Evaluation of the thermoelectric potential of the type-I clathrate Ba₈Ni_yZn_xGe_46−x−y

The electrical resistivities, ρ, of samples Ba₈Zn_xGe_46−x with x = 7.33, 7.62 and 7.78 were measured in the temperature range from 4 to 300 K with a QD PPMS-system. Figure 3 records the electrical resistivities normalized to their values at 300 K. The change in ρ(T) from metallic-like (for x = 7.33, 7.62) to semiconducting nature (for x = 7.78) can be clearly seen. The electrical resistivity as a function of temperature of the samples Ba₈Zn_7.33Ge_38.67 and Ba₈Zn_7.62Ge_38.38 below 300 K can be described in terms of the Bloch–Grüneisen model applicable for simple metals, resulting in residual resistivities of ρ₀ = 327 µΩ cm, and ρ₀ = 475 µΩ cm and Debye-temperatures of θ_D = 220 K and θ_D = 250 K, respectively. θ_D for both samples is close to θ_D = 225 K extracted from heat capacity measurements on Ba₈Zn_7.7Ge_38.3 [14]. Figure 4 presents the electrical resistivity data above room temperature. The resistivity is increasing with increasing Zn-content. Not only are the absolute values changing, but also the temperature dependence is changing from a more or less linear to a more complex behaviour. Whereas the resistivity does not modify too much between 7.3 and 7.62 Zn-atoms per unit cell, there is a dramatic increase for higher Zn-contents. Taking also into account the room-temperature value of 750 mΩ cm for Ba₈Zn₈Ge₃₈ revealed by Christensen and Iversen [16], the resistivity at 300 K increases by a factor of more than 1300 for substituting only 0.7 Ge-atoms/u.c. by Zn. This small change in composition is severely affecting the electrical resistivity, evidencing the proximity to a metal-to-insulator transition, close to the Zintl-balanced composition Ba₈Zn₈Ge₃₈.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The corresponding Seebeck coefficients, S, are plotted in figure 5. Also, here an increase in the Zn-content is accompanied by an increase in the absolute S-values. All samples exhibit an almost linear temperature dependence from 300 to 700 K. The electrical resistivity as well as the Seebeck coefficient of Ba₈Zn_7.78Ge_38.22 shows a maximum close to 800 K.

Zoom In Zoom Out Reset image size

Thermal conductivity was measured for samples with x = 7.33, 7.62 and 7.78 from 300 to 830 K revealing little change with temperature (figure 6). The total thermal conductivity (λ_tot) is decreasing with increasing Zn-content, which can be mainly attributed to a reduction in the electron contribution (λ_e). The Wiedemann–Franz relation was employed to estimate λ_e using the electrical resistivity data (with L = 2.44 × 10⁻⁸ W Ω K⁻²). As expected for a small change in composition the phonon thermal conductivity (λ_ph) for the samples with x = 7.33 and 7.62 is practically similar. Only for Ba₈Zn_7.78Ge_38.22 was a significantly higher λ_ph observed. There are two possible explanations: either (i) the Wiedemann–Franz law, in a strict sense only valid for a free electron gas, does not hold any more, as the resistivity data of this sample clearly reveal a non-metallic-like behaviour (see figures 3 and 4), and/or (ii) λ_ph is increased due to a reduced scattering of phonons by electrons, as the absolute charge carrier density of this sample is the smallest (see figure 7). The theoretical lower limit of the lattice thermal conductivity according to Cahill and Pohl [23] was calculated taking a Debye-temperature of 225 K, gained from earlier heat capacity measurements [14] and is shown as a dashed black line in figure 6.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Hall-effect measurements were performed for three samples with a Zn-content of 7.33, 7.62 and 7.78, respectively. In agreement with the Seebeck coefficients and with results reported previously [15], all samples indicate electrons as the predominant charge carriers. All three samples reveal almost temperature-independent charge carrier densities, and, as expected, a decrease in the absolute value with increasing Zn-content leading from −10.3 × 10²¹ cm⁻³ for Ba₈Zn_7.33Ge_38.67 to −3.2 × 10²¹ cm⁻³ for Ba₈Zn_7.78Ge_38.22 (figure 7). Although Alleno et al [15] also observed a carrier concentration invariant with temperature for the sample Ba₈Zn_7.83Ge_38.17, the absolute value of −2.5 × 10²¹ cm⁻³ is slightly smaller than the values measured within this work.

