Shantanu Misra1

Bartlomiej Wiendlocha2
email: wiendlocha@fis.agh.edu.pl

Janusz Tobola2

Florian Fesquet1

Anne Dauscher1

Bertrand Lenoir1

Christophe Candolfi1
email: christophe.candolfi@univ-lorraine.fr

Jean Lamour

Band Structure Engineering in Sn1.03Te Through In-induced Resonant Level

Narrow-band-gap IV-VI semiconductors represent a historically important class of thermoelectric materials. As one of the representative compound of this class, SnTe has been reinvestigated over the last years demonstrating its potential as a high-temperature p-type thermoelectric material. Here, we present a detailed study of the influence of very low doping levels of In, from 0.05% up to 2%, on the high-temperature transport properties of the selfcompensated Sn1.03Te compound. Our results evidence a strong impact of In on the transport properties, consistent with the presence of an In-induced resonant level (RL) in the valence bands of Sn1.03Te. This peculiar behavior is confirmed by electronic band structure calculations performed using the Korringa-Kohn-Rostoker method with the coherent potential approximation (KKR-CPA) revealing a narrow and sharp peak in the density of states (DOS) induced by the hybridization of the In s-states with the electronic states of Sn1.03Te. This 2 distortion in the DOS results in a spectacular increase in both the thermopower and electrical resistivity at 300 K. Although the influence of the RL is somewhat lessened at higher temperatures, a significant enhancement in the ZT values is nevertheless achieved with a peak ZT of 0.75 at 800 K which represents an increase of 35% over the values measured in Sn1.03Te.

Of relevance for practical applications, the weak dependence of the RL on temperature leads to enhanced average ZT value.

Introduction

Thermoelectric materials constitute a class of compounds that can convert a thermal gradient into an electric potential and vice versa, allowing for any waste heat to be harvested into useful electrical energy.1,2 The efficiency with which this energy conversion is realized is mainly determined by the material's dimensionless thermoelectric figure of merit defined as 𝑍𝑇 = 𝛼 2 𝑇/𝜌(𝜅 𝑒 + 𝜅 𝑝ℎ ) = 𝑃𝑇/(𝜅 𝑒 + 𝜅 𝑝ℎ ) where 𝛼 is the thermopower (or Seebeck coefficient), 𝑇 is the absolute temperature, 𝜌 is the electrical resistivity, 𝑃 = 𝛼 2 /𝜌 is the power factor and 𝜅 𝑒 and 𝜅 𝑝ℎ are the electronic and lattice thermal conductivity, respectively.1,2

Increasing the efficiency of a thermoelectric device relies on designing materials with high ZT values at the operating temperature. The optimization of this material's parameter is however notoriously challenging due to the intertwined nature of the above-mentioned transport properties. Most strategies developed can be divided into two main categories: the optimization of the power factor through band structure engineering and the minimization of the phononmediated heat transport.2

One possible way of optimizing the power factor is via the introduction of an impurity that gives rise to a so-called resonant level (RL) at the edge of either the valence or conduction bands. [START_REF] Heremans | [END_REF][4][5] The impurity atoms distort locally the electronic band structure beyond a rigid-band model. A significant increase in the thermopower and hence, in the power factor is then observed when the chemical potential resides in this distortion due to the increased density of states (DOS). In addition, a subtle balance between the localized nature of the RL and its hybridization with the states of the host structure must be achieved. This requires a proper selection of the resonant impurity, as well as its concentration. For example, a 3d transition metal impurity in IV-VI semiconductors may form a large DOS peak. However, due to a poor hybridization with the s-p host electronic states, the resonant level tends to localize by forming a narrow impurity band. On the other hand, too large concentration of the s-p-like RL may lead to too large smearing of the resonant level, suppressing its beneficial influence on the thermopower. Examples of these two extreme cases are provided by the PbTe:Ti and Sn-doped Bi2Te3 systems, respectively.6-9

The presence of a RL has been demonstrated in several chalcogenide compounds, most of them classified as state-of-the-art thermoelectric materials. Prominent examples include Sndoped Bi2Te3, Tl-doped PbTe or, more recently, Sn-doped -As2Te3.3,5,8-10 P-type SnTe has been intensively reinvestigated over the last years as a Pb-free analogue of PbTe with hightemperature thermoelectric properties rivaling those achieved in PbTe upon proper optimization through substitutions, nanostructuration or a combination of both.11-32 This compound recently entered the above-mentioned list of known RL with the identification of a RL induced by In substitution on the Sn site. 33 Intriguingly, In induces superconductivity over a wide range of concentrations with critical temperatures as high as 4.5 K,34-45 bearing resemblance with the anomalous, low-carrier-density superconductor Pb1-xTlxTe. [46][47][48] The possible connection of the In resonant behavior in SnTe with superconductivity is however yet to be determined.

The initial investigation of the high-temperature thermoelectric properties of the series Sn1-xInxTe for x = 0.0025, 0.005 and 0.01 identified the lowest In content studied as the optimum concentration. 33 Combined with reduced lattice thermal conductivity achieved by ball-milling, the resonant-driven enhancement of the power factor led to an improved ZT value of 1.1 at 873 K.33 Compared to other RL systems for which the optimum doping level is typically around 2%, this In concentration is significantly lower, possibly due to the large amount of Sn vacancies inherently present in pristine SnTe. Because the heavy hole doping induced by these vacancies may shift the chemical potential away from the RL, it is of interest to determine in more detail their influence on the thermoelectric properties of In-substituted SnTe. The introduction of excess Sn in starting compositions has been widely shown to be an effective way of reducing the vacancy concentration in SnTe, making the series Sn1.03-xInxTe worthy of further experimental and theoretical investigation.

Here, we investigate in detail the combined impact of Sn self-compensation and In substitution on the high-temperature thermoelectric properties of Sn1.03-xInxTe by electronic band structure calculations coupled with transport properties measurements (300 -800 K).

Through systematic variations in the In content from x = 0.0005 up to 0.02, we observe a stepwise increase in the 𝛼 and 𝜌 values at 300 K which evidences the strong impact of In on the transport properties despite the very low concentrations probed. A careful comparison of the experimental data with the Ioffe-Pisarenko plot, predicting the hole dependence of 𝛼, confirms that In also acts as a resonant impurity in Sn1.03Te. The excess Sn introduced leads to a shift of the optimum In concentration at which the highest thermoelectric properties are achieved.

Although the beneficial influence of the resonance is lessened at high temperatures, the Ininduced RL yields a peak ZT value of 0.75 at 800 K for x = 0.0035, which represents an improvement of 35% compared to the self-compensated sample Sn1.03Te.

Experimental and computational details

Synthesis and sample characterization

Polycrystalline samples Sn1.03-xInxTe (x = 0, 0.0005, 0.0015, 0.0025, 0.0035, 0.0045, 0.0075, 0.01 and 0.02) were prepared by powder metallurgy using stoichiometric quantities of Sn shots (99.999%), In shots (99.999%) and Te powders (99.999%). Prior to use, both Sn and Te were purified to eliminate possible traces of oxygen. These starting elements were placed in evacuated silica tubes sealed under secondary vacuum. Due to the high vapor pressure of SnTe at high temperatures that may influence the actual chemical compositions, all samples were carefully prepared in silica ampules of similar dimensions to ensure a nearly equivalent sampleto-volume ratio. The sealed ampules were heated in a rocking furnace to 1100 K, kept at this temperature for 5 h and then quenched in room-temperature water. The ingots were handground into fine powders that were consolidated by spark plasma sintering in graphite dies of 10 mm in diameter at 750 K under a uniaxial pressure of 65 MPa for 10 minutes. All the resulting cylindrical pellets were found to have geometrical densities exceeding 96% of the theoretical density.

Powder X-ray diffraction (PXRD) was performed at room temperature to check the crystal structure and the possible presence of secondary phases using a Bruker D8 Advance diffractometer (CuKα1 radiation).

