Thermoelectric materials taking advantage of spin entropy: lessons from chalcogenides and oxides

ABSTRACT

The interplay between charges and spins may influence the dynamics of the carriers and determine their thermoelectric properties. In that respect, magneto-thermoelectric power MTEP, i.e. the measurements of the Seebeck coefficient S under the application of an external magnetic field, is a powerful technique to reveal the role of magnetic moments on S. This is illustrated by different transition metal chalcogenides: CuCrTiS₄ and CuMnTiS₄ magnetic thiospinels, which are compared with magnetic oxides, Curie-Weiss (CW) paramagnetic misfit cobaltites, ruthenates, either ferromagnetic perovskite or Pauli paramagnet quadruple perovskites, and CuGa_1-xMn_xTe₂ chalcopyrite telluride and Bi_1.99Cr_0.01Te₃ in which diluted magnetism is induced by 3%-Mn and 1%-Cr substitution, respectively. In the case of a ferromagnet (below T_C) and CW paramagnetic materials, the increase of magnetization at low T when a magnetic field is applied is accompanied by a decrease of the entropy of the carriers and hence $\left|S\right|$ decreases. This is consistent with the lack of MTEP in the Pauli paramagnetic quadruple perovskites. Also, no significant MTEP is observed in CuGa_1-xMn_xTe₂ and Bi_1.99Cr_0.01Te₃, for which Kondo-type interaction between magnetic moments and carriers prevails. In contrast, spin glass CuCrTiS₄ exhibits negative MTEP like in ferromagnetic ruthenates and paramagnetic misfit cobaltites. This investigation of some chalcogenides and oxides provides key ingredients to select magnetic materials for which S benefits from spin entropy.

Graphical abstract

KEYWORDS:

• Thermopower
• chalcogenides
• oxides
• spins
• entropy
• magneto-resistance

CLASSIFICATIONS:

• 50 Energy Materials
• 203 Magnetics / Spintronics / Superconductors
• 206 Energy conversion / transport / storage / recovery
• 210 Thermoelectronics / Thermal transport / insulators, chalcogenides, magneto-thermopower

1. Introduction

Thermoelectric (TE) materials have the potential to be one of the green solutions towards todays’ global energy crisis [1,2]. Over the years, to make TE devices a commercial success, many efforts have been made first to optimize material parameters like Seebeck coefficient (S) and electrical resistivity (ρ) through band engineering approaches, including band distortion [3], band convergence [4,5], and band nesting [6], to find a candidate with a large power factor PF as PF enters the TE Figure of merit ZT (= S²T/(ρκ) = (PF)T/κ), where к and T are the thermal conductivity and absolute temperature, respectively. Many chalcogenides having transition metal cations with partially filled d orbitals in them exhibit fascinating TE properties. In most cases, their interesting transport properties cannot be described by standard models. Strong correlations between carriers and interplay between charges and spins can influence the dynamics of the carriers and determine their TE properties [7–9]. Magnetic field-dependent thermoelectric power (MTEP) along with magnetoresistance (MR) studies provide a tool to unveil the possible mechanisms of unconventional transport properties of such magnetic systems and subsequently find new TE materials with suitable parameters. As the Seebeck coefficient depends on both transport coefficients and entropy [10], the application of a magnetic field can affect both terms. A significant MTEP usually signifies large orbital or spin degrees of freedom of the carriers, which through increasing entropy can result in large S, which is required for TE application [11]. At first, considering metal transition oxides, Wang et al., through MTEP measurement of Na_xCoO₂ samples with mixed valency of low spin Co³⁺ and Co⁴⁺, demonstrated that S is dominated by spin entropy in these layered oxides [8]. As for the latter system, the misfit cobalt oxides, another class of compounds containing CoO₂ layers of the CdI₂-type (Figure 1), exhibit a temperature dependence of S, which is not metal like. The low temperature slope of S in BiCaCoO misfit is strongly enhanced by the presence of paramagnetic spins as evidenced by its large negative MTEP below 20 K [12]. However, in oxides, the MTEP is not limited to CdI₂-type structure in which the transition metal forms hexagonal layers but is also found in square lattices of 3D perovskites (Figure 1). A giant negative MTEP between −80% and −100% under 5 T in the 60–225 K temperature range was observed in the Nd_0.75Na_0.25MnO₃ perovskite manganite and was accompanied by a large MR, which can be correlated with magnetic field-induced collapse of antiferromagnetism (AFM) [13]. It is noteworthy to mention that large MTEP and MR can also be observed without the presence of any magnetic cation in some topological insulators and phases having Dirac states [14–16]. Such effects arise from non-trivial band structure and are not in the scope of this study.

