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

Dorra Ibrahim

Jean-Baptiste Vaney

Selma Sassi

Philippe Masschelein

Anne Dauscher

Bertrand Lenoir

SnSe: Breakthrough or Not Breakthrough?

Keywords: thermoelectric, chalcogenide, SnSe, semiconductor, thermal conductivity

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The design of highly-efficient thermoelectric materials represents a formidable challenge. Such a compound should indeed strike a subtle balance between electronic and thermal properties [START_REF] Goldsmid | Thermoelectric Refrigeration[END_REF][START_REF]Thermoelectrics and its Energy Harvesting[END_REF][START_REF] Tan | [END_REF]. A high thermopower 𝛼 and low electrical resistivity 𝜌 typical of heavily-doped semiconductors should be reconciled with low total thermal conductivity 𝜅, usually seen in glassy systems, in order to maximize the dimensionless thermoelectric figure of merit 𝑍𝑇 = 𝛼 2 𝑇/𝜌𝜅 = 𝑃𝑇/𝜅 at an absolute temperature 𝑇 [START_REF] Goldsmid | Thermoelectric Refrigeration[END_REF][START_REF]Thermoelectrics and its Energy Harvesting[END_REF][START_REF] Tan | [END_REF]. Adding another degree of complexity, these mutually incompatible properties cannot be optimized independently since they are all interdependent through the charge carrier concentration. Most of the best thermoelectric materials possess favourable energy band structure with multiple valleys that helps to achieve high power factors P [START_REF] Goldsmid | Thermoelectric Refrigeration[END_REF][START_REF]Thermoelectrics and its Energy Harvesting[END_REF][START_REF] Tan | [END_REF]. Concomitantly, they also harbour specific crystallographic features that strongly limit the ability of acoustic phonons to propagate such as, for instance, localized lattice vibrations ("rattling modes") or chemical bonds that vibrate anharmonically [4][5][6][7][8]. Additional geometric architecturing such as nanostructuring or inducing nanoscale domains through controlled synthetic conditions represents another strategy to limit the heat transport and thus, achieve better thermoelectric performances [9][10][11][12].

For decades, the family of lead telluride (PbTe) compounds and its selenide analogue PbSe have been the thermoelectric materials of choice for operating at medium temperatures (typically near 500 K) [START_REF] Goldsmid | Thermoelectric Refrigeration[END_REF][START_REF]Thermoelectrics and its Energy Harvesting[END_REF][START_REF] Tan | [END_REF]. These semiconductors gather many important characteristics to be a prospective area of research for achieving high thermoelectric performances [START_REF] Goldsmid | Thermoelectric Refrigeration[END_REF][START_REF]Thermoelectrics and its Energy Harvesting[END_REF][START_REF] Tan | [END_REF].

Despite a simple cubic rock-salt crystal structure, the lattice thermal conductivity is very low due to significant anharmonicity in the lattice that originates from a strong anharmonic coupling between transverse optic modes and longitudinal acoustic modes, as shown by neutron spectroscopy techniques [13][14][15]. The electronic band structure with non-parabolic bands and multiple valleys is the key ingredient that explains the good thermoelectric properties achieved by proper doping. In comparison to other chalcogenide semiconductors, PbTe-based compounds have received significant attention in the 1960s and 1970s and are still the subject of numerous studies nowadays [START_REF] Goldsmid | Thermoelectric Refrigeration[END_REF][START_REF]Thermoelectrics and its Energy Harvesting[END_REF][START_REF] Tan | [END_REF]. Tin selenide SnSe is another representative of the vast family of chalcogenide semiconductors whose basic crystallographic and physical properties have also been investigated in these decades [16][17][18]. Although chemically similar to PbTe, SnSe crystallizes with a distorted lattice described in the orthorhombic space group Pnma [19][20][21]. In retrospective, it is perhaps not so surprising that this compound has received less attention in this period. Its orthorhombic crystal structure gives rise to anisotropic transport properties, even in polycrystalline samples. In addition, the number of extrinsic dopants that can be introduced in SnSe seemed more restricted than in PbTe-based compounds. For these reasons, significant efforts were devoted to optimize the thermoelectric properties of PbTe and its derivatives.

In 2014, this forgotten, "ugly-duckling" compound made a remarkable comeback to the forefront of research in thermoelectricity with the announcement of record-breaking ZT values of up to 2.6 at 900 K in single crystals [22,23]. What was even more surprising was the fact that such high values have been achieved in pristine single crystals while high ZT values are usually reached once a careful optimization of the electronic and thermal properties has been realized. Another striking outcome of this report was the very low lattice thermal conductivity measured at high temperatures ranking this compound among the best thermal insulators known to date [22]. Such extremely rare combination of transport properties raised a number of questions: Can one optimize even further the thermoelectric performances of single-crystalline SnSe through doping? Is it possible to achieve similar ZT values in polycrystalline samples more suitable for integration in thermoelectric generators? What microscopic mechanisms impede so efficiently the propagation of the heat-carrying acoustic phonons in such a simple crystal structure? In addition to these important questions of both practical and fundamental interest, the difference between the thermal conductivity measured in this study [22] and that reported in the 1960s [16][17][18] has rapidly become another source of interrogation and debate within the thermoelectric community.

Although consensual answers to some of these questions are yet to be obtained, this announcement stirred up strong interest to revisit in detail the physical properties of this compound by modern techniques.

In this chapter, we will provide a brief review of the accumulated knowledge acquired on this compound based on the experimental and theoretical results obtained over the last few years. We will focus more particularly on the low-temperature allotrope of SnSe referred to as the  form which shows the best thermoelectric performances. Our goal is to highlight the important structural and chemical aspects that influence its transport properties and on the strategies employed to optimize its thermoelectric performances. As excellent reviews on this compound and, more generally, on tin chalcogenides already exist [24][25][26], we will only focus on the main aspects and issues that recently emerged and remain to be elucidated in future studies. These slabs are bound by weak Sn-Se bonding along the a axis and are corrugated, creating zig-zag chains when the structure is projected along the b direction (see Figure 1). Interestingly, this creased honeycomb structure is similar to that of black phosphorus whose sheets, called phosphorene, are currently attracting attention for fast-response electronic and photonic devices [27]. Upon warming, SnSe undergoes a displacive structural transition near 800 K which leaves the orthorhombic symmetry of the unit cell unchanged [22,28]. The high-temperature allotrope, referred to as -SnSe, is described in the space group Cmcm (No. 63) with roomtemperature lattice parameters a = 4.31 Å, b = 11.71 Å and c = 4.32 Å [22,28]. Other allotropes have been identified at high pressures with structural modifications above 27 GPa that gives rise to a topologically-non-trivial semi-metallic state and to superconductivity at 39 GPa with a critical temperature of 3.2 K [29][30][31][32][33][34][35][36][37]. The -to- transition, which has been directly imaged by high-resolution transmission electron microscopy [22], leaves clear signature on both the transport and thermodynamic properties. As we shall see below, the anisotropic crystal structure of SnSe results in a significant anisotropy in the transport properties, should the samples be in single-crystalline or polycrystalline form.

