To further understand the structural details of Ba₃Co₂O₆(CO₃)_0.7, we study the elastic diffuse scattering behavior of it. Figures 1C, D show the contour plot of neutron intensity at 50 K in the (H, K, 0) plane of Ba₃Co₂O₆(CO₃)_0.7 with L = 0 and L = −2.85. The sharp spots in Figure 1C demonstrate the good crystallinity of the single crystal. The signals of strong diffuse scattering are clearly observed at L = −2.85. Figure 1F, corresponding to the red solid line in Figure 1E, shows the elastic intensity along the [H, H, 0] direction and the relative diffuse intensity of each diffuse spot. Morgan et al. [30] have analyzed the short-range diffuse scattering using reverse Monte Carlo method and concluded that the diffuse scattering has short-range correlation between the disorder vibrations of polyatomic CO₃ ²⁻.

Figure 1G shows the temperature dependence of the superlattice (1/3, 1/3, 2.85) reflection. Upon warming, the intensity starts to decrease at 150 K and approaches a T-independent constant greater than 250 K. Thus, the vibrations of polyatomic CO₃ ²⁻ should be affected by the thermal effect, and the localization of the CO₃ ²⁻ cluster should be relaxed.

To obtain the electrical transport properties of Ba₃Co₂O₆(CO₃)_0.7 single crystal, Hall carrier concentration n _H was collected at a reversible magnetic field of 2 T, and Hall mobility μ _H was determined with zero-field resistivity, as shown in Figure 2. The exponential increase of carrier concentration with rising temperature indicates the thermal excitation of carriers. The charge polarity is dominated by holes in the measured temperature range, which is consistent with the positive Seebeck coefficients reported by Igarashi et al. [31]. The low carrier concentrations suggest that the further acceptor doping is required for TE applications. As marked in Figure 2A, there are two anomalies at 150 and 250 K, which agrees very well with the diffuse data and confirmed that CO₃ ²⁻ ion is a conductor in the system.

The temperature versus μ _H of Ba₃Co₂O₆(CO₃)_0.7 single crystal is nonmonotonic, as shown in Figure 2B. The calculated mobility involving both the ionized impurity scattering and acoustic phonon scattering can well match the experimental values over a wide temperature range. μ _H follows a temperature dependence closing to T ^1.5 at low temperature range, implying that ionized impurity scattering dominates at low temperature in Ba₃Co₂O₆(CO₃)_0.7 single crystal. We believe that the ionized impurity scattering relates to polyatomic CO₃ ²⁻. As the temperature increases, the localized ordering of CO₃ ²⁻ weakens, and the scattering of ionized impurity also diminishes. On the other hand, the lattice vibrations become stronger with increasing temperature. These two factors together lead to the acoustic phonon scattering, gradually overshadowing the ionized impurity scattering and becoming the dominant scattering mechanism in the material, which causes a negative temperature-dependence slope of μ _H greater than 250 K. The CO₃ ²⁻ is delocalized when T > 250 K, which might introduce other scattering mechanisms affecting the carriers transports and make the carrier mobility lower than the theoretical value.

To investigate the thermal effect in Ba₃Co₂O₆(CO₃)_0.7, the specific heat C _P was measured in the temperature range from 2 to 250 K at zero field. As shown in Figure 3 A, the profile of data is very smooth, and no obvious peak is detected, indicating the absence of structure and magnetic transitions. The inset of Figure 3A displays the data as a C _P/T versus T ². According to the Debye model, the lattice term of the total specific heat C _P/T is linear with T ² at low T, but the data demonstrate a deviation from the linear relationship at low temperature and a nonzero intercept (approximately 0.29 J · mol⁻¹ · K⁻²) at 0 K. The nonzero intercept is an order of magnitude larger than that for NaCo₂O₄, which is a strongly correlated electronic cobalt system [32], and suggests Ba₃Co₂O₆(CO₃)_0.7 as also a strongly correlated electronic system.

In order to further study the specific heat of Ba₃Co₂O₆(CO₃)_0.7, a fitting was gained by using the Debye–Einstein model, which can fit the lattice specific heat data well at both low and high temperature. The model is written as follows [33]: C p ( T ) = C D [ 9 R ( T θ D ) 3 ∫ 0 x D x 4 e x ( e x − 1 ) 2 d x ] + ∑ i C E i [ 3 R ( θ E i T ) 2 exp ( θ E i T ) [ exp ( θ E i T ) − 1 ] 2 ]   , (1) where the first term is the contribution from the acoustic branch (Debye term), and the remaining terms are related to the contribution of the optical branch (Einstein terms). C_D and C_Ei are the relative weights of the Debye and Einstein terms, respectively. R is the universal gas constant. θ _D and θ _Ei represent Debye and Einstein temperatures. There are 13.8 atoms per formula in our system. The best fitting for our data results is one Debye term and three E terms with a ratio 1:6:2.8:4 for C_D:C_E1:C_E2:C_E3 and θ _D = 163 K, θ _E1 = 816 K, θ _E2 = 179 K, and θ _E3 = 290 K, respectively.