Figure 1. Schematic structural representation example of a) CuCr₂S₄ Thiospinel structure-type, b) Misfit Cobaltite structure, c) SrRuO₃ Perovskite structure-type and d) CuGaTe₂ Chalcopyrite structure-type

Apart from oxides, there exists other metal transition magnetic compounds containing chalcogen anions like S, Se and Te, which show excellent TE parameters with or without MTEP effect. For sulphides, considering the spinel structure first found in oxides, this is exemplified by the CuCrTiS₄ thiospinel (Figure 1), characterized by a large TEP absolute value (S_300K = −180 µV/K), and for which MTEP was reported [17]. This large S absolute value is remarkable if one considers the low S values reported for the metallic behaviours of both end members CuCr₂S₄ and CuTi₂S₄ [17]. In some magnetic chalcopyrites also, where all metals are tetrahedrally coordinated to sulphur anions (Figure 1), magnetism was invoked to explain the large TEP. Compounds derived from AFM CuFeS₂ chalcopyrite may exhibit a large power factor, by doping at the Cu site, or by creating S vacancies [18]. Indeed, a maximum power factor of 10⁻³ Wm⁻¹K⁻² was found for Cu_0.97Zn_0.03FeS₂ where large TEP was explained through an enhancement of the effective mass m* due to the interaction between delocalized carriers and localized antiferromagnetic spins [19,20].

Here, we present an MTEP investigation of oxides and sulphides possessing different structures schematically described in Figure 1. The variety of their crystallographic structures, 2D or 3D, with magnetic cations forming hexagonal or cubic arrays and with very different metal coordinations, octahedral in misfit cobaltites and perovskite ruthenates, tetrahedral in chalcopyrites or both octahedral and tetrahedral coordinations in thiospinels, is at the origin of different electronic and magnetic states. In the following, the relationship between magnetism and thermopower is demonstrated in this diversity of thermoelectric compounds. Though fundamental, our comparative study will help to understand and correlate the interplay between spin and thermal transport in systems having different magnetic properties, which might be important for their further development towards better TE materials.

2. Experimental details

According to our previous studies [21–23], among magnetic chalcogenides, sulphides like CuCrTiS₄ thiospinel and tellurides like CuGa_1-xMn_xTe₂ chalcopyrite (0 ≤ × ≤ 0.03) and Bi_1.99Cr_0.01Te₃ were chosen. The synthesis of these materials was carried out using solid state reaction technique by heating the constituent elements under vacuum. A CuMn_0.5Ti_1.5S₄ ceramic sample was also prepared replacing Cr by Mn/Ti metal and using the same synthesis method as in Ref [21]. As for oxides, three families were investigated with different magnetic properties: SrRuO₃ perovskite, DCu₃Ru₄O₁₂ (D= Na, Ca, Ca_0.5La_0.5, La) quadruple perovskites, and misfit cobaltites like [BiA₂O₄][CoO₂]_b1/b2 (A = Ca²⁺, Sr²⁺, Ba²⁺; b₁/b₂ = crystallographic misfit ratio) were taken for investigation. These oxides were prepared by standard ceramic method. The details of synthesis of the samples along with their in-depth structural characterization and experimental details about magnetic property measurements can be found elsewhere [21–26].

Thermopower (TEP) measurements were performed using a homemade sample puck in a physical property measurement system (9 T-PPMS, Quantum Design, San Diego, USA) [27]. Measurements are performed using a four-point steady-state technique with separate measuring and power contacts. Bars of typical dimension of 2×2×10 mm³ were mounted using GE varnish on the heat sink of the cryostat, and two chromel-constantan thermocouples were attached to monitor the temperature gradient. The thermoelectric voltage was measured through the chromel wires. For MTEP measurements (Figure 2), magnetic field perpendicular to the temperature gradient was applied (Figure 2(a)). Measurements were done either in isothermal conditions by varying the magnetic field or as a function of temperature in an applied magnetic field. The thermoelectric properties of CuGa_1-xMn_xTe₂ and Bi_1.99Cr_0.01Te₃ were measured using the TTO option from Quantum Design. For CuGa_1-xMn_xTe₂, magnetic fields were applied both parallel and perpendicular to the temperature gradient. As shown below, the two different configurations gave essentially the same results. Thus, for Bi_1.99Cr_0.01Te₃, the magnetic field was applied parallel to the direction of heat gradient (Figure 2(b)).