Crystal structure, synthesis and defect chemistry

Synthesis of polycrystalline and single-crystalline SnSe

Since elemental Sn and Se are commercially available with very low impurity levels, both highly-pure polycrystalline and single-crystalline samples can be easily synthesized.

Large polycrystalline specimens can be obtained by direct melting the powders or shots of the elements at high temperatures in sealed silica ampoules. Unlike other thermoelectric materials for which such technique does not necessarily yield homogeneous ingots due to their inherent defect chemistry, polycrystalline SnSe samples appear to be well homogeneous.

Large single crystals can be synthesized by more complex growth techniques. While early studies on SnSe used vapour phase techniques in sealed silica tubes [16][17][18], another elegant technique is the Bridgman technique. This method implies to pull the ampule with a carefully shaped end through a large temperature gradient with the highest temperature above the melting point of the compound. The crystal then solidifies from a seed at the shaped end of the ampoule and progressively forms along its length. This process has been carried out successfully in both horizontal and vertical configurations to obtain large, centimetre-sized single crystals of SnSe. However, one major issue regarding the growth of single-crystalline SnSe is related to the phase transition undergone near 800 K, that is, below the melting point of this compound (1134 K). Together with the high vapour pressure of Se, the volume increase across the -to- transition (around 2.5%) can lead to breakage of the ampoule during the process. This problem can be mitigated by sealing the first silica ampoule into a second one to prevent loss of Se vapour and oxidation of the elements. The single crystals obtained by such techniques have typically centimetre-size lateral dimensions and can be easily cleaved along the a and c axes [22,[38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54]. Of note, other techniques related to hydro-and solvothermal syntheses have been also successfully used to obtain singlecrystalline plates or nanobelts [55,56]. Obtaining large crystals is a necessary prerequisite to study in detail the components of the transport properties tensors and allows for more sophisticated spectroscopic techniques to be employed.

Defect chemistry

As many thermoelectric materials [START_REF] Goldsmid | Thermoelectric Refrigeration[END_REF][START_REF]Thermoelectrics and its Energy Harvesting[END_REF][START_REF] Tan | [END_REF], defect chemistry plays an important role in determining the transport properties of SnSe. Although this predisposition to defects is not as critical in SnSe as in the well-known Bi2Te3-based alloys for instance, their presence can nevertheless result in single crystals with physical properties varying along the growth direction. In particular, such variations can be problematic when combining transport properties measured on samples taken at different positions in the crystal to calculate the ZT values.

Pioneering works on single crystals of SnSe grown by vapour techniques have probed the full range of carrier concentrations showing that only p-type crystals can be obtained with hole concentrations varying between 310 15 and 210 18 cm -3 at 300 K [16][17][18].

The hole concentration strongly depends on the details of the synthetic procedure and possible subsequent annealing step employed (a process referred to as saturation annealing). Although this p-type nature was supposed to be due to an excess of Se atoms [16][17][18], it was not clear at that time whether Sn vacancies, antisite defects or both types of defects were responsible for this behaviour. Furthermore, laboratory structural and chemical probes such as X-ray diffraction or energy-dispersive spectroscopy are not sensitive enough to discriminate between different amounts and types of defects. The determination of the main type of defects has been recently addressed by detailed electronic band structure calculations [57]. These theoretical studies are consistent with native p-type doping primarily due to Sn vacancies which are energetically favoured over antisite defects under both Sn-and Se-rich conditions. Another possible source of extrinsic p-type doping has been recently advanced based on the presence of SnSe2 microdomains in single crystals grown under various experimental conditions [58]. These inhomogeneities are believed to yield interfacial charge transfer to occur from SnSe to SnSe2 resulting in an overall hole-doped SnSe matrix. Yet, as mentioned by the authors, the high cooling rate employed during the growth process is likely the key parameter that leads to the nucleation of this secondary phase. Several other recent studies performed on single crystals have indeed not reported the presence of such domains while these pristine crystals remain intrinsically hole-doped.

Electronic band structure and electronic properties

3.1 Electronic band structure of -SnSe Significant efforts have been devoted to determine both theoretically and experimentally the main features of the electronic band structure of SnSe that govern its transport properties. While some dispersion in the computed band gap exists in the literature due to the details of the calculations and the method used, most of the studies indicate a band gap on the order of 0.6 -0.9 eV [22,43,[59][60][61][62][63][64][65][66][67][68][69][70][71][72]. This value is consistent with the experimental band gap values of 0.86 and 0.92 eV obtained by optical absorption spectroscopy [22,54].

The band gap is indirect with a valence band maximum (VBM) lying along the -Z direction while the conduction band minimum (CBM) is located along the -Y direction (Figure 2) [60]. The overall structure of the valence and conduction bands is rather complex with several maxima and minima, respectively, close to each other in energy. In particular, the highest VBM shows a "pudding-mould-like" shape, that is, a band that shows a dispersive part and a flatter portion nearby [60]. Such a shape has in particular been proposed to be beneficial for achieving good thermoelectric performances [73]. Furthermore, a second valence band maximum lies just below the main valence band maximum. The indirect nature of the band gap implies anisotropic density-of-states effective masses that differ significantly between the three crystallographic axes. The density-of-states effective mass ellipsoids were calculated to be equal to 0.74m0, 0.31m0, 0.16m0 (m0 is the bare electron mass) for the highest VBM, and 2.40m0, 0.11m0, 0.15m0 for the lowest CBM along the a, b and c axes, respectively [59]. The second VBM shows similar values along the a and c directions (0.90m0 and 0.15m0, respectively) while it differs more significantly along the b direction (0.12m0) [59]. The presence of multiple local extremum near the band edges is an important aspect to understand the good electronic properties achieved in this compound as we will see below.

Based on the computed band structure, the electrical conductivity and thermopower have been computed using the Boltzmann transport approach for a wide range of temperatures and p-and n-type doping levels [60]. The results evidenced significant anisotropy in the transport properties between the different crystallographic axes, as expected for such layered crystal structure. The anisotropy depends on the carrier concentration and while at relatively high concentrations the thermopower shows only little variations between the three directions, more significant differences may appear at carrier concentrations on the order of 10 17 cm -3 . One of the main results of these calculations is related to the prediction that n-type SnSe should potentially exhibit higher thermoelectric performances with respect to p-type analogues [60].

Angle-resolved photoemission spectroscopy

Because the details of the electronic band structure can depend on the particular computational method used, direct experimental probes of the electronic band structure are particularly relevant to unveil its main characteristics in both pristine and hole-doped crystals. The growth of high quality, large single crystals and the ease with which they can be cleaved make angle-resolved photoemission spectroscopy (ARPES) a powerful tool for such investigations. The experimental investigation of the valence band structure of SnSe by this technique has been recently addressed by several groups as a function of the doping level (undoped and Na-doped single crystals) and temperature [27,58,[74][75][76]. Probably due to the low doping level studied, the presence of Na had little influence on the main characteristics of the band structure of SnSe [76]. Regarding pristine SnSe, all these studies consistently evidenced the presence of highly anisotropic valence bands with multiple valleys in agreement with available calculated band structures (Figure 3). The density-of-states effective masses have been derived from the collected ARPES k-E maps along the main directions (-X, -Y and -Z). The results obtained along the -Z and -Y directions indicated effective masses on the order of 0.2𝑚 0 along the b and c axes for both directions. The values along the a axis are slightly higher as predicted theoretically.

