Disorder by design: high-entropy oxides as next generation thermoelectric materials†

Subhra Sourav Jana ^ab, Ritwik Banerjee ^b and Tanmoy Maiti *^b
aInstitute of Materials Research, German Aerospace Center, Linder Höhe, 51147 Köln, Cologne, Germany
^bPlasmonics and Perovskites Laboratory, Department of Materials Science and Engineering, IIT Kanpur, Kanpur, U.P. 208016, India. E-mail: tmaiti@iitk.ac.in; Tel: +91-512-259-6599

First published on 24th July 2025

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Subhra Sourav Jana

Subhra Sourav Jana is a postdoctoral researcher at the German Aerospace Center (DLR). He received his PhD from the Indian Institute of Technology (IIT) Kanpur, where he was a recipient of the prestigious Prime Minister's Research Fellowship. His doctoral work earned him the Best Thesis Award from both the Department of Atomic Energy and the Materials Research Society of India, in recognition of his contributions to high-temperature thermoelectric power generation. He holds a B.Tech. and an M.Tech. in Ceramic Engineering from the National Institute of Technology (NIT) Rourkela. His research interests include materials development—particularly high-entropy materials—and device fabrication in the field of thermoelectrics.

Ritwik Banerjee

Ritwik Banerjee is an integrated M.Tech-PhD student at IIT Kanpur. He holds a B.Tech in Metallurgical Engineering from National Institute of Technology (NIT) Durgapur. His natural inclination is towards mathematical formulation. He also does simulation and modelling to support his mathematical formulations. His research interests include thermodynamic modelling, thermoelectric materials, plasmonic structures, modulating orbital angular momentum of light, etc.

Tanmoy Maiti

Tanmoy Maiti is Rahul and Namita Gautam Chair Professor in the Department of Material Science and Engineering at IIT Kanpur, India. He received his PhD degree from Pennsylvania State University. He earned his Master’s in Ceramic Engineering from IIT-BHU and Bachelor’s in Ceramic Technology from the College of Ceramic Technology, University of Calcutta. He did his post-doctoral research at Lawrence Berkeley National Laboratory and Penn State University. His research interests span the areas of thermoelectrics, plasmonics and oxide electronics. However, a common thread in his research is to address the global energy problem and designing next-generation chip-scale technology.

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Introduction

High entropy alloys have gained immense interest¹ since they were first proposed in 2004 by Cantor et al.² and Yeh et al.³ High entropy materials are formed by mixing five or more elements, each occupying a single crystallographic site, in an equimolar or a nearly equimolar ratio. The term ‘high entropy’ is derived from the high configuration entropy (ΔS_config) as expressed in eqn (1).

	(1)

R is the universal gas constant and X_i is the molar fraction of the i-th element. The favorable Gibbs free energy is obtained by increasing the configuration entropy due to the increase in the number of elements, which leads to the formation of an entropy-stabilized single-phase solid solution instead of phase separation caused by a high enthalpy effect.⁴ The maximum value of configuration entropy that can be achieved for a 5-element system is 1.61R, when all the elements have an equal molar fraction. It is somewhat accepted that a material system with ΔS_config ≥1.5R can be denoted as a high entropy material.⁵

The idea of high entropy alloys was translated to high entropy oxides (HEOs) by Rost et al.,⁶ where they successfully developed single phase (Mg_0.2Zn_0.2Co_0.2Cu_0.2Zn_0.2)O with a rock salt structure by mixing five oxides at equimolar ratios. The configuration entropy in the case of an oxide can be given by eqn (2).

	(2)

where X_i, and X_j are the molar fractions of the i-th and j-th elements in cation and anion sites, respectively. The presence of multiple cations in a particular crystallographic site primarily contributes to the effect of high configuration entropy. However, the anion site is occupied by only oxygen atoms, so the contribution from the anion site is negligible. Using the very same concept, different types of high entropy ceramics such as nitrides, carbides, and borides have also been developed.⁷ Nevertheless, HEOs have shown great promise for various functional applications in view of their magnetic properties,^8–10 enhanced mechanical properties,¹¹ colossal dielectric properties,¹² low thermal conductivity,¹³ and so forth. The wide range of these functional properties of HEOs offers a tremendous possibility, especially in the field of thermoelectrics, since thermoelectricity necessitates the optimization of various interdependent physical quantities simultaneously. Thermoelectricity has emerged as one of the most prominent research areas in recent years due to its potential to address one of the most significant challenges of the 21st century, i.e., meeting increasing energy demand without negatively impacting the environment. A thermoelectric generator (TEG) has the ability to convert waste heat directly into useful electricity without creating any carbon footprint. Besides, TEGs can be adopted in many other applications because of their compact size, lack of moving parts, absence of gas emissions, etc.¹⁴ The conversion efficiency of a thermoelectric material is related to the dimensionless figure of merit , where S is the Seebeck coefficient, σ is electrical conductivity, and κ_e and κ_l are electronic and lattice contributions of thermal conductivity. Oxides, such as SrTiO₃,^15,16 ZnO,^17–19 In₂O₃,²⁰ Ca₃Co₄O₉,^21,22 and CaMnO₃,²³ are considered potential thermoelectric materials due to their high-temperature stability, abundant availability, and low processing cost compared to state-of-the-art materials such as chalcogenides. However, these oxides suffer from several drawbacks, one of which is their higher thermal conductivity, resulting in poor thermoelectric performance (low ZT).¹⁴ The thermal conductivity of oxides is typically found in the range of 5–10 Wm⁻¹ K⁻¹, which is quite higher than that of state-of-the-art materials such as chalcogenides (<1 Wm⁻¹ K⁻¹). Since overall thermal conductivity of these materials is dominated by lattice thermal conductivity, it is of the utmost importance to develop a strategy for inducing enhanced phonon scattering in oxide systems. This will help attain higher ZT values in oxide thermoelectrics. Several strategies, including doping,^24–27 composite formation,^28–33 and grain refinement,^34,35 have been adopted by various research groups to suppress the thermal conductivity. However, these strategies have not proven successful in achieving thermal conductivity below <1 Wm⁻¹ K⁻¹. Nanostructuring is considered one of the successful strategies for reducing the thermal conductivity of chalcogenides and intermetallics. However, the very same strategy doesn't work well for oxides, since oxides inherently possess phonon mean free paths in the nanoscale range.^36–38 This motivates the thermoelectric community to explore novel design strategies to develop oxide thermoelectrics, possessing a smaller phonon mean free path.

The concept of the high entropy effect was first introduced in oxide thermoelectrics by Banerjee et al.³⁹ where they were able to achieve ultralow thermal conductivity for Sr(Fe_0.2Ti_0.2Mo_0.2Nb_0.2Cr_0.2)O₃ oxide perovskite. Colonizing the B site in the ABO₃ perovskite structure with multiple cations leads to atomic disorder and lattice distortion, which eventually helps in attaining reduced thermal conductivity. Besides, electrical transport can also be tuned using the concept of a high entropy effect. High configuration entropy favors a large Seebeck coefficient, because it may increase crystal symmetry.^40–42 High crystal symmetry, local lattice distortion or orbital hybridization of multiple elements can always increase the possibility of band degeneracy. A recent study on high entropy AgSbPbSnGeTe₅ (ref. 43) chalcogenides has shown multi-band convergence, which eventually improves the Seebeck coefficient. Another investigation demonstrated that increasing the configurational entropy—such as by incorporating AgSbSe₂ (ref. 44) or AgBiSe₂ (ref. 45) into GeSe—can drive a structural transition from a rhombohedral phase to a cubic phase. This structural modification is associated with an enhancement in the Seebeck coefficient. Notably, while high symmetry can facilitate better electronic transport, it may also promote phonon propagation if not properly balanced with lattice-level disorder.⁴⁶ Entropy optimization is very important towards the goal of attaining a ‘phonon glass electron crystal’ system, which in turn results in enhanced thermoelectric performance as schematically shown in Fig. 1. Nevertheless, predicting the final properties of high-entropy oxides (HEOs) remains challenging, as the synergistic interactions among the multiple constituent elements can lead to complex and often non-linear effects—commonly referred to as the “cocktail effect.”¹

Several studies on HEO thermoelectrics have been published in recent years. However, very few have reported the thermodynamic calculation implying the high entropy effect present in the system. Moreover, a comprehensive review on the effect of incorporating various cations in oxides influencing their thermoelectric performances is not available in the literature. Here, we present a unified thermodynamic model for calculating the free energy of high entropy oxide systems. Furthermore, we have provided a perspective on the thermoelectric properties of high entropy oxides with future directions.