Figure 2. Schematic diagram of the MTEP set up in (a) homemade sample puck where direction of magnetic field is perpendicular to the direction of heat flow, as represented by the black circles (b) TTO option from Quantum Design where direction of heat flow and magnetic field is parallel to each other

MR was measured in the same instrument by a conventional four probe method by applying magnetic field perpendicular to the current flow. For CuGa_1-xMn_xTe₂ samples, MR measurements were performed by applying magnetic field both parallel and perpendicular to the current.

3. Results and discussions

3.1. Paramagnetic systems

For spinel chalcogenides, magnetic and magnetotransport properties are determined by distribution and localization of metal ions in their tetrahedral as well as octahedral cationic sites and their oxidation states [28–30]. In this family, cubic thiospinel CuTi₂S₄ having Ti³⁺ and Ti⁴⁺ mixed valence exhibits a Pauli paramagnetic behaviour whereas a double exchange between high spin Cr³⁺ and Cr⁴⁺ is reported to be responsible for ferromagnetism observed in CuCr₂S₄ [31,32]. A solid solution exists between these two members, and the CuCrTiS₄ thiospinel having half-filled t_2g Cr³⁺ and empty t_2g Ti⁴⁺ shows a paramagnetic behaviour down to low temperature (Figure 3). A spin glass transition occurs at T = 8 K [33]. Fitting susceptibility data with Curie-Weiss law shows a slight upward deviation of the experimental data below T = 75 K (inset of Figure 3). This points to small antiferromagnetic fluctuations occurring due to magnetic interaction among Cr³⁺ ions. While its magnetic properties are governed by Cr³⁺, Ti ions mainly contribute to carriers conduction. To describe transport properties for T < 75 K, a variable range hopping model (VRH) was employed for this system [21]. Despite the metallicity and rather small S value of both end members, i.e. ferromagnet CuCr₂S₄ (S = +16 µV/K) and Pauli paramagnet CuTi₂S₄ (S = −12 µV/K) [34], a thermopower as high as −140 µV/K at 300 K is observed in CuCrTiS₄ (Figure 4). Due to a small departure from the stoichiometry of the compound, a small fraction of Ti³⁺ ions having slightly filled t_2g orbitals remains in the system, which leads to large negative S. A giant MR, reaching −95% at 5 K for 9 T is observed [21] (top panel of Figure 5). This could be attributed to the gradual alignment of neighboring magnetic moments of Cr³⁺ ions with the application of external magnetic field. Such a large negative MR in the VRH regime is similar to the behaviour observed in colossal magneto-resistive (CMR) manganites [35]. So MR in the whole T range was scaled as a function of $\left(1-B_J^2\right)$, where B_J is the Brillouin function (in [21]), with S = J = 3/2. This confirms that paramagnetic Cr³⁺ ions progressively align themselves as the temperature is lowered and give rise to giant negative MR. The effect of external magnetic field on S can be observed below 45 K as shown in Figure 6. A large MTEP with absolute S decreasing from 34 µV/K at 0 T to 25 µV/K at 9 T for a typical temperature T = 20 K is obtained (inset of Figure 6). This S lowering can be attributed to the gradual alignment of Cr³⁺ spins, which decreases the magnetic entropy. For comparison, the properties of the metallic thiospinel CuMn_0.5Ti_1.5S₄ are reported. This compound was found to be isostructural to the cubic CuCrTiS₄ thiospinel. As shown in Figure 3, the susceptibility is smaller in the whole T range, with a rapid increase at low T. A Curie-Weiss fitting to this T-dependent susceptibility in between 100 K and 300 K yields µ_eff = 5.7 µ_B/f.u. Such a value is consistent with S = 0 for Ti⁴⁺ and S = 5/2 for high spin Mn²⁺ as a theoretical value µ_eff = 5.9 µ_B is calculated for a high spin d⁴ cation. Thus, in both CuCrTiS₄ and CuMn_0.5Ti_1.5S₄, the Ti cations are tetravalent, whereas Cr is trivalent for the former and Mn divalent for the latter. CuMn_0.5Ti_1.5S₄ is metallic with a small upturn of electrical resistivity below 50 K (inset of Figure 4), the temperature below which the χ values strongly increase. In this T range, below 50 K, a small negative MR appears, reaching ~ −2% in 9 T at 5 K (bottom panel of Figure 5). For this metallic thiospinel, the thermopower, with values in between those of CuCrTiS₄ and CuTi₂S₄, exhibits negative values, varying almost linearly with temperature as expected for a metal (Figure 4). In clear contrast with CuCrTiS₄ which exhibits VRH transport, only a very small MTEP is detected (not shown), showing the major role of localized paramagnetic spins on MTEP.

