Estimation of the Grüneisen Parameter of High-Entropy Alloy-Type Functional Materials: The Cases of REO_0.7F_0.3BiS₂ and MTe

Abstract

In functional materials such as thermoelectric materials and superconductors, the interplay between functionality, electronic structure, and phonon characteristics is one of the key factors to improve functionality and to understand the underlying mechanisms. In the first part of this article, we briefly review investigations on lattice anharmonicity in functional materials on the basis of the Grüneisen parameter (γ_G). We show that γ_G can be a good index for large lattice anharmonicity and for detecting a change in anharmonicity amplitude in functional materials. Then, we show original results on the estimation of γ_G for recently developed high-entropy alloy-type (HEA-type) functional materials with a layered structure and a NaCl-type structure. As a common trend for those two systems with two- and three-dimensional structures, we found that γ_G increased with a slight increase in the configurational entropy of mixing (ΔS_mix) and then decreased with increasing ΔS_mix in the high-entropy region.

1. Introduction

1.1. Thermoelectric Materials and Superconductors

In the last two decades, the development of effective energy-creating and energy-saving technologies has supplied interesting tools to solve the energy problem and to prevent climate change. One of the solutions is represented by thermoelectric (TE) materials and modules [1,2]. TE modules allow the direct conversion of unused thermal energy into useful electrical power, which can help to reduce carbon dioxide emissions and contribute to a more sustainable society. However, the performance of TE devices needs to be further improved for their practical application, and thus, the development of new TE materials that can allow us to fabricate high-performance TE devices is needed. To estimate the performance of TE materials, dimensionless figure-of-merit (ZT), which is calculated by the following Formula (1), is essential.

$$ZT={\frac{S^2\sigma T}{\kappa }}^{ }={\frac{S^2\sigma T}{{\kappa }_{\mathrm{el}}+{\kappa }_{\mathrm{ph}}}}^{ }$$

(1)

where S, σ, κ, and T are Seebeck coefficient, electrical conductivity, thermal conductivity, and absolute temperature, respectively. Generally, κ is considered as the sum of κ_el and κ_ph, which are contributed by electron (mobile carrier) and phonon. Since κ_el can be controlled by the modification of the electronic transport properties, as described by the Wiedemann–Franz law [3], finding a material with essentially low κ_ph with a large S²σ has been one of the strategies for developing high-ZT materials, typically with ZT ˃ 1 [1]. One of the key strategies for improving the TE properties is creating a layered structure. Several layered compounds, such as Bi₂Te₃, Co oxides, and CsBi₄Te₆, exhibit high thermoelectric performance [4,5,6,7]. In layered systems, low-dimensional electronic states, a structure composed of a stacking of sheets, and/or a large unit cell could be advantageous for producing high TE properties. In addition, nano-structuring and band convergence have been found to be effective to improve ZT in TE materials such as PbTe [2,8]. Another way to reduce κ_ph is the use of lattice anharmonicity [9,10,11,12]. This strategy is relatively new, but various TE materials have been discovered to exhibit a high TE performance [13,14,15]. Recently, the introduction of high-entropy states in TE materials has been developed [2,16,17,18,19,20]. Due to the introduced disorder, κ_ph is expected to be largely suppressed, and a high ZT~2 is observed in high-entropy alloy-type (HEA-type) chalcogenides [19]. However, a strategy to obtain a low κ_ph by entropy control or the effect on anharmonicity has not been established. Therefore, in this article, we addressed this issue on HEA-type TE materials using the Grüneisen parameter (γ_G) [21], which will be reviewed in the next section.

