The bridge between the materials and devices of thermoelectric power generators

Abstract

While considerable efforts have been made to develop and improve thermoelectric materials, research on thermoelectric modules is at a relatively early stage because of the gap between material and device technologies. In this review, we discuss the cumulative temperature dependence model to reliably predict the thermoelectric performance of module devices and individual materials for an accurate evaluation of the p–n configuration compared to the conventional model used since the 1950s. In this model, the engineering figure of merit and engineering power factor are direct indicators, and they exhibit linear correlations to efficiency and output power density, respectively. To reconcile the strategy for high material performance and the thermomechanical reliability issue in devices, a new methodology is introduced by defining the engineering thermal conductivity. Beyond thermoelectric materials, the device point of view needs to be actively addressed before thermoelectric generators can be envisioned as power sources.

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Hee Seok Kim

Dr Hee Seok Kim is a postdoctoral fellow in the department of physics and TcSUH at the University of Houston and was recently appointed as an assistant professor at the University of South Alabama. He received his PhD degree from the department of mechanical engineering at the University of Washington and his Master and Bachelor degrees in mechanical engineering from Sungkyunkwan University in Korea. His current research focuses on energy conversion devices and power/efficiency measurement systems from material to device level. He was awarded a Research Fellowship from Japan (2010) and has published over 20 peer-reviewed papers.

Weishu Liu

Dr Weishu Liu is an associate professor of materials science and engineering at Southern University of Science and Technology (SUSTec). He obtained his PhD degree from the University of Science and Technology Beijing in 2009. He was a post-doc at UW and BC, an associate researcher at UH, and a principal scientist at Sheetak Inc. His research interests include thermoelectric materials and their applications in waste heat harvesting, solar heat utilization, and wearable devices. He was a “China's National 1000-Youth Talent” fellow in 2016. He has published 62 journal papers with total citations of >2700 and an H-index of 29.

Zhifeng Ren

Zhifeng Ren, PhD, is M. D. Anderson Chair Professor in the Department of Physics and TcSUH at the University of Houston. He is experienced in materials synthesis, physics, and applications and has made seminar contributions to high temperature superconductors, carbon nanotubes, ZnO nanowires, thermoelectric materials, nano-bio materials and sensors, flexible transparent electrodes, efficient catalysts for hydrogen evolution reactions, 2D nanosheets for enhanced oil recovery, etc. He has published ∼400 journal papers and has 38 issued patents. He received the 2008 R&D 100 Award and the 2014 Edith and Peter O'Donnell Award. He is an elected APS fellow, AAAS fellow, and NAI fellow.

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Broader context

Thermoelectric devices generate electricity directly from thermal energy, and they are composed of dissimilar layers, such as electrodes, insulators, and bonding interfaces, and thermoelectric materials. Because of the differences in their physical characteristics, fabricating thermomechanically reliable devices is challenging even though various thermoelectric materials have been developed over the last decades with enhanced figures-of-merit (ZT). To use thermoelectric generators as primary energy sources, the device technology needs to be balanced with the material fabrication technology. This review summarizes and discusses a reliable prediction model for thermoelectric efficiency and output power, direct thermoelectric indicators representing practical performance, and the methodology of how to improve a material's ZT and device reliability. This will provide important guidance to estimate the realistic performance of materials and devices, which will help in the development of thermomechanically robust and thermoelectrically high-performing power generators.

1. Introduction

Thermoelectric technologies enable clean, sustainable power generation because of their direct conversion of heat to electricity with no global warming gas exhaust.^1–3 This solid-state energy conversion technology is being applied to a wide range of applications, such as solar-thermoelectric power generators,^4,5 ground^6–9 or aerial^10–12 vehicles, power plants,^13,14 and personal power sources.¹⁵ The performance of a thermoelectric material is indicated by the dimensionless figure-of-merit, ZT, defined as:where S, σ, κ, T, and PF are the Seebeck coefficient, electrical conductivity, thermal conductivity, absolute temperature, and power factor (=S²σ), respectively. The total thermal conductivity κ is expressed as κ = κ_lat + κ_carr + κ_bipolar, where κ_lat, κ_carr, and κ_bipolar are the lattice thermal conductivity, carrier thermal conductivity, and bipolar thermal conductivity, respectively. ZT indicates how efficiently a thermoelectric material can convert thermal energy to electric energy according to its operating temperature. In order to figure out a material's practical performance, direct measurements of the efficiency and output power generation are desired, but the experimental results vary depending on the measurement techniques and conditions or instruments. It is also important to correctly estimate the material's intrinsic efficiency and output power by an analytical or numerical method. Numerical methods for efficiency and output power predictions of thermoelectric materials and devices have been well developed based on a one-dimensional model with adiabatic conditions^16–18 and a three-dimensional finite element analysis via commercial packages such as ANSYS^19–21 or COMSOL.^22–24 These numerical analyses can lead to practical predictions of efficiency and output power depending on the proper boundary and initial conditions, but the analyses are complicated and need simulation resources. In addition, this method does not have any explicit indicators for thermoelectric materials or devices, such as ZT and PF.

On the other hand, an analytical model for efficiency and output power prediction is simple and consistent. It also has less uncertainty depending on the various resources. The conventional analytical model was formulated in the 1950s based on the assumption that all physical properties are constant according to temperature, i.e., temperature independent, and the model has been commonly used. However, because of this assumption, the conventional model often overestimates or underestimates a material's efficiency, which may lead to the wrong design for thermoelectric module devices. Snyder et al.²⁵ discussed the concept of a compatibility factor for a spatial temperature gradient when thermoelectric properties vary with temperature by matching the relationship between the heat flow and electrical current. Recently, to avoid a misunderstanding of the efficiency and output power, cumulative temperature dependence (CTD) models for thermoelectric materials²⁶ and devices²⁷ were developed and were based on an overall temperature dependence of the material properties at given temperature boundary conditions. The CTD models give rise to a reliable prediction of thermoelectric efficiency and output power compared with the conventional model. Additionally, the Thomson effect was ignored in the conventional model. Studies to account for the Thomson effect were reported under the condition of a constant Thomson coefficient,^28–30 which is not practical in some cases, but the CTD model analytically takes into account the Thomson effect without any restrictions on the Thomson coefficient behavior. This provides practical fractions of Joule and Thomson heat returning to the hot end. It also shows direct indicators for thermoelectric efficiency with the engineering figure of merit (ZT)_eng and for output power with the engineering power factor (PF)_eng.