Designing HEOs

Selection criteria for elements

The majority of the oxide thermoelectrics that have been exploited contain a perovskite crystal structure. Perovskites have a formula of ABO₃, where the B-site is occupied by transition metal elements and the A-site corresponds to alkaline earth metals. High entropy oxides are designed by colonizing A and/or B sites with various elements. The crystal structure and the space group are determined by the ordering and tilting of BO₆ octahedra. The stability of the crystal structure is determined using the tolerance factor (t) in eqn (3).

	(3)

where r_A, r_B and r_O are radii of cations A and B and anion O, respectively. When ‘t’ is between 0.9 and 1, a cubic phase is expected to occur; an orthorhombic or rhombohedral phase may occur when ‘t’ is less than 0.9, and a tetrahedral or hexagonal phase may emerge when ‘t’ is more than 1. For high entropy oxides, the average radius is usually calculated considering the concentration and radius of each of the elements. It is expected that HEOs are very likely to form a crystal structure with cubic symmetry due to entropy-driven stabilization. High configuration entropy can cause the reduction of Gibbs free energy, which might allow them to retain the highest crystal symmetry. However, this is not always the case; Kumar et al.⁴⁷ showed that the tilting of the CoO₆ octahedron, caused by lattice distortion, gives rise to the formation of an orthorhombic crystal structure with a Pbnm space group. Jana et al.⁴⁸ showed that optimization of entropy can result in the highest Seebeck coefficient (among HEOs) in non-perovskite high entropy (Sr_0.2Ba_0.2Li_0.2K_0.2Na_0.2)Nb₂O₆, possessing a tungsten bronze crystal structure with a P4bm space group. On the other hand, Banerjee et al.³⁹ and Bhattacharya et al.⁴⁹ found out that the crystal structures of respective HEOs follow the tolerance factor criteria. Meanwhile, Kumar et al.⁵⁰ identified the crystal structure as orthorhombic, despite the t value (t = 0.912) remaining within the cubic structure prediction bracket. The reason for this discrepancy has been attributed to the effect of vacancies and bond deformations. In the case of defective perovskites, the Goldsmith tolerance factor (t) is susceptible to error, as it ignores the effect of vacancies and bond deformations. Additionally, as the calculated t value is close to 0.9, it is possible that, due to structural distortions, it has led to a non-cubic phase. This is a common phenomenon for the structures possessing the tolerance factor in the marginal territories (i.e., in which the t values are closer to 0.9 and 1). The applicability of the tolerance factor to high-entropy oxides is limited due to the multicomponent nature of the cation sites. In such systems, average ionic radii are sometimes used to estimate the effective tolerance factor, but this approach can obscure local distortions and site preferences. The structures of HEOs are often attributed to configurational entropy stabilization rather than geometric factors alone. Therefore, while the tolerance factor is an interesting phenomenon to consider for the selection of cations of HEOs, it is not a strict criterion for structure prediction in HEOs.

However, the size disorder parameter (δ), as expressed in eqn (4), should also be taken into account while designing a high entropy oxide by populating a particular crystallographic site with multiple cations.

	(4)

where is the average atomic radius and r_i and c_i are the radius and atomic percentage of the ith element. The size disorder parameter (δ) in the range of (0 <δ < 5) is favorable for single-phase solid solution formation.^51,52 Zhang et al.⁵³ showed the evolution of secondary phases in (Ca_0.25Sr_0.25Ba_0.25RE_0.25)TiO₃ [RE is Ho, Gd, Dy], when δ is very large (>15%). Although this crystallographic computation is crucial for the selection criteria of different elements in designing HEO, it cannot solely predict the formation of entropy-driven single phase solid solution. It is also necessary to determine the thermodynamic phase stability of the HEO.

Thermodynamic calculation

Cantor et al.² are often credited with the pioneering discovery of high entropy alloys in 2004, comprising Co, Cr, Fe, Mn, and Ni. The suppression of intermetallic phases and obtaining a single phase in these alloys consisting of elements possessing distinct crystal structures intrigued the scientific community. This feat was attributed to the high configurational entropy of multicomponent alloys. However, it's worth noting that merely increasing the number of elements and configurational entropy doesn't always guarantee the formation of single-phase alloys, suggesting the influence of additional factors.⁵⁴ While there are instances of successfully forming single-phase high entropy alloys,^55–60 there are also numerous cases where phase separation leads to multi-phase high entropy alloys.^61–66 Both single-phase and multi-phase high entropy alloys have their own advantages and disadvantages. Intermetallic phases in high entropy alloys can enhance mechanical properties by serving as strengthening mechanisms, with multi-phase high entropy alloys often displaying higher ultimate tensile strength compared to conventional alloys or single-phase HEAs.⁶⁷ However, phase separation can detrimentally affect electrical properties, particularly in high entropy oxides (HEOs).^68,69 As a result of that, there has been growing interest in calculating the calcination or sintering temperature for single phase HEOs having no phase separation to save time and energy. Although CALPHAD (acronym for calculation of phase diagrams) is the widely accepted way of calculating various phases at room temperature of HEAs,^70,71 there is not much literature available on the thermodynamic calculation of high entropy oxides (HEOs) or high entropy perovskites (HEPs) until the last decade. To address this gap, recently, Rost et al.⁶ calculated the transition temperature to a single phase for Mg_xNi_xCo_xCu_xZn_xO as a function of composition, as well as configurational entropy, which has garnered significant attention. However, truly predicting the transition temperature of high entropy oxides requires consideration of other factors in addition to configurational entropy. In the quest to design next generation high entropy oxides, Banerjee et al.³⁹ introduced a comprehensive thermodynamic analysis combining the concept of thermodynamic free energy change and configurational free energy change to determine the transition temperature into a single phase for HEPs, which has since become a widely utilized method adopted by researchers for thermodynamic calculation of high entropy oxides.