The charge carrier mobility, μ, of the selected samples is shown in figure 8. The mobility was calculated from the relation μ = σ/ne, with μ being the mobility, σ the electrical conductivity, n the charge carrier density, and e the carrier charge (in this case the electron charge). Due to the different behaviour in electrical resistivity (see figure 3) different scenarios are found for the samples with lower Zn-content in contrast to the one with x_Zn = 7.78. A detailed analysis reveals a temperature independent mobility, μ(T), below 100 K, which is, assuming non-degeneracy for the charge carriers and a distribution function given by Maxwell–Boltzmann statistics, usually attributed to neutral impurity scattering in semiconductors [24]. In the temperature range from 100 to 300 K for both samples (x_Zn = 7.33, 7.62) a decrease in the mobility close to a T^−1/2-behaviour (indicated by a black solid line in figure 8) is observed. This temperature dependence is expected in the case of alloy scattering, which occurs whenever different atoms in semiconductor compounds are not arranged periodically in the crystal structure [24]. Yan et al observed a similar behaviour for Ba₈Cu₅Si₁₈Ge₂₃ [25]. Room temperature values of μ are well within the range of other Ba–Ge-based clathrate compounds listed in [9].

Zoom In Zoom Out Reset image size

The electron effective mass can be derived from equation (1), usually valid for a single parabolic band at high n and/or low T [26],

$$\begin{equation} S=\frac{8\pi ^2k_{\rm B}^2 }{3eh^2}m^\ast T\left( {\frac{\pi }{3n}} \right)^{2/3} \end{equation} \tag{ 1 }$$

where n is the charge carrier density, T is the temperature and m^* is the effective carrier mass. An evaluation revealed a significant difference in behaviour for samples with a Zn-content of 7.33, and 7.62 in contrast to that with 7.78. Whereas the calculated values of the effective mass for the former samples are almost similar (7m_e for Zn7.33 and 6.8m_e for Zn7.62), m^* for Zn7.78 appears significantly increased (10.5m_e). This observation indicates that the assumption of a single parabolic band is not valid in the case of Ba₈Zn_xGe_46−x, which was already demonstrated for Ba₈Au_xGe_46−x [27]. Furthermore, a comprehensive compilation of effective masses [9] clearly reveals that p-type compounds, in general, exhibit larger m^*-values than n-type clathrates. As Ba₈Zn_7.78Ge_38.22 is located close to the n/p-transition, an enhanced effective mass can be observed. The Pisarenko plot (figure 9) shows the Seebeck coefficients at 300 K as a function of the charge carrier concentration; the dashed lines indicate the predicted values for a single parabolic band behaviour for 6.8m_e and 10.5m_e according to equation (1).

Zoom In Zoom Out Reset image size

In order to evaluate in detail the thermoelectric potential of the ternary clathrate-I Ba₈Zn_xGe_46−x, electrical resistivity and Seebeck coefficient data at 300 K and 730 K were plotted as a function of the Zn-content and least-squares fits were applied (figure 10). In addition to the data gained within this work, electrical resistivities and Seebeck coefficients reported earlier [14, 16] were used. As these Seebeck coefficients [14, 16] were not evaluated up to 730 K, simple linear extrapolations to the measured data were used to estimate the values. From figure 5 one can see, that the S-values up to 730 K follow an almost linear temperature dependence, thus the extrapolated values from literature data should not deviate too much from the real values. Electrical resistivity data, unfortunately, do not exhibit a simple T-dependence, thus no extrapolated data were used in this case, but the same exponential function used for the data at 300 K was fitted to the data at 730 K.

