Materials’ design for high-performance thermoelectric oxides is discussed. Since chemical stability at high temperature in air is a considerable advantage in oxides, we evaluate thermoelectric power factor in the high temperature limit. We show that highly disordered materials can be good thermoelectric materials at high temperatures, and the effects of strong correlation can further enhance the figure of merit by adding thermopower arising from the spin and orbital degrees of freedom. We also discuss the Kelvin formula as a promising expression for strongly correlated materials and show that the calculation based on the Kelvin formula can be directly compared with the cross-layer thermopower of layered materials.

## INTRODUCTION

Thermoelectrics is an energy conversion technology that converts heat flow into electric power, or vice versa through the thermoelectric phenomena in solids.^{1} The materials’ performance is characterized by the figure of merit *z* = *α*^{2}*σ*/*κ*, where *α* the thermopower, *σ* the conductivity, and *κ* is the thermal conductivity of a material. Although there is no theoretical limitation on the *z* value, *z* is optimized through a delicate interplay among the three parameters. Namely, thermoelectric materials should show a high conductivity, a large thermopower, and a low thermal conductivity at the same time, and such materials are rarely found owing to conflicting requirements for the three parameters.^{2} In this respect, oxides have been regarded as out of the question for good thermoelectric materials, because they are mostly poor conductors due to high ionic nature and good heat conductors due to light oxygen atoms which are tightly bound with each other.

Oxide thermoelectrics arrived at a turning point when good thermoelectric properties were reported in the layered Co oxide Na_{x}CoO_{2} in 1997.^{3} This oxide showed a large thermopower of 100 *μ*V/K together with a low resistivity of 200 *μ*Ω cm at 300 K, exhibiting the power factor *α*^{2}*σ* as good as or even better than Bi_{2}Te_{3}. A unique feature is that the carrier concentration is large enough to be of the order of 10^{22} cm^{−3}, which could reduce the resistivity, but it is unlikely that the large thermopower can be understood within a simple one-electron picture.^{4} Theoretical explanations beyond the conventional design rules have been proposed,^{5–9} and good thermoelectric oxides are still being actively sought world-wide.^{10}

The dimensionless figure of merit *zT* determining the conversion efficiency at temperature *T* is plotted as a function of *T* for various oxides in Fig. 1.^{11–22} Some oxides indeed show reasonably large values of *zT* at high temperatures, and at least we can say that oxides are *not* out of the question. In this article, we will briefly review the unique features of oxide thermoelectrics by mainly focusing on *α*^{2}*σ* at high temperatures. *α*^{2}*σ* is closely related to the unconventional electronic properties, which require new design rules for materials development. Of course *ab initio* calculations on the thermoelectric parameters are powerful for oxide semiconductors. A recent good review on this issue is recommended to the readers.^{23} We wish to also note that we do not discuss the lattice thermal conductivity mainly because of the page limitation. Another reason is that the strategies on how to reduce the lattice thermal conductivity, e.g., rattling^{24} and nano-structuring,^{25} are essentially applicable to oxides as well as other materials.

## BRIEF SUMMARY OF CONVENTIONAL THERMOELECTRICS

On the basis of the Boltzmann transport,^{26} the thermopower of single-type carriers in a parabolic band is expressed as

where *d* is the dimension of the system and *r* is the power of the energy dependence of the scattering time in *τ*(ε) = *τ*_{0}ε^{r}. *F _{p}*(

*η*) is called the Fermi integral defined by $ F p ( \eta ) = \u222b 0 \u221e dx x p / ( exp ( x \u2212 \eta ) + 1 ) $. The value of

*η*is the reduced chemical potential of (

*μ*− ε

_{0})/

*k*, where

_{B}T*μ*the chemical potential, and ε

_{0}is the energy at the bottom [top] of the conduction [valence] band for electrons [holes]. Pichanusakorn and Bandaru

^{26}have found that the thermopower takes a value of 130-187

*μ*V/K, when

*α*

^{2}

*σ*is maximized. They further state that the optimized thermopower is almost independent of the system dimension and the scattering mechanism. Mahan

^{1}has also pointed out in his review that the thermopower takes a universal value of 2

*k*/

_{B}*e*= 170

*μ*V/K under the maximized

*α*

^{2}

*σ*within the framework of semi-classical statistics. In both cases, the reduced chemical potential

*η*is found to be of the order of unity, meaning that the magnitude of

*μ*− ε

_{0}is in the same order of

*k*. In such situation, the carrier concentration is far smaller than that of metals, and this is the reason why good thermoelectric materials are found in semiconductors.