For the last decade, a reduction in thermal conductivity through nanostructure technology has been an effective route to improve ZT. This approach has shown dramatic ZT enhancement mainly due to reducing the lattice thermal conductivity κ_lat by intensive phonon scattering.^31–36 Recently, high peak ZT values were reported in several materials, which were caused by a very low κ_lat, such as the peak ZT of 1.86 in Bi_0.5Sb_1.5Te₃ with κ_lat = 0.3 W m⁻¹ K⁻¹,³⁷ the peak ZT of 2.2 in PbTe:SrTe with κ_lat = 0.5 W m⁻¹ K⁻¹,³⁸ the peak ZT of 2.3 in Cu₂Se with κ = 0.15 W m⁻¹ K⁻¹,³⁹ and the peak ZT of 2.6 in SnSe with κ_lat = 0.23 W m⁻¹ K⁻¹.⁴⁰ Such low thermal conductivities lead to higher ZTs, but new challenges arise in designing thermoelectric generators at the device level. The reduced thermal conductivity allows for a higher thermal resistance in the thermoelectric material across the hot and cold side. This generates a larger temperature difference with a fixed length of the thermoelectric components or allows for a shorter length of the components corresponding to the given thermal boundary temperatures. A shorter length may lead to larger power generation when a cross-sectional area is fixed because of the reduced electrical resistance, and less consumption of materials leads to a larger specific power density, such as Watt per kg, Watt per volume, or Watt per dollar (cost). This is a superior design direction for thermoelectric devices. However, the shortened length because of the lowered thermal conductivity may cause a larger shear stress at the joining interfaces, a smaller thermal stress resistance, and a larger relative electrical or thermal contact resistance. Such thermomechanical vulnerability can cause position realignment of the thermoelectric components,⁴¹ cracks on the thermoelectric legs,⁴² degradation of the bonding strength between the thermoelectric component and metal electrodes,⁴³etc., which eventually result in a deterioration of the output power performance. To avoid such vulnerability, various approaches such as a device design with tapered thermoelectric legs,⁴⁴ various leg geometries,⁴⁵ a linear structure with a dovetail-shaped electrode,¹² an angled linear structure,⁴⁶ and optimized component dimensions^47,48 were investigated. However, those approaches focused on an alternative design for thermoelectric devices without considering a thermoelectric material's tuning strategy. Kim et al. demonstrated a methodology for how to balance between the thermal conductivity reduction in thermoelectric materials and the thermomechanical reliability of fully assembled thermoelectric devices.⁴⁹

In this review, we summarize the analytical model for reliable prediction of thermoelectric efficiency and output power from the material to the device level based on the CTD model. In addition, the correlation of the efficiency vs. the engineering figure of merit (ZT)_eng and the output power density vs. the engineering power factor (PF)_eng are discussed. (ZT)_eng and (PF)_eng are direct thermoelectric indicators for efficiency and output power generation, respectively. We then review the methodology of how to reconcile the ZT improvement strategy by reducing the thermal conductivity and thermomechanical device reliability to create a thermoelectrically high-performance and thermomechanically stable system.

2. Cumulative temperature dependence model for homogeneous materials

The conversion efficiency of a thermoelectric generator was first derived by Altenkirch in 1909 based on the assumption of constant material properties, and this constant property model (CPM) identified the three desired material properties for good thermoelectric materials. These are a large Seebeck coefficient (S), low electrical resistivity (ρ), and low thermal conductivity (κ),⁵⁰ where the electrical resistivity ρ is the reciprocal of the electrical conductivity, (ρ = σ⁻¹). In 1957, Ioffe derived the thermoelectric efficiency formula as a function of the figure of merit ZT based on the CPM, and the formula is:⁵¹where T_h, T_c, and T_avg are the hot side, cold side, and their average temperature, respectively. η_c is the Carnot efficiency, η_c = (T_h − T_c)/T_h. Z is defined as S²σ/κ in eqn (1). The peak ZT has been used to represent the performance of thermoelectric materials since the 1950s,^1,52,53 but it only shows the performance at a specific temperature, which is not enough to represent a material's characteristics over the broad thermal boundary conditions used in a practical thermoelectric system operating environment. However, the efficiency prediction in eqn (2) based on the CPM is not reliable when Z is temperature-dependent. Because of the assumption of temperature independence in CPM, eqn (2) only correctly predicts the thermoelectric efficiency in limited materials that show a constant Z behavior or for materials with a small temperature gradient across the hot and cold side of the material.^18,54,55 CPM often overestimates a material's efficiency.^56,57S, ρ, and κ are temperature dependent in most thermoelectric materials, and the ZT profile is not linear with temperature.^{21,40,58–60} Additionally, thermoelectric materials operate at large temperature differences, so the efficiency prediction by eqn (2) is not reliable. To use eqn (2), the average Z can be obtained in two ways: (1) by integrating a temperature dependent Z with respect to temperature, , and (2) by taking a Z value corresponding to the average temperature, Z_Tavg = Z(T_avg). The Z_intT_avg and Z_TavgT_avg are called the average ZTs. Fig. 1a shows the temperature-dependent ZT and average ZTs of the p-type Ni-doped MgAgSb,⁶¹ and their corresponding efficiencies from eqn (2) shown in Fig. 1b match with the numerical simulation because Z vs. T is not must different from a constant (Fig. 1c). The numerical simulations were carried out based on the finite difference method to account for the temperature dependence of S, ρ, and κ over a large temperature difference between the cold and hot sides.^16,17 In Fig. 1d, however, the n-type In₄Se_3−x⁵⁷ has larger variations in the average ZTs depending on how they were averaged because of the strong temperature dependence of Z (Fig. 1f), which results in different efficiency predictions (Fig. 1e). Even though the peak ZT of In₄Se_3−x is higher than that of the Ni-doped MgAgSb, the average ZTs and the predicted efficiency are much lower, which means that the peak ZT value is not the correct indicator of a material's efficiency. In addition, any average ZT cannot be a correct index because of the lack of consistency from the following reasons: (1) average ZTs vary based on the averaging technique (Fig. 1a and d), (2) Z_Tavg is estimated to be larger than Z_int when the Z curve is in a convex upward shape (Fig. 1c) and vice versa (Fig. 1f), and (3) analytical prediction of some materials based on the average ZTs is far from the numerical analysis (Fig. 1e). For this reason, eqn (2) often misleads and gives rise to an impractically high efficiency prediction, and it is desirable to reliably predict the conversion efficiency of thermoelectric materials and module devices for a large temperature difference based on their temperature-dependent thermoelectric properties.