These results are thus consistent with each other's and in line with the predicted values.

Among these studies, one of them [76] has nevertheless found values significantly higher than those obtained in other studies by one order of magnitude (effective masses of up to 2.23𝑚 0 ) even at the lowest temperature probed. This study also evidenced that the effective masses tend to decrease with increasing temperature up to 600 K. which is significantly smaller due to the less dispersive valence bands, resulting in higher density-of-states effective mass 𝑚 𝑎𝑖 * compared to 𝑚 𝑏𝑖 * and 𝑚 𝑐𝑖 * . Hence, these experimental results evidence the beneficial contribution of several valence bands to the transport properties and provide an insightful guidance for developing relevant transport models to describe their variations with the hole concentration (in the so-called Pisarenko-Ioffe plot).

Electronic transport properties of pristine and doped single-crystalline SnSe

Because of the inherent defect chemistry of SnSe, the hole concentration of the synthesized single crystals can vary from sample to sample. Since the discrimination of such small amounts of defects is very difficult, the most effective metric is the characterization of its electrical properties and notably, its carrier concentration. Pristine single-crystalline SnSe has been always obtained so far as a lightly-doped p-type semiconductor [22,[38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54]. This state is reflected by the large electrical resistivity and thermopower values measured with hole carrier concentration of the order of 10 17 cm -3 at 300 K. As expected from its orthorhombic crystal structure, the transport properties are anisotropic and their study in single crystalline specimens requires a careful examination along the three crystallographic axes. Due to the low hole concentrations, the electrical resistivity values are typically on the order of few .m at 300 K and show a semiconducting behaviour with increasing temperature. Concomitantly, the thermopower can reach values as high as 800 V.K -1 at 300 K [22,[38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54]. While the anisotropy remains moderate in the thermopower values, the electrical resistivity is more anisotropic following the sequence a > c > b along the a, b and c crystallographic axes. However, this is not to say that the thermopower cannot show some significant anisotropy as predicted theoretically [60] and demonstrated experimentally [42].

Only few studies on doped single crystals have been undertaken so far [38,43,45,46,[START_REF] Duong | [END_REF][79][80]. While substitutions of Na, Ag and Bi for Sn have been considered, only the solid solution SnSe-SnS has been considered as a mean to substitute on the Se site.

Percentages as low as 3 at% Na are sufficient to significantly enhance the hole concentrations with maximum values higher than 10 19 cm -3 at 300 K [38,43,45]. Although this increase in the hole concentration induces a decrease in the thermopower values, the net result of the dopants is positive with an overall increase in the power factor at high temperatures [38,43,45]. Introduction of similar amount of Ag acts similarly with similar trends in the electrical properties as a function of the Ag content [43,46]. Intriguingly, the substitution of Sn by 6 at% of Bi switches the electrical conduction to n-type over the whole temperature range covered (300-800 K) [START_REF] Duong | [END_REF]. With power factors that rival those obtained in p-type samples, these results show that good thermoelectric performances can be equally achieved in both n-and p-type crystals. Without exceptions, for all these dopants, the best thermoelectric performances have been achieved along the b axis as in pristine single crystals [38,43,45,46,[START_REF] Duong | [END_REF][79][80]. Because electronic band structure calculations have indicated that better performances could be achieved in n-type SnSe [60], further studies on single crystals with dopants known to induce an n-type conduction such as Br for instance (see below) would be worthwhile.

Electronic transport properties of pristine and doped polycrystalline SnSe

Although the study of single crystals is illuminating in determining the basic physical mechanisms governing the transport properties and their possible evolution as a function of the doping level, the carefully-controlled environment required for their growth and their mechanical properties make them poor candidates for direct integration in thermoelectric generators. The assessment of the thermoelectric performances in polycrystalline samples thus rapidly became a central topic within the thermoelectric community.

The first reports on these aspects [81,82] have demonstrated that pristine SnSe behaves as a lightly-doped p-type semiconductor with values of the electrical resistivity and thermopower in line with those achieved in single crystalline samples (Figure 4). The anisotropy seen in single crystals is not fully suppressed and survives in polycrystals. The transport properties thus need to be measured both parallel and perpendicular to the pressing direction to estimate correctly the ZT values. The average value of the data measured on single-crystalline SnSe [22] (filled circle symbols)

have been added for comparison. Material reproduced from Sassi et al. [81], Appl. Phys.

Lett., 104, 212105 (2014).

The most surprising result evidenced by these investigations [81,82], and widely confirmed in following studies, is the higher thermal conductivity values measured with respect to those reported in single crystals. Even though the values reached at high temperatures are very low for a compound with such a simple crystal structure (on the order of 0.5 W.m -1 .K -1 at 800 K [81,82]), the difference between polycrystalline and singlecrystalline samples is particularly unexpected since the presence of grain boundary scattering in the former should in principle result in lower values. The origin of this discrepancy will be specifically addressed in detail below. In pristine polycrystalline samples, maximum ZT values of 0.5 have been achieved at 823 K [81,82]. Noteworthy, the orthorhombic-to-orthorhombic structural transition undergone by SnSe near this temperature leads to significant loss of Se and to irreversible plastic deformation of the samples [81]. This clearly precludes any use of this compound at temperatures above 800 K, a conclusion further corroborated by the rapid degradation at temperatures above 873 K observed in oxidation studies on polycrystalline SnSe [83].

With the aim of further tuning the carrier concentration, numerous dopants have been considered resulting in either p-type or n-type electrical conduction. For p-type doping, the main elements known to effectively dope PbTe and PbSe have been primarily investigated. This includes alkali metals (Na [83][84][85][86][87][88][89], K [85,87,90], Li [87]) and alkaline-earth elements (Ca, Sr and Ba [91]). In addition, various transition metals (Ag [82,[92][93][94][95], Cu [91,96,97], Zn [91,98], Al [97], Tl [99], Ti [100] and In [97,101]), metalloids (Pb [89,97], Te [102][103][104] and S [105,106]), Ge [91,[107][108][109], as well as some lanthanides (Sm [110] and LaCl3 [111])

have also been investigated. In contrast, the halogens Cl [112,113], Br [114] and I [106] as well as Bi [113] were shown to induce n-type electrical conduction. Only few studies have used so far a multiple-doping approach on either a single site (n-type Sn0.74Pb0.20Ti0.06Se [100])

or on both sites (such as in p-type Sn0.99Na0.01Se0.84Te0.16 [102]). In most cases, the introduction of the foreign element increases the carrier concentration which helps to optimize the power factor. Due to the low doping levels, most of these dopants have little influence on the lattice thermal conductivity, giving rise ultimately to enhanced ZT values near 800 K. So far, peak

20

ZT values ranging between 0.8 and 1.3 have been achieved in either p-type or n-type samples.

