#### 3.1 Introduction

The common parameters of Na_{x}CoO_{2} and misfit cobaltates is the presence of CoO_{2} layers of edge shared CoO_{6} octahedra which favor the stabilization of low spin states Co^{3+} and Co^{4+}. In the so-called “misfits”, the CoO_{2} layers are separated by 2 18, 3 5, 6, or 4 15 RS-type layers, while in Na_{x}CoO_{2}, depending on *x*, a layer of randomly filled or ordered Na^{+} separates the CoO_{2} layers 19. In misfits, the two different sublattices (CoO_{2} and NaCl-like layers) are alternately stacked along the *c*-axis (see Fig. 1), which causes the “misfit” ratio *p* (*p* = *b*_{1}/*b*_{2}, where *b*_{1} and *b*_{2} represent *b*-axis parameters of the two incommensurate monoclinic sublattices RS layer and CoO_{2} layer, respectively).

In Na_{0.7}CoO_{2}, large thermopower *S* (100 µV K^{−1} at 300 K), low resistivity *ρ* (200 µΩ cm at 300 K) 1 and also reduced lattice thermal conductivity *κ* 20 are observed, leading to the high thermoelectric performance defined by the figure of merit *Z* = *S*^{2}/*ρκ* 21. The misfit cobaltates possess similar properties, with large thermopower (between 60 and 200 µV K^{−1} at 300 K) and metallicity observed for *T* > 100–200 K. Only four misfits have been shown so far to be metallic in the whole *T* range, from 2 to 400 K, as Na_{0.7}CoO_{2} 22–25. However, in all misfit cobaltates, the Seebeck coefficient presents a common evolution as a function of *T*: a rapid increase at low *T* followed by a plateau for *T* > 100–150 K. The coexistence of these two classically antagonistic properties (large *S* and metallic behavior) has stimulated a lot of interest for this class of materials.

Two models have first been proposed to explain the large thermopower in these materials. A high temperature limit of *S* has been calculated from the Hubbard model, taking into account the large spin and orbital degeneracy associated to Co^{3+} and Co^{4+} in low spin states 8. This is the so-called generalized Heikes formula which states that

- ((1))

with *x* the Co^{4+} concentration, and *g*_{3} and *g*_{4} the spin and orbital degeneracy associated to Co^{3+} and Co^{4+}. For low spin states Co^{3+} and Co^{4+}, *g*_{3} = 1 and *g*_{4} = 6. This formula reproduces well the large values of *S* observed at 300 K and above, even if the *g*_{3}/*g*_{4} term has been questioned and should be reduced to 1/2 26. This localized picture is however difficult to reconcile with the origin of metallicity in these oxides.

From band structure calculations 2, the importance of the rhombohedral symmetry has been evidenced. This symmetry induces a lifting of the *t*_{2g} degeneracy into two orbitals, the broad band responsible for metallicity and a narrow *a*_{1g} band responsible for large Seebeck coefficient, due to the large derivative of the density of states (DOS) at the Fermi level. With this model, the classical Mott equation is used to calculate the thermopower (*S* ∼ (dln*σ*(*E*)/d*E*)_{E=EF}, with *σ*(*E*) = *n*(*E*)*µ*(*E*) where *n*(*E*) is the DOS and *µ*(*E*) the mobility), and correlation effects are not considered.

More recently, Sriram Shastry and coworkers have examined a *t*–*J* model to explain the complex transport properties of Na_{x}CoO_{2} system, combining the Heikes formula and a diffusive term for *S* 27. They show that the Heikes term is dominant in a wide range of temperature and (*t*, *J*) values. The same conclusion has been obtained by LDA + DMFT calculations 28, which shows that the Heikes formula can be a good approximation at high *T* in the incoherent regime.

Depending on the (*t*, *J*) values, the Heikes formula could thus be valid even at room temperature in this metallic system. Furthermore, the doping should play a major role in tuning *S* and we have decided to investigate this effect in the misfit layered-cobalt oxides [Bi_{2}A_{2}O_{4}]^{RS}[CoO_{2}]_{p} (A = Ba, Sr, and Ca, 1.65 ≤ *p* ≤ 2) 15, 16, 29–31 family. These compounds were also found to be good thermoelectric materials. Especially, the whisker crystal of BiSrCoO was shown to exhibit a large dimensionless-figure-of-merit *ZT* > 1 at 973 K 32. In misfit cobaltates, the [CoO_{2}]_{p} mono layer with Co triangular lattices is separated by a block of four RS-type layers with square lattice. In this system, the magnitude of the thermopower changes from 90 to 150 µV K^{−1} according to A cation (Ba^{2+}, Sr^{2+}, and Ca^{2+}) at 300 K 16, 29, 31. Since the oxidation state of the A ion is always 2+, there is in principle no doping effect by changing A. However, the A ion causes a modification of the misfit ratio *p* because of different ionic radii of A ions. As a result, the change of the misfit ratio affects carrier concentration 33. Indeed, due to electroneutrality between the two block layers, the formal Co valency in the CoO_{2} planes can be written as

- ((2))

with *α* the positive charge in the NaCl-like layers. A systematic combination of thermopower and Hall coefficient measurements performed on single crystals of these three different misfits has been used to investigate the importance of both misfit ratio *p* and doping on the thermoelectric properties.