Fig. 1 ZT vs. T for Ni-doped MgAgSb⁶¹ (a) and In₄Se_3−x⁵⁷ (d), in which open symbols are measured data, and the black lines are fitted curves. The solid blue and dashed red lines are average ZTs by Z_intT_avg and Z_TavgT_avg, respectively. Efficiencies at T_c = 25 °C from a numerical simulation (solid diamonds) and CPM using integration (solid blue line) and average temperature (dashed red line) for Z_avg of Ni-doped MgAgSb (b) and In₄Se_3−x (e). Z vs. T for Ni-doped MgAgSb (c) and In₄Se_3−x (f). Reproduced from ref. 26 with permission from the National Academy of Sciences.

2.1. Efficiency by the cumulative temperature dependence (CTD) model

The CTD model takes into account the overall effect of the temperature dependence of S, ρ, κ, and τ for a large temperature difference, where τ is the Thomson coefficient defined as TdS(T)/dT. No assumption is required to specify the type of temperature dependence of S, ρ, κ, and τ. In addition, the practical fractions of Joule and Thomson heat returning to the hot end are evaluated. The maximum efficiency by the CTD model is expressed as:²⁶where (ZT)_eng is the engineering figure of merit, which is dependent on the temperature difference (ΔT), and is defined as:α_i, W_J, and W_T are a non-dimensional parameter and dimensionless weight factors of the Joule and Thomson heat, respectively.²⁶ Calculating W_J and W_T is essential to obtain a reliable prediction for the maximum efficiency from eqn (3). Since the cumulative effect of the temperature-dependence of ρ and τ is considered for a large temperature gradient in this model, the lumped approximation of 1/2 by the CPM,⁶²i.e., half of the Joule and Thomson heat to the hot side, is not reasonable anymore, and the practical Joule and Thomson heat influencing the hot side should be evaluated. In the CTD model, if ρ(T) and τ(T) are assumed to be temperature independent, the weight factors become 1/2, indicating a lumped fraction of half of the Joule and Thomson heat on the hot side. This means that W_J and W_T are a good general expression for the practical contribution of the Joule and Thomson heat. Moreover, eqn (3) is a generic expression that can be converted to the efficiency formula by the CPM (eqn (2)). Thus, the formula from the CTD model can be unrestrictedly applied to any case regardless of the temperature dependence of S, ρ, κ, and τ.

Fig. 2a shows the efficiency prediction for K doped PbTe_1−ySe_y⁵⁸ and SnSe⁴⁰ with the temperature boundary condition that T_h increases while T_c is kept at 50 °C. Fig. 2b shows how relatively accurate the efficiency calculations by the CPM and CTD models are compared with the efficiency from the numerical simulation (solid line), η_n, which is the most accurate. The calculation by CPM (eqn (2)) overestimates the efficiency for K doped PbTe_1−ySe_y and SnSe compared with the numerical prediction, and the efficiency from the CTD model (eqn (3)) is close to the value from the numerical simulation. Fig. 2c shows the efficiencies for half-Heusler (HH: Hf_x(ZrTi)_1−xCoSb_0.8Sn_0.2)¹⁸ and skutterudite (SKU: Ce_0.45Nd_0.45Fe_3.5Co_0.5Sb₁₂),⁵⁴ and Fig. 2d shows their accuracy with respect to η_n. The calculated efficiencies for both materials from the CPM and CTD models are in good agreement with those from the numerical simulation. The small variations in HH and SKU compared to K doped PbTe_1−ySe_y and SnSe are because of the linear-like behavior of S, ρ, and κ in HH and SKU based on temperature.

Fig. 2 (a) Predicted efficiencies as a function of ΔT and (b) relative accuracy with respect to the numerical results of K-PbTeSe (blue circles) and SnSe (red squares). (c) Predicted efficiency and (d) relative accuracy of HH (blue circles) and SKU (red squares). Reproduced from ref. 26 with permission from the National Academy of Sciences.

The accuracy is partially associated with the temperature dependent τ(T) representing the intensity of the Thomson heat (Fig. 3a). In order to examine the cumulative effect of the Thomson heat for a large temperature difference, the overall Thomson coefficient at a given thermal boundary condition is defined as , which is shown in Fig. 3b, and the positive values of τ_ΔT lead to an increased efficiency compared to that without the Thomson effect. Therefore, it is closer to the efficiency predicted by the numerical analysis. The reasons for the relative difference between the CTD model and numerical model are the linearized expression of dT/dx, and the different types of temperature dependence, i.e., a cumulative (or overall) effect in the CTD model and an instantaneous effect in the numerical model.

Fig. 3 (a) Calculated Thomson coefficient τ at each temperature and (b) the overall Thomson coefficient τ_ΔT as a function of ΔT at T_c = 50 °C. ΔT-Dependent weight factor of W_J for the Joule heat (c), and W_T for the Thomson heat (d), where T_c is fixed at 50 °C. Reproduced from ref. 26 with permission from the National Academy of Sciences.

Fig. 3c and d shows ΔT-dependent weight factors for the Joule and Thomson heats, respectively, based on the CTD model. In Fig. 3c, the W_J above 1/2 indicates that a higher fraction of the Joule heating returns to the hot side than that calculated by CPM because of the increased trend of ρ(T). In Fig. 3d, the W_T below 1/2 means that the Thomson heat on the heat flux at the hot end is smaller than the simply lumped fraction from the CPM since the dS/dT decreases.^18,54,58 The diverged W_T (inset in Fig. 3d) means τ_ΔT becomes zero (Fig. 3b), indicating that no overall effect of the Thomson heat at the temperature difference is considered even though the Thomson heat exists at each temperature.

2.2. Output power by the cumulative temperature dependence (CTD) model

The output power density, W m⁻², at the maximum efficiency, including the Thomson effect based on the CTD model, is expressed as:²⁶where the external load resistance is equal to the internal resistance of a thermoelectric leg, and (PF)_eng is the engineering power factor in the unit of W m⁻¹ K⁻¹, which is defined as:The output power density is dependent on the thermoelectric leg's length (L) as well as the material properties. Fig. 4a and b shows the output power densities of HH and SnSe according to ΔT at T_c = 50 °C, in which a cubic-shape thermoelectric leg (2 × 2 × 2 mm³) is considered for the prediction of the output power generation. For the output power density calculation by the CPM, the conventional power factor PF is calculated from a simply averaged S and ρ at a given temperature difference. The (PF)_eng based prediction of the power density for HH and SnSe has a better agreement with the numerical prediction than the conventional PF. In Fig. 4c, (PF)_eng shows a similar tendency to the power density by the numerical simulation (Fig. 4a and b), indicating an intrinsic power generation performance of a thermoelectric material, which has different trends from PF (Fig. 4d). Thus, (ZT)_eng and (PF)_eng enable one to directly compare the level of a material's performance at any operating temperature difference without the detailed calculations for efficiency and output power generation.