Other functional materials important for solving the energy problem are the superconductors (SC). As it is well known, SCs exhibit zero-resistivity states at temperatures lower than their SC transition temperature (T_c). For most superconductors, SC states emerge through the formation of electron pairs called Cooper pairs, which is achieved via electron–phonon interactions [22]. Therefore, the mechanisms of superconductivity mediated by phonons for most superconductors are classified as being of the conventional type. Since 1986, high-T_c SCs were discovered in Cu-based [23] and Fe-based [24] SCs, and their SC mechanisms are believed to be unconventional, namely, not mediated by phonons. However, conventional mechanisms have been regarded as a promising way to achieve a high T_c due to the recent development of hydrogen-based SC materials under extremely high pressure [25,26]. In addition, lattice anharmonicity has been considered as a key factor for SC in hydrogen-based materials [27]. Therefore, the understanding of lattice anharmonicity in SCs is also an important issue. As in HEA-type TE materials, the effect of high entropy has recently been introduced in various SC [28,29,30,31,32,33,34,35,36,37]. In HEA-type SCs, the effect of disorder on electronic and phonon characteristics and the modification of anharmonicity have not been addressed yet. Therefore, knowledge about the relationship between high-entropy states and anharmonicity would be useful for further design of SCs with a high T_c.

Motivated by this background, we studied the anharmonicity of two-dimensional and three-dimensional systems with different configurational entropy of mixing, from zero entropy to HEA states. To investigate the evolution of anharmonicity with the increase in configurational entropy, we used γ_G in this study. As a conclusion, we propose that anharmonicity in both two- and three-dimensional structures can be modified by the increase in configurational entropy. The suppression of anharmonicity in HEA states would be a common feature in HEA-type materials with different structural dimensionalities.

1.2. Grüneisen Parameter (γ_G)

The γ_G of inorganic materials is calculated using the following formula; γ_G = β_VBV_mol/C_V, where β_V, B, V_mol, and C_V are the volume thermal expansion coefficient, the bulk modulus, the molar volume, and the specific heat, respectively. The parameters needed for the estimation of γ_G are calculated as follows

$${\beta }_{\mathrm{V}}=\frac{1}{V\left(300 \mathrm{K}\right)}\frac{dV}{dT}$$

(2)

$$B=\rho \left(v_{\mathrm{L}}^2-\frac{4}{3}{v}_{\mathrm{s}}^2\right)$$

(3)

$${\theta }_{\mathrm{D}}=\left(\frac{h}{k_{\mathrm{B}}}\right){\left[\frac{3n}{4\pi }\left(\frac{N_{\mathrm{A}}\rho }{M}\right)\right]}^{\frac{1}{3}}{v}_{\mathrm{m}}$$

(4)

$$v_{\mathrm{m}}={\left[\frac{1}{3}\left(\frac{2}{v_{\mathrm{s}}^3}+\frac{1}{v_{\mathrm{L}}^3}\right)\right]}^{-\frac{1}{3}}$$

(5)

In the formulas, dV/dT, ρ, v_L, v_s, v_m, θ_D, h, k_B, n, N_A, and M denote the temperature gradient of the lattice volume, the density of the material, the longitudinal sound velocity, the shear sound velocity, the average sound velocity, the Debye temperature, the Plank’s constant, the Boltzmann’s constant, the number of atoms in the molecule (formula unit), the Avogadro’s constant, and the molecular weight (per formula unit). Although the absolute value of γ_G of functional materials depends on their crystal structure or constituent elements, at least, the value becomes a good index for discussing the evolution of lattice anharmonicity by doping, pressurizing, or element substitution in similar compounds.

For example, in the metal telluride system Pb_1−xSn_xTe, which is a TE material family [8] and a parent phase of SCs with topological electronic states [38,39], the solution of Sn and Pb results in an increase of γ_G: γ_G = 1.5, 2.5, 2.8, and 2.1 for x = 0, 0.25, 0.5, and 1 [40]. By the enhanced lattice anharmonicity in the alloy phase (x = 0.25 and 0.5), κ_ph is clearly suppressed, and the effect was explained by the changes in γ_G [40]. In our previous study on LaOBiS_2−xSe_x, which is a TE system [15] and parent phase of layered SCs [41], we revealed that the anharmonic lattice vibration is the origin of a low κ_ph, using neutron inelastic scattering [12]. By partial Se substitution in LaOBiS_2−xSe_x, phonon softening was observed, and κ_ph decreased with the decrease of the low-energy phonon energy. This trend was reproduced by the Se concentration dependence of γ_G in LaOBiS_2−xSe_x [42]. As described above, estimation of γ_G is a good way to investigate the evolution of lattice anharmonicity in functional materials. Therefore, in this article, we investigated γ_G for HEA-type compounds, because these materials, including TE and SC materials, have recently been drawing attention as electronic materials. Here, we estimated γ_G for the layered BiS₂-based system REO_0.7F_0.3BiS₂ with an HEA-type RE (rare-earth) site and for Mte with an HEA-type M (metal) site.