Numerous studies have adopted this approach to evaluate the thermodynamic viability of their respective oxides. Examples include compositions such as (Sr_0.2Ba_0.2La_0.2Eu_0.2Pb_0.2)Nb₂O₆,⁷² Sr(Ce_0.05Sn_0.08Zr_0.2Ti_0.16Y_0.3Nb_0.11Al_0.1)O_2.9,⁷³ (Sr_1/3Ba_1/3Ca_1/3)TiO₃,⁷⁴ SrCe_1−xSn_xO₃ (0.1 ≤ ×≤ 0.5),⁷⁵ (Sr_0.2Ba_0.2Li_0.2K_0.2Na_0.2)Nb₂O₆,⁷⁶ and SrCe_0.95M_0.05O₃ (M = La, Pr, Y, Sn).⁷⁷ In this review, a more generalised and extensive form of calculation has been provided supplementing the initial study in this area.²⁴

The calculation procedure initiates by fitting the temperature-dependent specific heat (C_P) of the desired oxide using a polynomial equation, as represented by eqn (5). Prior to fitting, the specific heat values are converted from J gram⁻¹ K to J mol⁻¹ K by multiplying with the molecular weight:

	(5)

The polynomial equation described in eqn (5) offers a versatile framework capable of modelling a diverse array of plots depicting the variation of specific heat with temperature, as illustrated in Fig. 2. The equation features five coefficients—A through E—each capable of being zero, positive, or negative. This yields up to 3⁵ = 243 possible coefficient combinations and, thus, 243 potential curve profiles. Yet, not all yield distinct behaviors. For instance, varying only A and B (with C, D, and E set to zero) results in simple straight lines—different slopes and intercepts, but fundamentally similar shapes. As more coefficients become non-zero, the curve diversity grows richer. Rather than exhaustively plotting every case, our goal is elegantly simple: to demonstrate how this compact polynomial form can flexibly capture a wide spectrum of C_p vs. T profiles with minimal complexity.

Using MATLAB, eqn (5) was plotted under several constraints, each represented by a distinct colored line:

• The red line corresponds to conditions where a is positive, b is negative, and c, d, and e are all set to zero.

• The magenta line indicates scenarios where a is positive, b is positive, and c, d, and e are all set to zero.

• The green line represents scenarios with a being positive, b being negative, c being negative, and d and e both set to zero.

• The blue line indicates cases where a is positive, b is negative, c is negative, d is positive, and e is set to zero.

• The cyan line depicts situations where a is positive, b is negative, c is negative, d is positive, and e is positive.

• The yellow line corresponds to conditions where a is positive, b is negative, c is positive and d and e are both set to zero.

• Lastly, the black line represents conditions with a being positive, b being negative, c being negative, d being positive, and e being negative.

This graphical representation clearly showcases the broad applicability of the equation in capturing the specific heat variations with temperature observed across various oxide materials.

From the C_P value of the oxides, the thermal part of the change of enthalpy (ΔH_thermal) and thermal part of the change of entropy (ΔS_thermal) of the oxides from room temperature (298 K) to the calcination temperature (T) can be expressed as respectively and two new equations have been derived from eqn (5).

ΔHthermal = H0T − H0298 = AT + BT2/2 + CT3/3 + DT4/4 − E/T + F − Hf	(6)

ΔSthermal = AlnT + BT + CT2/2 + DT3/3 − E/2 T2 + G	(7)

A, B, C, D and E are the fitting parameters of C_P, H_f is the enthalpy of formation of a solid under standard conditions at 298 K, and F and G are the integration constants of eqn (6) and (7). By applying the boundary conditions, ΔH_thermal = 0 at T = 298 K and ΔS_thermal = 0 at T = 0, we derive the values of F and G.

There is another way to represent C_P as a temperature dependent polynomial function, which has been applied in the cases of CaCO₃,⁷⁸ NaAlSi₃O₈,⁷⁹ KAlSi₃O₈,⁷⁹ CaAl₂Si₂O₈ (ref. 79) etc., as demonstrated in eqn (8):

CP = A + BT + CT2 + DT−0.5 + E/T2	(8)

In this case, eqn (9) and (10), which represent the thermal change of enthalpy and entropy, are modified accordingly:

ΔHthermal = H0T − H0298 = AT + BT2/2 + CT3/3 + 2 × DT0.5 − E/T + F − Hf	(9)

ΔSthermal = S0T − S0298 = AlnT + BT + CT2/2 − 2 × DT−0.5 − E/2 T2 + G	(10)

Subsequently, the enthalpy of formation (H_f) for the prepared High-Entropy Perovskite (HEP) was calculated using Aronson's formula,^80,81 specifically designed for A_aB_bO_y-type oxides.

Hf = −96.5 × y[xA(χA − χO)2 + xB(χB − χO)2] + 108.8y	(11)

where χ represents the electronegativity taken from the Pauling electronegativity series;⁸² x is the mole fraction of the respective cations and y is the number of oxygen atoms (O) in the oxide molecule under consideration.

This formula was first utilized by Banerjee et al.³⁹ for a Sr(Ti_0.2Mo_0.2Fe_0.2Nb_0.2Cr_0.2)O₃ high entropy oxide perovskite system to calculate formation enthalpy, denoted as H_f in eqn (9) and represented in eqn (12).

Hf = −96.5 × y[xSr(χSr − χO)2 + xTi(χTi − χO)2 + xMo(χMo − χO)2 + xFe(χFe − χO)2 + xNb(χNb − χO)2 + xCo(χCo − χO)2] + 108.8y = −987.93 KJ mol−1	(12)

Here, While there exists a method to enhance the accuracy of the calculation, it inevitably introduces more complexity. Initially, the Pauling electronegativity values were solely considered for the predominant valence state of the cation, e.g., the electronegativity value for only Ti⁴⁺ has been taken into consideration corresponding to titanium in eqn (12). However, it's essential to acknowledge that electronegativity values are contingent upon the chemical environment, the oxidation state and the coordination number of the element.⁸⁰ Despite Ti⁴⁺ being the primary constituent derived in the high-entropy perovskite Sr(Ti_0.2Mo_0.2Fe_0.2Nb_0.2Cr_0.2)O₃, Ti³⁺ and Ti²⁺ may also coexist with Ti⁴⁺. This is validated by X-ray photoelectron spectroscopy (XPS) analysis of Sr(Ti_0.2Mo_0.2Fe_0.2Nb_0.2Cr_0.2)O₃, revealing the presence of Ti²⁺ 2p₃/₂ and Ti²⁺ 2p₁/₂ at 11.3% and 7.7%, respectively, while Ti³⁺ 2p₃/₂ and Ti³⁺ 2p₁/₂ are present at 35.6% and 22.7% and Ti⁴⁺ 2p₃/₂ and Ti⁴⁺ 2p₁/₂ are present at 17.2% and 5.4%, respectively.³⁹ So, it is more accurate to write in eqn (12) instead of 0.2 × (X_Ti − X_O)², where denote the electronegativities of Ti⁴⁺, Ti³⁺ and Ti²⁺, respectively. The same rationale applies to the other cations present in Sr(Ti_0.2Mo_0.2Fe_0.2Nb_0.2Cr_0.2)O₃. The modified formula has the potential to be used extensively in similar calculations in the future.

Subsequently, substituting ΔH_thermal,298K = 0 and ΔS_thermal,0K = 0, the values of F and G are derived. While deriving the value of F is quite straight forward, finding the value of G is a little bit tricky, as substituting T = 0 in eqn (10) results in ΔS_thermal being undefined due to ln 0. To address this issue, ΔS_thermal is tabulated at different temperatures as a function of integration constant F and by the process of extrapolation, the value of F has been derived.

The NIST database stands out as the most comprehensively documented source for obtaining thermodynamic values of oxide precursors.⁸³ However, there remain some other sources, which also cover some of the oxide thermodynamic data, such as CaO,⁸⁴ SrCO₃,⁸⁴ BaO,⁸⁴ BaCO_3,⁸⁴ CaCO_3,⁷⁸ Pr₆O₁₁,⁸⁵ La₂O₃,⁸⁵ Nd₂O_3,⁸⁵ Bi₂O₃,⁸⁶ Mn₃O₄,⁸⁷ Mn₂O₃,⁸⁷ MnO₂ (ref. 87), etc. For certain oxides, abrupt variations in enthalpy or entropy may occur due to phase transitions, as observed in SrCO₃ (ref. 84) or BaCO₃.⁸⁴ In such cases, enthalpy and entropy can be expressed as follows:

	(13)

	(14)

These equations account for the complexities arising from phase transitions and provide a framework for calculating enthalpy and entropy over temperature intervals T₁ to T₄.