#### 3.2 Results

Figure 2 shows in-plane resistivity *ρ* for A = Ba, Sr, and Ca crystals. (Hereafter we denote A = Ca, Sr, and Ba crystals by BCCO, BSCO, and BBCO, respectively.) At 300 K, the values for BBCO, BSCO, and BCCO are 4.1, 8.4, and 10.1 mΩ cm, respectively. With increasing *p*, *ρ* systematically decreases in the whole *T* range. The 8.4 mΩ cm for BSCO is consistent with other data for crystals 30, 31, 34. The 10.1 mΩ cm for BCCO is also similar to other data for crystals though temperature dependence is rather different 26. The 4.1 mΩ cm for BBCO is very close to the behavior reported in Ref. 29. BCCO and BSCO exhibit metallic conduction above = 202 and 78 K, respectively. The most interesting feature is no upturn of the resistivity in BBCO at low temperatures, as previously reported in Ref. 29, with a residual resistivity below 1 mΩ cm. As shown in the inset of Fig. 2, the resistivity of BCCO at 2.5 K is 7 orders of magnitude larger than that of BBCO, and the resistivity of BSCO is 2 orders of magnitude larger than that of BBCO. The metallic behavior observed for BBCO is completely suppressed at low *T* in BCCO and BSCO, and replaced by a strongly localized behavior: the larger d*ρ*/d*T* is obtained for the smaller *p*. It must be noticed that in the case of misfits, only a few metallic compounds have been reported so far 22–25. In Na_{x}CoO_{2}, metallicity is observed down to low temperatures except for *x* = 0.5, where charge ordering is observed 35. In misfits as well, the block layer might play a role on the localization observed at low *T*.

Figure 3 shows the in-plane thermopower *S* as a function of temperature. Even if strong differences of *ρ*(*T*) are observed at low *T* between the three different crystals, the three *S*(*T*) exhibit the same trend. The temperature dependence is metallic-like (d*S*/d*T* > 0) at low temperatures, with a large slope, and a very small *T* dependence is observed for *T* > 200 K. At 300 K, the slope d*S*/d*T* is almost zero for BCCO, and increases as *p* increases. The magnitude of *S* for BBCO, BSCO, and BCCO is 94, 123, and 149 µV K^{−1} at 320 K, respectively. With increasing misfit ratio *p*, *S* systematically decreases. The value is almost the same as that of polycrystalline samples 16, 29. In BBCO crystals, *S* is smaller than in Ref. 34, with 94 µV K^{−1} to be compared to 110 µV K^{−1}.

Figure 4(a) shows in-plane Hall coefficient *R*_{H} as a function of temperature. For BSCO, the magnitude at 300 K is 1.3 × 10^{−2} cm^{3} C^{−1}, which is almost the same as the previously published data 30, 31. The magnitude for BCCO and BBCO is 0.7 and 2.6 × 10^{−2} cm^{3} C^{−1}, respectively. According to the equation of *R*_{H} = 1/*ne*, where *n* and *e* represent carrier concentration and unit of charge, respectively, we estimated *n* value in Fig. 4(b). BBCO crystal shows the largest carrier concentration in the whole temperature range, which is consistent with the low magnitudes of the resistivity and thermopower. Compared to previously reported data, the value of *R*_{H} is larger (7 × 10^{−3} cm^{3} C^{−1} to be compared to 5 × 10^{−3} cm^{3} C^{−1} in Ref. 34). All the samples exhibit linear-like temperature dependence above 100–200 K, the larger linear regime being observed for the smaller *p*, and BCCO and BSCO exhibit upturn below 90 K, while BBCO does not show such an upturn. We do not observe here the low *T* increase of *R*_{H} reported in other BBCO crystals 34, it rather saturates at low *T* in our crystals. Finally, it should be noted that in the case of BCCO, there is a downturn of *R*_{H}, with a maximum at *T* ∼ 300 K, a behavior which is not observed in BSCO and BBCO.

#### 3.3 Discussion

The resistivity, Hall effect and Seebeck coefficient measurements show that when *p* decreases, *ρ*, *R*_{H} and the thermopower increase (see the data at 320 K plotted in Fig. 5a and b). In a first approach, the change of *p* has only a quantitative effect on *S*, with a similar *S*(*T*) dependence and only a shift of the values in the whole *T* range. On the other hand, it has a much more drastic impact on the resistivity, with strong variations of *ρ*(*T*) observed at low *T*, and a metallic-to-insulator behavior when going from BBCO to BCCO below 100–200 K. This localized behavior also strongly modifies Hall effect at low *T*. Let us focus on the results obtained at high temperature, *i.e*., for *T* > 200 K.

