The Interplay of Magnetism and Thermoelectricity: A Review

Abstract

The introduction of magnetism into a solid material might significantly modulate its electronic transport behavior, thereby serving as a means to tune the thermoelectric properties that have attracted considerable research attention in recent years. In this review, an introduction to recent studies on the interplay of magnetism and the thermoelectric Seebeck and Nernst effects is given. Concerning the Seebeck effect, the influence of superparamagnetic nanoparticles on the electronic and phonon transport of conventional nonmagnetic thermoelectric materials, as well as the spin‐related thermoelectric transport phenomenon in magnetic materials, are discussed. Then, the Nernst effect‐related transverse thermoelectric transport properties in nonmagnetic and magnetic topological materials are summarized, followed by a short introduction to the Nernst devices. Last, a further outlook on this new research direction is offered.

Introduction

Thermoelectric (TE) technology that can realize the direct conversion between thermal energy and electricity has exhibited important applications for electricity generation and solid-state refrigeration in various fields, including but not limited to deep space exploration, the Internet of Things, 5G communications, and flexible/wearable electronics.^[1] For a TE material, the dimensionless TE figure of merit zT is usually used to characterize its conversion efficiency: zT = S²σT/(κ_e + κ_L), where S, σ, T, κ_e, and κ_L are the Seebeck coefficient, electrical conductivity, absolute temperature, electronic thermal conductivity, and lattice thermal conductivity, respectively.^[2]

Since the first TE effect was discovered by Thomas Seebeck around 1821, TE research has experienced several major advances and many strategies have been proposed to enhance the performance of TE materials. Thanks to the development of solid-state physics and semiconductor theory in the middle of the 20th century, narrow bandgap semiconductors have been found to hold good TE candidates,^[3] exemplified by the discovery of Bi₂Te₃, PbTe, and SiGe that have exhibited good TE performance in the temperature ranges of near room temperature, mediate temperature, and high temperature, respectively. Since the 1990s, there are a series of new ideas proposed to enhance the performance of TE semiconductors by regulating electron and phonon transport, such as low-dimensionality,^[4] “phonon-glass lectron-crystal” paradigm,^[5] nanoengineering,^[6] and band engineering.^[7]

For a TE semiconductor, the optimization of its carrier concentration, by tuning either the native defects or extrinsic doping, is always an important and foremost step to achieving enhanced TE performance.^[2,8] This is because the transport parameters related to the charge carriers, i.e., σ, κ_e, and S, usually exhibit opposite trends as the carrier concentration raises, leading to the peak zT only occurring at a certain carrier concentration. The optimal carrier concentration of most good TE semiconductors is in the range of 10¹⁹–10²¹ cm⁻³, depending on their density of state effective mass and working temperature.^[9] The electrical properties of TE materials are fundamentally determined by their band structures. This leads to the development of a series of band engineering strategies, including the convergence of electronic bands,^[10] modification of band effective mass,^[11] distortion of the density of states,^[12] the weakening of deformation potential,^[13] and utilization of band anisotropy,^[14] aiming at the improvement of the electrical power factor, S²σ. In addition, the introduction of multiscale scattering sources to suppress the heat-carrying phonons has become a commonly used strategy as another route to improve TE performance.^[15] As well-concluded in the “phonon-glass electron-crystal” paradigm, the basic idea to enhance the TE performance of a certain material lies in the suppression of its phonon transport while maintaining, or beneficially, improving the electronic transport. With this paradigm in mind, in this review, we discuss the interplay between magnetism (or spin degree of freedom) and TE transport, which has attracted increasing research attention in recent several years, providing a potentially useful way to enhance TE performance.

