Segmentation of thermoelectric (TE) materials is a widely used solution to improve the efficiency of thermoelectric generators over a wide working temperature range. However, the improvement can only be obtained with appropriate material selections. In this work, we provide an overview of the theoretical efficiency of the best performing unicouples designed from segmenting the state-of-the-art TE materials. The efficiencies are evaluated using a 1D numerical model which includes all thermoelectric effects, heat conduction, Joule effects and temperature dependent material properties, but neglects contact resistance and heat losses. The calculations are performed for a fixed cold side temperature of 300 K and different hot side temperatures of 700, 900, and 1100 K. We confirm that without taking into account the compatibility of TE materials, segmentation can even decrease the total efficiency. Choosing the TE materials carefully, one is, however, rewarded by a significant improvement in the total efficiency.

**physica status solidi (a)**

# Towards high efficiency segmented thermoelectric unicouples

## Abstract

## 1 Introduction

A large amount of thermal energy associated with many industrial and fuel combustion processes is available as waste heat. Waste heat recovery provides an opportunity to significantly improve the overall energy efficiency of many industrial processes. One approach for recovering energy from the system is to generate electrical power through thermoelectric (TE) conversion using thermoelectric materials. Studies on thermoelectric materials have been flourishing recently, resulting in a large variety of materials with figure of merit *zT* = *α*^{2}*T*/*ρκ* (with *α*, *κ*, and *ρ* being the Seebeck coefficient, thermal conductivity and electrical resistivity, respectively) exceeding 1, such as bismuth tellurides [1-3], skutterudites [4-6], Zinlt phases [7-12], lead tellurides [13-15], silicon germanium [16, 17], zinc antimony [18], copper selenide [19], Cu–Se derivatives [20-23], (AgSbTe)_{0.15}(GeTe)_{0.85} (TAGS) [24], AgPb_{m}SbTe_{2+m} (LAST) [25], lanthanum telluride [26], and CuGaTe_{2} [27]. Together with the development of high performance TE materials, TE device fabrication is also a growing area [28]. To achieve high thermal-to-electrical energy conversion efficiencies, it is desirable to operate thermoelectric generator devices over large temperature spans and to maximize the thermoelectric performance of the materials. However, no single thermoelectric material possesses a high performance over large temperature intervals. Therefore, it is necessary to combine different materials, which operate optimally in different temperature ranges. This can be achieved by, e.g. cascaded or segmented generators where the p- and n-type legs are formed from different materials joined in series [29-32].

A cascaded system consists of different stages with each stage comprised of p- and n-type legs made from single p- and n-type TE materials. The stages have their own independent electrical circuit. As a consequence, the optimum load resistance for each stage can be achieved individually, however, with the cost of both needing to do several load resistance optimizations and suffering from extra heat losses through the additional wires connecting each stage. In a segmented system, each p- and n-type leg is subdivided into segments made from different materials. Unlike cascaded systems, segmented systems use only a single electrical circuit, but here a high figure of merit *z* is not the only prime importance. Selecting compatible materials for combination is also a critical issue for the optimal performance. An example of an ineffective segmentation is the combination of TAGS with SiGe which – despite a high figure of merit of each material - results in a decrease of the overall efficiency (9.89%) when segmented as compared to a non-segmented leg of TAGS (10.45%) subjected to the same temperature span. Despite the importance of the material compatibility and that the necessary theoretical framework has already been developed [1, 33, 34], so far the compatibility factors of only these materials have been reported: TAGS, p- and n-type PbTe, SnTe, CeFe_{4}Sb_{12}, CoSb_{3}, La_{2}Te_{3}, p- and n-type SiGe [35]. In this work, we broaden the materials of consideration, show how to select materials for high performing segmented systems and evaluate how efficient the best segmented TE elements and unicouples are expected to be. The evaluation of efficiency was carried out using a 1D numerical method and the materials used in this study are state-of-the-art TE materials reported in the literature.