The variations of the electronic properties with the carrier concentration, and more particularly, the evolution of the thermopower with the hole concentration has been successfully modelled by considering a simple parabolic band model assuming that either one or two valence bands govern the transport properties [87]. For this analysis, several sets of available data in the literature on both polycrystalline and single-crystalline samples have been considered (Figure 5). corresponds to that determined theoretically and the value of 1.1 represents the best fit to the data. The references 14, 15, 19, 20 and 27 corresponds to the references [22], [38], [82],

[84] and [94] in the present chapter, respectively. Reprinted (adapted) with permission from Further attempts at optimizing the thermoelectric properties of SnSe have been considered through the formation of solid solutions with SnS [105], SnTe [115] or PbSe [116,117]. Solid solutions are usually a convenient way to further control the carrier concentration and to decrease the lattice thermal conductivity via enhanced point defect scattering. Although this last mechanism is less pronounced in the present case due to the very low lattice thermal conductivity already exhibited by this compound, these solid solutions resulted in slightly enhanced ZT values with a peak value of 0.8 achieved at 823 K for SnSe0.8S0.2 [105]. A notable exception is the system SnSe-PbSe for which very high ZT values of up to 1.7 at 873 K have been reported [116]. In this last study, the introduction of PbSe beyond the solubility limit led to the formation of nanoprecipitates, the influence of which on the transport properties has been proposed as the main key factor. However, the results reported in another study [117] on the same system show much lower values with a maximum twice as low as the above-mentioned value (around 0.85 at 800 K). Further work seems therefore necessary to determine the reasons behind such significant discrepancy between the two studies.

Finally, some studies on composite samples have been reported recently where incorporation of MoS2, MoSe2 or graphene into SnSe has been studied [118,119]. However, none of them have led to a significantly higher ZT values compared to those achieved in pristine SnSe.

Thermal properties

Specific heat of SnSe

The specific heat 𝐶 𝑝 provides valuable information on a given material below and above room temperature. At low temperatures, the specific heat represents an energyaveraged response that encodes the low-energy features of the phonon subsystem associated with optical phonons or zone-boundary excitations. At high temperatures, this thermodynamic property is an essential component to be measured accurately in order to correctly assess the thermal conductivity. The specific heat of SnSe has been studied on both polycrystalline and single-crystalline specimens (Figure 6). At low temperatures (Figure 6a), the results are consistent with the semiconducting ground state observed in pristine SnSe giving rise to a null electronic contribution to within b) a)

experimental uncertainty [120,121]. Near 10 K, the specific heat data, when plotted as 𝐶 𝑝 /𝑇 3 versus T, deviate from a conventional Debye behaviour typified by a pronounced peak. This peak has been interpreted as either Einstein contributions or as a Schottky contribution [120,121]. In this last model [121], the origin of this peak has been discussed in terms of hard and soft substructures related to short and weaker bonds within the layers, respectively.

These results thus show that the lattice dynamics of SnSe is dominated by two energy scales, in line with the two main regions observed in the phonon density of states [44,66,122].

The temperature dependence of the specific heat at high temperatures steadily increases with increasing temperatures up to about 600 K (Figures 6a and6b). Above this temperature, the nearly linear increase in 𝐶 𝑝 turns into a non-linear variation. Near 800 K, a clear peak is observed which corresponds to the orthorhombic-to-orthorhombic transition [22,81,122]. The transition appears as a broad peak between 650 up to 800 K in agreement with its continuous nature [22]. In order to determine the specific heat values over this temperature window for the calculation of the total thermal conductivity, it is therefore necessary to extrapolate linearly the values measured below 600 K up to 800 K.

Thermal conductivity of single-crystalline and polycrystalline SnSe

One of the main ingredients that led to the very high ZT values claimed in singlecrystalline SnSe is the very low lattice thermal conductivity values measured along the different axes of the crystal structure [22]. At 300 K, the values were reported to be 0.46, 0.70 and 0.67 W.m -1 .K -1 which further decreases to reach 0.24, 0.34 and 0.31 W.m -1 .K -1 at 700 K along the a, b and c axes, respectively [22]. Of note, due to the high electrical resistivity values measured in pristine single crystals, the electronic contribution to the thermal conductivity 𝜅 𝑒 = 𝐿𝑇/𝜌, where 𝐿 is the Lorenz number, is negligible so that 𝜅(𝑇) ≡ 𝜅 𝐿 (𝑇).

Soon after this report, the question regarding the lattice thermal conductivity of single crystalline SnSe has rapidly become a central issue for several reasons. First, compared to the lattice thermal conductivity values measured in the 1960s [16][17][18], the values reported by Zhao et al. [22] are significantly lower. Second, the subsequent studies on the transport properties of polycrystalline pristine samples have reported higher lattice thermal conductivity values than in single crystals [81,82]. This result is counterintuitive since the presence of grain boundaries is expected to provide an additional phonon scattering channel which should lead to the opposite trend, that is, lower values should be observed in pristine polycrystalline samples. Third, the anisotropy observed between the three crystallographic axes was not following the sequence 𝜅 𝐿 𝑎 < 𝜅 𝐿 𝑐 < 𝜅 𝐿 𝑏 predicted by abinitio phonon calculations [123]. All these elements raised two fundamental questions: What could be the origin of such discrepancies between polycrystalline and single-crystalline samples? Were the values measured in single crystals by Zhao et al. [22] underestimated?

A first important point to answer these questions has been raised by Wei et al. [124] and is related to the density of the samples used in the study of Zhao et al. [22]. Using the thermal conductivity, the specific heat and the thermal diffusivity values reported, these authors pointed out that the density of the samples measured was corresponding to only 88% of the theoretical density of SnSe from crystallographic data [124]. This lower density could provide a natural explanation of the lower values measured in [22] and would thus point to a significant underestimation of the lattice thermal conductivity of single-crystalline SnSe. With higher 𝜅 𝐿 values, the contradiction between the values measured in polycrystalline and single-polycrystalline samples would then vanish, thereby recovering a more conventional trend in 𝜅 𝐿 from single to polycrystals.

Beyond this simplest explanation, several mechanisms have been advanced to try to explain the difference between single-crystalline and polycrystalline specimens as well as the lower values measured in the former [125]. Notably, air sensitivity under ambient conditions has been evoked as a possible problem. However, several studies have pointed out that SnSe is not air sensitive enough to consistently explain the observed differences in 𝜅 𝐿 . Another possible reason discussed is the presence of "vast off-stoichiometry" in the single crystals studied by Zhao et al [22,45]. Since SnSe inherently possesses defects in the form of Sn vacancies, large amounts of such defects would tend to decrease the density in comparison to more stoichiometric crystals and would act as an efficient source of phonon scattering leading to lower 𝜅 𝐿 values. However, the estimated concentration of Sn vacancies should be as large as 16.5% to reconcile these different aspects [45,124]. This concentration is much higher than those obtained on single crystals grown under various synthetic conditions in the 1960s [16][17][18] and it remains unclear whether the structure of SnSe can accommodate such amounts, or equivalently, whether SnSe remains stable with such offstoichiometry.