_{B}TNa_{x}CoO_{2} has a larger carrier concentration than semiconductors and obviously requires unconventional explanations. Another feature in Fig. 1 is that *zT* continues to increase up to the highest temperature measured, indicating that *z* is weakly dependent on temperature. This is again incompatible with the conventional semiconductors where minority carriers are thermally excited and cancel the thermopower to cause rapid reduction of *zT*.

## HIGH-TEMPERATURE THERMOELECTRICS

Now let us discuss how the power factor behaves in the limit of high temperature because a vitally important feature of oxides lies in their high-temperature chemical stability in air. As mentioned above, the *z* values of oxides weakly depend on temperature, and thus we seek theoretical expressions for temperature-independent transport parameters at high temperatures.^{27}

Temperature-independent thermopower has been analyzed using the Heikes formula.^{28} In this formula, entropy per site is associated with the thermopower. In a simplest case, the thermopower of the Heikes formula *α*_{H1} can be given by

where *x* is the carrier concentration per unit cell. This formula is valid, when the thermal energy *k _{B}T* is much larger than the transfer energy

*t*or the band width

*w*. The on-site Coulomb repulsion

*U*is neglected (

*U*≪

*t*≪

*k*) in Eq. (2), and

_{B}T*α*

_{H1}is valid for weakly correlated electrons.

We seek for a proper expression for conductivity in the same condition. One extreme case is known as the Ioffe-Regel limit, where the electron mean free path ℓ is equal to the lattice parameters. In this picture, the Ioffe-Regel conductivity *σ*_{IR} is given by

for a three-dimensional case, where *a* is the lattice parameter.^{29} In a quasi-two-dimensional case, it equals

where *a*_{⊥} is the lattice constant along the cross-layer direction. $ \sigma IR 2 d $ is the two-dimensional Ioffe-Regel conductance per unit square. By comparing Eqs. (3) and (4), we find that the pre-factors are almost identical, but the *x* dependence is milder in quasi-two dimension than in three dimension. A Drude-type conductivity is proportional to *x*, and the power factor is maximized at an optimum value of *x*_{0} = *e*^{−2} for *d*[*x*(ln*x*)^{2}]/*dx* = 0. Thus the milder dependence of $ \sigma IR q 2 d $ will give a smaller *x* and a larger *α* for the maximized power factor.

Here we focus on the quasi-two dimensional systems in keeping the layered Co oxides in mind, but the essence is the same as in three dimensional cases.^{27} The thermopower is plotted as a function of carrier number per site *x* by the solid curve in Fig. 2(a). The thermopower is proportional to ln*x* for *x* ≪ 1, which is the same dependence as *α* in a semi-classical statistics.^{1} The sign is negative for *x* < 1, being consistent with a simple band picture. $ \alpha H1 2 x $ of a quasi-two-dimensional case is plotted as a function of *x* by the solid curve in Fig. 2(b). This value corresponds to the power factor when the pre-factor of 0.40*e*^{2}/*ħ a*_{⊥} is ignored. One can find that the maximum of the power factor lies at *x* = *e*^{−4} ∼ 0.02, owing to the mild *x* dependence of $ \sigma IR q2d $.

Let us evaluate the power factor for *x* = 0.02 and *a*_{⊥} = 5 Å. We get $ \sigma IR q2d =$280 S/cm and *α*_{H1} = − 400 *μ*V/K. Then the power factor equals 43 *μ*W/cmK^{2}. Assuming a low thermal conductivity *κ* of 20 mW/cmK, we estimate $z= \alpha H1 2 \sigma IR q2d /\kappa $ to be 2.1 ×10^{−3} K^{−1}, which gives *zT* = 2.1 at 1000 K. Note that the optimized *α* is much larger than 200 *μ*V/K, making a stark contrast to the conventional rules mentioned above. Although this value is the theoretical upper limit that is difficult to be achieved, we should emphasize that this estimation is done *in the limit of low mobility*. This is again incompatible against the conventional design rules, where high mobility is always required for high *zT* values. Equations (2)–(4) tell us that highly disordered materials can be fairly good thermoelectric materials at high temperature. This is one plausible strategy suitable for weakly correlated oxides.

## CORRELATED-ELECTRON THERMOELECTRICS

In many of transition-metal oxides, conduction electrons strongly interact with each other through poorly screened Coulomb repulsion owing to the narrow *d* bands. In such situations, one-electron picture may often be broken down to show exotic properties such as high-temperature superconductivity and colossally large magnetoresistance. Such systems are referred to as strongly correlated electrons, which have occupied a central position in solid state science for decades.