Transport property measurements

The consolidated pellets were cut with a diamond-wire saw into bar-shaped ( 1.5×1.5×8 mm3) and disc-shaped ( 10 mm in diameter,  1 mm in thickness) samples.

Hall effect measurements were performed at 300 K with a five-probe configuration using the ac transport option of a physical property measurement system (PPMS, Quantum Design).

Good electrical contacts were realized on the bar-shaped samples by brazing copper wires onto the samples with a low-melting point braze. The Hall resistivity 𝜌 𝐻 was determined by measuring the antisymmetric component of the transverse 𝜌 𝑥𝑦 voltage under magnetic field reversal following the formula 𝜌 𝐻 = [ 𝜌 𝑥𝑦 (+𝜇 0 𝐻) -𝜌 𝑥𝑦 (-𝜇 0 𝐻)]/2. The Hall coefficient 𝑅 𝐻 was derived from the slope of the 𝜌 𝐻 (𝜇 0 𝐻) data for fields -1 ≤ 𝜇 0 𝐻 ≤ 1T. The Hall hole concentration 𝑝 and the Hall mobility 𝜇 𝐻 were determined from the single-carrier relations 𝑝 = 1/𝑅 𝐻 𝑒 and 𝜇 𝐻 = 𝑅 𝐻 /𝜌 where 𝑒 is the elemental charge.

The electrical resistivity and thermopower were simultaneously measured on bar-shaped samples between 300 and 800 K by a conventional four-probe method under a low helium pressure. The thermal diffusivity a was measured on the disc-shaped sample between 300 and 800 K with a Netzsch laser flash instrument (LFA 467) under continuous argon flow. Prior to measurement, the samples were spray-coated with a thin layer of graphite to ensure a homogeneous signal absorption and emission on the respective surfaces. The thermal diffusivity was used to calculate the total thermal conductivity 𝜅 via the formula 𝜅 = 𝑎𝐶 𝑝 𝑑 where 𝐶 𝑝 is the specific heat and 𝑑 is the experimental density determined from the weight and geometric dimensions of the consolidated pellets. 𝐶 𝑝 was approximated by the Dulong-Petit law 𝐶 𝑝 = 3𝑁𝑅 where 𝑁 is the number of atoms per formula unit and 𝑅 is the gas constant. The temperature dependence of the density was not taken into account in the present case.

Electronic band structure calculations

Electronic structure calculations were performed using the Korringa-Kohn-Rostoker (KKR) method with the coherent potential approximation (CPA), applied to account for the chemical disorder,49,50 as implemented in the Munich SPR-KKR package. [START_REF] Ebert | The Munich SPR-KKR package[END_REF] The experimental rock-salt crystal structure and lattice parameter (6.32 Å) were used. Regular k-point mesh was used in calculations, using 1500 -5000 k-points for the self-consistent cycle and 4 -20105 for the density of states (DOS) and Bloch spectral density functions (BSF) calculations (number of points given in the irreducible part of the Brillouin zone). The crystal potential was constructed in the framework of the local density approximation (LDA), using Vosko, Wilk and Nussair formula for the exchange-correlation part.52 For all atoms, angular momentum cut-off lmax = 3

was set and full-potential full-relativistic calculations were performed. High convergence limits were put on the self-consistent cycle (10-5 Ry for the Fermi level EF and for the total energy).

The position of EF was obtained using the Lloyd formula. 49 To analyze the influence of the impurities on the electronic bands, Bloch spectral density functions were computed. This technique allows to take into account the effect of electron scattering, due to presence of impurities, on the electronic structure, which can be intuitively visualized by plotting the smeared electronic bands. This approach has been already successfully applied to study the influence of resonant impurities on the band structure of several thermoelectric materials, such as PbTe:Tl,53,54 PbTe:Ti,6 Mg2Sn:Ag.55 This method further allows to distinguish between localized (e.g. PbTe:Ti) and delocalized (e.g. PbTe:Tl) resonant level,7 as well as between resonant and non-resonant impurity behavior.55 More details about the Bloch spectral functions can be found in Refs. 49 and 50.

Results and discussion

Electronic band structure

The electronic structure of the binary SnTe compound was widely discussed in literature. [56][57][58] SnTe is a narrow-band-gap semiconductor, with a band gap 𝐸 𝑔 = 0.18 eV at 300 K. 16 Its valence bands are mainly characterized by a light and a heavy-hole band, just like in its sister compound PbTe. Strictly speaking, the light and heavy-hole bands are not two different bands, but two maxima of the last valence band, one occurring at the L point and the second in the so-called  point (located half way between  and K in the 𝑘 𝑧 = 0 plane). The experimentally-determined energy offset between these two maxima is about 0.3 -0.4 eV. 16 As the presence of Sn vacancies is an inherent property of SnTe, making it naturally heavily hole doped, Sn vacancies were added in our electronic structure calculations to simulate the system close to the experimental one. Figure 1a shows the density of states of SnTe containing 0.32% of Sn vacancies. Sn vacancies act, as expected, as two-hole acceptors, shifting the Fermi level into the valence bands. This vacancy concentration corresponds to a hole concentration of 1020 cm-3, as typically observed in real SnTe samples, and is very close to the experimental value estimated in Sn1.03Te (0.25% corresponding to a hole concentration of 8.01019 cm-3).59 As a first approximation, vacancies do not visibly modify the electronic structure of SnTe, the effect of which is thus mostly a rigid-band-like shift in 𝐸 𝐹 . The situation is radically different when considering the DOS of SnTe doped with 0.5% of In and without Sn vacancies (Figure 1b) where the In-induced formation of the resonant level can be clearly seen.

The carrier concentration of this system is 7.91019 cm-3, as In is a monovalent acceptor (that is, in a In1+ valence state). The mechanism of In (usually a trivalent element with the 5s25p1 valence shell configuration) becoming an acceptor against Sn (divalent in SnTe as shown by We further considered the DOS of 0.5% In-doped SnTe containing 0.32% of Sn vacancies (Figures 1e and1f,) in order to analyze the influence of Sn vacancies on the formation of the resonant level. Three effects are observed: (i) RL on In is also formed in the presence of vacancies; (ii) 𝐸 𝐹 is deeper in the valence bands due to additional holes given by the vacancies;

(iii) RL is slightly smeared resulting in a slightly lower partial In DOS (Figure 1f). As in such a case, in calculations, three "atoms" (Sn, In and vacancy) occupy the same crystallographic site, one might expect some additional RL smearing effect due to enhanced disorder, compared to the vacancy-free compound.

To analyze the influence of vacancy and RL on the electronic bands, Bloch spectral density functions (BSF) were computed for the same compositions. Figure 2 shows the twodimensional projections of BSF plotted in a way similar to the usual dispersion relations in several BZ directions, with the color marking the value of BSF. Figure 2a presents BSF of ptype SnTe containing 0.32% of Sn vacancy. The general features of the electronic structure of SnTe are in agreement with previous density-functional calculations and experimental studies.56-58 What is worth noting is that the minimum of the conduction band appears to be slightly off the L point, which is better seen when bands in the off-high-symmetry L - direction are plotted. While signatures of this effect may be seen in other studies, (e.g. in Ref.

56 in the W-L direction), no direct comparison is possible, since bands in the L - direction were not previously computed in literature. Moreover, this effect may be related to the inaccuracy of the local density approximation as well. In contrast to PbTe, we are not aware of any calculation going beyond the local density approximation (LDA) or the generalized gradient approximation (GGA) for SnTe, that is, using the GW method. Nevertheless, this potential error would be more important for n-type compounds than the p-type compositions studied herein. The L-and -band separation is here about 0.15 eV which is lower than for the energy offset obtained in prior studies. This underestimation is a direct consequence of the inaccuracy of the local density approximation. Of note, this error will manifest in calculations of the effective mass 𝑚 𝐷𝑂𝑆 * as a function of the hole concentration, discussed below, as a too early increase. We note that a similar situation was reported for PbTe. 54 As far as the effect of vacancies on the band structure is concerned, we may observe a minor band smearing, leaving sharp, well-defined bands (Figure 2a). In contrast, as shown in Figure 2b, strong band smearing is observed for 0.5% In-doped SnTe containing 0.32% of Sn vacancies. This effect is obviously related to the presence of resonant In impurities and is very similar to what has been observed in PbTe:Tl.53,54 In the presence of In-induced RL, electronic states are shifted towards the valence band edge and form a "cloud" around the L - band.