Another point of concern related to this explanation is rooted in the band gap measured on these single crystals. Assuming that such large amounts of defects are indeed possible, we may expect these to have a significant influence on the electronic band structure. In particular, a significant variation in the band gap could be anticipated.

However, direct measurements of the band gap width by absorption spectroscopy [22,54] yielded results consistent with a band gap of around 0.9 eV expected for stoichiometric SnSe. Both results could perhaps still be reconciled but this would require a rather surprising set of circumstances since the large amount of defects should indeed result in modifications of the electronic band structure but with a net result of a constant band gap.

Of note, investigations on the influence of off-stoichiometry in polycrystalline Sn1-xSe (with x = 0.01, 0.02 and 0.04) are in agreement with the binary Sn-Se phase diagram indicating that the initial Sn-deficiency is accommodated by the formation of the secondary phase SnSe2 rather than by creating Sn vacancies [126]. Hence, whether significant Sn deficiency is possible in SnSe remains to be demonstrated and would require further theoretical and experimental investigations. In particular, calculations of the electronic band structure with large amounts of Sn vacancies should provide insights into the expected band structure.

Moreover, systematic measurements of the band gap by absorption spectroscopy should be performed on single crystals with different carrier concentrations to determine whether varying the concentration of defects indeed can lead to modifications of the band gap.

Recent studies on pristine single-crystalline SnSe with densities close to the theoretical density for stoichiometric SnSe have been reported. These investigations have found lattice thermal conductivity higher than those reported by Zhao et al [22]. For instance, Ibrahim et al. [42] measured 𝜅 𝐿 values of 1.2, 2.3 and 1.7 W.m -1 .K -1 at 300 K along the a, b and c directions (Figure 7). [42]. The data are compared to those reported by Zhao et al. [22].

These values, between two and three times higher than those reported in prior studies, are consistent with early reports in the 1960s and with the sequence predicted by abinitio calculations [16][17][18]123]. Upon warming, these values decrease to reach a value of 1 W.m -1 .K -1 at 700 K in the b and c directions. As a consistency check, the values obtained by low-temperature measurements match very well with those obtained by a laser flash technique at high temperatures [42]. Since these measurements were carried out using a different measurement technology, it avoids any bias related to a particular technology.

Below 300 K, 𝜅 𝐿 exhibits a crystalline-like behaviour with a pronounced Umklapp peak (also referred to as a dielectric maximum) that reaches 52, 27 and 20 W.m -1 .K -1 at 15 K along the b, c and a directions, respectively (see Figure 7) [42,127]. Because the magnitude of this peak is sensitive to crystal imperfections, it usually can be considered as an effective metric of the quality of the grown crystal. The amplitude of this peak follows the sequence seen in the 𝜅 𝐿 values with the lowest values observed along the a axis [42]. This result is not surprising since this direction corresponds to the interlayer direction along which the phonon mean free path is expected to be the shortest due to a stronger phonon scattering at the interface between the layers.

Another recent study on Na-doped and Ag-doped single crystals further corroborate all these results [127]. The measurements of the thermal conductivity at low temperatures is consistent with the above-mentioned main features. In particular, the Umklapp peak reported along the b axis for pristine crystal is similar to that measured by Ibrahim et al. [42].

Upon doping, the amplitude of this peak is reduced due to additional point defect scattering induced by the doping atoms. What makes these results particularly interesting is that these measurements have been performed on crystals for which very high ZT values had been reported at high temperatures (with a maximum ZT of 2 at 800 K [43]). This astonishing difference between both studies has been discussed in terms of inhomogeneous distribution of dopants in the crystals. Such discrepancy clearly warrants further experimental investigations to determine whether the ZT values of up to 2 reported [43] are really achievable in SnSe or are rather due to an erroneous combination of transport properties measured on samples with different hole concentrations due to the inhomogeneous distribution of the doping species.

In polycrystalline SnSe, measurements need to be performed both parallel and perpendicular to the pressing direction due to the anisotropic crystal structure [81,82].

Compared to single crystals, the literature data on polycrystals exhibit less dispersion and most of these studies agree on the values in pristine SnSe. At 300 K, the measured 𝜅 𝐿 values are 0.9 and 1.4 W.m -1 .K -1 in the parallel and perpendicular direction, respectively [81,82]. The large Umklapp peak observed in single crystals survive in polycrystals with respective peak values of 12 and 6 W.m -1 .K -1 near 25 K [120]. Upon increasing temperature, the 𝜅 𝐿 values decrease to reach very low values that are close to 0.5 W.m -1 .K -1 in both directions (Figure 8). [120], Materials Today: Proceedings 2, 690 (2015).

One important aspect when estimating the lattice thermal conductivity of doped polycrystalline samples is related to the determination of the Lorenz number L whose values sensitively depend on temperature and on the scattering mechanism. The L values can thus significantly deviate at high temperatures from the conventional degenerate limit of the free electron or hole gas (2.4510 -8 V 2 .K -2 ). As mentioned earlier in this chapter, a single parabolic band model is sufficient to explain the trends in the electronic properties with the carrier concentration. This model can be also used to compute the temperature dependence of the Lorenz number in doped samples. This approach typically leads to Lorenz numbers that fall within the range 1.510 -8 -210 -8 V 2 .K -2 at high temperatures [87,127]. When subtracting the electronic contribution with these L values, similar 𝜅 𝐿 values are then observed in doped compounds regardless of the element used for doping. Some differences can nevertheless be observed since solid solutions or heavy doping can provide an additional phonon scattering source [105,116,117].

Microscopic origin of the low lattice thermal conductivity of SnSe

Thanks to the possibility to grow large, high-quality single crystals of SnSe, advanced spectroscopic tools can be used to pinpoint the microscopic origin of the very low lattice thermal conductivity of this compound. To this end, inelastic neutron scattering is the experiment of choice since this allows to directly probe the acoustic and optical phonon dispersions. Such experiment carried out on SnSe has revealed several striking features that make it distinct from other rock-salt semiconductors [44,122]. These results have shown that the continuous structural phase transition near 800 K is associated with the condensation of a phonon mode that strongly softens upon warming (Figure 9). Intriguingly, this mode remains strongly anharmonic below the temperature of the phase transition over a large temperature window. This anharmonicity, beyond that associated with the conventional thermal expansion of the unit cell, gives rise to anharmonic phonon-phonon interactions which effectively scatter the heat-carrying acoustic phonons and naturally leads to the very low lattice thermal conductivity values measured. This type of lattice-instability-driven suppression of the heat transport is reminiscent to those seen in the related rock-salt semiconductors PbTe, SnTe and GeTe [13][14][15]. However, some important differences exist between these three compounds and SnSe. PbTe is close to a ferroelectric transition but its crystal structure remains stable due to the fact that the softening of the transverse optic mode is only partial [13][14][15]. As for both SnTe and GeTe, they exhibit a displacive distortion of their cubic crystal structure giving rise to a ferroelectric rhombohedral structure while SnSe is not ferroelectric in its low-temperature orthorhombic phase. SnSe thus represents a model system to study in more detail the role of this type of lattice instability and its related anharmonicity on the thermal transport.