Zoom In Zoom Out Reset image size

The power factor (PF) at 730 K was calculated using the fit-values, with PF = S²/ρ, where S is the Seebeck coefficient and ρ the electrical resistivity. In order to estimate then the dimensionless thermoelectric figure of merit ZT = PF × T/λ_tot, where T is the temperature and λ_tot the total thermal conductivity, the electronic contribution to the thermal conductivity was calculated via the Wiedemann–Franz law and the phonon contribution was assumed to be 0.85 W mK⁻¹. This value is derived taking the average of all three λ_ph-values (for x_Zn = 7.33, 7.62 and 7.78) at 730 K (see figure 6). The results are shown in figure 10 depicting the dependence of PF and ZT on the zinc content in Ba₈Zn_xGe_46−x. Both curves exhibit a maximum, which for ZT is shifted to a higher Zn-content, attributed to the beneficial influence of a further decrease in the electron thermal conductivity. The maximum value of the thermoelectric figure of merit in the ternary clathrate-I system Ba₈Zn_xGe_46−x is expected to be 0.46 at 730 K for x being close to 7.7 (figure 11). In figure 12 the ZT-values of the measured samples (x = 7.33, 7.62 and 7.78) are plotted as a function of temperature. ZT is increasing with temperature and the results are in sound agreement with the estimated values, demonstrating that an estimation using the procedure described above can serve as a helpful tool to find the optimum composition with respect to the thermoelectric performance. It should be noted that obviously the maximum TE performance has already been found for the Ba₈Zn_xGe_46−x system. Results of the physical property measurements are summarized in table 4.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The electrical resistivity of all samples was measured in a temperature range from 4.2 to 800 K (figure 13) and in general shows a metallic-like behaviour. In several cases a flattening of the slope of the resistivity curve at temperatures above 650 K is observed (figure 13), which might indicate the proximity of the system to a metal-to-insulator transition. A suitable description for ρ(T) throughout the whole temperature range investigated can be derived from a two-band model developed and described in detail in [28, 29]. The model assumes a constant density of states (N₀), where the valence band and the conduction band are separated by a gap of width E_g. The Fermi energy (E_F) is located slightly above the band edge of the conduction band, which guarantees predominantly electron-like transport (an illustration of this assumption is shown in figure 14). This scenario is confirmed by theoretical calculations for a few ternary Ge/Si-based clathrates incorporating elements such as Cu [30], Zn [31, 32], Ag [29] and Au [33]. The two-band model is based on a temperature-dependent charge carrier density, which is combined with the Bloch–Grüneisen formula. In order to describe deviations from linearity also at moderate temperatures according to the theory of Mott and Jones a T³-term was added. Details on the fitting procedure are given in [34] for all samples, and results are listed in table 4. The fit to the most representative sample Ba₈Ni_0.8Zn_6.4Ge_38.8 is shown in figure 14, resulting in a band gap of 4040 K (0.35 eV) and a Debye-temperature θ_D = 300 K. In general, band gaps of all samples are in the range 4030 (0.35 eV) to 4500 K (0.39 eV) and θ_D gained from the fits are all close to 300 K. A closer inspection of the low-temperature behaviour revealed a logarithmic increase in the resistivity below 20 K (figure 15). A similar behaviour was also reported for Ba₈Ni_3.5Ge_42.1□_0.4 [20]. This unexpected increase of ρ(T) towards low temperatures might be due to some localization phenomena, corroborated by the fact that this feature appears more pronounced for the samples with the largest overall resistivities. Although localized systems can be frequently described in terms of Mott's hopping conductivity [35], an analysis of the data according to this model (right panel figure 15) clearly demonstrates that this is not the case for these compounds.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Seebeck values of all samples are negative throughout the whole temperature range investigated (4.2–800 K, figure 16), revealing n-type conductivity. In similarity to the electrical resistivity, increasing absolute values of the Seebeck coefficient of most samples as a function of temperature exhibit a slight flattening above 650 K, whereas an almost linear slope is observed at lower temperatures. The samples Ba₈Ni_0.4Zn_7.0Ge_38.7 and Ba₈Ni_0.5Zn_6.9Ge_38.6, revealing the maximum electrical resistivity of this set of samples, also have the most negative Seebeck values.