## 2 Calculation model

The efficiency of segmented and non-segmented TE elements is calculated using the 1D model described in details by Snyder [1]. This method takes into account all thermoelectric effects (the Peltier, Seebeck, and Thomson effect), heat conduction, Joule heating and the temperature dependence of the resistivity *ρ*(*T*), thermal conductivity *κ*(*T*) and Seebeck coefficient *α*(*T*). Using these material properties together with the hot (*T*_{h}) and cold (*T*_{c}) side temperatures and the expected heat flux from the heat source, a large number of properties can be calculated for a single leg, unicouple or module. These properties include the efficiency, the output power, the optimal load resistance and the optimal geometry (cross sectional area ratio of p- and n-TE legs and length of the legs).

Here, we use the 1D model to evaluate the efficiency of segmented and non-segmented TE elements under the optimal conditions, i.e. we disregard heat losses, thermal and electrical contact resistances and only report the efficiency for the optimal geometry and load resistance. The calculations can therefore be regarded as an upper limit for the actual efficiency obtainable in experiments. Under these optimal conditions, the efficiency solely depends on the hot and cold side temperatures and the selected TE materials; the expected flux from the heat source needs not to be specified. The 1D model seeks to define and use intrinsic variables that are independent on the system size. Rather than using the electrical current (*I*) as a fundamental variable, the reduced current density *u* *=* *J*/*κ▿T* is defined as the ratio between the current density (*J*) and the heat flux by conduction. The reduced current density is both independent on the length and cross-sectional area of the leg, and thus merely a function of the material parameters and the temperature along the leg. When *u* is defined at a single point – e.g. at the hot side, *u*(*T* *=* *T*_{h}) – the value of *u* at any temperature along the leg is fixed by the differential equation

with an approximate recursive solution given elsewhere [1]. Note that the properties in the model (such as *u*) has a direct reference to the temperature along the leg rather than a spatial coordinate *x*, conversion to the spatial coordinate can be made by calculating the temperature profile *T*(*x*). The differential equation is valid for both non-segmented and segmented legs if – for the latter – the material properties are changed from one material to another at a certain interface temperature (*T*_{i}) where the segments meet.

From the reduced current densities evaluated at the cold and hot side temperatures, *u*_{c}* = u*(*T* *=* *T*_{c}) and *u*_{h}* = u*(*T* *=* *T*_{h}), the efficiency of a segmented or non-segmented single leg can be found as

The value of *u*_{h} is a free parameter varied in the model to get the highest efficiency. In practice, *u*_{h} is varied by changing the load resistance. When an n-type leg and a p-type leg are connected electrically in series and thermally in parallel to form a unicouple, the efficiency is calculated by

where the subscripts n and p denote the n- and p-type legs. Numerically, both *u*_{p,h} and *u*_{n,h} are optimized simultaneously. Since *u*_{p,h} and *u*_{n,h} are mutually dependent by current conservation *I*_{p} *=* *A*_{p} *J*_{p} =* A*_{n}*J*_{n} = *I*_{n} (with *A*_{p} and *A*_{n} being the cross-sectional areas of p- and n-type legs), this is done in practice by optimizing both the load resistance and the area ratio *A*_{p}/*A*_{n} (see Ref. [1]).

As the profile of the reduced current density along the leg (*u*(*T*)) is fixed by Eq. (1), it will generally differ from the optimal reduced current density called the compatibility factor (*s*):

For the optimal case where *u*(*T*) = *s*(*T*) at all temperatures, the maximum efficiency for a single leg (*η*_{u = s}) is calculated as

