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where E c is the minimum energy of the conduction band, E v is the maximum energy of the valence band, and a c and a v are constants and they depend on materials. In terms of the Mott formulae in semiconductors, the Seebeck coefficient has the maximum value when a semiconductor is lightly doped.
Table 1.1 Seebeck coefficient of some metals, semiconductors, flexible materials at room temperature.
| Metal | S (μV K−1) | Semiconductor | S (μV K−1) | Flexible materials | S (μV K−1) |
|---|---|---|---|---|---|
| Sb | +47 | Se | +900 | PEDOT:PSS | +14–20 |
| Mo | +10 | Te | +500 | PEDOT:Tso | +30 |
| Ag | +6.5 | Si | +440 | PA | +10 |
| Cu | +6.5 | Ge | +300 | PPy | +10 |
| Hg | +0.6 | Sb2Te3 | +185 | PANi | +10–20 |
| Pt | +0.0 | Pb3Ge39Se58 | +1670 | Poly(Ni‐ett) | −125 |
| Ni | −15 | Pb15Ge37Se58 | −1990 | graphene | −10 |
| Bi | −72 | Bi2Te3 | −230 | Carbon nanotube | +20 |
For degenerated semiconductors, the Seebeck coefficient is given by
(1.6)
(1.7)
where n(e) and μ(E) are the charge carrier density and mobility at energy E. Hence, a semiconductor with a large variation in the carrier density near the Fermi level can have a high Seebeck coefficient. This can be achieved by changing the band structure. This way to increase the Seebeck coefficient is called “band structure engineering”. In addition, the charge carrier density n(E) is related to the density of states (DOS), g(E),
(1.8)
Thus, the Seebeck coefficient is related to the density of states near E F. Controlling the density of states can vary the Seebeck coefficient, and this is called “DOS engineering”.
Another important formula is the dependence of the Seebeck coefficient on the difference between the average energy (E J) of the transporting electrons and the E F,
(1.9)
where q is the elementary charge. A simple picture of an n‐type semiconductor is shown in Figure 1.4. The charge carriers are the electrons in the conduction band. Their average energy is E J. For n‐type semiconductors, E J − E F > 0, the Seebeck coefficient is negative. In contrast, E J − E F < 0 for p‐type semiconductors, the Seebeck coefficient is positive. When the |E J − E F| value is large, the material will have a high Seebeck coefficient. Increasing the doping level of an n‐type semiconductor can shift the Fermi level upward to the conduction band and thus lower the |E J−E F| value. Similar argument can be applied for p‐type semiconductors. Hence, the Seebeck coefficient decreases with the increase in the doping level.
Although increasing the doping level can decrease the Seebeck coefficient, it increases the electrical conductivity. The power factor (S 2 σ) is dependent on the Seebeck coefficient and the electrical conductivity (σ). As shown in Figure 1.5, there is an optimal power factor in terms of the charge carrier density.
Figure 1.4 Band structure of an n‐type semiconductor. VB and CB are for the valence band and conduction band, respectively. The dots at the CB stand for electrons.
Figure 1.5 Dependence of the Seebeck coefficient, electrical conductivity, and power factor on the charge carrier concentration.
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