Research Update: Oxide thermoelectrics: Beyond the conventional design rules

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 = α²σ/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_xCoO₂ in 1997.³ This oxide showed a large thermopower of 100 μV/K together with a low resistivity of 200 μΩ cm at 300 K, exhibiting the power factor α²σ as good as or even better than Bi₂Te₃. A unique feature is that the carrier concentration is large enough to be of the order of 10²² 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 α²σ at high temperatures. α²σ 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., rattling²⁴ and nano-structuring,²⁵ are essentially applicable to oxides as well as other materials.

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) = τ₀ε^r. F_p(η) is called the Fermi integral defined by $F p ( η ) = ∫ 0 ∞ dx x p / ( exp ( x − η ) + 1 )$⁠. The value of η is the reduced chemical potential of (μ − ε₀)/k_BT, where μ the chemical potential, and ε₀ is the energy at the bottom [top] of the conduction [valence] band for electrons [holes]. Pichanusakorn and Bandaru²⁶ have found that the thermopower takes a value of 130-187 μV/K, when α²σ is maximized. They further state that the optimized thermopower is almost independent of the system dimension and the scattering mechanism. Mahan¹ has also pointed out in his review that the thermopower takes a universal value of 2k_B/e = 170 μV/K under the maximized α²σ 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 − ε₀ is in the same order of k_BT. 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.

Na_xCoO₂ 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.

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_BT is much larger than the transfer energy t or the band width w. The on-site Coulomb repulsion U is neglected (U ≪ t ≪ k_BT) in Eq. (2), and 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. $σ IR 2 d$ 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 x₀ = e⁻² for d[x(lnx)²]/dx = 0. Thus the milder dependence of $σ IR q 2 d$ 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 lnx 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. $α H1 2 x$ 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 0.40e²/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 $σ IR q2d$⁠.

Let us evaluate the power factor for x = 0.02 and a_⊥ = 5 Å. We get $σ IR q2d =$280 S/cm and α_H1 = − 400 μV/K. Then the power factor equals 43 μW/cmK². Assuming a low thermal conductivity κ of 20 mW/cmK, we estimate $z= α H1 2 σ IR q2d /κ$ 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 μV/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.

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_BT (t ≪ k_BT ≪ 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 Maekawa⁷ 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 Co⁴⁺ and g_B = 1 for the low-spin-state Co³⁺. 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 $x ( 1 − x )$ 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_Bln2/|e| and k_Bln6/|e|. The additional thermopower of k_Bln6/|e| is thought to be an origin of the large thermopower in the layered cobalt oxides⁷ 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 $1 − x$⁠. 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 $σ IR q2d =$430 S/cm and α_KTM = 410 μV/K. The power factor equals 72 μW/cmK², indicating a 60% increment from the weakly correlated case.

We should note here that the conductivity of $x ( 1 − x )$ 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 $x ( 1 − x )$⁠. For comparison, we plot the calculation of $α H2 2 x ( 1 − x )$ by the dotted-dashed curve in Fig. 2(b). One can see that essential features are the same as $α H2 2 x ( 1 − x )$⁠: 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.

One serious concern is that the (extended) Heikes formula is valid only for k_BT ≫ t. A typical value of t ranges from 10⁻¹ to 10⁰ eV, and thus the condition of k_BT ≫ t is rarely satisfied in real measurements. Peterson and Shastry³⁴ 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_BT. Then the carrier concentration n given by n = ∫dεD(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 v²τ = ℓ²/τ in the integrals, where $v=v ( k → )$ 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 $k →$ dependence. Eventually the quantity $v ⊥ 2 τ$ equals $ℓ ⊥ 2 /τ= c 2 /τ$⁠, and is justified to be removed from the integrals of the numerator and denominator in Eq. (9) as a $k →$-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_1.6Pb_0.4Sr₂Co₂O_8+δ and the high-temperature superconducting Cu oxide Bi₂Sr₂CaCu₂O_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 2p 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 v²τ = a²/〈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

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.

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_⊥.