## 3 Results and discussion

### 3.1 Compatibility factor: proof-of-concept

The features of the calculation model are described in Fig. 1, which consider an example of a segmented leg consisting of Bi_{0.6}Sb_{1.4}Te_{3}, Ba_{8}Au_{5.3}Ge_{40.7}, and MnSi_{2} that is subjected to *T*_{c} = 300 K and *T*_{h} = 900 K. The interface temperature is 524 K between Bi_{0.6}Sb_{1.4}Te_{3} and Ba_{8}Au_{5.3}Ge_{40.7} and 772 K between Ba_{8}Au_{5.3}Ge_{40.7} and MnSi_{2}. Figure 1a demonstrates an example of the profiles of optimum relative current density *u*_{segmented} (Eq. (1)) and efficiency *η*_{leg} (Eq. (2)) along the segmented leg. The accumulated efficiency at any temperature *T'* is found from Eq. (2) by replacing *T*_{h} with *T*', *α* with *α* (*T = T*') and u_{h} with *u*_{segmented} (*T = T'*). *η*_{leg} builds up along the Bi_{0.6}Sb_{1.4}Te_{3} and Ba_{8}Au_{5.3}Ge_{40.7} segments, and starts to decrease from MnSi_{2} part. Comparing the almost constant relative current density with the widely varying compatibility factors of each segment (see Fig. 1a) reveals that each segment operates far from their individual optimal conditions even though the whole segmented leg has been optimized for efficiency. The total efficiency of the entire leg is *η*_{leg }= 12.9%. If *u* were not constrained by Eq. (1), its best performing value is the compatibility factor *s*. This would result in the efficiency *η*_{u=s} = 18.3%, which is the maximum obtainable efficiency as calculated from Eq. (5) with the replacement *T = T'*. The difference between the actual efficiency and the maximum obtainable efficiency diverges mainly in Bi_{0.6}Sb_{1.4}Te_{3} and MnSi_{2}. In particular, the decrease in the overall efficiency by adding the MnSi_{2} segment confirms the importance of choosing compatible materials as even replacing the incompatible MnSi_{2} with a passive material (thermal insulator) will result in a better performance. Figure 1b explains in more detail the incompatibility of the three materials. The optimum relative current density for each segment – when working as a single material leg – is 5.7, 4.6, and 1.2 V^{−1} for Bi_{0.6}Sb_{1.4}Te_{3}, Ba_{8}Au_{5.3}Ge_{40.7} and MnSi_{2}, respectively. Putting them into a segmented leg, with the cold and hot end fixed at 300 and 900 K, the segmented leg as a whole reaches its highest efficiency at *u*_{h} = 3.01 V^{−1}. From the profile of optimized *u*_{segmented} along the leg (Fig. 1a), this corresponds to the *u*_{h} of 2.61, 2.59, and 3.01 V^{−1} for the Bi_{0.6}Sb_{1.4}Te_{3}, Ba_{8}Au_{5.3}Ge_{40.7}, and MnSi_{2} segments, respectively (highlighted as (1), (2), and (3) in the figure). At these reduced current densities, the efficiencies are 7.77% for the Bi_{0.6}Sb_{1.4}Te_{3} segment, 6.16% for Ba_{8}Au_{5.3}Ge_{40.7} and −0.65% for MnSi_{2}. That is, albeit the Bi_{0.6}Sb_{1.4}Te_{3} and Ba_{8}Au_{5.3}Ge_{40.7} segments are not working at their optimal conditions, they both contribute to a positive efficiency, while the MnSi_{2} segment consumes power in order to achieve the required current density in the segment.

### 3.2 Materials in consideration and their TE efficencies

In order to study the possible combinations of segmented TE couples, the state-of-the-art n- and p-type TE materials have been considered. Thermal conductivity, Seebeck coefficient and resistivity as functions of temperature were collected from the literature. Note that these properties are not shown here. Figure 2 shows the dimensionless figure of merit *zT* and the compatibility factors as functions of temperature for both p- and n-type materials. It is obvious from Fig. 2a and c, that below 500 K bismuth tellurides (both n- and p-type) are hitherto the best performing materials. The majority of the materials such as PbTe, Zn_{4}Sb_{3}, TAGS, skutterudites, Cu_{2}Se, half-Heusler alloys show high *zT* at a medium-high temperature range of 500–1000 K. LAST is also a competitive material, with a reported *zT* of 2.2 at 800 K [25]. However, this high *zT* material so far has not yet been reproduced due to the difficulties in controlling its microstructure [36, 37]. To avoid a false impression, we do not include this reported data in the present calculation. At the higher temperature range (1000–1200 K) the Zinlt (p-type), lanthanum telluride (n-type) and silicon germanium (p- and n-type) are the best available materials (without including some possible high temperature oxides such as ZnO:Al,Ga [38] and Ca_{3}Co_{4}O_{9} [39]).

Most of the collected materials are compatible with each other over a large temperature range. Both p-type and n-type bismuth tellurides, however, are not compatible with the other materials at the low temperature region (300–400 K) but become more compatible at higher temperatures (Fig. 2b–d).