However, these electronic states do not form an isolated impurity band due to hybridization between resonant In and host SnTe states. These conclusions hold true when the In concentration reaches 2% (Figure 2c): the number of electronic states around the L - band increases (which is responsible for the larger DOS "hump" in Figure 1d), band smearing is enhanced without forming an impurity band. To quantify the level of hybridization between the resonant and host states, the In atomic contribution to the DOS(𝐸 𝐹 ) has been calculated. The results indicate that, even for 2% In, it does not exceed 10%. Thanks to this hybridization, the localization of the RL or the formation of a narrow impurity band is avoided. The strong band smearing, induced by RL, causes enhanced carrier scattering, which will dominate the transport properties of the system at low temperatures. By analyzing the width of the spectral function 𝛥 (that is, the width of the smeared band), the electronic life time 𝜏 = ℏ 𝛥 ⁄ may be estimated. For k-points in the middle of L - path, this approach yields 𝜏 ~ 310-14 s for 0.5% In-doped SnTe and 𝜏 ~ 310-13 s for SnTe containing 0.32% of Sn vacancies. Similar magnitude of the carrier life time was obtained for PbTe:Tl.54 These results highlight the importance of carrier scattering on the resonant level. Since these calculations correspond to 𝑇 = 0 K, this predicted reduction of the lifetime due to RL, compared to self-doped SnTe, will lead to enhanced residual resistivities at low temperatures, which could be experimentally verified, if very good quality samples of pristine SnTe are available in order to avoid microstructure-related effects. Note that the link between the evolution of the residual resistivity and the carrier lifetimes should remain qualitative due to the fact that a direct relation between carrier lifetime and relaxation time that appears in transport quantities is not straightforward. Such increase in the residual resistivity was observed for the PbTe:Tl system when compared to PbTe:Na,54 and may be used as an independent, experimental verification of the existence of the RL. Extending the present measurements to liquid-helium temperatures will be the subject of a forthcoming study.

Enhanced carrier scattering due to the presence of RL is also responsible for the larger, concentration-dependent resistivity of SnTe:In samples at 𝑇 > 300 K, as discussed below in the experimental section. It should be stressed here that scattering on RL, which is strong in SnTe:In samples, is much less temperature-dependent than scattering of carriers on phonons. Because of this difference, the transport properties of "resonant" samples are expected to exhibit weaker dependence on temperature than rigid-band-impurities-doped samples. At higher temperatures where the carrier relaxation time due to scattering on phonons is on the same order of magnitude as the scattering on RL (𝜏 ~ 10-14 s), differences in 𝜌 for samples with and without resonant impurities is not as large as the difference in carrier life times at 𝑇 = 0 K might have suggested.

The ratio of the electrical resistivities for the most (2% In) and least (0% In) resistive samples in our studies (see below) changes from 5.4 at 300 K to 1.3 at 800K. This weaker temperature sensitivity is expected to give rise to enhanced peak power factor and ZT values, but also to higher ZT over a broader temperature range, yielding higher average ZT, which is an important parameter for practical applications.

A qualitative link between the RL-induced increase in DOS and its effect on thermopower can be inferred from the calculation of the DOS effective mass 𝑚 𝐷𝑂𝑆 * of the studied materials, using the method discussed in Refs. 54 and 65. The results of these computations as a function of the carrier concentration (obtained by rigidly shifting 𝐸 𝐹 in the DOS computed for the actual material's composition) are shown in Figure 3. Absolute mass values are affected by the LDA inaccuracy in two ways, as can be analyzed for the "self-doped" SnTe case. Firstly, the lightband mass, which has been determined to be about 0.168𝑚 0 (𝑚 0 is the bare electron mass) in prior studies, is about 0.4𝑚 0 in our case. Secondly, the rapid increase in 𝑚 𝐷𝑂𝑆 * near 71019 cm-3 in the case of SnTe with 0.32% Sn vacancies is caused by the too high position of the top of the heavy-hole  band, the effect of which should appear at higher hole concentrations.

Nevertheless, the region of interest is the region before the heavy-hole band is reached. In spite of these inaccuracies, a relative increase of the 𝑚 𝐷𝑂𝑆 * due to In doping is clearly observed.

Hence, higher In concentrations will result in higher 𝑚 𝐷𝑂𝑆 * , and ultimately, in higher thermopower values.

Summarizing this theoretical part, KKR-CPA calculations confirm the formation of a resonant level when In is substituted on the Sn site in SnTe. Most importantly, this effect is not affected by the presence of Sn vacancies, which have nevertheless a small, but visible effect on the details of the RL. Electronic bands are smeared but no separated impurity band is formed which would have been detrimental to electrical conduction. Finally, the In-induced RL increases the DOS effective mass, proportionally to the In concentration. As we will see below, the experimental results lend strong support to these predictions.

Crystal structure and sample purity

The phase purity of the densified Sn1.03-xInxTe samples has been assessed by PXRD (Figure 4).

These patterns show that all the synthesized samples are single phased within the detection limits of PXRD. All the peaks can be indexed to the NaCl-type cubic structure of SnTe described in the 𝐹𝑚3 ̅ 𝑚 space group. However, two additional peaks at 30.7° and 32.1° that correspond to elemental Sn can be observed. The presence of these peaks is consistent with the Sn excess systematically introduced in the initial compositions and located at the grain boundaries, as shown by transmission electron microscopy. 61 The absence of In-rich secondary phases up to the highest In content of 2% is in agreement with prior studies that reported a much higher solubility limit of In of around 40%. 37,38,44 The lattice parameter of the x = 0.02 sample (a = 6.3197(2) Å), inferred from Rietveld refinements against the PXRD data, is slightly lower than that determined for the x = 0.0 sample (a = 6.3242(2) Å). For intermediate In concentrations below 1%, no evident shift in the lattice parameter was observed due to the very low In contents investigated. These results are consistent with prior studies evidencing that In substitution exerts a chemical pressure on the unit cell with a weakly shrinking for In concentrations up to about 9% before decreasing more rapidly up to the 40% substitution level.37,38 Despite this detrimental impact on the 𝜌 values, the best compromise near room temperature is achieved in the x = 0.075 and 0.01 samples which show the highest power factor (Figure 6). Upon warming, 𝑃 no longer evolves linearly with the In content above 550 K due to the significant increase in 𝛼 experienced notably by the x = 0 and 0.005 samples.

Electronic transport properties

Nevertheless, except for the x = 0.02 sample for which 𝑃 is clearly lowered, all the samples exhibit slightly enhanced, similar 𝑃 values which are shifted towards higher temperatures with a maximum value of 1.810-3 W m-1 K-2 achieved at 800 K. These results are consistent with those obtained by Zhang et al. 33 for the series Sn1-xInxTe. The enhancement of the power factor at high temperatures, as we already mentioned, is possible due to the fact that phonon scattering and resonant level scattering are on the same order of magnitude near room temperature. Upon warming, the scattering on RL is less temperature-dependent, although still strong, which allows to take advantage from the increase in 𝛼.