Figure 3. Temperature dependence of magnetic susceptibility (χ) of CuCrTiS₄ and CuMn_0.5Ti_1.5S₄ in a field of 10⁻² T. Inset: χ⁻¹ (T) of CuCrTiS₄ where the dashed line corresponds to the Curie-Weiss fitting

Figure 4. S(T) from 5 K to 325 K for CuCrTiS₄, CuMn_0.5Ti_1.5S₄ and CuTi₂S₄ thiospinels. Inset: Temperature dependence of electrical resistivity ρ for all the three samples

Figure 5. Isothermal Magnetoresistance curves of CuCrTiS₄ (top) and CuMn_0.5Ti_1.5S₄ (bottom) samples. Corresponding T values are shown in the graph

Figure 6. S(T) of CuCrTiS₄ measured at 0 T and 9 T. Inset: Isothermal S(H) of CuCrTiS₄ and BiCaCoO misfit oxide with the corresponding temperatures labeled in the graph

The positive impact of localized spins on thermopower had already been evidenced in Bi-based misfit cobaltites like [BiA₂O₄][CoO₂]_b1/b2 (A = Ca²⁺, Sr²⁺, Ba²⁺; b₁/b₂ = crystallographic misfit ratio). They consist of a single layer of CdI₂ type [CoO₂], which is stacked with four layers of rocksalt (RS) type structure [24]. Cationic substitutions in the RS type layers are responsible for a doping of the [CoO₂] layer by a mechanism of charge transfer and a modification of the positive charge in RS type layer, which is manifested through a change in the misfit ratio (b₁/b₂). These materials do not show any magnetic ordering and remain paramagnetic down to 2 K [36]. Figure 7 depicts S(T) data for [Bi_1.7Co_0.3Ca₂O₄][CoO₂]_1.67 abbreviated as BiCaCoO. Within the temperature range 20–125 K, a linear variation of S is observed. The large value of the slope in the linear region below 20 K reflects strong electronic correlations in accordance with the high value of the Sommerfeld coefficient γ and the universality of the ratio S/(Tγ) [7,12]. As far as MTEP is concerned, a strong magnetic field dependence of S is observed for BiCaCoO [24] below ~ 125 K: as shown in the inset of Figure 6, S is dramatically reduced as the magnetic field is applied. This decrease in S can be attributed to the fact that as magnetic field increases, paramagnetic spins gradually align and magnetization (M) is increased leading to the loss of entropy. A maximum decrease of 60% in S is observed for 9 T at 5 K. A large negative MR is also observed reaching −87% for 7 T at 2.5 K for BiCaCoO. All the S(H) curves below 20 K can be scaled down as a function of H/T and fitted by a Brillouin function [12]. Such a scaling behaviour confirms the freezing of spin fluctuations and hence the reduction of entropy in the system, which results in the reduction of S. It must be mentioned for comparison that in the metallic and Pauli paramagnet BiBaCoO misfit, no MTEP is observed [24,36], as discussed for thiospinels. MR and MTEP are observed only for localized carriers, with a large enough susceptibility associated with a Curie-Weiss behaviour.