The optimized efficiencies of the collected materials at different *T*_{h} = 700, 900, and 1100 K and *T*_{c} at 300 K were calculated using Eq. (2), and the result is presented in Table 1. p- and n-type bismuth tellurides have efficiencies of 9.9% (300–525 K) and 8.3% (300–580K), respectively. At 700 K, although PbTe–SrTe and PbTe have comparable *zT* values, the efficiency of PbTe (11.1%) over the working temperature range of 300–700 K wins that of PbTe–SrTe (9.5%). This is due to the fact that over the entire temperature range of 300–700 K, the average *zT* of PbTe ( = 1.18) is higher than that of PbTe-SrTe ( = 0.97). However, PbTe–SrTe has been reported to stand higher temperatures, and achieves the highest efficiency of 15.3% among all the selected materials at the working temperature range of 300–900 K. For the n-type materials in this temperature range, the half-Heusler alloy (Ti_{0.5}(Zr_{0.5}Hf_{0.5})_{0.5}NiSn_{0.998}Sb_{0.002} [40]) shows the highest efficiency, 13.8%. Zinlt, SiGe, Si_{0.78}Ge_{0.22}, and La_{3}Te_{4} can be used at high temperature of 1100 K, where the Carnot efficiency is significantly increased, but their calculated efficiency values are just within 10–11% resulting from their good performance only at high temperatures (>800 K). Therefore, if it is not the case of segmentation one should use, e.g. PbTe–SrTe joined with an insulator to bring down the hot side temperature, and reducing the capital cost of the TE devices.

materials | efficiency (%) | ||
---|---|---|---|

T_{h} = 700 K | T_{h} = 900 K | T_{h} = 1100 K | |

- aMaterial data is provided by Gerald Jeffrey Snyder, California Institute of Technology, USA.
| |||