Using the simple classical equation *R*_{H} = 1/*ne*, the Co valency can in principle be extracted. From the values measured at 200 K, in the linear *T* regime above the upturn, *v*_{co} increases from 3.06 to 3.08 and 3.15 for BCCO, BSCO, and BBCO, respectively. This means that as the Co valency increases by the *p* increase, *S* decreases, as expected in classical materials. From Eq. (2), the modification of the misfit ratio directly tunes the formal Co^{4+} content, thereby inducing a change in *S*. The larger *S* will thus be obtained for the smaller Co valency, *i.e*., the smaller *p*.

From Fig. 4a, it is obvious that this analysis of *R*_{H} is too simple as *R*_{H} is not constant. Compared to a classical material, *R*_{H} exhibits a strong *T* dependence in the whole *T* range, with a linear behavior at high *T*. This characteristic shape has also been observed in Na_{x}CoO_{2} 36 and different models developed for a triangular lattice can reproduce the high *T* linear dependence 37–39. The importance of the triangular lattice geometry being at the origin of this linear behavior is emphasized in Ref. 38. In these models, a linear slope is observed for *T* > *t*, and the slope depends on the carrier concentration. Using the *t*–*J* model enables a direct comparison with Na_{0.68}CoO_{2} 36. In the *t*–*J* model 37,

- ((3))

where *a* is a CoCo bond length, *d* the interlayer distance, |*e*| the magnitude of the electronic charge, *t* the hopping amplitude, and *x* is the doping away from half-filling. As parameters, we use *a* = 2 Å and *d* = 15 Å for each crystal. The positive slope of *R*_{H}(*T*) shows that t is negative in these misfits as in Na_{x}CoO_{2} with *x* ∼ 0.7 36.

To further analyze Hall effect, the *x* values of doping have to be extracted from other experiments. For exactly the same set of crystals, angle resolved photoemission spectroscopy (ARPES) measurements have been performed by Brouet et al. 40. A cylinder-like Fermi surface which has *a*_{1g} band characteristic has been observed in BBCO crystal as seen in Na_{x}CoO_{2}. The Fermi wave vector *k*_{F} of BBCO is almost the same as *k*_{F} of Na_{0.7}CoO_{2}. Thus, of BBCO is estimated to be +3.3 ± 0.05 40. Then using Eq. (2) with *v*_{Co} = +3.3 ± 0.05 of BBCO and misfit ratio *p* = 1.98, *α* is estimated to be 1.386 ± 0.1 for BBCO. Supposing this *α* value is the same for BSCO and BCCO, s are estimated to be +3.24 ± 0.055 and +3.18 ± 0.06 for BSCO and BCCO, respectively. These values are larger than the ones obtained from Hall effect, but are consistent with the ones obtained by a TEP scaling analysis on similar misfit polycrystals investigated by NMR 41. With the cobalt valence estimated to be 3.3+, 3.24+, and 3.18+ (*x* = 0.7, 0.76, and 0.82) from ARPES data, obtained t is 13, 21, and 41 K for BCCO, BSCO, and BBCO, respectively. It can be noted that another analysis of thermopower taking into account the magnetic field dependence of thermopower gave a doping level of 3.3 for BCCO 42. This would result in *t* = 10 K for BCCO, not far from 13 K. The main result here is therefore that, even if there is some uncertainty on the *x* values, the obtained *t* values are very small, with *t* smaller than 50 K. The increase of *t* as *p* increases is consistent with the fact that the smaller resistivity is observed in BBCO and also that this linear regime of *R*_{H} is evidenced at higher *T* in BBCO than in BCCO, as Eq. (3) should be valid only for *T* > *t*. The value of 41 K for BBCO with *x* = 0.7 is not so far from *t* = 50 K estimated for Na_{0.68}CoO_{2} with *x* = 0.68 by the same way 36. Also, the estimated *t* is of the same order as the one (8–12 meV) in Na_{x}CoO_{2} estimated by ARPES 43, 44, which shows that our result is reasonable. However, the *t* in misfits and Na_{x}CoO_{2} is much smaller than the value (130 meV) estimated by band calculation in Na_{x}CoO_{2} 2 implying strong electron correlation, which is consistent with rather large *γ* 20, 29. This very small *t* value might come from disorder within the separating planes 45.