Zoom In Zoom Out Reset image size

The temperature-dependent thermal conductivity, λ, from 4.2 to 670 K is displayed in figure 17. Common to all samples investigated is a low-temperature maximum of the thermal conductivity typically observed for crystalline materials with low defect concentrations. Nevertheless the absolute values of λ are rather low, predominantly due to the rattling effect, a well-known feature of cage compounds considered as PGEC-materials (phonon glass – electron crystal) [3]. A detailed study of the vibrational dynamics of the type-I Ba–Zn–Ge clathrate indeed revealed eigenmodes of Ba-atoms significantly contributing to the phonon spectral density at low energies (below 14 meV) [36]. In general, the position and magnitude of the low-temperature maximum of λ(T) depends on a subtle interplay of various interaction mechanisms and parameters. In order to disentangle these effects, determining the phonon contribution of the thermal conductivity, the low-temperature data were fitted to the Callaway model [37]. Applying the Wiedemann–Franz law allows one to separate the electron and phonon contribution of the thermal conductivity. Within the Callaway model the relaxation time of scattering processes can be described as a reciprocal sum $\tau _{\rm C}^{-1}$ of different scattering mechanisms:

$$\begin{equation} \tau _{\rm C}^{-1} =\tau _{\rm U}^{-1} +\tau _{\rm D}^{-1} +\tau _{\rm B}^{-1} +\tau _{{\rm el\mbox{--}ph}}^{-1} \end{equation} \tag{ 2 }$$

with U denoting scattering by Umklapp processes, D by defects, B by boundaries, and el–ph scattering by electrons. A detailed analysis presented in [34] indeed revealed a much smaller influence of defect and boundary scattering for the sample Ba₈Ni_0.8Zn_6.4Ge_38.8 compared with the other samples. Figure 18 shows a least-squares fit of the Callaway model to the sample Ba₈Ni_0.8Zn_6.4Ge_38.8. An additional T³-term accounts for radiation losses inherent of the steady-state heat flow method. The minimal thermal conductivity, λ_min (dashed line in figure 18), was estimated from the model of Cahill and Pohl [23], employing a Debye-temperature θ_D = 300 K, resulting in a room-temperature value of 5.7 mW cm⁻¹ K⁻¹, which is roughly half of the observed phonon contribution.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Hall-effect measurements were carried out for three different samples, namely Ba₈Ni_0.8Zn_6.4Ge_38.8, Ba₈Ni_0.5Zn_6.5Ge_39.0 and Ba₈Ni_0.6Zn_6.9Ge_38.5. In agreement with the negative Seebeck coefficients the Hall data refer to electrons as the main charge carriers, and to an almost temperature-independent charge carrier density from 10 to 300 K, typical for metal-like behaviour (figure 7). Although the room temperature electrical resistivity is increasing from Ba₈Ni_0.8Zn_6.4Ge_38.8 (1257 µΩ cm) via Ba₈Ni_0.5Zn_6.5Ge_39.0 (1400 µΩ cm) to Ba₈Ni_0.6Zn_6.9Ge_38.5 (1758 µΩ cm), the charge carrier density is increasing as well in the same sequence. In comparison with the results for the ternary clathrate-I Ba₈Zn_xGe_46−x the charge carrier density n is significantly reduced and does not vary too much with composition. Whereas for the ternary compounds an increase in the Zn-content by ∼0.4 atoms/unit cell leads to a reduction of n from −10.3 × 10²¹ to −3.2 × 10²¹ cm⁻³, compositional differences of the same order of magnitude in Ba₈Ni_yZn_xGe_46−x−y with y_nom = 1 cause almost no change in the carrier concentration. All samples, however, reveal a decrease in the charge carrier mobility with increasing temperature (figure 8). As the room-temperature mobility is decreasing from Ba₈Ni_0.8Zn_6.4Ge_38.8 and Ba₈Ni_0.5Zn_6.5Ge_39.0 to Ba₈Ni_0.6Zn_6.9Ge_38.5, the effective electron masses, calculated according to equation (1), exhibit a converse trend: m^*(Ba₈Ni_0.8Zn_6.4Ge_38.8) = 4.4m_e, m^*(Ba₈Ni_0.5Zn_6.5Ge_39.0) = 5.9m_e, m^*(Ba₈Ni_0.6Zn_6.9 Ge_38.5) = 6.2m_e. Adding a small amount of Ni to the compound enhances substantially the mobility μ of the charge carriers and the effective mass is slightly smaller. Nevertheless all three samples exhibit a similar μ(T) behaviour as observed for the ternary samples Ba₈Zn_7.33Ge_38.67 and Ba₈Zn_7.62Ge_38.38 indicating that the same scattering mechanisms are present. At low temperatures the T-dependence is governed by neutral impurity scattering and above 100 K a T^−1/2-dependence is observed, attributed to alloy scattering (shown by a black line in figure 8). At temperatures close to 300 K the decrease in the Hall-mobility is even more pronounced, indicating that another mechanism, the acoustic phonon scattering (μ ∝ T^−3/2, displayed by the dashed black line in figure 8), is dominating the electron transport [24].