As there is no distinct phase transition in the specific heat data, and the magnetic long-range order was not observed down to 2 K [34], the deviation of the experimental data from the fitted values indicates the presence of short-range magnetic fluctuations. Figures 3B, C display the magnetic specific heat, C _M, by subtracting the fitted lattice contribution from raw data, and the related magnetic entropy S _M, respectively. The S _M enhances with increasing temperature, and almost 100% recovered the theoretical value 17.6 J · mol⁻¹ · K⁻¹ at approximately 100 K. Both contributions of spin and orbital degrees of freedom are suggested to be considered [35]: Co⁴⁺ has high-spin state with S = 1/2 (3d ⁵), and Co³⁺ has intermediate-spin state with S = 1 (3d ⁶), leading to S _M = R(0.7 ln 6 + 0.3 ln 18) = 17.6 J · mol⁻¹ · K⁻¹, as shown by the blue dotted line in Figure 3C. There may be spin-phonon scattering in the huge magnetic entropy regimen that interferes with the thermal conductivity.

Figure 4 shows the temperature dependence of the total thermal conductivity κ _c and the lattice thermal conductivity κ _{c_L} of Ba₃Co₂O₆(CO₃)_0.7 parallel to the c-axis from 3 to 250 K κ _c consists of κ _{c_L} and the electronic thermal conductivity κ _{c_e}, where the electronic thermal conductivity can be determined by the Wiedemann–Franz law (κ _e = σL _n T, where the Lorenz number L _n = 2.45 × 10^–8 W · Ω · K⁻²) and subtracted. At low temperatures, κ _c and κ _{c_L} almost completely overlap due to the very small electronic thermal conductivity. As the temperature increases, κ _c and κ _{c_L} begin to separate because of the exponential increase in conductivity. Although κ _{c_e} is maximum at 250 K, it is only a few percent of κ _{c_L}. The contribution of phonon to κ _c is domain. κ _{c_L} increases rapidly at low temperature with the dominant boundary scattering, which displays a κ _{c_L} ∝ T ³ dependence. A maximum with a value of 13.5 W · m⁻¹ · K⁻¹ appears at approximately 12 K, where the Umklapp processes with exp (θ _D/bT) dependence become frequent and enough to compare with boundary scattering. One notes that the peak of κ _{c_L} is an order of magnitude smaller than that of other cobalt oxides, such as Ca₃Co₂O₆ [36]. This difference may be caused by the superlattice in Ba₃Co₂O₆(CO₃)_0.7, which can enhance the scatting of κ _{c_L}. As temperature increases further, κ _{c_L} drops very quickly and reflects that the Umklapp process scattering gradually becomes the main scattering mechanism. When the temperature is higher than 60 K, the trend of the curve gets flat. At approximately 250 K, the lattice thermal conductivity of Ba₃Co₂O₆(CO₃)_0.7 along the c-axis is 5.02 W · m⁻¹ · K⁻¹.

The data of κ _{c_L} were fitted to the formula given by the Debye model of phonon thermal conductivity[37]. κ c_ L = k B 2 π 2 ν p ( k B ℏ ) 3 T 3 ∫ 0 θ D / T x 4 e x ( e x − 1 ) 2 τ ( ω , T ) d x , (2) where k _B is the Boltzmann’s constant, ν _p is the average sound velocity, ħ is the Planck constant, x = ħω/k _B T, ω is frequency, and τ is the mean lifetime of phonon. The phonon relaxation is usually defined as follows: τ − 1 = ν p / L + A ω 4 + B T ω 3 ⁡ exp ( - θ D / b T ) . (3) These three items correspond with phonon boundary scattering, phonon point defect scattering, and the phonon–phonon Umklapp processes, respectively. L, A, B, and b are the fitting parameters. ν _p can be calculated by the Debye temperature θ _D = 163 K obtained by the fitting above and Eq. 4: θ D = ℏ ν p k B ( 6 π 2 N V ) 1 3 (4) where N is the number of atoms in crystal, and V is the volume of crystal, and then we obtain the average sound velocity of the sample ν _p ≈ 1,343 m · s⁻¹.

The best fitting is shown in Figure 4 as green solid line with L = 1.6 × 10^–5 m, A = 1.7 × 10^–41 s³, B = 7.4 × 10^–29 K⁻¹ s², and b = 2.7. The lattice thermal conductivity is mainly the contribution of phonons below approximately 18 K. As the temperature increases, the fitted values gradually deviate from the raw data. An extra contribution of the thermal conductivity has also been observed in Ca₃Co₂O₆ [38]. In combination with the heat capacity, the deviation of thermal conductivity might be due to the overestimation of the lattice thermal conductivity, because of the magnetic contribution to the total thermal conductivity: (1) The magnetic entropy increases rapidly after 18 K; in the meantime, κ _{c_L} starts to be higher than the theoretical value. (2) After the magnetic entropy reaches saturation at approximately 100 K, the κ _{c_L} also decreases slowly with growing temperature at the same rate as the theoretical one. As the electron concentration is very low, the electron–phonon scattering is not dominant in the system, and the CO₃ ²⁻ affects grain boundary and defect.