Now that we have all the necessary information, we can proceed with finding the value of the free energy change for heating the precursors to the calcination temperature and obtaining a single-phase high entropy oxide.

ΔGThermal = ΔGhigh entropy oxide − ΔGprecursors = (ΔHhigh entropy oxide − ΔHprecursors) − T(ΔShigh entropy oxide − ΔSprecursors)	(15)

Let's consider a high entropy oxide/perovskite denoted as A_aB_bC_cD_dE_eF_fO_y, prepared from various precursor oxides, such as A_a1O_x, B_b1O_x,…, F_f1O_x, etc. Here, a, b, c, d, e, and f are the mole fractions of cations in the high entropy oxide and a₁, b₁,…, and f₁ are the mole fractions of the precursors. In that case, ΔG_Thermal can be written as

	(16)

Using eqn (16), free energy changes for various high entropy oxides (HEOs) have been calculated as presented in Table S1 in the ESI†.

On the other hand, ΔG_mix is calculated from eqn (17).

ΔGmix = ΔHmix−TΔSmix	(17)

It is a well-known fact that the mixing enthalpy (ΔH_mix) of any alloy doesn't necessarily depend upon temperature, which means that the mixing enthalpy at 0 K and mixing enthalpy at calcination temperature bear the same value.^88,89 Due to temperature insensitivity, for any temperature, mixing enthalpy (ΔH_mix) of Sr(Ti_0.2Mo_0.2Fe_0.2Nb_0.2Cr_0.2)O₃ (ref. 39) can be calculated as:

ΔH1523Kmix = ΔH0mix = EHEP − (ESrO + EFe2O3 + EMoO3 + ETiO2 + ENb2O5 + ECr2O3)/6	(18)

where E denotes the energies of the different oxides after relaxation at 0 K and 0 Pa. The equilibrium lattice parameters and energy of HEPs have to be calculated from first principles. In order to find out the ground state energy of any system, a satisfactory way would be solving the time independent Schrodinger equation^90,91 for the system (many-body problem), which is illustrated as:

Hψ(ri,Ri) = Eψ(ri,Ri)	(19)

	(20)

where r_i and R_i are position vectors of electrons and nuclei, respectively, (eqn (19)) and N is the number of electrons. U(r_i,r_j) denotes the coulombic intra-electronic interactions (eqn (20)). Although the Born–Oppenheimer (BO) approximation simplifies the problem by treating electrons as an inhomogeneous cloud in a static potential field due to fixed nuclei, this doesn't offer a definitive solution for real materials.⁹² For example, solving the many-body Schrödinger equation for a lead nanocluster of 100 atoms entails a 24600-dimensional problem. To overcome this issue, a computational quantum mechanical modeling method called density functional theory (DFT) is used in investigating ground state properties of many-body systems. However, DFT analysis for multi-component high entropy oxide systems is complex and time-consuming, requiring extremely high computing power. Hence the amount of DFT reports on HEOs is scanty. Banerjee et al.³⁹ proposed a slightly deviated route as described below, which subsequent researchers^{48,50,72–75,77} have used in their reports.

Enthalpy of mixing (ΔH_mix) for high entropy alloys has been written using the Miedema approach⁹³ as

	(21)

where c_i and c_j represent the composition or mole fraction of the i-th and j-th elements respectively and Ω_ij is the regular melt–interaction parameter between the i-th and j-th elements and can be written as 4ΔH^AB_mix. The same approach has been applied for high entropy perovskite also. ΔH^AB_mix is the mixing enthalpy of all the possible equiatomic binary alloys. With N as the number of cations in the material, there are a total of ^NC₂ potential combinations of binary alloys. The enthalpy of mixing (ΔH_mix) for HEPs has been tabulated in an upper triangular matrix form, as conceptualized by Wang et al.⁹⁴

The mixing enthalpy of AaBbCcDdEeFfOy has been described in the above matrix form, where the diagonal elements are zero, as there is no mixing happening there. Element a12 or ΔHAmixa,Bb represents the mixing enthalpy between the ‘a’ mole fraction of element A and ‘b’ mole fraction of element B. As the mixing enthalpies of elements aij and aji are the same for equal composition, one of the elements between aij and aij has been taken as 0, resulting in the entire matrix being an upper triangular matrix.

Configurational entropy (ΔS_mix) of A_xB_yO_z type oxides can be modified from Boltzmann's fundamental entropy formula^95,96 as

	(22)

where x_a, y_b and z_o represent the mole fractions and M, N and P are the numbers of cations of the elements present in the A-site, B-site and O²⁻-site (or anion site), respectively. As O²⁻ ions consistently occupy the (1/2,1/2,0) site, the term always turns into 0. or can also be 0, depending on which site mixing is occurring, the A-site (0,0,0) or B-site (1/2,1/2,1/2).

The entropy of mixing (ΔS_mix) has been calculated under somewhat idealized conditions, assuming the absence of oxygen vacancies. This assumption implies that each unit cell of the high-entropy oxide perovskite contains 3 O²⁻ anions, which decreases with the introduction of defects in the lattice sites. While defects would ordinarily alter the value of ΔS_mix,^97,98 their contribution has been neglected due to the complexities involved in accurately determining their effect. Another limitation of using this formula is that configurational entropy is purely ideal, i.e., the mixing is completely random. This assumption may not hold true, depending on the tendency of a material to exhibit clustering or short-range ordering, which has also been neglected in the calculation.

Olivia et al.⁹⁹ introduced a novel approach of calculating entropy of high entropy materials, encompassing metals, carbides, carbonitrides, and perovskites, among others, introducing some modifications to the already established sublattice model^100,101 in the calculation. They introduced it as new entropy metric (EM) and defined it as

	(23)

S^config_{SL/mol atoms} represents the configurational entropy calculated using the sublattice model, R is the ideal gas constant, and L is the total number of sublattices. Adding up eqn (16) and (17), ΔG_total has been calculated for high entropy oxide/perovskite structures.^72–77

The negative value of total free energy change Sr(Fe_0.2Ti_0.2Mo_0.2Nb_0.2Cr_0.2)O₃ indicates that a single-phase oxide is favorable to form at elevated temperature (1500 °C). Adopting the same idea, Jana et al.⁷⁶ also found negative free energy change in the formation of (Sr_0.2Ba_0.2Li_0.2K_0.2Na_0.2)Nb₂O₆ and successfully developed a single-phase HEO with a tungsten bronze structure. In addition to thermodynamic calculations, Yao et al.¹⁰² also demonstrated that the optimal value of configuration entropy in Sr_0.8Ba_0.1La_0.1Ti_0.9Nb_0.1O_3−δ yields the highest thermoelectric performance.

Furthermore, the whole process of thermodynamic feasibility, as first conducted by Banerjee et al.,³⁹ is illustrated in Fig. 3. At 1500 K, all the precursors are transformed into Sr(TiFeNiCrNb)O₃. The thermodynamic free energy change for the entire process has been calculated by subtracting ΔG_{Sr(TiFeNiCrNb)O3} from the summation of ΔG_precursors. The intermediate compounds formed during the process are disregarded in the calculation, following Hess' Law,^103,104 which states that ΔH, ΔS, and ΔG are all state functions, as depicted using blank blocks in Fig. 3. Additionally, enthalpy of mixing (ΔH_mix) of Sr(TiFeNiCrNb)O₃ has been tabulated in an upper triangular matrix form, as conceptualized by Wang et al.⁹⁴ Any of the atoms between Ti, Fe, Ni, Cr, and Nb acquire the B site randomly, while only Sr occupies the A site and all the O sites are filled exclusively by O atoms.