Figure 7. S(T) of BiCaCoO misfit oxide (in 0 and 9 T)

At room temperature, no MTEP is observed in any of these misfit materials. However, the presence of magnetic cations still plays a role, as S at room temperature is mainly dependent on the formal valency of Co in [CoO₂] layers, and on the fact that these Co cations carry a spin as shown by the generalized Heikes formula [37,38]. The S data around 300 K can be described by:

$S=-\frac{k_B}{\left|e\right|}ln\left[\left(\frac{2S_n+1}{2S_{n+1}+1}\right)\left(\frac{1-x}{x}\right)\right]+\frac{{\mathrm{k}}_{\mathrm{B}}}{\left|\mathrm{e}\right|}\mathrm{l}\mathrm{n}\left({\mathrm{\Gamma }}_{orb}\right)$

where S_n and S_n+1 are the spins of the transition metal Mⁿ⁺ and M⁽ⁿ⁺¹⁾⁺, respectively, x is the carrier concentration and г_orb the orbital degeneracy. In the case of Bi-based misfits, both spin and orbital degeneracy terms related to low spin Co³⁺ (S = 0) and low spin Co⁴⁺ (S = ½) have to be taken into account to describe its room temperature S [39,40]. Taking into account the lifting of the t_2g degeneracy [40,41] the spin and orbital terms correspond to ln (2) = ~ 60 µV/K [8,40], a sizable fraction of the thermopower, ~ 140 µV/K at 300 K for x = 0.33 (Figure 7).

3.2. Ferromagnetic ruthenates

In the case of ruthenium oxides, the role of the spin entropy and magnetism can be investigated for different magnetic states due to the diversity of magnetic behaviour in these most often metallic oxides. The Seebeck coefficients of different ruthenium oxides exhibiting ferromagnetism (SrRuO₃), paramagnetism (CaRuO₃) and Pauli paramagnetism (quadruple perovskites) have been measured. SrRuO₃ is a ferromagnetic metal (T_C ~ 160 K) where Ru takes a d⁴ electronic configuration [42]. In this perovskite structure, a low-spin (S = 1) state of Ru⁴⁺ is favoured due to the large crystal field splitting between e_g and t_2g orbitals in the presence of octahedrally connected oxygens [43]. A close look at Figure 8 reveals that S(T) for SrRuO₃ does not exhibit the expected linear dependence for a typical Drude metal, and that magnetism plays a role with an accident observed near T_C ~ 160 K. In the case of the paramagnetic CaRuO₃ [44], the evolution is smoother up to high temperature. Calculation of the spin only term in Heikes formula using spins of Ru⁵⁺ (S = 3/2) in matrix of Ru⁴⁺ (S = 1) and Ru³⁺ (S = 1/2) in the matrix of Ru⁴⁺ gives S value of 25 µV/K and 35 µV/K, respectively, which is very close to the measured S for SrRuO₃ and related ruthenates around 300 K [25]. Above T_C, S is driven by a constant spin entropy. This model is much too simple, and a more rigorous calculation has shown that in the case of Sr₂RuO₄, the orbital entropy is quenched up to 1200 K, leading to a spin entropy-dominated Seebeck coefficient below 1200 K [45]. In addition, a negative MTEP is also observed below T_C for SrRuO₃ (inset of Figure 8 for T = 20 K). This field dependence of S follows the magnetization curve M(H) (Figure 9): a gradual increase of external magnetic field makes the spins more aligned up to its saturation magnetization (M_S) (to be compared to M(H), inset of Figure 8), leading to a decrease of S. From the inset of Figure 8, when the saturation magnetization  M_S is reached, the MTEP becomes constant. The magnitude of the SrRuO₃ MTEP is similar to the one measured in the ferromagnetic CaRu_0.8Sc_0.2O₃ (−15% at 30 K) [46].

Figure 8. S(T) of SrRuO₃ up to 800 K. Inset: Isothermal magneto-thermopower MTEP% = 100 x [(S(H) – S(H = 0))/S(H = 0)] and M(H) curve of SrRuO₃ at a typical temperature T = 20 K

Figure 9. Temperature dependence of χ for SrRuO₃ (in 0.1T) and CaCu₃Ru₄O₁₂ sample (in 1T)