The Heikes formula can describe the high-temperature thermopower in strongly correlated electrons as

where the on-site Coulomb repulsion *U* is larger than *k _{B}T* (

*t*≪

*k*≪

_{B}T*U*). In this situation, each site can accept only one electron owing to the large

*U*, and the factor of 2 represents the spin degree of freedom in the localized electron.

*α*

_{H2}is plotted as a function of

*x*by the dashed curve in Fig. 2(a), where it diverges for

*x*→ 0 or

*x*→ 1. This indicates that the states at

*x*= 0 and 1 are insulating. The insulating state at

*x*= 1 is called the Mott insulator phase.

Koshibae, Tsutsui and Maekawa^{7} have extended the Heikes formula to include the spin and orbital degrees of freedom in the multi-electron ions and have proposed the thermopower for a solid solution of A and B ions given by

where *g*_{A} and *g*_{B} are the degeneracy on the A and B ions, respectively. In the case of the layered cobalt oxides, we put *g*_{A} = 6 for the low-spin-state Co^{4+} and *g*_{B} = 1 for the low-spin-state Co^{3+}. The corresponding *α*_{KTM} is plotted as a function of *x* by the dotted curve in Fig. 2(a), where it diverges for *x* → 1 similarly to *α*_{H2}.

The power factors of the two cases are plotted in Fig. 2(b). Here we assume that the conductivity is proportional to $ x ( 1 \u2212 x ) $ to express the two insulating states at *x* = 0 and 1. In contrast to the weakly correlated case, the power factors take maxima around *x* = 0.95. This is because the spin and orbital degrees of freedom give additional thermopowers of *k _{B}*ln2/|

*e*| and

*k*ln6/|

_{B}*e*|. The additional thermopower of

*k*ln6/|

_{B}*e*| is thought to be an origin of the large thermopower in the layered cobalt oxides

^{7}and also explains why the layered cobalt oxides are of

*p*-type. We have controlled the spin state and the spin entropy in the perovskite Co/Rh oxide and have shown that the spin entropy dominates the thermopower.

^{30}

Figure 2(b) suggests a strategy for good oxide thermoelectrics. In the strongly correlated systems, a slightly doped hole has a large thermopower proportional to −ln(1 − *x*) and a fairly large conductivity proportional to $ 1 \u2212 x $. These situations are similar to the case of *x* ∼ 0 in the weakly correlated case. But now the doped hole carries additional entropy associated with the spin and orbital degrees of freedom. In fact, employing *x* = 0.95 and *a*_{⊥} = 5 Å, we get $ \sigma IR q2d =$430 S/cm and *α*_{KTM} = 410 *μ*V/K. The power factor equals 72 *μ*W/cmK^{2}, indicating a 60% increment from the weakly correlated case.

We should note here that the conductivity of $ x ( 1 \u2212 x ) $ is just an assumption. According to theoretical calculations based on the Hubbard model,^{31–33} the conductivity is likely to be proportional to *x*(1 − *x*) rather than $ x ( 1 \u2212 x ) $. For comparison, we plot the calculation of $ \alpha H2 2 x ( 1 \u2212 x ) $ by the dotted-dashed curve in Fig. 2(b). One can see that essential features are the same as $ \alpha H2 2 x ( 1 \u2212 x ) $: The maximum is located around *x* = 0.9 and shows that hole-doping is effective. Unlikely to the case of the Ioffe-Regel conductivity, we cannot say anything about the magnitude of the power factor, for the scattering time is not taken into account in this case.

## THE KELVIN FORMULA

One serious concern is that the (extended) Heikes formula is valid only for *k _{B}T* ≫

*t*. A typical value of

*t*ranges from 10

^{−1}to 10

^{0}eV, and thus the condition of

*k*≫

_{B}T*t*is rarely satisfied in real measurements. Peterson and Shastry

^{34}have recently found that the thermopower at reasonably high temperatures obeys the following relation:

which they call the Kelvin formula. They have further found that the Kelvin formula semi-quantitatively explains *α* for some correlated electron systems. Compared with the Heikes formula, the Kelvin formula includes the temperature dependence of the thermopower and is widely applicable to low-mobility materials regardless of the degree of electron correlation. In this respect, this is the first expression to compare experimental results.