Although the whole thermodynamic calculation has been shown for high entropy oxides and perovskites, the process for calculating the single-phase formation temperature has the potential to be used in other high entropy materials, such as carbides, nitrides, carbonitrides, borides, etc.

Electrical transport in HEOs

One of the requirements for being a good thermoelectric material is to have a large Seebeck coefficient (S) along with large electrical conductivity (σ), the combination of which can give rise to the enhanced thermoelectric power factor (S²σ). It is extremely difficult to obtain high S and high σ in the same material system, as they are inversely related to each other with respect to carrier concentration (n) and density of states electron effective mass (m*). Banerjee et al.³⁹ first reported a rare earth free high entropy Sr(Fe_0.2Ti_0.2Mo_0.2Nb_0.2Cr_0.2)O₃ oxide thermoelectric, which exhibits a maximum Seebeck coefficient of −95 μV K⁻¹ at 1100 K. Inspired by the work of Banerjee et al., subsequent studies have explored various SrTiO₃ based HEP systems, as tabulated in Fig. 4(a). Researchers have found that the Seebeck coefficient of these HEPs could be improved (>−100 μV K⁻¹) by incorporating rare earth elements such as La, Ce, Sm, Dy, Gd, Nd, and Ho.^53,105–108 This enhancement is primarily attributed to the presence of highly localized sharp 4f orbitals, which strongly influence the density of states near the Fermi energy.¹⁰⁹ Notably, the highest Seebeck coefficient that has been observed is −468 μV K⁻¹ in Sr_0.9La_0.1(Zr_0.25Sn_0.25Ti_0.25Hf_0.25)O₃. Despite their impact on thermopower, the 4f electrons are highly localized and do not contribute to electrical conduction in the same way as s or p electrons. As a result, the improvement in the Seebeck coefficient is often accompanied by a marked reduction in electrical conductivity.

Jana et al.⁷⁶ showed that optimization of entropy can give rise to a large Seebeck coefficient of 370 μV K⁻¹ at 1150 K in high entropy niobate with a tungsten bronze structure, which is the highest among niobate-based polycrystalline oxides. In contrast, high entropy manganites are not found to exhibit such a high Seebeck coefficient as titanates. Bi_0.2Ca_0.2Sr_0.2La_0.2Pr_0.2MnO₃ and La_0.2Ca_0.2Sr_0.2Ba_0.2Y_0.2MnO₃ exhibit a maximum S of −29 μV K⁻¹ (1060 K) and −40 μV K⁻¹ at 373 K,¹¹⁰ respectively. These observations suggest that the high-entropy effect does not universally lead to an enhancement of the Seebeck coefficient. Instead, the specific elemental composition plays a crucial role, as it significantly influences the density of states, carrier concentration, and, consequently, the Fermi energy. A deeper theoretical understanding is essential to elucidate these relationships and to enable more accurate predictions of compositions that optimize the carrier concentration and the electronic density of states for improved thermoelectric performance. The maximum Seebeck coefficients, along with the electrical conductivity of HEOs, are listed in Table 1.

---

The electrical conductivity of the above-mentioned HEOs, as shown in Fig. 4(b), is dominated by the semiconducting behavior. Some of the HEOs such as (Ca_0.25Sr_0.25Ba_0.25La_0.25)TiO₃, (Ca_0.25Sr_0.25Ba_0.25Nd_0.25)TiO₃ (ref. 53), (Ca_0.2Sr_0.2Ba_0.2La_0.2Pb_0.2)TiO_3,¹⁰⁶ and Sr_0.6Ba_0.3La_0.1Ti_0.9Nb_0.1O_3−δ,¹⁰² exhibit semiconductor behavior at low temperatures followed by a semiconductor to metal transition. The maximum electrical conductivity mostly remains in the range of 10³–10⁴ S m⁻¹ for all the HEOs.^{39,53,106,107} Sr_0.6Ba_0.3La_0.1Ti_0.9Nb_0.1O_3−δ¹⁰² has been found to exhibit a maximum electrical conductivity of 2.4 × 10⁴ Sm⁻¹ at 770 K and it has been attributed to high entropy engineering. However, this system can't be classified as a high entropy perovskite since the A site is occupied by three elements, while the B-site contains two transition metals, and these elements are not even present in equimolar ratios. Besides, Sr_0.6Ba_0.3La_0.1Ti_0.9Nb_0.1O₃ could be visualized as a Ba doped Sr_0.9La_0.1Ti_0.9Nb_0.1O₃ system, and its electrical conductivity is found to be deteriorated by more than two times after Ba doping.¹¹³ Hence, it is evident that the entropy effect doesn't effectively work for improving electrical conductivity. Interestingly, the electron concentration in most of these HEOs^{39,53,102,106,107} is found to be in the range of 10¹⁸–10²³ cm⁻³. Such a high electron concentration generally satisfies Mott criteria^{32,76,114–117} (>10¹⁸ cm⁻³) for most of the complex oxides(eqn (24)).

aBn1/3 ≈ 0.25	(24)

Here, a_B is the Bohr radius and n is the critical carrier concentration for the semiconductor to metal (S–M) transition to occur. Satisfying Mott criteria indicates that these HEOs should exhibit metallic-like behavior. Semiconducting behavior (dσ/dT > 0) has been observed by all the researchers in these HEOs in spite of satisfying Mott criteria. However, the S vs. T graph shows degenerate semiconductor or metallic like behavior. The band model cannot explain such an anomaly between the metallic Seebeck coefficient and semiconducting electrical conductivity. Yao et al.¹⁰² explained that ionic scattering dominates at low temperatures, which results in an increasing trend of conductivity with temperatures. Although the presence of ionic scattering cannot be entirely ruled out, it is difficult to accept the presence of ionic scattering at such elevated temperatures near 700 K. We have estimated the weighted mobility (μ_w) of different compositions in Fig. 4(c) using eqn (25).^118–121

	(25)

where k_B, m_e, and h are the Boltzmann constant, electron mass and Planck's constant, respectively. It is interesting to note that the μ_w of all compositions are following an increasing trend with temperature in the low temperature regime. The observation of such a low weighted mobility with a high carrier concentration has long been a subject of interest in oxide-based thermoelectrics. Banerjee et al.³⁹ first proposed the existence of Anderson localization of electrons in high entropy Sr(Fe_0.2Ti_0.2Mo_0.2Nb_0.2Cr_0.2)O₃, which is mainly responsible for poor electrical conductivity even after possessing electron concentration in the order of 10²² cm⁻³. The presence of multivalent cations induces fluctuations in the local electrical field, and the substantial lattice strain is introduced resulting in the localization of electrons—a phenomenon termed Anderson localization. This kind of localization is commonly observed in many complex disordered oxides.^32,122–132 This explains the thermally activated mobility in the low temperature regime. Jana et al.⁷⁶ further used the Mott and Cutler model¹³³ (eqn (26)) for (Sr_0.2Ba_0.2Li_0.2K_0.2Na_0.2)Nb₂O₆ to support the existence of Anderson localization.

	(26)

According to the Mott and Cutler model, when the Fermi energy of a material is surrounded by a finite number of localized states and mobility edge situated above E_F, thermopower is observed to have a linear temperature dependence. This model is only valid in low-temperature regions. However, linear dependence, even at high temperatures (above 600 K), confirms the presence of localized states surrounding E_F.⁷⁶ Such behavior corroborates the above hypothesis on Anderson localization in HEOs. The existence of electron localization has also been reported in other HEOs such as (Ca_0.2Sr_0.2Ba_0.2La_0.2Pb_0.2)TiO₃¹⁰⁶ and Sr_0.9La_0.1(Zr_0.25Sn_0.25Ti_0.25Hf_0.25)O₃.¹⁰⁵ As the temperature is increased, the localized electrons are promoted from one defect site to another. This thermally activated charge transport is well explained with the help of small polaron hopping (SPH) as described in eqn (27).