Conclusion

The simple orthorhombic semiconductor SnSe has focused attention over the last years owing to the record-breaking ZT values at high temperatures reported in Bridgmangrown single crystals. This announcement has revitalized the interest for this compound to gain an in-depth understanding of its structural, chemical and physical properties. Thanks to the presence of multivalleys in the valence bands revealed by various electronic band structure calculations and confirmed by ARPES experiments, SnSe exhibits good electronic transport properties near 800 K. Combined with very low lattice thermal conductivity due to some peculiarities in its bonding scheme, this results in thermoelectric performances that rival those achieved in PbTe-based thermoelectric materials.

Several extrinsic dopants have been successfully used to tailor its thermoelectric properties revealing that SnSe is in fact as chemically flexible as other rock-salt semiconductors. These investigations have raised the ZT values from 0.5 at 800 K in pristine samples up to around 1.0 at 800 K in properly-optimized polycrystalline samples. These lower values with respect to those announced in single crystals come from the significant difference in the lattice thermal conductivity measured. The fact that higher values have been measured in polycrystals challenges the usual trend that single crystals should possess higher lattice thermal conductivity due to the absence of grain boundary scattering. This counterintuitive finding has raised concern about the true lattice thermal conductivity values of single crystalline SnSe, an issue which is still debated at the time of writing this review.

As a concluding remark, does the (re)discovery of SnSe be considered as a breakthrough? On one hand, the announcement of record-breaking ZT values has led to a plethora of theoretical and experimental investigations of its physical properties, should it be in bulk form or as thin film down to only one single layer. In this sense, our knowledge on this so far largely overlooked material has been significantly advanced over the last years thanks to the modern techniques employed to probe its basic properties. On the other hand, the lattice thermal conductivity values required to reach these high ZT values seem to have been significantly underestimated, casting serious doubt on the possibility to experimentally achieve such high thermoelectric performances. Thus, the honorific title "best thermoelectric material ever discovered" that SnSe is sometimes dubbed is likely overemphasized. Although very intriguing on its own, this simple binary compound probably does not represent a real breakthrough in thermoelectricity. Yet, this compound as well as its Te-and S-based analogues remain a fascinating area of research to explore in years to come.

2. 1

 1 Crystal structure At room temperature, SnSe adopts an orthorhombic crystal structure (space group Pnma, No. 62; a = 11.37 Å, b = 4.19 Å and c = 4.44 Å at 300 K) which can be thought of as a three-dimensional distortion of the rock-salt cubic structure of PbTe (Figure 1) [19-21]. The basic building block is composed of two-atom-thick SnSe slabs that run along the b-c plane.

Fig. 1

 1 Fig. 1 Projection of the crystal structure of SnSe (space group Pnma, No. 62) along the b and c

Fig. 2

 2 Fig. 2 Electronic band structure along high-symmetry directions of the low-temperature

Fig. 3

 3 Fig. 3 Energy dispersion of the highest pockets in the valence bands of SnSe measured by

  These results help to understand the beneficial role of this multivalley band structure on the ZT values. Indeed, the overall ZT values along the c axis of the crystal structure is related to band parameters through the following relation [77] 𝑍𝑇 ∝ ∑ 𝛾 𝑖 𝜏 𝑖 √ 𝑚 𝑎𝑖 * 𝑚 𝑏𝑖 * 𝑚 𝑐𝑖 * 𝑖 where 𝑚 𝑎𝑖 * , 𝑚 𝑏𝑖 * and 𝑚 𝑐𝑖 * are the density-of-states effective masses along the a, b and c axis of the band i, 𝜏 𝑖 is the relaxation time of the charge carriers along the transport direction for the band i and 𝛾 𝑖 is the degeneracy of the band i which is equal to two in SnSe. This simple formula explains, for instance, the poorer values observed along the a axis. In this direction, the ZT values are proportional to the coefficient √ 𝑚 𝑏𝑖 * 𝑚 𝑐𝑖 * 𝑚 𝑎𝑖 *

Fig. 4

 4 Fig. 4 Temperature dependence of the (a) thermopower and (b) electrical resistivity of

Fig. 5

 5 Fig. 5 Pisarenko-Ioffe plot (thermopower as a function of the hole concentration) at 300 K.

Wei( 1 . 1 𝑚 0

 110 et al.[87], J. Am. Chem.Soc. 138, 8875 (2016). Copyright 2016 by the American Chemical Society.The average density-of-states effective mass 𝑚 𝐷𝑂𝑆 * used in transport equations can be calculated from the values along the three axes determined theoretically or experimentally by ARPES using the formula 𝑚 𝐷𝑂𝑆 * = (𝑚 𝑎𝑖 * 𝑚 𝑏𝑖 * 𝑚 𝑐𝑖 * ) 1/3 𝛾 2/3 (with the degeneracy factor 𝛾 equal to 2 for SnSe). Each of these models (with one or two valence bands) describes equally well the variations in the thermopower with the hole concentration at 300 K for concentrations varying between 10 17 cm -3 and 10 20 cm -3[87]. The best fit value of 𝑚 𝐷𝑂𝑆 * ) agrees very well with the value predicted by electronic band structure calculations or by ARPES experiments. These results consistently show that a single parabolic band model is sufficient to describe satisfactorily the evolution of the thermopower as a function of the doping level in p-type SnSe.

Fig. 6

 6 Fig. 6 (a) Temperature dependence of the specific heat of SnSe measured at low and high

Fig. 7

 7 Fig. 7 Temperature dependence of the lattice thermal conductivity measured on single-

Fig. 8

 8 Fig. 8 Lattice thermal conductivity as a function of temperature measured on pristine

Fig. 9

 9 Fig. 9 Dynamical structure factors calculated by DFT at 100 K (panels a and b) and measured


Jérome Long

Alexander Selikhov

Ekaterina Mamontova

Konstantin A Lyssenko

Yannick Guari

Joulia Larionova

Alexander A Trifonov

Single-molecule magnet behavior in luminescent carbazolyl Dy( iii ) octahedral complexes with a quasi linear N --Dy-N -angle