One theoretical advantage in the Kelvin formula is that *α*_{K} can be directly obtained from the thermodynamic quantity *μ* through *ab initio* calculations. *α*_{K} is associated with specific heat or entropy, which is compared with thermodynamic measurements.^{35} We have found that *α*_{K} can be also directly obtained experimentally in low dimensional metals. Let us consider a metal whose Fermi energy is much larger than *k _{B}T*. Then the carrier concentration

*n*given by

*n*= ∫

*d*ε

*D*(ε)

*f*(ε) is independent of temperature, where

*D*(ε) the density of states, and

*f*(ε) is the Fermi-Dirac distribution. Then we calculate

*dn*/

*dT*= 0 using the relation given by

and immediately arrive at

We compare the above equation with the standard expression of *α* in the Boltzmann equations given by

and find that Eqs. (8) and (9) differ only by the factor of *v*^{2}*τ* = ℓ^{2}/*τ* in the integrals, where $v=v ( k \u2192 ) $ is the Fermi velocity.

Now we show that the cross-layer thermopower of the quasi-two-dimensional metal equals *α*_{K}. Owing to the layered structure, the Fermi velocity along the cross-layer direction *v*_{⊥} is much smaller than along the in-layer direction, and the cross-layer mean free path ℓ_{⊥} is often close to the cross-layer spacing *c* above room temperature.^{36} In such situations, the cross-layer hopping becomes incoherent, and *τ* loses the $ k \u2192 $ dependence. Eventually the quantity $ v \u22a5 2 \tau $ equals $ \u2113 \u22a5 2 /\tau = c 2 /\tau $, and is justified to be removed from the integrals of the numerator and denominator in Eq. (9) as a $ k \u2192 $-independent value. In such cases, we can identify *α*_{⊥} with *α*_{K}. In this derivation, we did not assume the strong correlation, and we have elsewhere discussed to what extent this relation holds in the presence of strong correlation.^{37}

Figure 3 shows the cross-layer thermopower *α*_{⊥} experimentally measured for the layered Co oxide Bi_{1.6}Pb_{0.4}Sr_{2}Co_{2}O_{8+δ} and the high-temperature superconducting Cu oxide Bi_{2}Sr_{2}CaCu_{2}O_{8+δ}. The temperature dependence is similar, but the magnitude differs by a factor of three. In the localized picture, the Co oxides show the thermopower due to the spin and orbital degrees of freedom expressed by Eq. (6), which is absent in the Cu oxides. In the band picture, on the other hand, the valence bands of the Co oxides are narrower than those of the Cu oxides, because the formers consist of the *t*_{2g} orbitals and the latters the *e _{g}* orbitals strongly hybridized with oxygen 2

*p*orbitals. Narrower bands give a heavier effective mass and a larger thermopower. Yet another reason may be that the conduction layer is a triangular lattice in the Co oxides and a square lattice in the Cu oxides. The former does not hold an electron-hole symmetry, but the latter can. Thus the electrons and holes can be equally excited in the Cu oxides, making the thermopower smaller in magnitude.

^{38}The solid curve represents

*α*

_{K}for a quarter-filled triangular lattice.

^{34}In order to fit the data with the curve, we employed a value of 500 K for the hopping parameter

*t*. The curve is semi-qualitatively consistent with the observed

*α*

_{⊥}for the layered Co oxide.

The Kelvin formula may also be applicable to disordered materials, where *v*^{2}*τ* = *a*^{2}/〈*τ*〉. If the *k* dependence of 〈*τ*〉 is neglected, Eq. (8) can be equalized with Eq. (9). The thermopower of such materials then equals *α*_{K}. In this respect, the Kelvin formula is a good measure for the thermopower of the disordered materials. Theoretical studies using the Kelvin formula have revealed that the strong correlation enhances the thermopower around the temperature of *T* ∼ *t*/*k _{B}*.

^{39}By properly choosing the crystal structure, the on-site Coulomb repulsion, and the carrier concentration,

*zT*can exceed unity.

^{40,41}

## SUMMARY AND OUTLOOK

In summary, we have proposed three strategies for oxide thermoelectrics. First we have shown that the dimensionless figure of merit can be above unity even for the low-mobility materials showing the Ioffe-Regel conductivity at high temperature. Secondly, we have proposed that hole-doped Mott insulators can be superior to the weakly correlated materials discussed in the first strategy. Thirdly, we have found that the Kelvin formula is useful both for theoretical and experimental materials search.

## ACKNOWLEDGMENTS

The authors would like to thank A. J. Schofield, B. S. Shastry, and D. J. Singh for fruitful discussion and also appreciate T. Fujii for the measurements of *α*_{⊥}.