The influence of In on the Hall hole concentration 𝑝 𝐻 and Hall mobility 𝜇 𝐻 has been probed by Hall effect measurements performed at 300 K. Of note, 𝑝 𝐻 is related to the chemical hole concentration 𝑝 by the relation 𝑝 𝐻 = 𝑟 𝐻 𝑝 where 𝑟 𝐻 is the Hall factor usually taken equal to unity. However, it was shown in Pb1-xTlxTe that the RL affects this factor in degenerate samples, the extent to which depends on the concentration of the resonant impurity and temperature. 69 The deviations from unity were predicted to be the largest at low temperatures and low Tl concentrations while the effect on 𝑟 𝐻 remains moderate near room temperature. In the present Sn1.03-xInxTe series, assuming 𝑟 𝐻 ≈ 1, and hence 𝑝 𝐻 ≈ 𝑝, near 300 K should therefore provide a reasonable estimate of 𝑝. The dependence of 𝑝 𝐻 and 𝜇 𝐻 on x, shown in Figure 7, confirms a rapid and strong reduction in 𝜇 𝐻 with increasing x despite the very low doping levels considered herein. Meanwhile, 𝑝 𝐻 tends to slightly but continuously increase with

x suggesting that In behaves as an acceptor impurity in SnTe in agreement with our electronic band structure calculations. This is further confirmed by the slope of 𝑝 𝐻 versus the concentration of In atoms per cm-3 (not shown) which corresponds to about 0.6 holes per In atom. The evolution of these two parameters with x confirms that the reduction of 𝜇 𝐻 is the main factor that drives the observed increase in 𝜌. Such a reduction in 𝜇 𝐻 is expected in the presence of a RL, as mentioned above.

One straightforward and convenient way of revealing the resonant nature of an impurity atom is tied to the carrier concentration dependence of the thermopower known as the Ioffe-Pisarenko plot. The resonant behavior manifests itself by significant deviations from the conventional dependence followed by a rigid-band-like impurity. In the present case, this analysis is complicated by the presence of two valence bands that govern the transport of SnTe and the number of band parameters that need to be taken into account. Moreover, the peak value of 𝛼 seems to be enhanced and shifted towards lower hole concentration compared to the extremum predicted slightly below 1021 cm-3. The marked difference between a rigid-band model and experiment provides another strong evidence of the resonant character of In in Sn1.03Te.

As a consistency check, we tried to determine whether changes in the band parameters used to calculate the theoretical curve could explain the observed trend. The results, presented in Figure S1 in the ESI file, show that the density-of-states effective masses of the heavy-hole and light-hole bands are the two main parameters that control the peak in 𝛼 and its position.

However, an increase in the magnitude of the peak and its shift towards lower hole concentration are observed when both effective masses significantly decrease, in contradiction with the evolution of 𝑚 𝐷𝑂𝑆 * predicted by our electronic band structure calculations. It seems therefore difficult to reconcile these theoretical results with the experimental data without considering the presence of an In-induced resonant RL. Since the main role of the RL may be qualitatively understood as an increase in the density-of-states effective mass, its influence on the hole concentration dependence of the thermopower cannot be captured by the abovementioned two-valence-band model using a single set of band parameters.

Thermal transport properties

Figure 9 shows the temperature dependence of the total thermal conductivity 𝜅. For all samples, 𝜅 decreases with increasing temperature and tends to saturate above 700 K to values ranging between 2.1 and 2.7 W m-1 K-1. In contrast to the roughly similar values observed at the highest temperatures, 𝜅 strongly varies with the In concentration near room temperature from 7.9 W m-1 K-1 in the x = 0 sample down to 3.9 W m-1 K-1 in the x = 0.02 sample. While the temperature dependences of 𝜅 up to x = 0.0045 are very similar, clear differences emerge for higher In concentrations. In addition to an overall decrease with temperature strongly lessened, a crossover near 500 K is clearly visible. Above this temperature, 𝜅 no longer decreases with increasing x for x  0.0045 but surprisingly shows the opposite trend. This reversed variation with x likely reflects significant differences in the electronic thermal conductivity 𝜅 𝑒 . This Considering as valid the hypothesis that the effect of alloying with In up to x = 0.02 hardly affects 𝜅 𝑝ℎ , the strong increase in 𝜌 with x then provides an alternative approach to estimate the temperature dependence of 𝐿 in the present case. The weak temperature dependence of 𝜅 and the absence of significant upturn in 𝛼 above 500 K in the x = 0.02 sample further suggests a negligible role of the second heavy-hole valence band. Assuming that the 𝐿 values calculated for the x = 0.02 sample yield a correct estimate of 𝜅 𝑝ℎ , we used the same 𝜅 𝑝ℎ values to calculate the temperature dependence of 𝜅 𝑒 expected for the other samples.

Following this approach, the expected temperature dependences of 𝐿 were derived from 𝜅 𝑒 (𝑇) for all samples (Figure 10). Except for the x = 0.02 sample, the 𝐿 values are strongly temperature dependent and drop well below the non-degenerate limit of 1.510-8 V2 K-2 expected for acoustic phonon scattering. For the x = 0.0035 sample, 𝐿 even reaches 0.810-8 V2 K-2 above 600 K. The upturn in 𝐿 observed at high temperatures for low doping levels is likely an artifact, possibly due to minority carrier activation across the band gap of 0.18 eV. These results show that the Lorenz numbers should be significantly lower than those predicted by simple transport models to obtain consistent estimates of 𝜅 𝑝ℎ across the entire series of samples.

Such low values likely reflect the multiband character of the valence bands which has been shown to give rise to significant deviations from the 𝐿 values expected in single-band systems.72,73 In addition, because the relaxation time strongly depends on energy when the chemical potential nears the resonant state due to its significant width in energy (see Figure 2), scattering of the carriers may become strongly inelastic, thereby leading to significant variations in the Lorenz number. In particular, this effect has been discussed in the PbTe:Tl compounds for which, large deviations of 𝐿 from its degenerate limit has been shown to occur depending on the position of the chemical potential with respect to the resonant-band center. 69 The fact that the largest deviations of 𝐿 are observed in the x = 0.0035 sample, which features the most pronounced RL effect, is an intriguing indication that the RL has a strong impact on the Lorenz number. It is thus likely that these two combined effects are responsible for the anomalous Lorenz numbers inferred and provide a consistent explanation of the large differences in the 𝐿 values observed from sample to sample despite the very narrow In composition range covered in this series.

Dimensionless thermoelectric figure of merit ZT

The temperature dependence of the ZT values, shown in Figure 11, evidences a clear increase at 800 K upon substituting In for Sn. A maximum ZT value of 0.75 is achieved for x = 0.0035 which represents an increase of 35 % compared to the ZT value of the pristine sample Sn1.03Te.

This peak value is lower than that achieved by Zhang et al.33 due to the lower total thermal conductivity achieved by a reduction of the grain size via ball-milling. In our case, the significant increase observed in the 𝜌 values with increasing x is the main limiting factor that prevents higher peak ZT values from being achieved at higher In concentrations. However, interestingly, the average ZT value 𝑍𝑇 𝑎𝑣𝑔 , an important parameter for practical applications determined by integrating with respect to temperature over an extended temperature range,74 is maximized in the x = 0.0035 sample. Compared to the pristine sample where 𝑍𝑇 𝑎𝑣𝑔 = 0.45 over the 600 -800 K temperature range, a concentration as low as x = 0.0035 is sufficient to increase this value to 0.58 over the same temperature range (Table 1). This result shows that the weak temperature dependence of the RL helps to enhance 𝑍𝑇 𝑎𝑣𝑔 even though the increase in the peak ZT value is moderate.

Conclusions

We reported on a detailed joint theoretical and experimental study of the influence of In on the high-temperature thermoelectric properties of Sn1.03Te. KKR-CPA calculations have demonstrated the formation of a resonant state upon substituting In for Sn. The RL formation is only weakly affected by the presence of Sn vacancies, which have a small, but visible effect on the details of the resonant state. The computed Bloch spectral functions show that electronic bands are strongly smeared in SnTe:In but without forming a separated impurity band. The In-induced resonant state increases the DOS effective mass, proportionally to the In concentration.

Our extensive collection of experimental data obtained on samples with In concentrations spanning the range 0  x  0.02 are all consistent with the presence of a RL disturbing the valence band structure of Sn1.03Te. Despite the very low In contents of the present series, significant differences were observed with a stepwise increase with x in the thermopower and electrical resistivity. The latter trend originates from the strong impact of the RL on the hole mobility, the x dependence of which cannot be captured within a rigid-band-like scenario.