Single-molecule magnet behavior in luminescent carbazolyl

Coordination complexes based on lanthanide(III) ions have recently emerged as an exciting field in molecular materials due to their tremendous physical properties related to the combination of a strong spin-orbit coupling with a crystal-field effect, which results in a large anisotropy for some lanthanide ions. For instance, such lanthanides complexes could exhibit a magnetic bistability, usually called Single-Molecule Magnets (SMMs) behaviour, making them as potential candidates for applications in data storage, spintronics or quantum computing. [1][2][3][4] Yet, such feature could not be observed in all lanthanide complexes since the origin of the phenomenon requires the presence of a large anisotropic barrier that opposes two magnetic states (± m J ), which can be controlled by the coordination environment of the lanthanide ion. It is commonly accepted that in performing SMMs, the energy barrier should be as large as possible and the retention of the magnetization should be achieved in zero field under a socalled blocking temperature. Among several synthetic strategies aiming to increase the anisotropic barriers, stabilizing the oblate electronic density of some lanthanide ions, such as the Dy 3+ , by coordination of two negatively charged ligands with short bond lengths along an axial direction and the absence of any ligands in the equatorial plane has been recognized as a powerful approach. [5][6] This strategy permits to maximize the crystal-field splitting when a suitable symmetry is obtained. Hence, major achievements have been reached in the last few years by using coordination and organometallic chemistry concepts, [7][8][9][10][11][12] pushing back the limits of the magnetic anisotropy in the dysprosium metallocene family [13][14] with magnetic hysteresis observed up to 80 K. 15 On the other hand, the relaxation may be highly impacted by the presence of the Quantum Tunnelling of the Magnetization (QTM), as well as Raman and direct relaxations, which reduce the SMMs performances. Nowadays, upgrading SMMs require determining the parameters affecting the spinphonon coupling that stand in need to be grasped. Among the different ligands used in coordination chemistry, those involving oxygen or nitrogen donor atoms appear interesting since they may not only stabilize the oblate electronic density of Dy 3+ , but may also act as antennas to design luminescent SMMs. [20][21] Surprisingly, we realized that the number of high anisotropic barrier SMMs based on amido/imido ligands is still relatively scarce with respect to their alkoxy counterpart, 2 although recent theoretical calculations have predicted an effective energy barrier larger than 4000 K in a two-coordinate imido complex. 22 In this sense, carbazole-based ligands are highly appealing since: i) they provide rigidity owing to the presence of multiple aromatic rings; ii) benefit from great tunability to introduce novel functionalities; iii) they could act as a simple monoanionic amido ligand but also could form -complexes;

23 iv) the negatively charged character makes them very promising to design SMMs with axial crystal-field. Moreover, carbazole, featuring a rigid plane biphenyl core with wide band gap and high luminescent efficiency presents great opportunities for the synthesis of new emitters. [24][25][26] Although, derivatives of carbazolyl ligands (mostly tridentate) have been investigated towards lanthanide(III) ions, 27 simple carbazole has mainly been used with divalent lanthanide ions. [28][29][30] With this in mind, we report herein the synthesis, structure, photoluminescent and magnetic studies of two octahedral Dy(III) carbazolyl-based complexes with a quasi linear (carbazolyl)N  -Dy-N  (carbazolyl) arrangement, exhibiting a Dy 3+ luminescence associated with a SMM behaviour.

Our synthetic strategy relies on the formation of a cationic dicarbazolyl complex [DyR 2 ]

+ moiety (R  = carbazolyl) in association with a bulky non-coordinating anion. X-Ray diffraction analyses indicate that 1 and 2 crystallize in the orthorhombic Pnma and trigonal space groups, respectively, with a unique crystallographic complex within the asymmetric unit (Table S1). In 1, both cation and anion occupy the special position on the mirror plane that coincides with one carbazolyl ligand and crosses the second one. Both complexes exhibit an octahedral geometry (see Shape Analysis, Table S2) with two R  located in the trans axial position and four solvates molecules (py and THF for 1 and 2, respectively) in the basal plane (Fig. 1). The shortest distances involve two carbazolyl ligands in axial position with Dy-N(R) distances of 2.333(3)/2.335(3) and 2.318(4)/2.332(4) Å for 1 and 2, respectively. As expected, the Dy-O distances in 2 are relatively close (ranging from 2.318(3) to 2.344(3) Å) to those involving the carbazolyl ligands, while the Dy-N(py) distances in 1 are much longer (2.470(2) and 2.486(2) Å). Hence, a compressed octahedral geometry could be observed for 1 whereas 2 exhibits a much more regular one. Remarkably, the N-Dy-N angle of 178.6(1)° in 1 is closer to the linearity than the one found in 2 (172.3(1)°). The crystal packing analysis indicates that the two uncoordinated solvate pyridine in 1 strongly interact with the carbazolyl ligands through - interactions with the shortest C•••C contacts equal to 3.38 Å. The shortest Dy-Dy intermolecular distances in the crystals are equal to 9.947 and 10.619 Å for 1 and 2, respectively (Fig. S1).

The magnetic properties of 1 and 2 were investigated in both, static and dynamic modes by using a SQUID MPMS-XL magnetometer. The room temperature T values of 14.14 and 14.51 cm 3 .K.mol 1 for 1 and 2, respectively, are in accordance with the theoretical value of 14.17 cm

3 .K.mol 1 ( 6 H 15/2 )
expected for a unique Dy 3+ ion. Upon cooling, both T vs. T curves show a monotonous decrease which reflects the thermal depopulation of the m J levels (Fig. S2). Then, below 12 K, a dramatic decrease could be observed. The field dependence of the magnetization at 1.8 K and under 70 kOe shows a value of 5.90 and 5.66 Nβ for 1 and 2, respectively, without any saturation, indicating the presence of a magnetic anisotropy (Insert of Fig. S2). The sigmoidal shape of the M vs. H curves at low fields may suggest a flip of the Dy 3+ spins under an applied magnetic field or the occurrence of dipolar interactions. Additionally, a magnetic bistability could be evidenced in the hysteresis loops at low temperature (Fig. S3). Hence, the dynamic of the relaxation was investigated by the measurements of the magnetic susceptibility in alternating current (ac) mode. For both complexes, the frequency dependence of the out-of-phase susceptibility component (") under a zero dc-field reveals a series of single temperature dependent peaks, pointing out the occurrence of a slow relaxation of the magnetization (Fig. 2, Fig. S4). The corresponding Cole-Cole plots (Fig. S5) can be fitted with a generalized Debye model giving moderate  parameter values (< 0.2) indicating a narrow distribution of the relaxation times (Table S3-S4). Remarkably, out-of-phase signals could be observed up to 60 and 50 K for 1 and 2, respectively (Fig. S6). From these data, the investigation of the relaxation dynamics could be achieved by extracting the temperature dependence of the relaxation time, . The canonical ln vs. T 1 plot (Fig. 3) reveals a deviation from the linearity upon lowering the temperature indicating the presence of additional relaxation processes. The relaxation at low temperature becomes temperature independent indicating a QTM regime. Remarkably, the relaxation time measured at 2 K for 1 is six times greater with respect to that found for 2, pointing out a noticeable difference in the relaxation dynamics. The overall data range could be modelled using the following equation:

 1 =  0 1 exp(/kT) + CT m +  1
QTM (Eq. 1). 31 The first term accounts for a thermally activated process, while the second and third ones stand for two-phonon Raman and QTM respectively. The m value was fixed to 5, which is one of the possible values observed for Kramers ions. 32 The best fit parameters (Table 1) indicate that the magnetization relaxes trough these three processes. Noticeably, while the  values are rather close for 1 and 2, strong differences in the Raman coefficients and QTM rates could be observed.

