneous thermal conductivity.
Assuming p-and n-type legs have same material properties, we only need to consider a p-type branch with length L and cross section area A as shown in Fig. 1(a) to evaluate device performance [11]. The device is operated with the temperature of 1 T and 2 T at the cold and hot end, respectively. When an electric current I flows across the device along x-direction, heat can be absorbed at rate 1 IT q ab α = at the cold end due to Peltier effect. As shown in Fig. 1(a), the absorbed heat can be partially cancelled by the heat leakage due to the temperature difference between the hot and cold ends DT q and the flow of a portion of Joule heat ( R I q Joule ) generated inside the device where R is the electrical resistance. The net cooling power can then be expressed as where γ is defined as inhomogeneity factor of asymmetric Joule heat dissipation.
It is rather straightforward that to enhance the device performance max ) ( c q , max ) ( T ∆ and max φ [11,12], one needs to either enhance the Seebeck coefficient α or suppress DT q and Joule q . It is interesting to note that most past studies assume, by default, symmetric flow of the Joule heat to the cold and hot ends, namely, γ = 1/2 in Eq. (1) [11,12].
However, this assumption is valid only when all the transport coefficients are not spatial-dependent. The factor γ can be very different from 1/2 in inhomogeneous materials, which indeed gives rise to a great design freedom to improve the TE cooling performance. Indeed, the devices made of functional graded TE materials (FGTM) with inhomogeneous transport properties was first proposed by Ioffe [13] in 1960 and then be widely studied by many researchers to enhance the device performance [14,15,16,17,18,19]. For example, Bian et al. [20] found that an enhancement of max ) ( T ∆ can be achieved in FGTM with spatial-dependent Seebeck coefficient.
In this Letter, we investigate the performance of TE devices made of inhomogeneous materials with varied transport coefficients. By assuming spatial-and and thermal conductivity ) , ( T x λ , the following equation will be solved to analyze the device performance: where x is the distance from the cold end. The boundary conditions are chosen as In Eq. (2), the left term is the divergence of the Fourier heat current, while the first term on the right is the Joule heat generated by an electric current I flowing through the device, and the second term is the Thomson heating or cooling due to the temperature-and spatial-dependent Seebeck coefficient.
The cooling power can be then obtained as with the temperature profile. ) as a function of parameter c when the inhomogeneous thermal conductivity is ) Without losing generality, we study here the enhancement on the cooling performance of TE devices by utilizing the inhomogeneous materials with spatial-and  [21,22], has recently been developed to realize thermal rectification effect or thermal diode [23,24,25]. Table I lists several common analytical expressions of spatial-and temperature-dependent thermal conductivities considered in this work. The first example is the inhomogeneous materials with linear spatial-dependent thermal Here 0 λ is the reference thermal conductivity at the cold end and the slope c denotes the strength of the spatial dependence or inhomogeneity. The cooling power for this kind of material can then be derived as: where 1 2 Here we have introduced two new parameters of β and γ . The which now becomes, when the maximum electric current In comparison with the homogeneous thermal conductivity case, when with a as an arbitrary coefficient, there is one more factor γ 2 / 1 in m I and in the first term on the right side of Eq. (4), shown in Table I. The maximum It is obvious that both max ) ( c q and max ) ( T ∆ can be enhanced when In order to calculate the COP written as ) which is the ratio between cooling power and total input power, we now redefine an effective figure of merit as where ) is the mean temperature weighted by the inhomogeneity factor γ . Using such an effective figure of merit and weighted mean temperature and setting 0 / = dI dφ , the maximum COP is obtained as In the limit of homogeneous case when , the familiar results of , and the conventional expression of max φ with homogeneous thermal conductivity are recovered [12].
We perform the numerical calculations based on the above mentioned model for a TE element with L=5 mm and A=4mm 2 . The typical material properties of p-type BiSbTe alloy [ 26 ] have been adopted as follows: the Seebeck coefficient     intrinsic spatial-dependent thermal conductivities ) ,  Table I, the results with explicit spatial-dependent thermal conductivity can be briefly described as follows: i) the maximum cooling power and the maximum cooling temperature difference can be greatly enhanced while the maximum COP is only slightly enhanced in TE device; ii) to enhance the cooling performance, the thermal conductivity close to the cold end should be smaller than the thermal conductivity close to hot end, which results in a smaller fraction of the Joule heat flow towards the cold end, as noted by , and 0 > g as shown in Table I.
are present separately in Table I) and linear temperature-dependence where 0 T is the room temperature.
The detailed numerical results can be found in Supplemental Material [27]. One important observation is that the intrinsic spatial-dependent thermal conductivities due to its dependence on temperature do not lead to the asymmetric dissipation of Joule heat. In other words, γ is always equal to 1/2. The Joule heat flowing towards the cold end is exactly the same as the case with homogeneous thermal conductivity.
Therefore the maximum electric current m I is the same as that with homogeneous thermal conductivity. Only the normalized conducted heat β is modified.
Furthermore, there is no simple explicit forms of max ) ( T ∆ and Z for the case with which are noted as N/A in Table I.
We believe that there is a fundamental difference between the explicit spatial-dependent thermal conductivities case and the temperature-dependent thermal conductivities case. The physical explanation is that space inversion symmetry is broken for explicit spatial-dependent thermal conductivities, but conserved for temperature-dependent thermal conductivities. If we swap the boundary condition, 1 T T ↔ , the heat transport process and temperature profile after the reversion is exactly the same as that before the reversion. This might also be the reason why there is no thermal rectification effect for homogeneous materials with temperature-dependent thermal conductivities. Our earlier research shows that it is crucial to utilize some kind of symmetry breaking mechanism to realize a thermal diode [24,25].
Since the inversion symmetry is broken by the spatial-dependent thermal conductivities, the resulted asymmetric Joule heat flow can also be used for novel design of thermal diodes. In particular, without considering the Peltier effect, i.e. 0 → α , the heat current flowing out of the device changes from Therefore, the thermal rectification factor can be derived as [25]: It is obvious that any deviation from 1/2 for the inhomogeneity factor γ will induce a finite thermal rectification effect for nonzero Joule heat. Figure 3  To summarize, we have discovered that thermoelectric cooling performance can be significantly enhanced through the manipulation of Joule heat flow with explicit spatial-dependent inhomogeneous thermal conductivity. The flow of Joule heat towards the cold end can be suppressed when the thermal conductivity near the cold end is smaller than that near the hot end. We found that the maximum cooling power and the maximum cooling temperature difference can be significantly enhanced while the coefficient-of-performance is slightly enhanced. The intrinsic spatial-dependent thermal conductivity due to its temperature dependence cannot lead to such enhancement. Our findings suggest that the materials with inhomogeneous thermal conductivity used for thermal rectifier/diode can be also used to improve the performance of thermoelectric cooling, which in turn enriches the applications of thermal rectifier [25].
It should be pointed out that materials with inhomogeneous thermal conductivity can be now be readily achieved with nanotechnology. For example, the inhomogeneous nanotube [22], thin diamond film in which the inhomogeneity is due to spatially varying disorder associated with nucleation and grain coalescence [29], and thermal rectifier with pyramid shaped LaCoO 3 /La 0.7 Sr 0.3 CoO 3 [30]. We expect that our investigation will inspire many follow-up works in realizing inhomogeneous thermal conductivity and wide-spread applications of thermal rectifiers.