Compared to prior studies on In-substituted SnTe, the maximum power factor is shifted towards higher In concentrations due to the Sn self-compensation used herein. This difference highlights the needs to adjust the carrier concentration in RL-containing systems so that the position of the chemical potential is optimized against that of the RL. Although the beneficial influence of the RL observed at 300 K is somewhat lessened at 800 K, an increase in the ZT values is nevertheless obtained with a peak value of 0.75 at 800 K for x = 0.0035, which represents an increase of 35% compared to Sn1.03Te. Another important consequence of the RL for practical applications is its weak temperature dependence that yields a higher average ZT value compared to Sn1.03Te. The progressive increase in the In content has also a profound influence on the thermal transport with a clear evolution in the temperature dependence of the total thermal conductivity as x increases. In particular, the strong increase in 𝜌 with x enabled us to shed light on the difficult task to correctly evaluate the Lorenz number in SnTe due to the multiband nature of its valence bands. Based on a set of reasonable assumptions, our analysis unraveled very low Lorenz numbers that even drop below the non-degenerate limit. This result highlights that, despite the high degeneracy of the charge carrier gas, the multiband character of the electronic bands and the presence of the RL combine to significantly lower the Lorenz numbers above 300 K, making it difficult to adequately estimate the lattice thermal conductivity. Finally, the sensitivity of the transport properties to the concentration of the RL impurity further evidences Tables Table 1. Average ZT values calculated over the 300 -600 K and 600 -800 K temperature ranges for all samples.

In concentration , observed in the case of SnTe + 0.32% Sn vacancies near 71019 cm-3, is caused by the too high position of the top of the " band", which should appear at significantly higher hole concentrations.

  Mössbauer spectroscopy inRef 59, 60 and 61) is similar to that of In in Bi (Ref.62) or Tl in PbTe.53,54,63,64 One 5s electron of In forms a hyper-deep (HD) state in the low-energy region, below the p-like main valence band (VB), which bounds one 5p electron from Te taken from the main VB. One 5p electron of In contributes to the main valence band block, and the second 5s electron of In forms the resonant state. As a result, upon substituting In for Sn, the main VB is left with one electron less, making In an acceptor. The main difference with the Bi:In system is that In does not create there a resonant level near 𝐸 𝐹 .The formation of the DOS "hump", in agreement with previous reports,4,7,64 is related to the presence of In atoms. As shown in Figure1cwhere the partial DOS of In atoms is shown, the DOS peak formed by In 5s states defines the resonant behavior of In in SnTe. When the In concentration increases to 2% (Figure1d), the DOS "hump" broadens and the hole concentration increases.

Figures 5a and 5b

  Figures 5a and 5b show the temperature dependences of the electrical resistivity and

  Pioneering studies onSnTe byBrebrick et al. (Refs. 66, 67 and 68) have shown that a two-valence-band model using a combination of a parabolic heavy-hole band and non-parabolic light-hole band provides a good description of the experimental data over two orders of magnitude of the hole concentration. As shown in Figure8, the comparison of our experimental data at 300 K with the theoretical curve show that all the data of the In-substituted samples strongly deviate from the predicted variation. For the x = 0.0 sample, as discussed elsewhere,61 the deviation from theory may come from the single set of band parameters used to calculate the Pisarenko curve, some of which may likely vary with varying the Sn vacancy concentration. For the Incontaining samples, a steep rise in the 𝛼 values with increasing the In content is observed around 1020 cm-3 while the model predicts only little variations around this hole concentration.

  hypothesis is justified by the low In concentrations and the small difference in molar mass between Sn and In which are expected to result in weak mass, binding force and strain field differences induced by In alloying. Thus, according to the relation proposed by Callaway and Klemens to determine the evolution of 𝜅 𝑝ℎ in solid solutions,70,71 the difference between 𝜅 𝑝ℎ of pure Sn1.03Te and that of the In-containing samples should be very small. Although disentangling the electronic and lattice contributions to 𝜅 by estimating 𝜅 𝑒 via the Wiedemann-Franz law 𝜅 𝑒 = 𝐿𝑇/𝜌 would shed further light on this behavior, this subtraction requires to obtain a correct estimate of the Lorenz number 𝐿. However, this parameter is difficult to determine in SnTe due to its strong dependence on the details of the electronic band structure and on temperature. For most of SnTe-based compounds, a simple single-parabolic-band model with acoustic phonon scattering is usually used to determine the temperature dependence of 𝐿. While this model provides reasonable estimates of 𝜅 𝑝ℎ in samples exhibiting low hole concentrations, it fails to yield consistent results for heavily hole-doped samples. For the present series, neither a single-parabolic-band model nor a two-valence-band model with temperature-dependent band parameters (FigureS2in the ESI file) provides a satisfying approach to determine 𝐿. Both models tend to overestimate 𝐿 giving rise to large differences in 𝜅 𝑝ℎ among the samples (FigureS3in the ESI file), which are not expected given the very low Sn vacancy concentration in Sn1.03Te (estimated to be around 0.25%, Ref. 61) and the narrow In concentration range investigated herein. In addition, none of these models capture well the change in the transport regime observed above 500 K, the temperature above which the second heavy-valence band plays a significant role on the electronic conduction and hence, on 𝐿.

Figure 1 .

 1 Figure 1. Densities of states as a function of the energy for SnTe: a) with 0.32% Sn vacancies;

Figure 2 .

 2 Figure 2. Two-dimensional projection of the Bloch spectral functions in high symmetry

Figure 3 .

 3 Figure 3. Hole dependence of the density of states effective mass 𝑚 𝐷𝑂𝑆 *

Figure 4 .

 4 Figure 4. PXRD patterns of the samples in the series Sn1.03-xInxTe (0  x  0.02). For clarity,

Figure 5 .

 5 Figure 5. Temperature dependence of a) the electrical resistivity 𝜌 and b) thermopower 𝛼 of

Figure 6 .

 6 Figure 6. Power factor 𝛼 2 /𝜌 as a function of temperature for the series Sn1.03-xInxTe (0  x 

Figure 7 .

 7 Figure 7. Variations in the hole concentration 𝑝 and Hall mobility 𝜇 𝐻 as a function of the In

Figure 8 .

 8 Figure 8. Ioffe-Pisarenko plot showing the thermopower 𝛼 as a function of the hole

Figure 9 .

 9 Figure 9. Total thermal conductivity 𝜅 as a function of the temperature for the series Sn1.03-

Figure 10 .

 10 Figure 10. Temperature dependence of the Lorenz number 𝐿 for the Sn1.03-xInxTe samples (0 

Figure 11 .

 11 Figure 11. Dimensionless thermoelectric figure of merit ZT as a function of temperature for the

  Figure 1

Acknowledgements B. W. was supported by the National Science Center (Poland), project No. 2017/26/E/ST3/00119. J. T. was supported by the AGH UST statutory tasks no. 11.11.220.01/5 within subsidy of the Ministry of Science and Higher Education.

that the combination of an RL impurity with a co-doping element should be carefully chosen to avoid the shift of the chemical potential away from the RL or the disappearance of the RL due to band structure modifications induced by the second element.

Electronic Supplementary Information (ESI) available.

Conflicts of interest

The authors declare no competing financial interest.


A Zhadan

V Sarou-Kanian

L Del Campo

L Cosson

M Malki

C Bessada

Transport properties in molten carbonates: self-diffusion and conductivity measurements at high temperature

Keywords: Molten carbonates, high temperature, PFG NMR, electrical conductivity, CO2, selfdiffusion

A better understanding and use of molten carbonate fuel cells (MCFCs) requires more detailed consideration on transport properties in melt. The combination of different methodological developments can be one solution to improve our comprehension. Here we present 23 Na and 7 Li self-diffusion coefficients measured by pulsed field gradient (PFG) NMR technique combined with electrical conductivity obtained by a 4-electrod set up, in the eutectic mixture Li2CO3-Na2CO3 (52:48 %mol) at high temperature (up to 1050 K), and under pure CO2 atmosphere. The results were compared with known experimental data from literature obtained by radiotracers techniques and 2-electrod set up and also with some calculations of the transport properties.