Fig. 4 Output power densities of (a) HH and (b) SnSe based on ΔT for a cubic-shape thermoelectric leg. (c) ΔT dependent (PF)_eng for HH and SnSe at T_c = 50 °C showing the same tendency for the power density P_d curves (inset). (d) Temperature dependent power factor PF for HH and SnSe.

The units of (PF)_eng from the CTD model are different from that of the conventional PF. By this definition, PF gives the power density output for a given T, where the Seebeck coefficient S in PF is measured at a certain temperature T with a very small ΔT (a few degrees) compared to T (hundreds T). So, PF represents the potential for output power at a temperature, not ΔT, i.e., there is no ΔT information (T_h and T_c) in the conventional PF. The concept of (PF)_eng is the total amount of potential for output power at a given ΔT (T_h and T_c), which is imposed by the integral. That is why the units of (PF)_eng (W m⁻¹ K⁻¹) are PF (W m⁻¹ K⁻²) times the temperature (K) that is the units of accumulation under ΔT condition.

3. Direct thermoelectric indicators

To describe the overall performance of a material in its operating temperature range, an average ZT (ZT)_avg of a material with given thermal boundary conditions has been reported.^61,63–65 However, the (ZT)_avg is insufficient because is varies depending on how it is averaged. Alternatively, an effective ZT (ZT)_eff has been used,^66–68 which is more reliable than the (ZT)_avg, but it does not account for the effect of a temperature difference and only indicates an equivalent value at an averaged temperature. The previous section showed that (ZT)_eng and (PF)_eng represent the cumulative performance of a material for given boundary temperature conditions. Fig. 5a shows the ZT for selected p-type and n-type thermoelectric materials. ZT is not a direct indicator of the thermoelectric conversion efficiency since it does not indicate which material performs better without additional complicated calculations or experimental results at practical operating temperatures. However, Fig. 5b shows the (ZT)_eng, which is a ΔT dependent indicator for efficiency at given temperature boundary conditions, for materials. In this section, the one-to-one correspondence between (ZT)_eng and the conversion efficiency of a material is discussed, and it is shown that the output power density has a linear relationship with (PF)_eng.

Fig. 5 (a) ZT dependence on temperature and (b) (ZT)_eng dependence on the ΔT of selected n-type and p-type materials. For (ZT)_eng, a cold side temperature T_c is assumed to be 50 °C for all materials. p:BiTe, unpublished data; p:MgAgSb, ref. 69; p:BiCuSeO, ref. 70; p:half-Heusler, ref. 71; n:InSe, ref. 57; n:AlZnO, ref. 72; n:LAST, ref. 73; n:SiGe, ref. 74.

3.1. (ZT)_engvs. efficiency

The efficiencies of the state-of-the-art p-type and n-type materials at their corresponding maximum operable temperature T_h,max are plotted in Fig. 6a in terms of their average ZT, , where T_c is fixed at 50 °C for all materials. The efficiency η_n is calculated based on a finite difference analysis,^16,17,49 which takes into account the temperature dependence of S, ρ, and κ as well as the Thomson effect. The p-type and n-type materials and their maximum operable temperatures are listed in Table 1. The scattered pattern in Fig. 6a shows that (ZT)_avg is not a direct indicator of a material's maximum efficiency. For example, p-type Bi₂Te₃ has a 30% lower efficiency with a 40% higher (ZT)_avg compared to BiCuSeO. Thus, (ZT)_avg is not adequate to represent a material's overall performance. Instead of (ZT)_avg, (ZT)_eff gives rise to proper characteristics, which are .⁶⁶Fig. 6b shows the efficiency corresponding to the (ZT)_eff of the materials. The relationship of η_nvs. (ZT)_eff still shows a scattered pattern, so (ZT)_eff is also not effective for reliable prediction of a material's efficiency under operable temperature conditions. Fig. 6c plots the efficiencies of the materials according to their (ZT)_eng. From the correlation between η_nvs. (ZT)_eng, the conversion efficiency of a material can be recognized by evaluating (ZT)_eng without efficiency calculations. For example, a (ZT)_eng of 0.6 represents about a 10% conversion efficiency regardless of the type of material, which makes it reasonable to compare the performances of different materials. This relationship has the benefit of taking into account each accumulated property based on ΔT rather than T_avg.

Fig. 6 The predicted efficiency of materials as a function of (a) (ZT)_avg, (b) (ZT)_eff, and (c) (ZT)_eng with all efficiencies evaluated by a numerical simulation accounting for the temperature dependence of each property. T_c is fixed at 50 °C for all efficiencies.

Table 1 The maximum operable temperature for p-type and n-type materials at the hot side

p-Type			n-Type
Material	T h,max (°C)	Ref.	Material	T h,max (°C)	Ref.
a Unpublished data.
Bi0.4Sb1.6Te3	250		Bi2Te2.7Se0.3S0.015	250
β-Zn4Sb3	250	75	Ba8Ga14Cu2Sn30	275	76
Ba8Ga15.75Cu0.25Sn30	275	76	AgPbmSbTem+2	400	73
MgAgSb	295	69	In4Se3−x	400	57
Pb0.98Te0.75Se0.25K0.02	450	58	Mg2Sn0.75Ge0.25	450	77
SnSe	450	40	Pb0.995SeCr0.005	450	78
MnSi1.78	500	79	Ba0.08La0.05Yb0.04Co4Sb12	550	80
Ce0.45Nd0.45Fe3.5Co0.5Sb12	550	54	Mg2Si-0.5 at% Sb/1.0 at% Zn	550	81
Hf0.19Zr0.76Ti0.048CoSb0.8Sn0.2	650	18	Hf0.25Zr0.75NiSn0.99Sb0.01	650	82
Bi0.875Ba0.125CuSeO	650	70	ZnO-0.25 at% Al	700	72
Si80Ge20B5	850	32	Si80Ge20P2	850	74

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3.2. (PF)_engvs. output power density

The thermoelectric leg length was determined to satisfy a constant temperature gradient of 200 °C mm⁻¹, e.g., 3 mm length of a material for ΔT = 600 °C. In Fig. 7a, the average power factor (PF)_avg has a scattered relationship to P_d by the numerical simulation, similar to η_nvs. (ZT)_avg, as shown in Fig. 6a. This shows that (PF)_avg is insufficient for directly indicating the output power performance of materials. In contrast, Fig. 7b illustrates the linear correlation between P_d and (PF)_eng under a constant temperature gradient. This indicates that a material's (PF)_eng represents its output power density for a given temperature condition by accounting for the cumulative effect of the temperature dependent S(T) and ρ(T). Fig. 7c shows the relative P_d and the ratio of P_d from a numerical computation to P^CTD_d by the CTD model, which explains that the variation comes from the fact that the CTD model accounts for path independence according to temperatures while the numerical model accounts for an instantaneous temperature dependence.

