Compromise between E g and Bipolar Effect

The choice of the optimum bandgap involves balancing the two effects that the bandgap of a given semiconductor has on its transport properties. On the one hand, it follows from the elemental construction of bands in solids that the smaller the energy gap, the lower the effective mass of the conduction- and valence-band carriers around that gap. Therefore, carriers in a solid with a small gap will have higher mobilities than carriers in one with a large gap. Of course, this also involves the

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discussion of the DOS mass versus the transport mass, but that is outside the discussion here (assuming that the semiconductor has a high band degeneracy Nv). On the other hand, the smaller gap limits the maximum temperature of operation of a TE semiconductor. Indeed, at a high-enough temperature, minority carriers will be excited across that gap, and, since they necessarily have a charge polarity opposite that of the of the majority carrier, they will decrease the Seebeck coefficient and increase the ambipolar thermal conductivity. The details of these effects are being examined further. The question is “what is the optimum bandgap for TE materials?”

Optimum Bandgap: Chasmar and Stratton^[224] first investigated the optimum gap of a TE semiconductor. They found that the best bandgap was about 6kBT for indirect-gap TE materials.

Later, Mahan et al.^[250] suggest that 6–10kBT should be suitable for good TE materials. Under an SPB model, Nolas^[225] stated that the bandgap should be larger than 8kBT to suppress the contributions of minority carriers to the TE transport. Recently, Dehkordi et al.^[77] compared the bandgap of some of the best-reported TE materials. It is observed that most of the bandgaps indeed fall between 6kBT and 10kBT. However, there are some systems where the bandgaps are lower than 5kBT, for example, the n-type skutterudite (Yb0.2Co4Sb12). In many samples, the presence of intrinsic defect states in the gap can significantly reduce the optical absorption edge, which is often interpreted to be the “optical bandgap”. When present in a low enough concentration, these defect states are often localized and do not affect the band conduction or act like minority carriers, they decrease the Seebeck coefficient, and increase the ambipolar conductivity; essentially, they are inert. When defects are present in high concentrations, they form an impurity band. The conduction mechanisms are then through hopping or tunneling, and none of the models described previously hold. The reader is referred to the specialized literature for further details.^[251]

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Suppressing the Bipolar Effect: For most TE materials, the bipolar effect is still a sufficiently

influential effect to prevent them from reaching a higher zT value. There are three common methods to suppress the bipolar effect:

i) Increasing the carrier concentration. A higher majority carrier concentration shifts the Fermi level to deeper states and pushes the onset of intrinsic excitation to an elevated temperature. However, the increased carrier concentration will decrease the Seebeck coefficient. The compromise here leads to a peak in the Seebeck coefficient vs temperature curve. The effectiveness of this method is limited for systems in which the carrier concentration has already been optimized.

ii) Band engineering, including expanding the bandgap and reducing the minority effective mass. An enlarged bandgap can shift the thermal excitation temperature to a higher temperature.^[252] Forming solid solutions is the commonly adopted strategy to manipulate the bandgap. Due to the suppression of intrinsic excitation, higher zT can be obtained at higher temperatures; good examples include Bi2−xTe3[123]

and Fe(V,Nb)Sb^[226]. However, an increase in the bandgap will sometimes increase mb* and thus decrease the carrier mobility, which is not favorable for high zT. It is reported that a higher mb* in La-doped PbTe can be partially ascribed to the enlarged bandgap, according to the Kane band model.^[125]

For extrinsic semiconductors, the bipolar thermal conduction b depends on the minority carrier conductivity and partial thermopower. Techniques to minimize b focus on reducing one or the other. Both are determined by the minority carrier effective mass and Eg.^[253] A small minority effective mass is beneficial for a small minority conductivity, and thus, a small b. In p-type Ba-filled

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skutterudites, by increasing the Fe/Co ratio from 2:2 to 1:3, a smaller minority effective mass was obtained in BaxCo3FeSb12 as compared to BaxCo2Fe2Sb12, leading to a smaller b in BaxCo3FeSb12.^[253]

iii) Preferentially scattering the minority carriers will reduce their mobility. By comparing the b of p-type zone melted and nanostructured Bi0.5Sb1.5Te3 samples, Wang et al.^[253] reported that a significant b reduction in the nanostructured sample was observed, which can be ascribed mainly to the large reduction in the minority carrier mobility (n). Since the estimated minority electron wavelengths are much higher than that of minority carrier, it was postulated that nanoprecipitates will scatter the minority carrier preferentially. Similar b reductions in nanostructured Bi2Te3-based alloys have also been reported by other researchers, and the preferential scattering was ascribed to the existence of interfacial charged defects.^[14,254]