3.3. Pauli paramagnetic ruthenates

To compare with these perovskites, S and ρ measurements of a group of quadruple perovskite (QP) samples DCu₃Ru₄O₁₂ (D= Na, Ca, Ca_0.5La_0.5, La) having Ru oxidation state ranging from 3.75 (D = La³⁺) to 4.25 (D = Na⁺) and presenting Pauli like behaviours (Figure 9) were performed. Density of states calculations in these materials shows that the influence of Cu states can only be observed far below E_F, and hence have no effect on the transport properties [47]. ρ(T) measurements up to 900 K for LaCu₃Ru₄O₁₂ exhibit a monotonic increase of ρ, almost linearly with T without any saturation (Figure 2 in [26]), consistently with the bad metal behaviour observed in many ruthenates such as SrRuO₃ or Sr₂RuO₄ [48–50]. Even though S values of all these QP and single perovskite SrRuO₃ are similar (~32 µV/K) at the highest temperature of 900 K, the evolution of S(T) differs with a gradual increase in the case of QP, with a much smaller slope than for SrRuO₃ (Figures 10 and Figures 8). In fact, the saturation S value of 32 µV/K for SrRuO₃ is reached at T ~ 200 K whereas around this temperature all the S curves of the QP group almost merge to 12 − 15 μV/K. At lower temperatures, there is a substantial change in the value of S of DCu₃Ru₄O₁₂ and the values of S depend on the M value of the samples, with the smallest Seebeck coefficient measured for NaCu₃Ru₄O₁₂ (Figure 11) which possesses the smallest Pauli susceptibility. For T > 200 K, all S curves converge and almost a similar S value of all the samples is observed in the temperature range 200–900 K. So formal Ru oxidation state has no effect on S in these samples. A monotonic increase in S up to high temperature can be rather related to classical metal-like picture following Mott’s formula, where band structure plays a significant role. MTEP measurements show no significant effect of external magnetic field on S in these Pauli paramagnets as exemplified for CaCu₃Ru₄O₁₂ in the inset of Figure 10.

Figure 10. T-dependent Seebeck coefficient S for all the quadruple perovskite samples under investigation. Inset: Isothermal S(H) of CaCu₃Ru₄O₁₂ at 20 K showing negligible MTEP

Figure 11. S of quadruple perovskites at 50 K and 900 K

3.4. Diluted paramagnetism in a telluride, a Kondo effect?

The last example is CuGa_1-xMn_xTe₂ which is a paramagnetic system with no evidence of magnetic order observed at least down to 5 K (Figure 12). Mn²⁺ doping in the pristine system increases hole concentration. As a result, smaller electrical resistivity ρ is observed in x = 0.03 as compared to x = 0. But in spite of increasing in carrier concentration, relatively high TEP is observed in Mn-doped sample. Figure 13 shows the effective mass enhancement m*/m₀ of CuGa_1-xMn_xTe₂, which has been derived from the Seebeck coefficients and the carrier concentrations at T = 325 K based on a parabolic band model. The effective mass enhances from 0.6m₀ for x = 0 to 1.5m₀ for x = 0.03 [51]. It has been demonstrated that the interactions between holes and the magnetic ions play a pivotal role in enhancing m*. Strong correlation between magnetic ions and holes is inferred from the magnetoresistance (MR) shown in Figure 13. At T = 10 K, transverse magnetoresistances MR_T shown in Figure 14(a) decrease significantly, reaching almost −40% for x = 0.01 and −20% for x = 0.02 and 0.03. Shown in Figure 14(b) is the longitudinal magneto resistance MR_L of x = 0.03 with field parallel to current, measured at various temperatures. MR_L at T = 10 K almost agrees with MR_T of x = 0.03 in Figure 14(a), which means that the large MR is intrinsic to the coupling of carriers and magnetic moments. The strong coupling has also manifested itself as the unusually large anomalous Hall effect (AHE), observed in x = 0.03 for T ≤ 20 K [22]. Calculation revealed a negative value of AHE constant R_S which also indicates antiferromagnetic coupling between Mn²⁺ and carriers. It is notable that the magnetoresistance in Figure 14(a) is most significant in the diluted limit (x = 0.01), suggesting that the on-site interaction between hole and Mn²⁺ moment is responsible, rather than the Mn-Mn inter-site couplings.