	(27)

The activation energy for hopping, E_a, is determined from the linear relationship between ln(σT) and 1/K_BT, as shown in Fig. 4(d). All the compositions are well fitted with respect to the SPH. E_a values of all the samples are listed in Table 2. This implies that the thermally activated charge transport in these HEOs is governed by SPH. However, the electrical conductivity and mobility eventually show the transition to metallic behavior above certain temperatures (500–800 K), demonstrating a decrease in conductivity with temperatures. This phenomenon occurs due to the dominance of acoustic phonon scattering. It is interesting to observe that the high entropy approach helps attain the rare combination of a metallic-like Seebeck coefficient and semiconductor-like electrical conductivity, causing an increase in both S and σ with temperature, which has made it feasible to obtain a higher power factor, as shown in Fig. 4(e).^{39,105,106,134}

---

Among p-type high-entropy oxide thermoelectrics, few reports are available, mostly on layered cobaltites and cobaltates. Monoclinic cobaltite (Ca_0.35Sr_0.2Ba_0.15Na_0.2Bi_0.1)₃Co₄O₉ (ref. 134) with a C2/m space group exhibits a maximum value of the Seebeck coefficient of 174 μV K⁻¹ at 973 K, which is comparable to that of pristine Ca₃Co₄O₉. The Seebeck coefficient of (Ca_0.35Sr_0.2Ba_0.15Na_0.2Bi_0.1)₃Co₄O₉ has been found to be increasing with temperatures. However, the electrical conductivity increases with temperatures, and the maximum σ has been found to be 9145 Sm⁻¹ at 823 K, which is about 70% higher than that of pristine Ca₃Co₄O₉. Such a remarkable enhancement of σ has been attributed to the increased hole mobility. It has been demonstrated that flatter and larger grain sizes of high entropy cobaltites, driven by preferential grain growth, facilitate easy carrier transport and reduced grain boundary scattering. Kumar et al.⁴⁷ reported a high entropy cobaltate-based (LaNdPrSmEu)_0.95Sr_0.05CoO₃ perovskite with an orthorhombic structure, where the Seebeck coefficient and electrical conductivity both follow semiconductor-type behavior. (LaNdPrSmEu)_0.95Sr_0.05CoO₃ exhibits a maximum Seebeck coefficient of 300 μV K⁻¹ at 300 K and it decreases to about 30 μV K⁻¹ at 950 K.⁴⁷ Consecutively, electrical conductivity increases from 5318 Sm⁻¹ at 300 K to 10⁵ Sm⁻¹ at 950 K. Although the electrical conductivity value is quite good, the low Seebeck values at elevated temperatures are not suitable for high temperature thermoelectrics.

Thermal transport

Thermal conductivity must be minimized to develop a better thermoelectric material. However, thermoelectric oxides such as SrTiO₃ exhibit thermal conductivity as high as 10 W m⁻¹ K.¹³⁵ As the thermal transport in oxides is mainly dominated by the phononic part, it is essential to find ways to block phonon transport in oxide systems. The transfer of heat is disrupted or reduced by different phonon-scattering mechanisms, mainly by grain boundary scattering, defect scattering, electron–phonon scattering, and phonon–phonon scattering.¹³⁶ Nanostructuring is widely used in chalcogenide systems to suppress the thermal conductivity. However, it doesn't work well for oxides, as they possess the phonon mean free path in the nanometer range. Banerjee et al.³⁹ were the first to attain an ultralow lattice thermal conductivity (κ_l) of 0.7 Wm⁻¹ K⁻¹ in Sr(Fe_0.2Ti_0.2Mo_0.2Nb_0.2Cr_0.2)O₃ by introducing the concept of the high entropy effect. This was a path breaking result to obtain κ_l below 1 Wm⁻¹ K⁻¹ in oxide thermoelectrics. This makes HEO systems appealing to the thermoelectric community in search of better composition. Subsequently, researchers have investigated various high entropy oxide systems, and a common observation has been found to be low thermal conductivity. The minimum thermal conductivity of various HEOs is presented in Table 1 and Fig. 5(a).

Zhang et al.¹⁰⁶ reported a minimum lattice thermal conductivity of 1.7 Wm⁻¹ K⁻¹ at 1073 K in (Ca_0.2Sr_0.2Ba_0.2La_0.2Pb_0.2)TiO₃, which is close to Cahill's minimum limit for the SrTiO₃ system. Zhang et al.,⁵³ in another study, reported rare earth-based high entropy oxides [Ca_0.25Ba_0.25Sr_0.25RE_0.25TiO₃ (RE = Nd, Sm, Gd, Dy, Ho)], where the κ_l is estimated to be between 2 and 3 Wm⁻¹ K⁻¹ in the entire temperature range of 300–1100 K. Shi et al.¹³⁷ were able to reach a lattice thermal conductivity of 0.9 Wm⁻¹ K⁻¹ at 1073 K in manganate-based HEOs. Among p-type oxides, Yang et al.¹³⁴ reached an ultralow lattice thermal conductivity of 0.66 Wm⁻¹ K⁻¹ at 973 K in (Ca_0.35Sr_0.2Ba_0.15Na_0.2Bi_0.1)₃Co₄O₉, which is found to be 25% lower in comparison to pristine Ca₃Co₄O₉. It has been reported that the low lattice thermal conductivity in these HEOs is primarily attributed to structural disorder including point defects, dislocations, stacking faults, and nanoscale grain boundaries.^53,106 The presence of these multidimensional lattice defects is evident in microstructural analysis of these HEOs. Researchers have identified that the reduced thermal conductivity at high temperatures is primarily due to point defect scattering, which is largely driven by the presence of oxygen vacancies. However, previous reports have already revealed that point defect scattering contributes less than 0.5% to the total thermal resistance of these kinds of complex oxides.^32,122,130 Furthermore, high dimensional defects such as grain boundaries and dislocations don't have a predominant impact on phonon scattering at high temperatures, as suggested by Casimir.¹³⁸ Hence, it is imperative to investigate the heat transport mechanisms in HEOs in a detailed manner. Besides, quantitative analysis of thermal conductivity is also essential for advancing our understanding of thermal transport.

In order to better understand such low thermal conductivity in HEOs, the phonon mean free path (L_Phonon) has been estimated using eqn (28).

	(28)

where v_m is the acoustic mean velocity in the medium and C_v is the heat capacity at a constant volume. The temperature dependence of L_Phonon for Sr(Fe_0.2Ti_0.2Mo_0.2Nb_0.2Cr_0.2)O₃ and (Sr_0.2Ba_0.2Li_0.2K_0.2Na_0.2)Nb₂O₆ is shown in Fig. 5(b). The minimum L_Phonon of Sr (Fe_0.2Ti_0.2Mo_0.2Nb_0.2Cr_0.2)O₃ has been found to be around 5 Å at 1100 K. Jana et al.⁷⁶ also achieved a minimum L_Phonon of 4 Å at 470 K in tungsten bronze (Sr_0.2Ba_0.2Li_0.2K_0.2Na_0.2)Nb₂O₆. The maximum L_Phonon remains below 10 Å. These extremely low values of the phonon mean free path suggest that phonon scattering occurs in the order of interatomic distance. In contrast to other systems, (Sr_0.2Ba_0.2Li_0.2K_0.2Na_0.2)Nb₂O₆ exhibits a slight increase in L_Phonon with temperature−a characteristic typically associated with glass-like behavior and that has already been reported in many complex oxides^139–141 including niobate systems.^{123,129,142–144} This peculiar glass-like thermal conductivity at higher temperature has been attributed to domination of diffusive phonons, which possibly originates from high structural disorder in the form of oxygen vacancies, chemical bond inhomogeneity and mass fluctuations due to the high entropy effect.¹⁴⁵ Besides, this slight increment in κ_l might also arise from weakening of the domain walls,¹²³ leading to the gradual reduction of phonon scattering centers. However, it is essential to evaluate the contribution of different phonon scattering mechanisms to understand the actual reason behind such a low phonon mean free path and, consequently, ultralow thermal conductivity.