p-type | |||

Bi_{0.6}Sb_{1.4}Te_{3} [3] | – | – | _{–} |

NdFe_{3.5}Co_{0.5}Sb_{12} [5] | 8.9 | – | – |

Hf_{0.5}Zr_{0.5}CoSb_{0.8}Sn_{0.2} [41] | 5.8 | 8.9 | |

Yb_{14}Mn_{0.2}Al_{0.8}Sb_{11} (Zinlt) [7] | 4.5 | 7.6 | 10.8 |

Zn_{4}Sb_{3} [18] | 10.4 | – | – |

Cu_{2}Se [19] | 6.2 | 9.8 | – |

PbTe [13] | 11.1 | – | – |

PbTe-SrTe [14] | 9.5 | 15.3 | – |

MoSb_{5.4}Te_{1.6} [42] | 2.1 | 4.2 | – |

YbCd_{1.6}Zn_{0.4}Sb_{2} [43] | 8.3 | – | – |

EuZn_{2}Sb_{2} [44] | 8.1 | – | – |

SiGe [16] | 5 | 7.8 | 10.4 |

(AgSbTe)_{0.15}(GeTe)_{0.85} [25] | 10.2 | – | – |

Ba_{8}Au_{5.3}Ge_{40.7} [45] | 10 | – | – |

MnSi_{2} [46] | 4.5 | 6.7 | – |

n-type | |||

Bi_{2}Te_{3} [4] | – | – | – |

Ba_{0.08}La_{0.05}Yb_{0.04}Co_{4}Sb_{12} [5] | 11.1 | – | – |

Ti_{0.5}(Zr_{0.5}Hf_{0.5})_{0.5}NiSn_{0.998}Sb_{0.002} [40] | 7.7 | 13.8 | – |

CoSb_{3}a | 6.2 | 9.6 | – |

PbTe_{1−x}I_{x} (x = 0.0012) [15] | 10.1 | – | – |

Si_{0.78}Ge_{0.22} [17] | 5.8 | 9.5 | – |

Ba_{8}Ga_{16}Ge_{30} [47] | 3.6 | 6.7 | – |

Mg_{2}SiSn [46] | 9.4 | – | – |

La_{3}Te_{4} [26] | 4.3 | 7.2 | 10.0 |

### 3.3 Segmented legs

Figure 3 demonstrates the efficiencies of non-segmented p*-* and n-type legs and legs with 2- or 3 segments with different hot side temperatures at *T*_{h} = 700, 900, and 1100 K. The listed TE materials given in Fig. 2 were paired with the bismuth tellurides to form segmented TE legs. For 3-segment legs, only the ones with highest efficiencies are plotted. The gain in efficiency obtained by segmentation is also illustrated. In general, segmentation of most p-type materials gives a higher efficiency boost than n-type materials. Among n-legs, segmented n-Clathrate leg with hot side at 700 K gives a significant improvement in efficiency of more than 50% compared to the non-segmented n-Clathrate one. PbTe-I, n-HH (half-Heusler), and n-Skutt. (Skutterudite) with their high *zT* are the three most efficient n-TE legs with efficiencies above 10% at 700 K (12% for segmented n-HH leg at 900 K). Among p-type segmented legs, Mo_{3}Sb_{5.4}Te_{1.6} exhibits the highest increment in efficiency, i.e. from 2.1 to 9.6%. Strikingly, segmentation of PbSrTe with Bi_{0.6}Sb_{1.4}Te_{3} provides an outstanding efficiency of over 19%. Yet, a better improved efficiency can also be obtained by segmentation of three materials of, e.g. Bi_{0.6}Sb_{1.4}Te_{3} [2] with Ba_{8}Au_{5.3}Ge_{40.7} [47] and PbSrTe [13]. Remarkably, an efficiency value of over 20% can be obtained for the segmented leg of Bi_{0.6}Sb_{1.4}Te_{3}/PbSrTe with Zinlt [45] or SiGe [15] for applications at temperature range of 300–1100 K, whereas the segmentation of Zinlt or SiGe with Bi_{0.6}Sb_{1.4}Te_{3} only results in an efficiency of ∼15%.

### 3.4 Segmented unicouples

The p- and n-type segmented TE legs were then paired together to make TE unicouples whose efficiencies are calculated by using Eq. (3) and the maximum values are presented in Table 2. The legs were chosen in correspondence with their working temperature ranges (300–700, 900, or 1100 K). At hot side temperatures of 900 and 1100 K, besides 2-segment TE legs, only 3-segmented TE legs with the highest efficiencies were considered for unicouples. At *T*_{hot} = 700 K, with the same n-leg, the unicouples with Bi_{0.6}Sb_{1.4}Te_{3} + TAGS p-legs give the highest efficiencies. Similarly, unicouples with Bi_{2}Te_{3} + PbTe n-segmented legs give the best performance among the segmented n-legs. The unicouple paired from segmented Bi_{0.6}Sb_{1.4}Te_{3} + TAGS and Bi_{2}Te_{3} + PbTe legs has the efficiency of 13.5%, which is the top value computed in 300–700 K range. For the same temperature range, the unicouple with non-segmented p-TAGS leg and n-PbTe leg gives an efficiency of 10.2% (not shown in Table 2), which can be improved by about 30% by the segmentation of them with Bi_{0.6}Sb_{1.4}Te_{3} and Bi_{2}Te_{3} to form segmented legs. Increasing the hot side temperature up to 900 K, for most cases, although segmentations of two materials can enhance the performance of the unicouple, the improvement is generally small. One example is: 2-segment unicouple of p-leg Bi_{0.6}Sb_{1.4}Te_{3} + PbSrTe and n-leg Bi_{2}Te_{3} + n-Clathrate possesses 15.2% efficiency, while the most efficient 3-segment unicouple of (p-leg Bi_{0.6}Sb_{1.4}Te_{3} + p-Clathrate + PbSrTe and n-leg Bi_{2}Te_{3} + PbTe + SiGe) gives 16.8%, that is, only a small enhancement. Increasing the hot side temperature up to 1100 K, designing unicouples with 3-material segments becomes more feasible. For example, the efficiency of 3-segmented unicouple comprised of p-leg Bi_{0.6}Sb_{1.4}Te_{3} + PbSrTe + SiGe and n-leg Bi_{2}Te_{3} + PbTe + SiGe reaches 18.2%, implying an improvement of 28% as compared to 2-segment unicouple (14.2%) of p-leg Bi_{0.6}Sb_{1.4}Te_{3} + SiGe and n-leg Bi_{2}Te_{3} + SiGe. The 3-segment unicouple with the highest calculated efficiency is found by segmenting Bi_{0.6}Sb_{1.4}Te_{3} + PbSrTe + SiGe as p-leg and either segmenting Bi_{2}Te_{3} + PbTe + Si_{0.78}Ge_{0.22} or Bi_{2}Te_{3} + n-Skutt. + La_{3}Te_{4} as n-leg. Bi_{2}Te_{3} + CoSb_{3} and Bi_{2}Te_{3} + n-Clathrate can offer similar efficiencies for unicouples whose p-legs are the same and n-legs are segmentations from either one of these two materials with Si_{0.78}Ge_{0.22} at the uppermost hot side.