Figure 12. Magnetic susceptibility of CuGa_1-xMn_xTe₂ per Formula-Unit mol

Figure 13. Effective mass m* with respect to the free electron mass m₀ of CuGa_1-xMn_xTe₂ derived from a parabolic band model. Broken line is guide for eye

Figure 14. (a) Transverse magnetoresistance (MR_T) of CuGa_1-xMn_xTe₂ measured at T = 10 K, where magnetic field is perpendicularly applied to current. (b) Longitudinal magnetoresistance (MR_L) of CuGa_0.97Mn_0.03Te₂ where magnetic field is parallel to current

This interpretation is in accordance with the results of magnetic susceptibility. Magnetic susceptibility was fitted with Curie-Weiss function, $\chi =C/(T-\theta )+{\chi }_0$ where C, θ and χ₀ is Curie constant, Weiss temperature and the temperature-independent term, respectively. The effective magnetic moment of 5.35μ_B obtained for x = 0.03 sample is close to the Mn²⁺ moment (5.92μ_B) with S = 5/2 and g = 2. In addition, the Weiss temperature comes out to be negative (−108.5 K), which indicates AFM interaction on Mn spins. One of the reasons of getting large negative θ can be AFM exchange between Mn²⁺ ions. But at such low Mn concentration, Mn-Mn distance is much longer, and it is difficult to imagine that Mn–Mn interaction would give rise to this large negative θ. In addition, the possibility of a direct overlap of 3d states of Mn is quite low in this case which rules out the contribution of Mn-3d impurity band to the large S (Figure 15). This large negative θ and large S can arise due to Kondo-type interaction between magnetic impurities and carriers [52]. The Kondo model was applied by Osinniy et al. to explain large TEP of ferromagnetic Ga_1-xMn_xAs [53]. Kondo interaction is a process of compensating the magnetic moments of impurities by forming spin-singlet states with carrier electrons. In the low temperature limit, this effect eliminates the magnetic entropies of impurities, while the entropies are transferred to carrier electrons with enhanced effective mass. As a result, the Seebeck coefficient increases with the Kondo-type interaction. Indeed, enhanced effective carrier mass m* has been observed in CuGa_1-xMn_xTe₂, as shown in Figure 12.

Figure 15. Seebeck coefficient as a function of temperature of CuGa_0.97Mn_0.03Te₂ measured under µ₀H = 0 and 7 T. The field is perpendicular to heat flow in (a), while parallel in (b) . Inset shows electrical resistivity under µ₀H = 0 T and 7 T with current perpendicular to field in (a), and parallel to field in (b). Broken line in (a) shows a fit with the Kondo-model

Figure 15 depicts the temperature dependence of S and electrical resistivity measured under zero and applied field of 7 T. Electrical resistivity ρ increases rapidly at low temperatures. Under magnetic field of µ₀H = 7 T, this increase is significantly suppressed, as shown in the insets of Figure 15(a,b). This infers that the contribution of magnetic moment and carriers through Kondo-like mechanisms plays an important role in the transport properties of CuGa_1-xMn_xTe₂. It was suggested that the Kondo-type interaction can lead the temperature dependent Seebeck coefficient as: S(T) = aT + S₀T/(T + T₀), with a, S₀, and T₀ are parameters [52,53]. The broken line in Figure 15(a) is the fit with the Kondo model. The model appears to explain the overall behaviour. On the other hand, as shown in the main panel of Figure 15, S is hardly affected by the application of field, possibly because of the too small Zeeman energy of the field of 7 T compared to the antiferromagnetic coupling between carrier and Mn²⁺ moment, of the order of θ = −100 K. This contradicting behaviour has recently also been observed in Sr₂Fe_1+xRe_1-xO₆ double perovskites [54]. It must be emphasized that the thermopower depends on the entropy and on a transport term associated with the band structure [10]. As shown in all the examples presented here, MTEP is associated with MR, but on the other hand the presence of MR is not sufficient to ensure a MTEP effect due to the complex interplay between the entropic and transport terms.