Fig. 7 The predicted output power density as a function of (PF)_avg (a) and (PF)_eng (b). The dashed lines in (b) represent P_dvs. (PF)_eng evaluated by the CTD model. (c) Relative P_d and a ratio of P_d from a numerical simulation to P_d by the CTD model. For all P_d computation, T_c is fixed at 50 °C.

4. Cumulative temperature dependence model for p–n module devices

The energy conversion efficiency of a typical thermoelectric module consisting of p-type and n-type materials connected thermally in parallel and electrically in a series is a function of the maximum device figure of merit [Z]* of a p–n pair, defined as:⁶²which is a function of S, ρ, and κ. The subscript p and n denote p-type and n-type materials, respectively. The square brackets and superscript asterisk indicate the p–n module and the maximum property, respectively. The maximum efficiency and output power density of a thermoelectric module have been analytically calculated based on the assumption that S, ρ, and κ are temperature independent as:⁶²This is reasonably accurate when the operating temperature difference ΔT is small, or when [Z]* is temperature independent over the operating temperature range, but most thermoelectric material properties are strongly temperature dependent. Hence, the conventional model by eqn (8) and (9) is not reliable for the prediction of the practical conversion efficiency of thermoelectric modules operating at large temperature differences between the hot and cold sides. As thermoelectric modules operate with a pairing of p-type and n-type materials, a balance of thermal flow and electric current for dissimilar p-type and n-type materials is also required. This section reviews the maximum efficiency of thermoelectric modules based on the CTD model, which leads to more reliable predictions than the CPM, the maximum device engineering figure of merit [ZT]_eng* and the maximum device engineering power factor density [PF]_eng,d* as direct indicators of a thermoelectric module at practical temperature differences. Two case studies of p–n pairings are shown: (1) chalcogenides: K_0.02Pb_0.98Te_0.75Se_0.25 (p-type)⁵⁸ and Pb_0.995SeCr_0.005 (n-type),⁷⁸ and (2) chalcogenides: SnSe (p-type)⁴⁰ and PbSe (n-type).⁷⁸

4.1. Maximum efficiency and output power density of p–n module by the CTD model

The maximum efficiency of a p–n module device by the CTD model and its corresponding [ZT]_eng* are defined as:²⁷where [α]_i is the non-dimensional parameter of the p–n module.²⁷ [η]_CTD* from eqn (10) is obtained by optimizing the device engineering figure of merit [ZT]_eng, in which each p-type and n-type material satisfies the geometry conditions for maximum efficiency. The maximum power density [P_d]_CTD* of a p–n module is defined as:²⁷where L = L_n = L_p. The maximum output power is assumed to be generated when the external load resistance is equal to the internal resistance, R_p + R_n.

[η]_CTD* for a p-PbTe/n-PbSe paired module is shown in Fig. 8a, and the prediction by CPM underestimates the efficiency over the whole ΔT range. The CTD model efficiency prediction is accurate over the whole temperature range compared to the simulation result. For the p-SnSe/n-PbSe combination shown in Fig. 8b, the CTD model predicts efficiency by a 3% relative difference at ΔT = 400 °C from the numerical analysis while the prediction by the CPM is a 17% overestimation. In some cases, however, the CPM analysis is more accurate than the CTD model,²⁷ but this is because the averaged material properties are accidentally close to each equivalent property. Thus, the analysis by the CPM is not reliable.

Fig. 8 The maximum efficiency as a function of ΔT for a p–n module of (a) p-PbTe/n-PbSe and (b) p-SnSe/n-PbSe. The maximum output power density as a function of ΔT for a p–n module of (c) p-PbTe/n-PbSe and (d) p-SnSe/n-PbSe. Reproduced from ref. 27 with permission from the American Institute of Physics.

The [P_d]_CPM* prediction for p-PbTe/n-PbSe (Fig. 8c) and p-SnSe/n-PbSe (Fig. 8d) shows a large relative difference at ΔT = 400 °C compared to the simulation results. Because the prediction by the CPM is only reliable when thermoelectric properties have a weak temperature dependence, such as SKUs^54,80 and HHs,^18,82 the CTD model leads to a more accurate prediction through the whole ΔT range (Fig. 8c and d) compared to the numerical simulation. The differences between the CTD model and numerical analysis could result from the type of p–n pairs, type of temperature dependence of each material, integration, simulation conditions and the assumption of a linear temperature profile. The differences are also caused by the fact that the CTD model accounts for the path-independence of material properties according to the temperature while the numerical analysis incorporates the path-dependence along with the temperature.

4.2. Direct indicator of the p–n module

For the p-PbTe/n-PbSe assembly, the simply averaged (ZT)_eng for the p-type and n-type materials is in good agreement with the [ZT]_eng* for their corresponding p–n module within a 1% relative difference, as shown in Fig. 9b, while the averaged (ZT)_eng of the p-SnSe/n-PbSe is overrated by 47% compared to the [ZT]_eng* of the module at ΔT = 400 °C, as shown in Fig. 9a. The module efficiency should be carefully predicted by simply averaging the ZTs of the p-type and n-type materials, which often overestimate the module efficiency (Fig. 9a). Fig. 9c shows the module efficiency [η]_CTD* for selected p–n configurations according to their corresponding [ZT]_eng*, showing a linear correlation similar to η_nvs. (ZT)_eng for the individual materials, as shown in Fig. 6c. From this correlation, a module with a [ZT]_eng* of 0.6 expects about 10% of the module efficiency regardless of the combination of the p-type and n-type materials or the operating temperature range. Fig. 9d shows the linear correlation between [P_d]_CTD* and [PF]_eng,d*, indicating that the [PF]_eng,d* of a p–n configuration represents the module's output power density at a given leg geometry, similar to P_dvs. (PF)_eng shown in Fig. 7b. Thus, [ZT]_eng* and [PF]_eng,d* directly represent a device's efficiency and output power density, respectively.