*T*

_{h}= 700 (blue), 900 (green), and 1100 K (red) made by legs of different p-type legs (rows) and n-type legs (columns). The legs are made by two segments (Table 2a, upper part) or three segments (Table 2b, lower part) with the segment at the cold side being fixed to be bismuth telluride. The red highlights mark the highest efficiencies

Besides the requirement of high efficiency, consideration on the practical aspects of material choices is inevitable when designing segmented unicouples. The combinations that give the highest efficiencies are made from materials that are toxic (such as telluride and lead), expensive (gold and telluride) or facing the issue of low supply (tellurides) [48]. For this reason, other materials that are relatively inexpensive (e.g. half-Heusler alloys, silicides) [49] and environmental-friendly (e.g. silicides and oxides) [39] are becoming more and more important despite their lower efficiencies. Figure 4 displays the most promising segmented unicouples with possible high efficiencies. From Fig. 4, it is clear that there exist many materials combinations resulting in an improved unicouple efficiency of over 10% upon the hot side temperature at 700, 900, and 1100 K. At *T*_{h} = 700 K, the unicouple with segmented p-leg Bi_{0.6}Sb_{1.4}Te_{3} + TAGS and n-leg Bi_{2}Te_{3} + PbTe yields in the highest efficiency of 13.5%. Unicouples built based on segmenting silicides with bismuth tellurides can achieve a calculated efficiency of 10.5%. With segmented half-Heusler-based unicouples, the calculated efficiency is 11.5%. At a hot side temperature of 900 K, the highest calculated efficiency, 15.3%, belongs to the segmented unicouple from p-leg Bi_{0.6}Sb_{1.4}Te_{3} + PbSrTe + Zinlt and n-leg Bi_{2}Te_{3} + PbTe + Si_{0.78}Ge_{0.22}. Increasing the hot side temperature to 1100 K, the efficiency can reach the highest value of 18.2% for unicouples of p-leg Bi_{0.6}Sb_{1.4}Te_{3} + PbSrTe + SiGe and n-leg either of Bi_{2}Te_{3} + PbTe + SiGe or Bi_{2}Te_{3} + PbTe + La_{3}Te_{4}.

## 4 Conclusions

The efficiency of different segmented thermoelectric single legs and unicouples comprising of one, two or three state-of-the-art materials for each leg was calculated. At *T*_{c} = 300 K and *T*_{h }= 700 K, the most efficient unicouple was found to be 13.5% in which the unicouples were *n*-leg Bi_{2}Te_{3} + PbTe and *p*-leg Bi_{0.6}Sb_{1,4}Te_{3} + TAGS ((AgSbTe)_{0.15}(GeTe)_{0.85}). At *T*_{c} = 300 K and *T*_{h }= 900K the p-leg Bi_{0.6}Sb_{1,4}Te_{3} + Clathrate + PbTe-SrTe with *n*-leg Bi_{2}Te_{3} + PbTe + SiGe unicouple has the highest efficiency of 16.6%. At 1100 K, the highest calculated efficiency is 18.2% which belonged to p-leg Bi_{0.6}Sb_{1,4}Te_{3} + Clathrate + PbTe-SrTe + SiGe and n-leg of either Bi_{2}Te_{3} + PbTe + SiGe or Bi_{2}Te_{3} + Skutterudite + La_{3}Te_{4} unicouples. Segmentation can provide significant improvements in efficiency. However, segmentation may also decrease the efficiency if incompatible materials are used.

## Acknowledgements

The authors acknowledge the support from Copenhagen Cleantech Cluster for the research work (in the project 48062 X-CCC) and the Programme Commisssion on Energy and Environment (EnMi), which is part of the Danish Council for Strategic Research (Contract No. 10-093971), for sponsoring the research of the OTE-Power Project.