The existing good TE materials are generally nonmagnetic semiconductors and adding magnetic ions to a semiconductor was conventionally recognized to severely deteriorate carrier mobility because of the enhanced magnetic scattering. Since around 2016, the introduction of superparamagnetic nanoparticles into several nonmagnetic semiconductors was reported to benefit their TE performance,^[16] attributing to the complex multiple scattering of carriers and phonons (Figure 1a). In magnetic materials, there are atomic spin interactions, spin waves, and magnetic transition,^[17] which can serve as an additional degree of freedom in regulating electron and phonon transport. Similar to the phonon drag, the presence of a temperature gradient in ferromagnets (FMs) and antiferromagnets (AFMs) might induce the magnon drag (MD) phenomenon, the momentum of the magnons will be transferred to the carriers through s–d electron interaction (Figure 1b).^[18] This effect could significantly contribute to the improved S of some materials, for instance, Fe and Co,^[19] Ni,^[20] MnTe,^[21] and Mn_1-xCr_xSb.^[22] Furthermore, the S of a material is determined by the average entropy flow per carrier and thus the spin entropy may contribute to the thermopower as reported in NaCoO₂,^[23] CuCr₂S₄,^[24] CaFe₄Sb₁₂,^[25] and EuMnSb₂.^[26]

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Besides the Seebeck effect, the Nernst effect which measures the transverse voltage induced by the applied orthogonal temperature difference and the magnetic field (Figure 1c) provides another means for potential heat-to-electricity conversion.^[27] For instance, the experimentally obtained lowest temperature through TE refrigeration was achieved by utilizing the reversible effect of the Nernst effect, i.e., Ettingshausen effect.^[27h] However, compared with the Seebeck effect, the Nernst effect has attracted relatively less research attention. In the past several years, the thriving of topological materials has brought many intriguing phenomena, including the anomalous Hall effect (AHE), anomalous Nernst effect (ANE, shown in Figure 1d),^[27a,g,28] and nonlinear optics.^[29] In ferromagnetic or antiferromagnetic topological materials, spontaneous magnetization breaks the time-reversal symmetry and brings about the nonzero Berry curvature (BC). BC is determined by the band structure and can be deemed as a pseudo-magnetic field in momentum space which could induce significantly enhanced ANE under small magnetic fields (about or below 1 T), providing a new platform for transverse TE research.

In this review, we first summarize the current progress on how the magnetic nanoparticles and spin wave/entropy affect the Seebeck effect of TE semiconductors with discussions of their beneficial and also adverse effects on electronic transport. Then, the ANE in nonmagnetic and magnetic topological materials with a potentially large BC is highlighted. In particular, the ways to tune the ANE both theoretically and experimentally are discussed. Further, the devices based on the Nernst effect are introduced aiming at practical applications. Last but not least, there are still many unanswered questions about the interplay between magnetism and thermoelectricity and thus, we regret not covering all the works and may omit important literature.

Magnetic Fundamentals

Before entering into the sections that discuss the interplay between magnetism and TE transport, we first give a short introduction to the fundamentals of magnetism in solids, which should be helpful for understanding when they are referred to in the following sections. The magnetic moment in a material is mainly contributed by electrons, especially the electron spin. Based on the macroscopic magnetism, solids can be divided into five main types, including diamagnets, paramagnets, FMs, AFMs, and ferrimagnets, distinguished by their magnetic susceptibility χ = M/H (shown in Figure 2a), where M and H are the magnetization and the applied external magnetic field intensity, respectively. Diamagnets belong to weak magnetism, the magnetic susceptibility is negative and tiny, usually in the order of 10⁻⁵. Paramagnets exhibit positive magnetization and usually have a magnetic susceptibility between 10⁻⁶ and 10⁻³. Most existing good TE materials without doping belong to either diamagnets or paramagnets and thus their transport properties might be significantly modulated after introducing magnetism.^[30]

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Ferromagnetism is a strong magnetism that originates from the parallel arrangement of atomic magnetic moments (Figure 2a), and the magnetic susceptibility of FMs is much larger than that of paramagnets.^[32] Particularly, there is a characteristic temperature in FMs above which ferromagnetism disappears, namely Curie temperature (T_C). Above T_C, FMs will turn into PMs, accompanied by abnormal phenomena in the specific heat, the coefficient of thermal expansion, and magnetic entropy.^[33] An applied external magnetic field can magnetize the FMs, and the magnetization can remain a finite value after withdrawing the external magnetic field. The magnetization can be dropped to zero only when the magnetic field is reversely strengthened to a coercive force (H_c). FMs can be classified as hard and soft magnetic materials according to the size of H_c. if its H_c <1000 A m⁻¹, the FMs can respond quickly to the external magnetic field and be easily magnetized, which is usually deemed as soft magnetic material. By contrast, the FMs with H_c above 1000 A m⁻¹ still hold a large magnetization even if the external magnetic field is removed, called a hard magnetic material.^[17] AFMs are also magnetically ordered while the magnetic moments of adjacent atoms are antiparallel. Similar to T_C, AFMs also have a transition temperature, called Néel temperature (T_N). The main exchange mechanism in FMs and AFMs involves the overlap of local atomic orbitals of adjacent atoms.