To estimate the contribution of different phonon scattering mechanisms, Banerjee et al.³⁹ used Debye–Callaway's model (eqn (29) and (30)).¹³⁶

	(29)

where

	(30)

where A, B, and C are constants and relaxation time (τ) depends on grain boundary scattering , defect or isotope scattering (Aω⁴), phonon–phonon Umklapp scattering and electron-phonon scattering (Cω²). ω, and θ_D are the phonon frequency and Debye temperature, respectively. Grain boundary scattering is generally a low-temperature phenomenon¹³⁸ and, hence, it is omitted from the calculation. Being the only temperature dependent term, dominates most of the fittings anyway, where thermal conductivity decreases sharply with the temperature. The effect of Aω⁴ and Cω² can be realised when thermal conductivity remains almost constant with temperature.

As suggested by Slack,¹⁴⁶ α is ≤ 2 for the Umklapp process. n is considered to be 3 for the three-phonon process in phonon–phonon scattering. However, Banerjee et al.³⁹ showed that κ_l of Sr(Fe_0.2Ti_0.2Mo_0.2Nb_0.2Cr_0.2)O₃ could not be fitted with the Callaway model when n = 3. The fitting has been found to be improved as the n value increases from 1 to 5 (Fig. 5(c)). Larger values of n suggest the manifestation of higher-order phonon interaction, which is generated due to the anharmonicity created by multiple elements, as shown schematically in Fig. 5(d). Multiple elements in the lattice are expected to generate phonon modes with different frequencies, finally facilitating phonon–phonon scattering at higher order (n > 3). This enables us to attain ultralow thermal conductivity in single-phase solid solutions, which opens up a new opportunity not only in thermoelectrics but also in many other functional applications which demand low thermal conductivity at high temperatures.

The ZT values of the HEOs are listed in Table 1. The maximum ZT values of all HEOs are way lower than those of conventional chalcogenides, intermetallics, etc. High entropy based thermoelectrics are still in their early stages of development. It holds immense potential to improve their thermoelectric properties. As discussed above, entropy engineering offers substantial control over both electrical and thermal transport. By carefully selecting elements and optimizing the entropy, it is possible to improve the electrical transport. Furthermore, achieving an ultralow phonon mean path can be facilitated by populating lattice sites with multiple elements. The synergistic effect between the charge transport and thermal transport allows these HEOs to achieve better ZT performance. However, unlocking this potential requires a good amount of theoretical and experimental efforts, coupled with well-defined strategies.

Strategies to improve the ZT

Although the reported ZT values of the above-mentioned HEOs are not very high from an application point of view, predicting the thermoelectric properties of HEOs is highly challenging due to the complex cocktail effect. Here, we have demonstrated two key strategies for how HEOs can be designed and their performance can be enhanced.

Introduction of AI

The idea of high entropy offers a vast compositional space for the development of thermoelectric materials. However, it is critical to know which composition is going to satisfy all the requirements for an ideal thermoelectric material. Artificial intelligence (AI) and machine learning (ML) can play a crucial role in the selection of elements, streamlining the discovery process.

Various computational methods have long been used to predict the thermoelectric properties of relevant compounds. A widely adopted approach for predicting electron transport properties involves solving the Boltzmann transport equation (BTE) under the rigid band approximation (RBA). However, the difficulties in producing the analytical solution of Boltzmann equations prompted scientists to devise various computation codes, such as BoltzTrap¹⁴⁷ and BoltzWann¹⁴⁸ etc. On the other hand, semi-empirical models, such as the Debye Callaway model¹⁴⁹ and quasi harmonic Debye model,^150,151 along with computational codes, such as ShengBTE,¹⁵² the Automatic Anharmonic Phonon Library (AFLOW-AAPL)¹⁵³ and phonon spectral calculation using symmetry and group theory¹⁵⁴ were developed to calculate and predict lattice thermal conductivity tensors. While these physics-based simulation methods offer reasonable accuracy in predicting electrical and thermal properties, they demand significant computational power and time. For instance, solving the Kohn–Sham equations ¹⁵⁵ for a multi-component high entropy oxide assembly to analyze electrical conductivity, or solving the phonon transport iterative Omini equations¹⁵⁶ , is significantly challenging.

This is where machine learning (ML) algorithms come into play. By analyzing vast databases of material properties—such as electronegativity, atomic size, crystal structure, and specific heat—ML models can efficiently predict high-entropy oxide (HEO) compositions with improved thermoelectric properties. Unlike traditional physics-based simulations, ML models rely on data-driven predictions, allowing them to bypass some of the complexities and intricacies of material physics and significantly speed up computations.

There are quite a few databases containing thermoelectric properties, both theoretical^157–160 and, experimental,^161–163 covering a range of materials with varying compositions and temperatures. After choosing the relevant material databases, data pre-processing is very much needed before selecting the relevant ML models. One critical pre-processing step is feature selection, which reduces dimensionality and complexity by eliminating noisy, irrelevant, and redundant features. Depending on the search strategy, feature selection methods are generally categorized into three types: filter methods, wrapper methods, and embedded methods.

Filter methods are independent of ML algorithms, as they select the most discriminative features based on the inherent characteristics of the data. The features are selected based on certain criteria and the selection methods include the chi-square test (χ²), analysis of variance (ANOVA), the Pearson correlation coefficient (PCC), the distance correlation coefficient (DCC) etc. χ² is one of the simplest feature selection methods and measures the relative difference between observed and expected data between two discrete variables at a time. χ² can be represented as

	(31)

where O_i is the observed value and E_i is the expected value.

While χ² is limited to comparing two discrete variables, ANOVA extends this capability by allowing comparisons across multiple groups or discrete variables.

The PCC, on the other hand, quantifies the linear correlation between continuous variable pairs (x, y), related by the equation

	(32)

where n is the number of samples in the dataset, x_i and y_i are the i-th sample point, and and ȳ are the mean values of the respective variables. The PCC takes values between −1 and +1, indicating negative to completely positive correlation. Additionally, two independent variables always result in the PCC being zero, although the converse is not true. This means that the value of the PCC zero doesn't always guarantee the independence of the variables. Moreover, non-linear correlation between variables disturbs the accuracy of the PCC.

The DCC is an alternate correlation coefficient that does not have this weakness, where dcor (x, y) = 0 always guarantees that x and y are independent. Edelmann et al.¹⁶⁴ proved that, for any fixed Pearson correlation coefficient strictly between −1 and 1, the distance correlation coefficient can take any value in the open unit interval (0,1). This can be represented as

	(33)

dcor (x, y) = 0, if and only if x and y are independent of each other or in other words,

dcov (x) dcov (y) = 0	(34)

This makes the DCC a more reliable metric for assessing relationships between variables, particularly in cases where nonlinear dependencies exist.