The *t* value is thus very small in these compounds. According to the numerical calculation for thermopower based on *t*–*J* model on the triangular lattices performed by Peterson et al. 27, thermopower systematically increases when the carrier concentration increases, and the thermopower shows high-temperature limit (*i.e*., Heikes formula or generalized Heikes formula) at around *T* ∼ 5*t*–6*t*. Indeed we have observed almost constant thermopower around this temperature range, for *T* > 200 K, as shown in Fig. 3 (since *t* is 13–41 K, the limit *T* ∼ 5*t*–6*t* is reached at 65–205 K < *T* < 78–246 K). This means that the high temperature limit of the Heikes formula can easily be reached in this family of oxides, even at 300 K. These reduced energy scales explain why an almost constant Seebeck coefficient is observed as soon as *T* > 200 K. This had already been evidenced in BCCO 43 and this study shows that *t* is not strongly modified by a change of *p*, with a small increase of *t* when *p* increases.

It can be noted that *R*_{H} seems to saturate at *T* ∼ 300 K in BCCO. Such a maximum of *R*_{H} has already been observed in thin films of [Ca_{2}CoO_{3}][CoO_{2}]_{1.62} 46 coexisting with a maximum of resistivity, and interpretated as a transition to incoherent excitations. This is not observed, at least for *T* ≤ 300 K, in the two other misfits and in Na_{x}CoO_{2}. Also, for BCCO, the maximum of *R*_{H} does not coexist with a maximum of resistivity in this range of *T*.

By using the experimental *S* values and *v*_{Co} obtained from ARPES experiments, the spin and orbital degeneracy term *g*_{3}/*g*_{4} of the Heikes formula can be estimated. We obtain *g*_{3}/*g*_{4} = 0.78/0.76/0.8 for BBCO, BSCO, and BCCO respectively, closer to 1/2 than to 1/6. As previously emphasized by Pollet et al. 26, this larger value is induced by the modification of the orbital filling of Co^{3+} and Co^{4+} due to trigonal distortion 47. For Co^{3+}, *g*_{3} is always equal to 1, all the *t*_{2g} orbitals are filled. On the other hand, for Co^{4+}, due to the splitting of the *t*_{2g} orbitals in *a*_{1g} and , there is only one hole in the *a*_{1g} orbital and the orbitals are filled so that *g*_{4} = 2. The enhancement of S by spin and orbital degeneracy is thus not so large as compared to the prediction of Koshibae et al. 8.

Finally, as shown in Fig. 5(c), the obtained power factor defined by PF = *S*^{2}/*ρ* is estimated to be ∼2 × 10^{−4} Wm^{−1} K^{−2} for all the crystals. This is, however, not expected in degenerated semiconductor thermoelectrics which show a bell-shape curve in PF = *f*(*v*_{Co}) 48. For this range of doping ∼10^{20}–10^{21} cm^{−3}, the slope of PF as a function of doping is very large. This shows that this classical description of PF in thermoelectric materials is not valid in these strongly correlated systems.

#### 3.4 Conclusion

By combining resistivity, Hall effect, and thermopower measurements on the same crystals of misfit cobaltates of the family [Bi_{2}A_{2}O_{4}][CoO_{2}]_{p} (A = Ca, Sr, and Ba, 1.69 ≤ *p* ≤ 1.98), the thermoelectric properties of these crystals have been investigated. As shown by the variation of *R*_{H}, the modification of *p* modifies the charge transfer between the block layers, and an increase of *p* induces an increase of the formal Co valency in the CoO_{2} plane. The *R*_{H} coefficients exhibit a complex evolution, with a linear high *T* dependence for *T* > 150–200 K. This linear dependence is characteristic of triangular lattices and has been interpretated using the *t*–*J* model developed by Kumar and Sriram Shastry 37. The *t* parameters are very small (13–41 K) and this justifies why the generalized Heikes formula can be used even at relatively low *T*. *t* increases as *p* increases, consistently with the better metallicity observed in BBCO. Also, compared to BSCO and BCCO, the linear regime of *R*_{H} and the almost constant Seebeck coefficient are observed at higher *T* due to the larger value of *t* in BBCO. Using the Co valency extracted from ARPES experiments and the *S* values, the enhancement of *S* induced by the spin and orbital degeneracy term *g*_{3}/*g*_{4} is found to be closer to 1/2, due to the splitting of the *t*_{2g} orbitals in *a*_{1g} and , as previously reported in Ref. 26.

These measurements on single crystals actually show that the power factor *S*^{2}/*ρ* remains constant for this range of doping, close to 2 × 10^{−4} Wm^{−1} K^{−2}. This is not expected as a strong dependence of PF on the carrier content is classically observed for this range of doping. This demonstrates that the classical approach followed for the search of new thermoelectric materials has to be completely modified in strongly correlated materials. To obtain larger values of power factor, one strategy is now followed by different groups 9, 10: by going from oxides to selenides or sulfides, the decrease of the ionic character could lead to a reduction of the electrical resistivity.