Fig. 9 Comparison of [ZT]_eng* (larger symbols), (ZT)_eng of individual p- (solid lines) and n-type (dashed lines) materials, and their averaged (ZT)_eng (smaller symbols) in (a) p-SnSe/n-PbSe and (b) p-PbTe/n-PbSe. (c) The predicted efficiencies of the module devices by the CTD model as a function of (ZT)_eng. T_c is fixed at 50 °C. Reference materials are Bi₂Te₃ of p-type,³⁷ HH of p-type¹⁸ and n-type,⁸² p-type PbTe,⁵⁸ n-type PbSe,⁷⁸ p-type SnSe,⁴⁰ SKU of p-type⁵⁴ and n-type,⁸⁰ SiGe of p-type³² and n-type,⁷⁴ and n-type Bi₂Te₃. Reproduced from ref. 27 with permission from the American Institute of Physics.

5. Engineering thermal conductivity κ_eng: balance between the thermal conductivity reduction of thermoelectric materials and device reliability

Because reducing the lattice thermal conductivity of thermoelectric materials is an effective strategy to improve ZT,^{31,33,35,37–40} one may have the following questions: is there a lower limit of thermal conductivity for a thermoelectric material? What is expected when the thermal conductivity continues to decrease? This section reviews how to figure this out by defining an engineering thermal conductivity (κ_eng) that is the allowable minimum thermal conductivity of a thermoelectric material to ensure a module device is thermomechanically reliable and thermoelectrically high-performing.⁴⁹κ_eng highlights the lower limit of the thermal conductivity by reconciling the reduction of thermal conductivity and device reliability in static operations, and it guides the sustainability of a thermoelectric material to transient thermal impacts. A methodology is presented to evaluate κ_eng for a steady state and transient operation. A case study for Bi₂Te₃ indicates that ZT improvement by reducing the thermal conductivity influences the device reliability, so how to improve the performance of thermoelectric materials in a reliable module device in a balanced way by considering κ_eng should be discussed.

5.1. κ _eng at steady-state operation

A reliable thermoelectric device at a steady-state operation requires an appropriate L range of thermoelectric legs to avoid failure at interfaces due to thermal shear stresses from shorter L, which is associated with a κ-independent L_min. A longer leg length gives rise to degradation of the output power generation because of the higher electrical resistance and material failure resulting from a temperature gradient larger than its maximum limit, which is associated with the κ-dependent L_max. The optimized L for the thermoelectric legs should be L_min < L < L_max to satisfy both device reliability and ZT improvement by reducing the thermal conductivity. Fig. 10 illustrates how to evaluate κ_eng based on a L_max and L_min analysis. Once materials of all the components, including bonding types for a module assembly, are determined, the thermal conductivity and allowable ΔT_max of thermoelectric legs are known. When the input heat flux (q_in) is given as a boundary condition, the κ-dependent L_max is determined. The explicit form of L_max is expressed as:⁴⁹where P̄_F is the averaged power factor of a thermoelectric material.

Fig. 10 Flow chart for κ_eng evaluation in a steady state mode.

L _min is determined based on the half width (w) of a thermoelectric leg, the maximum thermal shear stress (τ_max), and critical criteria (τ_Y). The maximum shear stress τ_max is expressed based on the 1-pair thermoelectric assembly model (Fig. 11) by Suhir et al.^83–85 as:

τ_∞ is the shear stress at the edge in an assembly of infinite w with continuous bonding,⁸⁵ and χ_r is a dimensionless reduction factor associated with the module design.⁴⁹f_ce is a fraction factor showing a ratio of cross-section areas of the total area to the thermoelectric legs. k is the interfacial shearing stress.^49,85 The critical criteria τ_Y can be the yield strength of solder or brazing materials, bonding strength at interfaces, or any types of material failure strength. For a conservative module design, the lowest value among the possible τ_Y should be selected. For a reliable design, the τ_max from eqn (14) needs to be less than the τ_Y, i.e., τ_max < τ_Y, and the L_min can be determined at τ_max = τ_Y. According to the L_max and L_min evaluation, if L_max < L_min, the κ_eng evaluation is not available, which means that this design is always vulnerable and the material geometry and/or bonding configuration should be modified. If L_max > L_min, L_max can decrease by reducing the thermal conductivity of a thermoelectric material for ZT improvement. Further reduction of κ makes L_max approach L_min. When L_max becomes equal to L_min, the corresponding κ is the κ_eng that is analytically expressed as:⁴⁹If L_max is still larger than L_min even though κ approaches zero, the thermoelectric module is robust enough for reducing the κ of a thermoelectric material with respect to the selected τ_Y. Thus, the κ_eng indicates the lower limit of the thermal conductivity by reconciling the material's tuning strategy (κ reduction) and device reliability.

Fig. 11 (a) Schematic illustration of the model for the shear stress evaluation. w is the half width of the leg, and D is the half width of the ceramic substrate in the 1-pair assembly. (b) An infinitely long assembly with a continuous bond. L_ce, L_el, and L_b are the lengths of the ceramic substrate, electrode, and solder/brazing layer, respectively.

5.2. κ _eng on transient operation

In some applications, thermoelectric devices operate under thermal boundary conditions that change over time and vary from place to place, such as waste heat recovery systems in automobiles or combustion-thermoelectric generators under irregular thermal fluctuations.⁸⁶ Such thermal loadings or impact at transient operations give rise to crack initiations or failure in thermoelectric materials because thermoelectric materials are mostly brittle. The failure of brittle ceramics by thermal loadings or shocks has been investigated through the thermal stress resistance parameter,^87–90 but not much attention has been paid to thermoelectric materials. For this issue, a homogeneous thermoelectric leg with an infinite width shown in Fig. 12 was examined,⁴⁹ and temperature-independent material properties were assumed at a constant heat transfer coefficient h with a mild thermal impact.⁸⁷ The maximum allowable temperature change ΔT_cr to avoid failure by a transient thermal loading is given as:⁹⁰where σ_u, ν, α, and E are the ultimate strength, Poisson's ratio, coefficient of thermal expansion (CTE), and Young's modulus of a thermoelectric material, respectively. The ratio of σ_u to E for a material usually has little variation because of their proportional relationship to each other, and α and h are the intrinsic material property and environmental factor of the heat exchangers or thermal boundaries, respectively, and the ratio of κ to L is only susceptible to the ΔT_cr against the transient thermal loadings. For a thermoelectric operation, ΔT_cr should exceed the ΔT_max to avoid immediate failure of a thermoelectric material. Based on eqn (16) and the condition of ΔT_cr > ΔT_max, the engineering thermal conductivity κ_eng in a transient state when ΔT_cr = ΔT_max at L = L_min is defined as:⁴⁹which is the minimum thermal conductivity of a thermoelectric material allowing for ΔT_max without failure from transient thermal boundary conditions. In this case, the κ_eng at a transient operation is evaluated using only the thermoelectric material, not the whole module assembly, but this indicates a material's intrinsic reliability against a dynamic thermal loading as a pertinent material property. Thus, the κ_eng at the transient mode is a type of design factor against failure depending on the dynamic thermal boundary conditions.