Wrapper methods involve training a model and evaluating subsets of features based on their performance. One common approach is recursive feature elimination (RFE) by sequential forward selection and backward elimination. A more advanced wrapper method is called the genetic algorithm, which borrows its core ideas from human evolution and genetics. In this approach, the problem is initially represented as a set of encoded strings. A fitness function is then assigned to evaluate each string, followed by the application of genetic operators such as reproduction, crossover, and mutation to refine feature selection.

On the other hand, embedded methods integrate feature selection within the model training process itself. Notable examples include lasso regression, which performs well with high-dimensional datasets by enforcing feature sparsity, and ridge regression, which is more effective when dealing with datasets containing correlated features.¹⁶⁵

After extracting the features, the refined dataset can be fed into several machine learning models. Gaultois et al.¹⁶⁶ built one of the initial machine learning models for thermoelectric property prediction. As achieving good ZT thermoelectric materials is a multi-objective optimization problem, subsequent studies have focused on optimizing individual thermoelectric parameters, such as electrical conductivity,^167,168 the Seebeck coefficient^169–171 and thermal conductivity^172–175 using ML models separately. The effectiveness of a particular ML model depends on the dataset's characteristics, size, and the feature selection method applied beforehand.

ML models in thermoelectric research have followed a similar trend. For instance, in the quest of optimizing the carrier concentration in diamond like superconductors, Miller et al.¹⁷⁶ found out that simple linear regression models¹⁷⁷ work best in terms of matching the experimental results. However, there have been other studies where more sophisticated models, such as random forest models,^{169,178–180} gradient boosting regression,^{160,181–183} kernel ridge regression,^184–186 Gaussian process regression,^187–190 neural networks^191–193 etc., have been proven to be better models than others.

One of the difficulties in applying ML models in thermoelectric research is the limited amount of data. Among the existing datasets, a significant portion consists of theoretical data, which can sometimes be crude approximations of actual values. Additionally, experimental data can exhibit considerable variability, as small changes in processing routes and parameters can lead to substantial differences in electrical conductivity (σ), Seebeck coefficient (S), and thermal conductivity (k).

This challenge is even more pronounced in the case of high-entropy alloys (HEAs) and oxides. The total number of possible high entropy alloys can be expressed with the formula

where r represents the total number of elements chosen to form the alloy from a possible n number of candidate elements. is an interesting function in the sense that the value of N increases up to r = n/2. The above argument can be proved mathematically with the concept of mathematical induction and is shown in ESI S2.†

As most of the HEAs are 5, 6 or 7 component alloys, r takes the value of 5, 6 or 7. Given that approximately 40 to 50 metal or non-metal elements are potential candidates for HEA formation, n is around 45.

This essentially boils down to r ≪ n/2 in the case of HEAs. From the above arguments, we can conclude that the number of potential high entropy alloys is much higher than that of the binary, ternary and quaternary alloys. Consequently, when applying ML to predict HEA properties, the available databases often prove more insufficient.

Another important parameter for the sufficiency of the data points in the database is model complexity.¹⁹⁴ In an n-component HEA, interactions occur not just between all n elements but also among subsets of components. The total number of interactions can be expressed as . For instance, in a 5-component alloy, the interactions among binary, ternary and quaternary subsets must also be considered, leading to a total iteration count of I_{(5−component alloy)} = C (5,2) + C (5,3) + C (5,4) = 25. This increasing complexity poses a significant challenge for ML models, often preventing them from achieving ideal predictive accuracy.

Despite the complexities, ML can also unravel the complex relations between the materials structure and thermoelectric properties and also ignite the exploration of alternative oxide systems, albeit with occasional fabricating challenges. Herein lies the importance of thermodynamic calculations: predicting transition temperatures for newly envisaged HEOs/HEPs in advance can significantly streamline research endeavors, saving considerable time and resources. However, achieving success in this domain necessitates robust collaboration of data scientists and material scientists. The optimization of machine learning models and realizing the intricate relations between descriptors can guide us in the pursuit of the best performing thermoelectric system.

Composite formation

One of the drawbacks this kind of complex oxide, including HEOs, suffers from is its poor electrical conductivity even after bearing such a large carrier concentration above 10²⁰ cm⁻³. As discussed above, the presence of Anderson localization of electrons due to huge lattice strain and multivalent metal ions is primarily responsible for suppressed electrical conductivity. Composite formation with HEOs has huge potential to curb this problem. Previous reports^32,122,195 have shown that forming composites with different carbon additives can significantly improve electrical conductivity without sacrificing the Seebeck coefficient. These additives are expected to impart sufficient strain for the delocalization of electrons, resulting in enhanced carrier mobility. In addition, carbon-based materials such as CNTs, graphene, and graphite are also expected to supply high momentum electrons, which knock localized electrons off to attain an itinerant state. Besides, the presence of a secondary phase would also help in the manifestation of multiple phonon modes, which could further enhance the phonon–phonon scattering. Hence, forming a composite of HEOs with different additives offers tremendous possibilities to synergistically improve electrical and thermal transport, which is essential for designing high-performance thermoelectric materials.

Using these approaches can possibly help us in developing high performance oxide based thermoelectric materials. Such materials could readily be adopted by industries, as oxide raw materials are inexpensive and non-toxic unlike traditional thermoelectric materials. Moreover, successful oxide thermoelectrics can address one of the greatest challenges − long term high temperature stability − since oxides can easily withstand temperatures above 1000 K, which has yet not been possible with current state-of-the-art thermoelectrics.

Conclusions and outlook

In this work, we have presented a comprehensive overview of thermoelectric transport phenomena in various n- and p-type high-entropy oxides (HEOs), including systems based on titanates, manganites, cobaltates, and cobaltites. For the first time, we have also introduced a detailed thermodynamic framework to guide the design of HEOs for thermoelectric applications. Our results demonstrate that entropy engineering—via compositional complexity—offers a promising route to optimize key thermoelectric parameters, such as enhancing the Seebeck coefficient and reducing lattice thermal conductivity.

The inherently strong phonon scattering in these materials, due to their configurational disorder, results in ultralow phonon mean free paths—often in the sub-nanometer range. We also find that compositional tuning and composite formation strategies provide additional levers to fine-tune the overall thermoelectric performance of HEOs.

Nevertheless, predicting thermoelectric behavior in these systems remains challenging due to the complex cocktail effect, where synergistic and non-linear interactions between multiple elements make property predictions non-trivial. To overcome this, AI-driven methods combined with thermodynamic modeling present a powerful pathway to accelerate discovery. Machine learning models can help identify stable, high-performance compositions without the need for exhaustive DFT calculations.

To advance the field of HEO-based thermoelectrics, we recommend the following:

Researchers should

• Use AI and machine learning to efficiently explore and screen the vast compositional space of HEOs.

• Combine thermodynamic calculations with AI models to predict phase stability and entropy contributions.

• Focus on structure–property relationships, particularly how elemental selection and site disorder influence electron and phonon transport.

• Validate ML predictions experimentally to establish robust, interpretable datasets for future models.

• Explore composite strategies to synergistically improve electrical and thermal transport.

Researchers should avoid

• Relying solely on empirical trial-and-error approaches, given the scale of compositional possibilities.

• Assuming that higher configurational entropy alone guarantees improved thermoelectric performance.

• Overlooking the role of the microstructure and defect chemistry, which are often as critical as composition.

• Applying AI without proper training datasets or feature engineering, which may lead to misleading predictions.

Data availability

The data supporting the findings of this study are available in the ESI†. Additional datasets generated and/or analyzed during the current study are available from the corresponding author upon reasonable request.

Conflicts of interest

There are no conflicts of interest to declare.

Acknowledgements

This work was supported by the grant from Science and Engineering Research Board, DST (SERB-DST), India (grant no: CRG/2022/003515).

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d5ta02713f

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