1.3. Motivation of the Study

In our previous work on RE(O,F)BiS₂, we synthesized polycrystalline samples of REO_0.5F_0.5BiS₂ with different configurational entropy of mixing (ΔS_mix) at the RE site [28], according to a compositional guideline established for alloy-based HEAs [43,44], where the HEA composition is defined as one containing five or more elements with a composition range of 5–35%; ΔS_mix is defined as ΔS_mix = −R Σ_i c_i ln c_i, where c_i and R are the compositional ratio and the gas constant, respectively [44]. Some of the alloy-based HEAs exhibit superconductivity, and the observed superconducting characteristics of HEAs were clearly different from those of pure or low-entropy alloys and amorphous [45,46,47,48,49,50,51,52,53,54,55,56]. Therefore, the investigation on the effects on local structure and superconducting properties by the introduction of high configurational entropy should be important for understanding superconductivity in HEA-type materials. In the layered compound RE(O,F)BiS₂, by changing the number of element and composition at the RE site, samples with different ΔS_mix were systematically synthesized [56]. Notably, the SC shielding fraction was improved via the suppression of in-plane Bi-S1 local disorder (local distortion) (see Figure 1e for crystal structure and the definition of the S1 site) by an increase in the ΔS_mix in REO_0.5F_0.5BiS₂ [56]. Since all the examined samples had similar lattice constants, we concluded that the increase in ΔS_mix resulted in the modification of the local structure and of the SC properties of REO_0.5F_0.5BiS₂ [56]. Therefore, the estimation of γ_G for RE(O,F)BiS₂ would be useful to understand the effect explained above. In addition, we investigated the effect of ΔS_mix on γ_G of metal telluride (Mte) to understand the generalizability of the effects.

2. Results

In this study, we designed and synthesized polycrystalline samples of REO_0.7F_0.3BiS₂ according to the material design concept described in [45]. We fixed F concentration to a uniform electron-doping level; the composition of REO_0.7F_0.3BiS₂ was that typical of the superconducting phase of the BiS₂-based family. Considering the in-plane chemical pressure, which was tuned by the a-axis in the BiS₂-based system [41], we chose the RE site composition to prepare samples with similar constant a after trial and error. As shown in Figure 1f, the lattice constant a for all samples was similar. Information for the REO_0.7F_0.3BiS₂ samples (sample #1–#4) is reported in

Table 1. Sample information, including nominal composition, ΔS_mix/R, relative density, sound velocity (v_L and v_S), volume thermal expansion coefficient (β_V), Debye temperature (θ_D), bulk modulus (B), and γ_G.

Sample No.	#1	#2	#3	#4
RE site	Pr	La0.3Pr0.4Nd0.3	La0.2Ce0.2Pr0.25Nd0.35	La0.2Ce0.2Pr0.2Nd0.2Sm0.2
ΔSmix/R (RE)	0	1.09	1.36	1.61
Relative density	99%	97%	97%	99%
VL (m/s)	3430	3400	3320	3260
VS (m/s)	1860	1730	1720	1850
βV (1/K)	0.0000369	0.0000356	0.0000365	0.0000388
θD (K)	221	207	205	219
B (GPa)	47.9	54.4	50.9	41.8
γG	0.94	1.02	0.98	0.86

. By changing the RE-site elements, ΔS_mix was systematically tuned, and sample #4 could be regarded as an HEA-type compound because of ΔS_mix > 1.5 R. The relative density and the values of v_L were measured using high-pressure annealed samples. We found that v_L slightly decreased with the increase of ΔS_mix. To evaluate β_V, the temperature dependence of the lattice volume was measured using synchrotron XRD (SXRD). Figure 1 shows the Rietveld refinement results of the SXRD patterns measured at temperatures near 300 K. Although small impurity peaks of RE₂O₂S and/or REF₃ were found, the single-phase analysis appeared sufficient for the estimation of the lattice volume using the Rietveld refinement. The temperature dependence of the estimated lattice volume (V) was determined by refining all the SXRD patterns collected at different temperatures and plotted in Figure 2. V linearly increased with the increasing temperature. By linear fitting of the data and Formula (2), β_V was estimated and is presented in Table 1.