Fig. 12 Thermoelectric leg with infinite width for κ_eng evaluation in a transient mode.

5.3. Case study: Bi₂Te₃ based module

5.3.1. L _min evaluation. The ratio of τ_Y to τ_∞ and the shear stress reduction factor χ_r of a Bi₂Te₃ based module is shown in Fig. 13a as a function of L/L_tot, and the thickness of the ceramic plates, electrodes, and solder layers are 0.7, 0.5, and 0.07 mm, respectively, and f_ce = 2.86 (35% filling factor). The mechanical properties of each component are listed in Table 2. The τ_Y can be the yield strength of a solder layer ranging from 22.8 to 27.8 MPa at room temperature for various solder alloys applicable to the Bi₂Te₃ based module assembly,⁹¹ the fracture strength of Bi₂Te₃, such as 34 MPa ultimate tensile strength at room temperature,⁹² or the bonding strength of the Ni metallization – Bi₂Te₃ leg configuration, which was measured up to 30 MPa at room temperature.⁹³ For a conservative design of the module, the yield strength (lower value of τ_Y) of the solder alloys leads to L_min of the thermoelectric leg with a cross sectional area of 1.6 × 1.6 mm² (w = 0.8 mm),⁶⁸ and L_min should be longer than 0.77 mm under the maximum ΔT = 150 °C at T_c = 50 °C when τ_Y = 25.7 MPa. The critical w–L_min relationship for the thermomechanical reliability is shown in Fig. 13b, where it is reliable when the selected w and L are placed in the upper zone of the border. If not, the interfacial structure in the module is vulnerable at the operating condition.

Fig. 13 (a) The τ_Y/τ_∞ and reduction factor χ_r of the shear stress as a function of non-dimensional length, L/L_tot. The various yield strengths of solders (23–28 MPa) are shown as error bars. (b) The w vs. L/L_tot relationship to avoid the yield criteria of the solder. (c) The hot side temperature with L according to [small kappa, Greek, macron]. The horizontal line indicates T_h,max. The solid and dashed curves indicate the numerical and analytical analyses, respectively, where [small kappa, Greek, macron]₁ = 1.13 and [small kappa, Greek, macron]₂ = 2.13 W m⁻¹ K⁻¹. (d) The L_max associated with [small kappa, Greek, macron] at a given thermal boundary. (e) The κ_eng at a steady state with L_min. (f) The κ_eng at a transient mode as a function of h. (A) h = 50 (forced convection by air), (B) h = 100 (natural convection by water), and (C) h = 6000 (forced convection by water) W m⁻² K⁻¹. Reproduced from ref. 49 with permission from the Wiley-VCH.

Table 2 Bi₂Te₃ based module: mechanical properties of each component as input data

Component	Young's modulus (GPa)	Poisson's ratio	CTE (×10^−6 °C^−1)	Yield strength at 25 °C (MPa)
AlN plate^94	320	0.237	4.2	—
Cu electrode^91	115	0.31	17	—
Sn–Sb solder^84,91	44.5	0.33	27	25.7
p-Type Bi2Te3^44,47	47.9	0.4	16.8	112

---

5.3.2. L _max evaluation. To figure out the hot side temperature of the thermoelectric leg when the input heat flux q_h is given, a numerical simulation by a finite difference method was carried out based on a realistic assembly model accounting for the temperature dependence of the material properties. Fig. 13c shows the L-dependent hot side temperature of the Bi_0.4Sb_1.6Te₃ leg according to the leg's average thermal conductivity where T_h,max = 200 °C. In Fig. 13d, L_max is 1.15 mm by the numerical analysis and 1.33 mm by eqn (13). If the thermal conductivity is further reduced from [small kappa, Greek, macron] = 1.13 W m⁻¹ K⁻¹ to [small kappa, Greek, macron] = 0.6 W m⁻¹ K⁻¹, L_max becomes 0.68 mm and 0.95 mm by the numerical and analytical models, respectively, where q_h is assumed to be 20 W cm⁻², which is in a typical heat flux density range (up to 40 W cm⁻²) for most thermoelectric applications.⁹⁵ The L_max according to [small kappa, Greek, macron] by the analytical model (eqn (13)) outweighs the L_max by the numerical analysis over the whole [small kappa, Greek, macron] range. This is mainly because of the absence of the Thomson heat as well as the assumption of the temperature independence of S, ρ and κ in the analytical model. This leads to an underrated κ_eng in a steady-state mode by the analytical model as 0.4 W m⁻¹ K⁻¹ at L_min = 0.77 mm while the κ_eng evaluated by the numerical analysis is more conservative with a larger κ_eng (0.7 W m⁻¹ K⁻¹ at L_min = 0.77 mm) as shown in Fig. 13e. According to the L_min of 0.77 mm, its corresponding κ_eng in the transient mode was 0.08, 0.15, and 9 W m⁻¹ K⁻¹ by eqn (17) at h = 50, 100, and 6000 W m⁻² K⁻¹, respectively (Fig. 13f).

5.3.3. κ _eng of Bi₂Te₃ materials. The thermal conductivities of some of state-of-the-art, p-type Bi₂Te₃ materials have been reported as 0.5–1.7 W m⁻¹ K⁻¹ for ZT ∼ 1.8.^{31,37,96–98}Fig. 14 shows the [small kappa, Greek, macron] for recent p-type Bi₂Te₃, and it is close to and/or below the κ_eng at steady state. This indicates that there is not much room for thermal conductivity reduction for ZT improvement. As discussed, the reduction of thermal conductivity may cause an immediate fracture of a material when it is below the transient κ_eng, and the failure of the interfacial structure or degradation of the output power occurs when the thermal conductivity is lower than the steady state κ_eng. Thus, the strategy of reducing thermal conductivity for ZT enhancement should be carefully addressed by considering the mechanical reliability issue.

Fig. 14 The averaged thermal conductivities of the state-of-the-art, p-type Bi₂Te₃ compared with the κ_eng of the p-type Bi_0.4Sb_1.6Te₃ for the steady state (solid line) and transient mode (dashed line). The measured property data of the reference materials are from ref. A,⁹⁶ ref. B,⁹⁷ ref. C,⁹⁸ and ref. D.³⁷ Reproduced from ref. 49 with permission from the Wiley-VCH.