Ferromagnetic crystals generally have a multidomain structure. By reducing the grain size, a new magnetic property can be formed at a certain temperature.^[34] With decreasing volume, the entire crystalline grain prefers to become one magnetic domain, and the size at which the grain becomes a single domain is called the critical size. Below the critical size, the magnetism of the grain will change significantly compared to the large FMs and the effect of thermal exercise will be enhanced. The magnetization curve along different crystal axis directions is different for a single crystal, and the magnetization energy required for the FMs can be defined as magnetocrystalline anisotropy energy.^[31] With the rising temperature, the thermal energy can overcome the magnetocrystalline anisotropy energy and thus, some magnetic nanoparticles will switch from ferromagnetic to superparamagnetic. Below the blocking temperature (T_B), atomic magnetic moments align in parallel along the same direction in each domain. Above T_B, the direction of the magnetic moment within the particle may change repeatedly from one direction to another direction in parallel over time, the susceptibility is much higher than that of general paramagnetic materials; above T_C, all materials transform into paramagnets (Figure 2b). The introduction of superparamagnets was found to be a feasible way to improve the performance of several material systems,^[16a,b,d] which will be discussed in Section 3.1.

Magnetism and Seebeck Effect

Superparamagnets

The presence of magnetic impurities in TE semiconductors might lead to significant suppression of carrier mobility, which was conventionally thought unbeneficial for the enhancement of TE performance. In recent several years, the introduction of magnetic nanoparticles, specifically superparamagnetic particles, into some TE systems, had been reported to benefit the enhancement of TE performance.^[16a,b,d] Typical 3D FMs, such as Fe, Co, Ni, its alloys,^[30c] and other magnetic oxides and nitrides, can keep superparamagnetic at specific sizes and temperatures, which are expected to bring about peculiar multiple carrier scattering effects.^[16a] Typically, Zhao et al.^[16b] introduced just a few superparamagnetic Co nanoparticles (SPNPs) into the Ba_0.3In_0.3Co₄Sb₁₂ matrix, which result in the simultaneous regulation of electron and phonon transport, improving the TE performance (Figure 3a–d).

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Specifically, the introduced Co particles that have average diameters of 5–10 nm were found randomly distributing on the grain boundaries as secondary phases and thus, were thought not to affect the band structure and effective mass of the matrix. Concerning their effect on thermal transport, it is imaginable that with the introduction of randomly distributed nanoscale particles, the scattering of phonons can be enhanced. As demonstrated by the measured κ_L (Figure 3a), the sample with Co nanoparticles exhibits lower κ_L compared with that of the Ba_0.3In_0.3Co₄Sb₁₂ matrix. Similar to the enhanced scattering of phonons, the introduced Co nanoparticles also suppress the carrier transport, thereby reducing carrier mobility. However, it is surprisingly found that the σ of the samples with SPNPs is increased, compared to the Ba_0.3In_0.3Co₄Sb₁₂ matrix (Figure 3b). The work function of Co nanoparticles is lower than the matrix's, suggesting that the 4s electrons of the Co nanoparticles can transform into the matrix. Hence, although the introduction of Co nanoparticles decreases carrier mobility, the increased carrier concentration due to the SPNPs eventually contributes to the increased σ. At room temperature, by introducing 0.2% Co nanoparticles (in weight ratio), the carrier concentration n increases from ≈3.5 × 10²⁰ to 4.8 × 10²⁰ cm⁻³.^[16b] Not just Co, the introduction of 0.2% Ni and Fe SPNPs into the Ba_0.3In_0.3Co₄Sb₁₂ matrix was also found to contribute to the increased σ, owing to the increased carrier concentration.