A second similar example is the Cr-doped Bi₂Te₃ tetradymite system. Bi₂Te₃ with diluted Cr doping, as well as with transition-metal TM atoms (V, Mn and Fe), have been found to exhibit ferromagnetic transitions, with Cr doping exhibiting relatively high transition temperatures near ~240 K. An effective magnetic moment of 3.53μ_B is obtained for Bi_1.99Cr_0.01Te₃ consistent with the Cr³⁺ moment (3.87 μ_B). In contrast to the Mn²⁺ doping in CuGa_1-xMn_xTe₂, the Cr³⁺ doping does not affect the carrier concentration, as predicted theoretically and confirmed experimentally [23,55]. But a TEP enhancement is observed in the case of Bi_1.99Cr_0.01Te₃ compared to the pristine composition and is associated with an increase of the weighted mobility μ_w from 73.6 cm² V⁻¹ s⁻¹ to 76.3 cm² V⁻¹ s⁻¹ in the doped sample with 1%mol Cr. This representative quality parameter of the charge transport can be defined as ${\mu }_W=\mu \left(m^{\ast }/m_O\right){}^{3/2}$, where $\mu$ is the electron mobility, $m^{\ast }$ is the effective mass and $m_O$ is the electron mass, and the increasing of this parameter supported that an additional mechanism, such as the carrier-magnetic moment interaction, was effective and increased the TEP. As evidence, the non-magnetic isovalent Ga³⁺ ions substitution for Bi³⁺ is compared with the same doping level with magnetic Cr³⁺. Only the magnetic doping with Cr led to an increase of the TEP. Bi_1.99Cr_0.01Te₃ is reported to have a narrow hysteresis loop for the magnetization curves [23]. The relatively small coercivity, (µ₀H_c ≈ 0.2 T), and saturation magnetization indicates that the sample is a weak ferromagnet similar to Mn-doped Bi₂Te₃ [56]. However, the temperature-dependent S(T) of Bi_1.99Cr_0.01Te₃ measured at 0 T, 5T and 9 T (Figure 16) revealed that the TEP is not affected by the magnetic field. It is suggested that the Bi_1.99Cr_0.01Te₃ TEP enhancement is not likely due to a spin fluctuation effect, as reported for the Fe₂V_0.9Cr_0.1Al_0.9Si_0.1 [57], but by a localized electron-magnetic moment coupling promoting a heavier effective mass m* as experimentally observed [23].

Figure 16. Seebeck coefficient as a function of temperature of Bi_1.99Cr_0.01Te₃ measured under µ₀H = 0 T, 5T and 9 T

4. Conclusions

A comparative study of MTEP in magnetically doped chalcogenides showing different magnetic behaviour was performed. A significant negative MTEP is observed in ferromagnetic perovskite ruthenates and paramagnetic misfit cobaltites. In the case of ferromagnetic material, below T_C, the increase of magnetization is accompanied by a decrease of the entropy of the carriers and hence $\left|S\right|$ decreases. On the other hand, no MTEP is observed in quadruple perovskites indicating that Pauli paramagnetism with a very small susceptibility value is less effective in inducing MTEP. In addition, significant MTEP is absent in CuGa_1-xMn_xTe₂ and Bi_1.99Cr_0.01Te₃ where Kondo-type interaction between magnetic impurities and carriers is indicated to prevail. Spin glass system CuCrTiS₄ thiospinel also exhibits negative MTEP like in ruthenates and misfit cobaltites. Variable range hopping behaviour of the thiospinel along with negative MR makes the system similar to mixed-valence manganites [35]. Our MTEP study reveals that, in Pauli paramagnetic materials having smallest magnetic susceptibility values, the effect of spin on S is negligible. In contrast magnetism contributes to enhance S in ferromagnetic and some paramagnetic systems, whereas, when diluted, paramagnetism induces MR through a Kondo effect at very low T, but does not induce MTEP. The low temperature magnetic susceptibility of the different materials ranges from ~ 4.10⁻³ emu/mol for Pauli quadruple perovskites and CuGa_1-xMn_xTe₂ to 0.06–0.2 emu/mol for paramagnetic misfits and thiospinels, respectively, up to 4 emu/mol in SrRuO₃. These very different values give a lower limit (~ 10⁻² emu/mol) below which MTEP has not been observed.

This investigation suggests that one route to enhance S for TE applications is to find new materials where localized or weakly coupled magnetic ions introduce spin entropy to the system. It is interesting to notice that for such a negative MTEP, the impact of magnetism is maximum in zero magnetic field which is very important for potential applications. The presence of MTEP is a strong indicator of the role these ions play. In that respect, ferrimagnetic systems would also be of interest for MTEP measurements as they combine non-compensated antiferromagnetically coupled ferromagnetic sublattices.

Acknowledgments

The authors acknowledge the support received through ‘Raman-Charpak Fellowship 2019ʹ jointly funded by the Department of Science and Technology (DST), Government of India and the French Institute in India (IFI), French Embassy in India, Ministry for Europe and Foreign Affairs, Government of France. CB, NT and TM acknowledge support from JST Mirai Program JPMJMI19A1.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the JST Mirai Program [JPMJMI19A1].