In general, regarding the approach to maximize the efficiency of a thermoelectric generator by maximizing the material ZT, the strategy depends on the system environment, target applications, generator design, etc. Assuming that no forced cooling is applied on the cold side, reducing the thermal conductivity can be the first priority for the material tuning process with larger ΔT (broad operating temperature) needs for a thermoelectric generator. Once a reasonably broad thermal boundary condition can be achieved, one can further reduce the thermal conductivity for more efficient systems. If the output power generation is more of a concern than the system efficiency, it is better to increase the power factor. For example:

(a) by reducing the electric resistivity: if a target area is limited or the thermoelectric materials are lightweight, a larger number of thermoelectric legs can be imbedded into a generator system, i.e., a higher filling factor is required or allowed. Such a design demands a high aspect ratio for leg length to width to avoid mechanical failure by trading off an optimum ratio and thermomechanical reliability even though the length and width are interrelated. In this design, the total resistance of a system should be minimized by decreasing the material's electric resistivity while an open circuit voltage is proportional to the number of thermoelectric legs.

(b) by increasing the Seebeck coefficient: if there is no limit on a target area or heavy thermoelectric materials are selected, a lower filling factor is allowed. Because a fewer number of legs are imbedded, the design decreases the device resistance, but requires a larger open circuit voltage because of the lack of legs, so an increase in the Seebeck coefficient is better to maximize ZT for a higher system efficiency.

As discussed in this section, if the thermal conductivity is saturated or below the κ_eng like Bi₂Te₃, increasing the power factor is a better material tuning strategy since the power factor does not affect the thermomechanical reliability. Additionally, tuning the temperature dependence of material properties is another concern in maximizing ZT according to the CTD model.

5.4. Engineering thermal conductivity in practical design

So far, κ_eng is evaluated based on the ideal interface bonding with respect to the yield and/or bonding strength measured at room temperature. The yield strength of the solder material is generally temperature dependent. For example, the yield strength of Sn–Pb is reduced to 75% from room temperature to 130 °C,⁹⁹ and Sn–Ag–Cu has a 67% reduction from room temperature to 160 °C.¹⁰⁰ Such temperature dependent yield strength results in an increasing κ_eng in the steady-state mode since L_min associated with a lower yield strength at higher temperatures becomes larger. Another important issue for the practical design is an actual interfacial structure. Usually, semiconducting thermoelectric materials cannot be directly bonded onto metal electrodes,⁶³ so thin metallization layers are introduced. In addition, a diffusion barrier is often required at the interface of the high temperature side to avoid the unwanted chemical reaction that causes a high contact resistance¹⁰¹ and degrades the mechanical properties of the joint.¹⁰² For Bi₂Te₃ materials, the bonding strength of the interface varies from 10 to 30 MPa at room temperature depending on the Ni metallization methods, such as sputtering, electroplating, and hot pressing.¹⁰¹ The weak bonding strength requires a high aspect ratio (L ≫ w) of a leg, resulting in a larger κ_eng. A secondary phase formed by the chemical reaction between the leg and metal alloy exhibits different characteristics, which may also weaken the bonding strength. Barako et al. demonstrated cracking at the hot side interface of a Bi₂Te₃ based commercial module under a thermoelectric generator mode.¹⁰³ This might be because the κ_eng associated with the module's bonding and/or yielding criteria is higher than the thermoelectric materials' thermal conductivity.

The device reliability of thermoelectric power generators depends on the yield or bonding strength of the interfacial structures as well as the thermal conductivity of the thermoelectric leg to ensure the mechanical stability and performance in a balanced way. Beyond the lower thermal conductivity of thermoelectric materials for higher efficiency and output power generation, the higher mechanical strength of interfacial structures and the study of mechanical characteristics in thermoelectric materials should be entailed. Besides the engineering thermal conductivity, the device reliability is also improved by the appropriate mechanical properties of each component. However, tuning the mechanical properties of a thermoelectric leg and/or interfacial structures without degrading the thermal and electrical properties is a challenge, and simultaneous achievement of a high bonding strength and low electric contact resistance at the metal/thermoelectric interface without deteriorating thermoelectric performance is key for balanced improvement of the reliability and output performance of a device.

6. Summary and outlook

We have reviewed the CTD model by examining the reliable prediction for the thermoelectric performance from material to device level and investigated direct thermoelectric indicators representing the efficiency and output power of a p–n pair module as well as homogeneous materials. Additionally, the methodology of how to balance between the material tuning strategy and device reliability through κ_eng was summarized. In the CTD model, (ZT)_eng and (PF)_eng are associated with the overall effect of thermoelectric materials on the temperature difference, and the corresponding efficiency prediction incorporates the corrected contribution of the Joule and Thomson heats. This efficiency formula is a generic expression applied without restrictions no matter whether the material properties are strongly temperature dependent or assumed to be temperature independent. Beyond the conventional ZT and PF, (ZT)_eng and (PF)_eng as direct performance indicators have the advantage that one can immediately expect the efficiency and output power density of a material at a given thermal boundary condition once (ZT)_eng and (PF)_eng are known, respectively, regardless of what the material system is and their temperature differences. In addition, the use of the efficiency vs. (ZT)_eng and power density vs. (PF)_eng relationships enables one to avoid misdirected thermoelectric material exploration. It is noted that reducing the thermal conductivity for ZT improvement should be carefully carried out for some thermoelectric materials as the thermal conductivity influences the device reliability. Based on the methodology, further reduction of the thermal conductivity of some materials without considering the mechanical characteristics of the interfacial structures in a module device would be a futile attempt, and it makes the device fragile.

For the last few decades, vast efforts have been devoted to developing new thermoelectric materials and improving the thermoelectric performance of the materials, but the research on design, fabrication, and demonstration of thermoelectric module devices is still in a relatively early stage compared to the material research. Such a gap in the technology between the material and device hinders the use of promising thermoelectric devices as primary power sources, especially in the intermediate and high temperature range. Beyond the performance improvement of thermoelectric materials, studies on the device point of view should be actively addressed to investigate the mechanical properties, electrode materials, thermal/electrical contact resistance, interface structures, and module packaging for a thermomechanically reliable and thermoelectrically high-performance design.

Acknowledgements

This work was partially supported by the U.S. Department of Energy under Contract No. DE-SC0010831 (It is not possible to indicate the exact parts since it is a perspective article.) and also partially by “Solid State Solar Thermal Energy Conversion Center (S³TEC)”, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Science under award number DE-SC0001299 (It is not possible to indicate the exact parts since it is a perspective article).

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