Introduction

The capture of CO2 and its exploitation has become a significant worldwide issue due to environmental reasons, and could supply an inexhaustible energy source [1]. Among the different methods proposed to take advantage of this CO2 storage , the electrolysis of CO2 in molten carbonate batteries, the molten carbonate fuel cells (MCFCs) [START_REF] Fuel Cell Today | The Fuel Cell Industry Review[END_REF], is one of the most efficient solution. Actually, molten carbonates are known to be a good CO2 solvent. These systems are liquids in very large temperature and pressure ranges. They have low viscosity and a significant capability to dissolve volatile compounds such as water and CO2 [START_REF] Kono | Ultralow viscosity of carbonate melts at high pressures[END_REF][START_REF] Jones | Carbonate melts and carbonatites[END_REF]. Molten carbonates are difficult to study experimentally because of their high melting temperatures: from 670 K for the eutectic composition Li2CO3-Na2CO3-K2CO3 (43.5-31.5-25 %mol) to 1164 K for pure K2CO3. In addition, these molten salts are corrosive toward different materials usually used for sample holder in different experimental techniques. Cell materials have to fulfil at least three conditions: resisting to high temperature, resisting to the corrosion by the salts and not perturbing the detected signal. The quite recent commercialization of high-quality ceramic materials able to resist to high temperature and to corrosion by molten carbonates has opened new perspectives for spectroscopic and conductivity measurements.

Two experimental techniques were used in this paper for a better understanding of the structure and the dynamics of molten carbonates: nuclear magnetic resonance (NMR) [START_REF] Bonafous | High temperature NMR observation of mobile phases up to 1500 ° C[END_REF] and electrical conductivity measurements both in the liquid state at high temperature [START_REF] Gaillard | Carbonatite melts and electrical conductivity in the asthenosphere[END_REF] up to 1300 K.

Pulsed field gradient NMR at high temperature (HT PFG NMR) particularly provide a valuable insight into the dynamics of molten salt through the measurements of self-diffusion coefficients [START_REF] Rollet | Measuring self-diffusion coefficients up to 1500 K: A powerful tool to investigate the dynamics and the local structure of inorganic melts[END_REF]. This technique is selective and give unique information for each observable nucleus, depending on the mobility of anions and cations in molten carbonates.

The self-diffusion coefficient D, is related to the ability of atoms or groups of atoms to move in their environment. The measurement of D is generally made by the capillary method and radioactive tracers [START_REF] Spedding | Trace-Ion Diffusion in Molten Alkali Carbonates[END_REF]. A radioactive salt is put in a platinum capillary and plunged in the same non-radioactive molten salt. After a selected time of diffusion, the capillary is cooled and solidified, and then is cut in different segments were the radioactivity is counted. This method is precise but difficult to handle. It requires the use of radioactive species whose period may be short. Because of its difficult implementation and the requirement of radioactive tracers, this technique have been limited and the self-diffusion coefficients are now commonly estimated through electrochemical experiments. The development of pulse field gradients (PFG) NMR at high temperature provides different advantages: the selectivity toward the isotope, the direct determination of D over a wide diffusion time range (10 -9 -10 -10 m 2 /s), and the possibility to perform in situ experiments up to 1000 K. The results will be connected with electrical conductivity measurements realized by the 4-electrode cell system more adapted for highly conductive liquids at high temperature.

Experimental

Samples preparation

Carbonate mixtures in the binary system Li2CO3-Na2CO3 were prepared from dried powders Li2CO3 (99.998%), Na2CO3 (99.997%) from Alfa Aesar. Sodium carbonate is hygroscopic, and all mixtures were prepared in a glove box under dried argon.

For NMR experiments, alumina (Al2O3) crucibles were directly filled by carbonates and stored in a gloves box at ambient temperature. For electrical conductivity measurements, the same mixtures of carbonates were melted in alumina crucible under air, and after cooling, stored in the electric oven at 423 K.

Materials characterization

NMR

High-temperature NMR experiments were performed with a laser heating system developed at CEMHTI, Orleans (France) [START_REF] Lacassagne | Study of the solide/liquid transition of the cryolite by nuclear magnetic resonance of 27 Al, 23 Na et 19 F[END_REF]. This system consists in a Bruker AVANCE I NMR spectrometer (magnetic field of 9.4 T), associated with two CO2 lasers (λ = 10.6 µm, 250 W), which allows heating the sample container (crucible) from top and bottom sides, insuring a good homogeneity of temperature inside the crucible.

For other molten salts systems such as fluorides or chlorides, the crucible is made usually from boron nitride (BN) [START_REF] Machado | Study of NaF-AlF3 Melts by Coupling Molecular Dynamics, Density Functional Theory, and NMR Measurements[END_REF] with no interaction and a good thermal conductivity during the experiments. However, in the case of molten carbonates, the salt reacts with BN at high temperature and create dangerous gases such as cyanamides [START_REF] Ostrovskaya | Crystallization of boron nitride from solution in a lithium borate melt[END_REF]. We have to find new materials for the container, fulfilling the three criteria for high temperature NMR experiments: non-metallic so not shielding the NMR radio frequency; inert toward molten carbonates; good absorption of CO2 laser for efficient heating; good thermal conductivity insuring temperature homogeneity. Three refractory materials were tested: alumina, zirconia and magnesium oxide.

Alumina has a moderate thermal conductivity and high melting temperature, but it can interact with molten carbonates at temperature higher than 1073 K to create meta-aluminate (MAlO2) [START_REF] Gál | Reactions of binary and ternary alkali metal carbonate mixtures with aluminium oxide[END_REF]. Zirconia (ZrO2) is inert to molten carbonates but has a low thermal conductivity.

Finally, magnesium oxide (MgO), has a good thermal conductivity and high melting temperature and don't interact with molten carbonates. The geometrical complexity of the crucible with plug, creates some difficulties in the fabrication, but now we find some compromise between the geometry and the quality of MgO, to design specific containers adapted to this experiments. Taking advantages of the properties of Al2O3 and MgO, the crucible is a new configuration associating a closed oxide magnesium crucible with an alumina crucible inside that can be filled in a gloves box under dried argon (Figure 1).

Figure 1.

The NMR spectra were collected for the Li2CO3-Na2CO3 (52:48 %mol) eutectic composition. The chemical shifts of 23 Na and 7 Li were referenced to 1M aqueous solutions of NaCl and LiCl respectively. Samples were heated up to 993 K. For every experiment, the acquisition started after 6 min of heating to be sure that the thermal equilibrium inside the crucible was reached.

Self-diffusion coefficients measurements were performed using PFG NMR technique specifically adapted for molten salts up to high temperature [START_REF] Rollet | Measuring self-diffusion coefficients up to 1500 K: A powerful tool to investigate the dynamics and the local structure of inorganic melts[END_REF][START_REF] Rollet | Self-diffusion coefficient measurements at high temperature by PFG NMR[END_REF]. A Bruker Diff30 probe specially modified for high temperature experiments was used allowing sufficiently high gradient strengths (up to 1000 G/cm for this study) in conjunction with a convection-free NMR pulse sequence proposed by Jerschow et al. [START_REF] Jerschow | Convection Compensation in Gradient Enhanced Nuclear Magnetic Resonance Spectroscopy[END_REF] in order to get a proper diffusion measurement.

The self-diffusion coefficient of both cations, 7 Li, and 23 Na are registered over at least 300 K above the melting point, by simply changing the resonance frequency. It is just to remark that this technique is more efficient and rapid than the radiotracers diffusion, and don't need to involve any additional treatment of the data, nor destruction of the sample.