With the aim to shortcut the contribution from the QTM, the field dependence of the relaxation time  at 20 K was investigated (Fig. S7) and modelled with the equation  1 = DH 4 T + B 1 /(1+B 2 H²) + K (Eq. 2, Fig. S8, Table S5), 31 for which the first term accounts for the direct process (for Kramers-ion), the second one for the QTM and the K constant for the fieldindependent Raman and thermally activated processes. The optimum field (field at which the relaxation time is the greatest) was estimated at 1000 Oe and 2000 Oe for 1 and 2, respectively. Hence, the ac susceptibility data under these dc fields confirmed the reduction of the QTM (Fig. S9-S10, Table S6-S7). The fitting of the temperature dependences of the relaxation time has been performed with the following model:

 1 =  0 1
exp(/kT) + CT m + AT (Eq. 3) (Fig. 3), in which the third term accounts for the direct process (Table S8). The obtained  values are greater with respect to the zero-field data, in particularly for 1, which may be explained by the presence of a noticeable QTM regime. For both, in-field and zero field data, attempts to fit the temperature dependence of the relaxation time was also performed by fixing the  values from the linear fit in the high-temperature region. However poor fitting or unrealistic parameters were obtained (Table S9).

To gain further insights, the orientation of the anisotropic axes for the ground doublet were evaluated using the MAGELLAN 33 software by assuming that a negative charge is localized on the nitrogen atom of the R ligand. As expected, and owing to the negatively charged character of the carbazolyl ligands, these axes are found almost collinear to the (R)N-Dy-N(R)) string (Fig. S11). The deviation angles between the anisotropic axis and (R)N-Dy bonds are equal to 0.05 and 1.14° for 1, while they are slightly larger for 2 with values of 3.13 and 3.63°. Despite the linear character of the (R)N-Dy-N(R) angle in both compounds, relatively modest anisotropic barriers are found in zero magnetic field, while important differences in the relaxation dynamics are readily observed between two complexes. These features may be rationalized taking into account different parameters. Firstly, whereas 1 exhibits slightly longer Dy-N(R) distances with respect to 2

(2.333(3)/2.335(3) vs. 2.318(4)/2.332(4) Å), its exhibits an almost linear (R)N-Dy-N(R) angle of 178.6(1)° against of 172.3(1)°) in 2, that may explain its higher energy barrier. Another advantage of the carbazolyl ligands, besides the design of genuine SMMs, consists in their ability to efficiently sensitize the Dy 3+ ion. Indeed, a dysprosium-based luminescence could be observed at low temperature (77 K) for both complexes (Fig. 4), making these complexes as Note also that at room temperature, the solid-state emission spectrum for 2 exhibits the typical emission-lines for Dy 3+ when excited at 370 nm (Fig. S12, Fig. S13), while, 1 presents a dual emission with both, a broad band of the ligands and the characteristic Dy 3+ emission lines in the 565-585 nm region.

  Dy(III) octahedral complexes with a quasi linear N  -Dy-N  angle. , THF = tetrahydrofuran) exhibiting a quasi linear N-Dy-N angle in the axial direction, suitable to provide a coordination environment allowing a zero-field slow relaxation of the magnetization.

Fig. 1 . 2 .

 12 Fig. 1. Left: Molecular structure of the complex [DyR2(py)4] + in 1. Right: Molecular structure of the complex [DyR2(THF)4] + in 2. Colour code: orange, Dy; red, O; grey, C. Hydrogen atoms and the [BPh4]  moieties have been omitted for clarity.

Fig. 2 .

 2 Fig. 2.Frequency dependence of the out-of phase (") susceptibilities for 1 and 2 under a zero dc-field.

Fig. 3 .

 3 Fig. 3. Temperature dependence of the relaxation time for 1 (blue) and 2 (red) using the ac data at 0 Oe and 1000 Oe. The solid line represents the fit with Eq. 1 (0 Oe) and Eq. 3 (1000 Oe).

Fig. 4 .

 4 Fig. 4. Low temperature (77 K) solid-state emission spectra for 1 and 2 (exc = 370 nm).The intra 4f9 Dy 3+ transitions are reported.

Table 1 :

 1 Fit parameters of the temperature dependence of the relaxation time for 1 and 2.

	Compound	 (cm 1 )	0 (s)	m	C (s 1 .K m ) QTM (ms)
						(8.3 ±
	1 (0 Oe)	50 ± 4	(7 ± 2)  10 5	5*		0.5) 	4.6 ± 0.4
						10 6
	2 (0 Oe)	40 ± 4	(2.0 ± 0.8)  10 5	5*	(0.5 ± 1)  10 4	0.76 ± 0.03
	Noticeably, the parameters for 1 may be compared with those
	of the pentagonal bipyramidal complex [Dy(OtBu) 2 (py) 5 ][BPh 4 ]
	presenting a comparable angle of 178.91° (O-Dy-O), but much
	shorter Dy-O distances (2.110(2)-2.114(2) Å).	11 This difference
	explains the greater energy barrier of 1250 cm 1 in comparison
	to 1. More generally, the axial Dy-N(R) distances appear larger
	in comparison with those found in some phenoxide based
	SMMs (2.1-2.2 Å) 12, 34 that may decrease the axiality. Secondly,
	the role of the coordinated solvates in the equatorial plane
	appears also very important: the Dy-N(py) distances (2.470(2)
	and 2.486(2) Å) in 1 are much longer than the Dy-O(THF) ones
	(2.318(3) to 2.344(3) Å) in 2. Consequently, the presence of
	these strongly interacting THF moieties in the equatorial plane
	in 2 provides a greater transverse component. Thus, both the
	more linear (R)N-Dy-N(R) angle associated with larger bond
	lengths in the equatorial plane may explain the greater slow
	relaxation performances of 1 with respect to 2. More
	generally, there are rare examples of six-coordinated
	lanthanide complexes exhibiting a zero-field slow relaxation of
	their magnetization.	31, 35-40 However, it could be emphasized
	that 1 and 2 constitute unique examples of octahedral systems
	with an axial crystal field generated by two negatively charged
	ligands arranged in trans.

  the crystal-field splitting with the anisotropic barrier. However, the greater number of transitions (12 or 13) with respect to the expected splitting into 8 components for the6 H 15/2 state, points out the presence of "hot bands" involving the first excited doublet of the

	multifunctional magneto-luminescent SMMs. The transitions
	7 F 9/2 	6 H 15/2 involving the ground state	6 H 15/2 are particularly
	relevant to correlate 7 F 9/2 levels, 20, 41 precluding an unambiguous determination of the crystal-field splitting.	-1	-5 -4 -3		1 (0 Oe) 1 (1000 Oe) 2 (0 Oe) 2 (1000 Oe)
		2 K	lns	-6
					-7
					-8
					0.1	0.2	T	0.3 -1 / K -1	0.4	0.5

Acknowledgements

The financial support of the Russian Science Foundation is highly acknowledged (Project № 17-73-30036). The French authors thank the University of Montpellier and CNRS for funding and PAC of ICGM for measurements.

Conflicts of interest