Figure 3 shows the results of specific heat (C) for samples #1–#4. The data were analyzed based on a model for the low-temperature region; C = γT + βT³, where γ and β are specific heat constants of electronic and phonon contributions, respectively. θ_D was calculated from β using the following formula

$$\beta ={\frac{12{\pi }^4{N}_{\mathrm{A}}{k}_{\mathrm{B}}}{5{\theta }_{\mathrm{D}}^3}}^{ }$$

(6)

where C_V was calculated using the Dulong–Petit law, which gave C_V = 3R. V_mol was calculated using ideal density and molar mass estimated from the compositions. The parameters needed for calculating γ_G for REO_0.7F_0.3BiS₂ are listed in Table 1. The estimated γ_G is plotted in Figure 4a. The γ_G of #2 was larger than that of #1. The trend of γ_G increasing with a small increase of ΔS_mix is similar to the trend reported for Pb_1−xSn_xTe [40]. However, with a further increase of ΔS_mix, γ_G decreased in the middle-to-high entropy region. These results would suggest that the anharmonicity in REO_0.7F_0.3BiS₂ was enhanced in the low-entropy region and was suppressed in the middle- and HEA regions. To explore the commonality of this trend, we plotted the data presented in [40] in Figure 4b and added γ_G for HEA-type metal telluride (AgInSnPbBiTe₅) in the plot.

The temperature dependence of SXRD and the low-temperature specific heat were measured for AgInSnPbBiTe₅. Figure 5a,b shows the results of Rietveld refinement of the SXRD pattern and the temperature evolution of V. We estimated β_V = 0.0000672 K⁻¹. Figure 5c shows the low-temperature specific heat and the fitting result using the formula C = β₁T³ + β₂T⁵, which was used in previous work on MTe [57]. Using β₁, θ_D was calculated as 136 K. The v_L obtained at high density (relative density of ~100%) was 2740 m/s. The calculated B was 33.8 GPa, and the obtained γ_G was 1.94. The trend of γ_G calculated for MTe (Figure 4b) is quite interesting. As highlighted by the green line, γ_G of MTe decreased in the HEA region; this trend is similar to that shown in Figure 4a (REO_0.7F_0.3BiS₂). We considered that this trend would be caused by entropy tuning and that this is possibly a universal feature for HEA-type functional materials. In the next section, we will briefly discuss the possible origin of the ΔS_mix dependence of γ_G.

3. Discussion

Entropy is a general concept related to various physical quantities. To discuss the phenomena observed in this work, we have to consider at least two different entropies, i.e., the above-mentioned configurational entropy of mixing and the vibrational entropy. In a field of glass transition, configurational and vibrational entropies have been separately considered and analyzed, and their interplay has been discussed [58,59,60]. On the basis of such a concept established for metallic glasses, we will discuss the origin of the results shown in Figure 4. In the low-to-middle-entropy region, the small disorder introduced by element substitution affected the average atomic positions, and weak displacements of the atoms and/or bonds were introduced. As revealed by the extended X-ray absorption fine structure of Pb_1−xSn_xTe [40], bond anharmonicity was enhanced in the alloyed region, resulting in a large γ_G. Basically, structural disorder should be enhanced by an increase in ΔS_mix for both REO_0.7F_0.3BiS₂ and MTe systems. Therefore, in the HEA region, the local structure (atomic positions and bonds) should approach glass-like states. In such a case, the total entropy in the system would be governed by the configurational entropy, resulting in the suppression of the vibrational entropy. Although the results in this study are not exhibiting direct evidence of the interplay between configurational and vibrational entropies, this concept would be useful to develop functional materials with lattice anharmonicity via entropy tuning. To establish this new concept, further investigation of γ_G and lattice anharmonicity of functional materials are necessary. Investigation on materials with harmonic lattice vibration is also needed.