For the 13 C and 17 O self-diffusion measurement, the challenge will be the enrichment of the sample by these two isotopes. 17 O, the only observable isotope of oxygen, with 0.037% of natural abundance, will not allow any direct measurement, while for the 13 C isotope of carbon, 1.11% abundant, some commercial Na2CO3 and Li2CO3 enriched by 13 C, are now available, and it will be the aim of another paper. For this study, at first we will use the experimental data obtained by Spedding et al. [START_REF] Spedding | Tracer Diffusion Measurements in Mixtures of Molten Alkali Carbonates[END_REF] for CO 3 2-by capillary method.

From our measurements we will extract the diffusion coefficients for both alkali, and the corresponding activation energies. They will be compared with experimental and theoretical data from literature.

Electrical conductivity

Impedance spectra were recorded using a SOLARTRON SI 1260 spectrometer in the frequency range of 1Hz to 1MHz. The conductivity cell used for our measurements, constituted of 4 Pt-electrodes, is described in [START_REF] Simonneta | Electrical conductivity measurements of oxides from molten state to glassy state[END_REF]. This set up is specifically adapted for measurements of the electrical conductivity of highly conductive liquids at high temperatures. The outermost electrodes are two plates imposing a current flow whereas the innermost electrodes are two wires measuring the voltage drop. The use of a four electrode set up minimizes the polarization effects at the sample/electrode interface. In theory, such polarization effects should disappear with 4 electrodes but classical plots of the real vs. the imaginary part of the complex impedance do show small residual polarization effects for frequencies below ~10 Hz.

Sample temperatures were measured by a platinum thermocouple (type S) covered by an alumina sheath that was inserted into the system of crucibles containing the melt. This system consists of two alumina crucibles and one steel crucible. The first alumina crucible contains the carbonate sample, the second metallic one protects the guard crucible from the molten salts which could get in, if the first alumina crucible breaks. The third (and last) alumina guard crucible protects the oven from both, the molten carbonates and the gazes. All measurements were made under 1 atm CO2 in the temperature range from about 373 K to about 1023 K. The conductivity cell was calibrated by the 1M KCl solution at room temperature for different depth of immersion of electrodes. Details of the measurement procedure have been reported in the literature [START_REF] Miyazaki | An Apparatus for Electrical Conductance Measurements with Molten Carbonates[END_REF][START_REF] Lair | Theoretical predictions vs. experimental measurements of the electrical conductivity of molten Li2CO3-K2CO3 modified by additives[END_REF].

Results and discussion

3.1 23 Na and 7 Li NMR spectra 23 Na and 7 Li spectra at different temperatures are presented in The evolution of 7 Li and 23 Na chemical shifts with temperature are presented in Figure 3. The chemical shifts decrease slightly with temperature in a very narrow range (~ 1 ppm) such that it is difficult to state if the present variation of chemical shift with temperature results from physical (density) or chemical (local structure composition) changes.

23 Na and 7 Li PFG NMR

As far as we know, this is the first time that the self-diffusion coefficient for both Li + and Na + ions in molten alkali carbonate Li2CO3-Na2CO3 (52:48 %mol) eutectic mixture has been reported in the literature. Experimental diffusivity data provided by Spedding et al. [START_REF] Spedding | Tracer Diffusion Measurements in Mixtures of Molten Alkali Carbonates[END_REF],

who measured the self-diffusion coefficients of Na + and CO 3 2-ions in molten binary alkali carbonate Li2CO3-Na2CO3 (52:48 %mol) by capillary diffusion apparatus [START_REF] Spedding | Trace-Ion Diffusion in Molten Alkali Carbonates[END_REF], and diffusivity data calculated by molecular dynamics simulations (Desmaele et al. [START_REF] Desmaele | Atomistic simulations of molten carbonates: Thermodynamic and transport properties of the Li2CO3 -Na2CO3 -K2CO3 system[END_REF]) were compared with the results of the present work in Figure 4.

Figure 4.

The values of DNa and DLi are in the same order of magnitude than other alkali salts (10 -9 m 2 /s) [START_REF] Rollet | Measuring self-diffusion coefficients up to 1500 K: A powerful tool to investigate the dynamics and the local structure of inorganic melts[END_REF][START_REF] Rollet | Self-diffusion coefficient measurements at high temperature by PFG NMR[END_REF]. For both cations, D increases with temperature increase, confirming the thermal activation. It is worth noting that in our case, sodium cations diffuse more rapidly than lithium ones, which contradicts the simulation results [START_REF] Desmaele | Atomistic simulations of molten carbonates: Thermodynamic and transport properties of the Li2CO3 -Na2CO3 -K2CO3 system[END_REF]. Given the smaller size of lithium, it would be expected that it diffuses more quickly than sodium. Nevertheless, lithium has a higher ionic potential than sodium [START_REF] Hai-Jun | Ion Potential Principle Summary[END_REF], such that the lifetimes of the Li solvation shell is longer than that of Na.

The Arrhenius plot (figure 5) of D vs T provides an activation energy of 0.22 ± 0.01 eV and 0.23 ± 0.02 eV for sodium and lithium, respectively. The activation energy for DNa given by Spedding et al. [START_REF] Spedding | Tracer Diffusion Measurements in Mixtures of Molten Alkali Carbonates[END_REF] is higher, of about 0.48 eV, and Desmaele et al. [START_REF] Desmaele | Atomistic simulations of molten carbonates: Thermodynamic and transport properties of the Li2CO3 -Na2CO3 -K2CO3 system[END_REF] reported values between 0.33 and 0.4 eV for the Na + and Li + cations. Spedding et al. published also some experimental values for the carbon diffusion, attributed to the CO 3 2-anion. The values reported for Na + and CO 3 2-are of the same order, and suggests some kind of joint action between cations and anions. The diffuse motion in this melt occurs as a result of some combined action that can be related to the vacancy mechanism, cluster movement or ion-pair motion. In the very next future, we will measure the 13 C self-diffusion coefficients for CO 3 2-, in order to validate this assumption.

Electrical conductivity measurements

Conductivity measurements of molten Li2CO3-Na2CO3 (52:48 %mol) eutectic mixture were performed in the temperature range from 750 to 1050 K. The lower limit in temperature is imposed by the solidification of the sample. Impedance measurements were performed for frequencies ranging from 1 Hz to 1 MHz. The method of determination of static conductivity consists in representing the impedance in the complex plane. In the case of an ideal solid electrolyte, the electrical properties can be simulated by a simple parallel RC circuit, and the presentation of the impedance in the complex diagram (Cole-Cole representation) is an arc whose intersection with the real axis gives access to the dc-resistance of the sample. In our case, the plot of the imaginary part of the impedance as a function of its real part corresponds to an arc of a circle, but whose centre is slightly below the real axis, indicating a deviation from Debye behaviour [START_REF] Abelard | Dielectric relaxation in alkali silicate glasses: A new intepretation[END_REF]. At high temperature, the impedance of the sample is too small and the spectra exhibit a vertical line caused by the electrode inductance at high frequencies [START_REF] Yoshino | Effects of pressure and water on electrical conductivity of carbonate melt with implications for conductivity anomaly in continental mantle lithosphere[END_REF].

This representation also makes it possible to check the absence of contribution of the sample-electrode interface in the studied frequency range. A typical example of the obtained graph is shown in Figure 6 for Li2CO3-Na2CO3 (52:48 %mol) eutectic mixture at different temperatures. In the absence of new contributions at lower frequencies, the intersection of the arc with the real axis gives a good approximation of the static resistance at the given temperature, and therefore provides the conductivity of the carbonate mixture by the multiplication with the geometric factor of the sample.

Figure 6.