The thermoelectric figure of merit ZT as a function of temperature for all samples is displayed in figure 19. In general ZT is increasing with temperature, but taking into account the uncertainties and error bars of all data sets used for the evaluation of the thermoelectric figure of merit, it is difficult to gain a clear conclusion on the compositional dependence. The main reason for the lower ZT of some samples is the slightly higher thermal conductivity.

Zoom In Zoom Out Reset image size

The improvement of the absolute values of ZT (0.65 at 800 K) in comparison with the ternary Ba₈Zn_xGe_46−x clathrates (ZT = 0.40 at 800 K for Ba₈Zn_7.78Ge_38.22) can be attributed to both a slight increase in the PF (S²/ρ) and a smaller thermal conductivity. All results of the physical property measurements as well as the fitting results (two-band model) are listed in table 4.

The temperature dependence of the electrical resistivity of all samples was investigated in the temperature range from 300 to 850 K (figure 20). All samples of this series exhibit a pronounced maximum shifting towards lower temperatures with increasing resistivity. As the temperature regions of the rise and descent of resistivity are sufficiently determined within the investigated range, the two-band model, including a Mott–Jones term described in section 3.2.2, can be applied. Results infer that the band gap is of the same order of magnitude as for the samples Ba₈Ni_yZn_xGe_46−x−y with y_nom = 1, but in most cases it is well below 4000 K (0.34 eV). The size of the band gap increases as the absolute resistivity decreases and the maximum is shifted to higher temperatures: Ba₈Ni_2.8Zn_2.4Ge_40.8 with a gap of 3590 K (0.31 eV), Ba₈Ni_2.3Zn_3.2Ge_40.5 3870 K (0.33 eV), Ba₈Ni_2.4Zn_3.6Ge₄₀ 3910 K (0.34 eV), Ba₈Ni_2.8Zn_1.5Ge_41.7 3930 K (0.34 eV) and Ba₈Ni_2.9Zn_1.2Ge_41.9 with a gap of 4310 K (0.37 eV), respectively. It is interesting to note that according to the fit, the Fermi-level of this set of samples is located much closer to the band gap compared with Ba₈Ni_yZn_xGe_46−x−y with y_nom = 1, which seems to be the main reason, why a maximum of ρ(T) is found already at temperatures around 500 to 600 K, whereas for the other two series of samples investigated in this work no features like this can be explored below ∼800 K. The Debye-temperatures extracted from the fits show larger scatter and range as compared with the samples with smaller Ni-content (table 4) between 211 K (Ba₈Ni_2.4Zn_3.6Ge₄₀) and 380 K (Ba₈Ni_2.8Zn_2.4Ge_40.8).