4. Materials and Methods

The polycrystalline samples of REO_0.7F_0.3BiS₂ (see Table 1 for the nominal composition of the examined four samples indicated as #1–#4) were synthesized by a solid-state reaction method. Powders of La₂S₃ (99.9%), Ce₂S₃ (99.9%), Pr₂S₃ (99.9%), Nd₂S₃ (99%), Sm₂S₃ (99.9%), Bi₂O₃ (99.999%), and BiF₃ (99.9%) and grains of Bi (99.999%) and S (99.99%) were used. The Bi₂S₃ powders were synthesized by reacting Bi and S in an evacuated quartz tube. The mixture of starting materials with a nominal composition was obtained by mortar and pestle, pelletized, and sealed into an evacuated quartz tube. The samples were heated at 700 °C for 20 h for #1 and at 750 °C for 20 h for #2–#4. The obtained products were ground, pelletized, sealed into an evacuated quartz tube, and heated under the same heating conditions for homogenization. Since dense samples are needed for sound velocity measurements, the obtained REO_0.7F_0.3BiS₂ powders were annealed at 400 °C for 15 min using a cubic–anvil-type high-pressure synthesis system under 1.5 GPa. The obtained samples had a relative density higher than 97%. For the AgInSnPbBiTe₅ sample, a precursor powder was synthesized by reacting Ag (99.9%) powder and In (99.99%), Sn (99.999%), Pb (99.9%), Bi, and Te (99.999%) grains with a nominal composition at 800 °C in an evacuated quartz tube. The obtained precursor was annealed at 500 °C for 30 min under a high pressure of 3 GPa.

The synchrotron XRD (SXRD) was performed with the beamline BL02B02, SPring-8 (under proposals Nos.: 2020A0068 and 2021B1175). The wavelength of the X-ray was 0.496118(1) Å for the experiments with REO_0.7F_0.3BiS₂ and 0.495259(1) Å for that with AgInSnPbBiTe₅. The SXRD experiments were performed with a sample rotator system; the diffraction data were collected using a high-resolution one-dimensional semiconductor detector (multiple MYTHEN system [61]) with a step size of 2θ = 0.006°. The temperature of the samples was changed by a N₂-gas temperature controller.

The crystal structure parameters were refined using the Rietveld method with the RIETAN-FP program [62]. The tetragonal P4/nmm (#129) model was used for the refinements for REO_0.7F_0.3BiS₂. For AgInSnPbBiTe₅, the NaCl-type (cubic Fm-3m; #225) model was used for the refinements. For the structural parameters, we reported them in earlier works [28,29,41]. The schematic images of the crystal structure were drawn using VESTA software [63].

The temperature dependence of specific heat was measured with a Physical Property Measurement System (PPMS, Quantum Design) by a relaxation method. The longitudinal sound velocity (v_L) of the sample was measured on dense samples using an ultrasonic thickness detector (Satotech-2000C). The v_L was corrected using the relative density of the polycrystalline sample [42], and the corrected v_L was used for the calculation of γ_G.

5. Conclusions

In the introduction, the characteristics of TE and SC materials and the importance of lattice anharmonicity in those materials were reviewed. Motivated by recent works on the analysis of anharmonicity and γ_G, we investigated the structural and physical properties of REO_0.7F_0.3BiS₂, which is a layered (two-dimensional) system, and of the NaCl-type (three-dimensional) metal telluride (MTe); for both systems, ΔS_mix was controlled by changing the solution elements at the RE and M sites. By plotting the estimated ΔS_mix, we found that an increase in ΔS_mix in the low-to-middle-entropy region resulted in the enhancement of anharmonicity, but a further increase in ΔS_mix in the middle-to-high-entropy region clearly suppressed anharmonicity. Since this trend was observed in both cases with two- and three-dimensional structures, we propose that this trend would be a universal feature of functional materials in which the configurational entropy of mixing is modified by alloying one or more sites. Further studies on γ_G for various functional materials are desired to confirm this concept and will open a new pathway for material development by entropy tuning.