The values of the static conductivity as a function of temperature are presented on figure 7 and are compared with previous results obtained by Kojima et al. [23]. These authors used the 2-electrode method requiring corrections on the measured resistance. Our data are slightly higher than those of Kojima et al., which is not unexpected as the 4-electrode method used in this work avoids the wire resistance contribution to the impedance of the sample contrary to the system used by Kojima et al. [START_REF] Kojima | Physical Properties of Molten Li2CO3-Na2CO3 (52:48 mol%) and Li2CO3-K2CO3 (62:38 mol%) Containing Additives[END_REF]. Our measurements are thus believed to be more accurate, even the difference is rather small (about 5% at 1000 K).

Figure 7.

The conductivity values were fit to Arrhenius equation:

𝜎 = 𝜎 0 exp (- 𝐸 𝑎 𝑘𝑇 ) (1)
where σ (S.cm -1 ) is the conductivity of the molten carbonate, σ0 (S.cm -1 ) a preexponential factor, Ea (eV) is the activation energy, k (J.K -1 ) the Boltzmann constant and T (K)

is the absolute temperature. Table 1 shows the obtained results of pre-exponential factor and activation energy. It can be noticed that the activation energy of the electrical conductivity is in close agreement with activation energies extracted from self-diffusion coefficients of Li + and Na + , suggesting that the conductivity is mainly controlled by the mobility of these cations. On the other hand, Ea and σ0 of this study are in very good agreement with the corresponding values reported by Kojima et al. [START_REF] Kojima | Physical Properties of Molten Li2CO3-Na2CO3 (52:48 mol%) and Li2CO3-K2CO3 (62:38 mol%) Containing Additives[END_REF].

Table 1.

Discussion

The electrical conductivity of molten carbonates is related to other transport properties such as self-diffusion. So as to compare these properties, the density measurements on molten Li2CO3-Na2CO3 (52:48 %mol) eutectic point at 1 atm performed by Kojima et al. [START_REF] Kojima | Physical Properties of Molten Li2CO3-Na2CO3 (52:48 mol%) and Li2CO3-K2CO3 (62:38 mol%) Containing Additives[END_REF] are used.

The "ideal" electrical conductivity σ NE being related to the self-diffusion coefficients by the Nernst-Einstein equation:

𝜎 𝑁𝐸 = 𝑞 2 𝑘 𝐵 𝑇𝑉 ∑ 𝑁 𝑠 𝑠 𝑧 𝑠 2 𝐷 𝑠 ( 2
)
where q is the elementary charge, Ns is the number of ions contained in the volume V, zs is the conduction charge and Ds are the experimental self-diffusion coefficients for the chemical species, and s = Li + , Na + and CO 3 2-respectively. In order to compare experimentally obtained conductivities to those obtained by the Nerst-Einstein equation, we have used our NMR self-diffusion measurements for sodium and lithium show in figure 4 and data given by Spedding et al. [START_REF] Spedding | Tracer Diffusion Measurements in Mixtures of Molten Alkali Carbonates[END_REF] for CO 3 2-(table 2). The results of the "ideal" electrical conductivity compared to the experimentally obtained values for three different temperatures are presented in table 2.

Table 2.

The Haven ratio contains information about the correlation and the whole atomic mechanism of ionic motion. For the Li2CO3-Na2CO3 (52:48 %mol) eutectic mixture the value of the Haven ratio is around 0.7 which is very close to the value obtained by Desmaele et al.

[19] by ab-Initio calculation. A value of H less than 1 is generally explained by a large contribution of the anticorrelations between ions of the same charge. Moreover, Table 2 shows that H is practically invariant with temperature suggesting that the transport mechanism of cations and anions does not change with temperature.

Conclusions

In this paper we reported, Na + and Li + self-diffusion coefficients measurements in molten Li2CO3-Na2CO3 (52:48 %mol) eutectic mixture were combined with electrical conductivity using 4-electrode experimental set up at temperatures ranging from 770 to 1000 K

Increasing temperature influence values of the conductivity and self-diffusion coefficients. In the first approximation, the value of Haven ratio calculated around 0.7, practically invariant with temperature indicate that transport mechanism of cations and anions does not change with temperature.

Nevertheless, to better understand the transport mechanism, the diffusion coefficients for CO 3 2-will measured using 13 C enriched carbonate. Other eutectic mixtures of carbonates such as Li2CO3-K2CO3, Li2CO3-Na2CO3-K2CO3 and Na2CO3-K2CO3 will be studied at high temperature to provide some additional data in order to constraint the different model given by molecular dynamic simulations. [23] (dashed line).

Figure 2 .

 2 At room temperature the NMR lines are very broad for both nuclei. It is explained by the presence of dipolar and quadrupolar NMR interactions in the solid state carbonate mixture. Before the melting temperature, the line becomes narrow due to the increase of the cations mobility, and the progressive averaging of the dipolar and quadrupolar couplings. When the salt is completely melted, the NMR interactions except the chemical shift one are averaged to zero, yielding to a single sharp line. At high temperature the different ionic species present in the system are in rapid exchange (few picoseconds), compared to the time scale of the NMR measurements (10 -9 s). As a result, the NMR line corresponds to the average of all possible ionic configurations experienced by the observed nucleus.

Figure 2 .

 2 Figure 2.

Figure 5 .

 5 Figure 5.

Figure captions:Figure 1 .

 1 Figure captions:

Figure 2 .

 2 Figure 2. a) 23 Na (I=3/2) and b) 7 Li (I=3/2) NMR spectra at different temperatures for the Li2CO3-Na2CO3 (52:48 %mol) eutectic point.

Figure 3 .

 3 Figure 3. Evolution of 23 Na (black filled squares) and 7 Li (black empty squares) chemical shifts with temperature for the Li2CO3-Na2CO3 (52:48 %mol) eutectic point.

Figure 4 .

 4 Figure 4. 23 Na and 7 Li diffusion coefficient in Li2CO3-Na2CO3 (52:48 %mol) eutectic mixture.

Figure 5 .

 5 Figure 5. Arrhenius plots from7 Li (green squares) and23 Na (black squares) NMR diffusion coefficients for Li2CO3-Na2CO3 (52:48 %mol) eutectic mixture.

Figure 6 .

 6 Figure 6. Cole-Cole representation of the complex impedance Li2CO3-Na2CO3 (52:48 %mol) eutectic point at different temperatures.

Figure 7 .

 7 Figure 7. Comparison of the electrical conductivity measured with the 4-electrode system (squares and plain line) and electrical conductivity obtained by calculations from Kojima et al.

Fig. 1

 1 Fig.1

  Fig.2

Fig. 7

 7 Fig.5

Table captions :

 captions Table1. Fit parameters of equation (1) and comparison with Kojima et al. values[START_REF] Kojima | Physical Properties of Molten Li2CO3-Na2CO3 (52:48 mol%) and Li2CO3-K2CO3 (62:38 mol%) Containing Additives[END_REF].

Table 2 .

 2 Ideal electrical conductivity ( NE ) calculated at different temperature from selfdiffusion coefficients Na + , Li + measured by PFG NMR ; D(CO 3 2-) measured by Spedding et al.[START_REF] Spedding | Tracer Diffusion Measurements in Mixtures of Molten Alkali Carbonates[END_REF], (S/m) measured by 4-electrode system in this work, and Haven ratio[START_REF] Desmaele | Atomistic simulations of molten carbonates: Thermodynamic and transport properties of the Li2CO3 -Na2CO3 -K2CO3 system[END_REF] (𝐻 = 𝜎 𝑁𝐸 /𝜎).

			This work	Kojima et al. [23]
	Ea (eV)	0.20 ± 0.01	0.20
	ln(𝝈 𝟎 ) (Ω -1 .cm -1 .K)	3.28 ± 0.05	3.21
	Table 1
	T (K)	D𝐂𝐎 𝟑 𝟐-(m 2 /s) [15]	σ NE (S/m)	σ (S/m)	H
	861	7 • 10 -10	212	177.3	0.8
	952	1.73 • 10 -9	329	230.3	0.7
	1008	2.24 • 10 -9	385	267.9	0.7

Table 2

 2

Acknowledgements:

This work was funded by the French National Research Agency (ANR-11-EAPX-0036 project MCEC).