Zoom In Zoom Out Reset image size

For the two samples with the largest (Ba₈Ni_2.8Zn_2.4Ge_40.8) and smallest (Ba₈Ni_2.9Zn_1.2Ge_41.9) electrical resistivities, data were also measured down to 4.2 K. The results for the entire temperature range investigated are plotted in figure 21. The two-band model shows very good agreement with the measured data; only the logarithmic rise observed below 40 K (similar to the observations in the series Ba₈Ni_yZn_xGe_46−x−y (with y_nom = 1)) cannot be described. This increase in resistivity is much more pronounced for Ba₈Ni_2.8Zn_2.4Ge_40.8, whereas the electrical resistivity of Ba₈Ni_2.9Zn_1.2Ge_41.9 increases only by less than 10 µΩ cm within the temperature range below 15 K. Ba₈Ni_2.8Zn_2.4Ge_40.8 exhibits a more dramatic change: starting at 40 K the resistivity increases by more than 100 µΩ cm to 4.2 K (figure 15).

Zoom In Zoom Out Reset image size

Seebeck coefficients (S) of all samples were measured from 300 to 850 K and are displayed in figure 22. Similarly for electrical resistivity the Seebeck data also exhibit maxima, but the corresponding temperature is shifted to lower values. At 300 K the absolute values of the Seebeck coefficient of the various samples are increasing in the same sequence as the electrical resistivity does and exhibit negative values in the whole temperature range investigated.

Zoom In Zoom Out Reset image size

According to [38], a maximum in the Seebeck coefficient (S_max) can be used to estimate the band gap of semiconductors:

$$\begin{equation} S_{\max } \approx \frac{E_{\rm g} }{2eT_{\max } } \end{equation} \tag{ 3 }$$

E_g is the band gap, e the elementary charge and T_max is the temperature at the maximum of S. An analysis of the data using equation (3) reveals significantly smaller band gaps compared with the fitting results of the electrical resistivity. Whereas the Seebeck values suggest band gaps in the range 1926–2530 K (0.17–0.22 eV), the two-band model applied to the resistivity data revealed band gaps with an almost two times larger width (see also table 4). At least, also within this analysis the sample Ba₈Ni_2.8Zn_2.4Ge_40.8 exhibits the smallest gap width.

The thermal conductivity of all samples was investigated in the temperature range from 420 to 820 K. The results are shown in figure 23. All samples show a rather uncommon increase in the total thermal conductivity λ_tot with temperature, whereas the ternary clathrates Ba₈Zn_xGe_46−x and the quaternary series Ba₈Ni_yZn_xGe_46−x−y with y_nom = 1 exhibit an almost constant thermal conductivity above room temperature. Another interesting observation is that λ_tot of the two samples with the lowest Ni-content, Ba₈Ni_2.4Zn_3.6Ge₄₀ and Ba₈Ni_2.3Zn_3.2Ge_40.5, is significantly lower than for the other samples. Separating the phonon (λ_ph) and electron (λ_el) contributions applying the Wiedemann–Franz law (see figure 23) indicates that the increase in λ(T) can be partially attributed to an increase in λ_el, as the electrical resistivity is decreasing after passing through a maximum. Interestingly, the remaining part of the thermal conductivity, denoted in figure 23 as λ_ph, still exhibits an increase at elevated temperatures. As the phonon thermal conductivity, in general, should decrease with increasing temperature, this additional contribution is presumably caused by a bipolar effect. This bipolar thermodiffusion effect causes an additional contribution to the electronic thermal conductivity, whenever there are different types of charge carriers present in a certain material [39]. The presence of a sufficiently large amount of holes as charge carriers at temperatures above 500 K is also established by the decrease of both, electrical resistivity and Seebeck coefficient above this temperature.

Zoom In Zoom Out Reset image size

The thermoelectric figure of merit ZT as a function of temperature, shown in figure 24, also exhibits a maximum, which follows the drop in the absolute Seebeck coefficients at higher temperatures and the increase in the thermal conductivity, as both cannot be compensated by the decrease in the electrical resistivity. Interestingly the sample Ba₈Ni_2.4Zn_3.6Ge₄₀ with a small Ni-content reaches the highest ZT of 0.41 at 570 K. Although this value is not higher than the maximum ZT in Ba₈Zn_xGe_46−x, the thermoelectric figure of merit at this temperature is slightly improved, as ZT for the ternary samples does not exceed values of 0.3 at 570 K. Thermoelectric properties at 700 K and band gaps as well as Debye-temperatures of all samples are listed in table 4.

Zoom In Zoom Out Reset image size