Although the carrier concentration is increased, the absolute S also exhibits an obvious increase (Figure 3c). Since the band structure was thought not to be changed, the possible reasons for the enhanced S were attributed to the modulated carrier scattering mechanism. First, the introduced nanoparticles will result in additional scattering of low-energy carriers due to the band bending in the interfaces between the matrix and metallic nanoparticles.^[35] In addition, when the temperature is above T_B and below T_C, the magnetic moment of Co nanoparticles in the superparamagnetic state will orient randomly, in contrast to the ferromagnetic state where the direction of the magnetic moment is rigid. In this condition, the electrons were thought to undergo temperature-dependent multiple scattering,^[16b] in contrast to the ferromagnetic state-induced single scattering (Figure 3e). Together, it was thought that the selective scattering of electrons due to the band bending and the multiple scattering contributes to the increased scattering factor and thus enhances the S, although the carrier concentration is also increased. Theoretically, the enhanced scattering of carriers will indeed benefit the S while also reducing carrier mobility. Thus, to experimentally identify how the scattering factor in Ba_0.3In_0.3Co₄Sb₁₂ changes after the introduction of SPNPs, for example, using the temperature-dependent measurements of the Nernst coefficient,^[36] might give more insights into how superparamagnetism interplays with the TE transport.

Spin Wave/Entropy

In the above, the effect of the extrinsic superparamagnetic nanoparticles on the transport properties of nonmagnetic TE semiconductors is discussed, while some peculiar phenomena can exist in the materials that have an inherent magnetic transition, i.e., FMs and AFMs. Among them, MnTe has attracted extensive attention in recent years. The undoped MnTe is a p-type antiferromagnetic semiconductor and its T_N is 307 K,^[37] The doped Li_xMn_1−xTe exhibits decent TE performance with a peak zT of about unity,^[38] providing a good platform to understand the spin-related TE transport.^[37,39]

The temperature dependence of S for Li_xMn_1−xTe samples was reported by Zheng et al.^[21b] For simplicity, here we only focus on three samples (x = 0.01, 0.03, 0.06), as shown in Figure 4a. In the antiferromagnetic region, below 150 K, the S increases slowly; above 150 K, S has a significant increase which can be fitted as a diffusion term (follows T¹ law) plus a contribution of spin (follows T³ law). The linear part represents the contribution of the carriers’ diffusion, which can be described as S_d by the Mott equation^[21b,40]1 $$$\begin{equation}{S}_{\rm{d}} = \frac{2}{3}{\left( {\frac{{{\pi}}}{3}} \right)}^{2/3}\frac{{{k}_{\rm{B}}}}{e}\frac{{{m}^*}}{{{\hbar }^2}}\frac{{{k}_{\rm{B}}T}}{{{n}^{2/3}}}\end{equation}$$$

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where e denotes the elementary charge, k_B is Boltzmann constant, ħ is the reduced Planck constant, m^* is the density of states effective mass and n is the carrier concentration. At 300 K, the theoretically calculated S_d for x = 0.01 is only 36.8 μV K⁻¹ while its actual S is 182 μV K⁻¹.^[21b] Above T_N, the S exhibits a linear increase in the PM region and the spin-related contribution does not vanish. Hence, both in the antiferromagnetic and paramagnetic regions, the S of Li_xMn_1−xTe was thought to have an important enhancement related to the inherent spin.

However, the increase in S is accompanied by a deterioration in σ. For the studied sample, from the Hall effect measurements, the carrier concentration n remains the same below and above T_N.^[21b] Hence, the variation of σ is inconsistent with that of μ which exhibits a huge drop near T_N. Below T_N, similar to the electron scattering by acoustic phonons, the itinerate carriers could be scattered by the random motion of the spin of lattice ions. Figure 4b shows the logarithmic curves of μ-T, μ decreases rapidly with increasing temperature. Assuming the phonon scattering dominates under the carrier transport, μ usually drops with increasing T by following a trend of T^−3/2. The deviation below the transition temperature might result from that spin brings an additional scattering of charge carriers. The reduction of carrier mobility is more significant for x = 0.01. In the case, where only phonon scattering is considered, μ is estimated to be 13.1 cm² V⁻¹ s⁻¹, whereas the actual value is 5.2 cm² V⁻¹ s⁻¹.^[21b] For x = 0.06, μ is inherently low owing to the enhanced phonon scattering, and the adverse effect of magnetic scattering becomes weaker. Above T_N, the coupling between the magnetism and the carriers disappears due to the strong thermal rise and fall, and the μ exhibits a weak temperature dependence.^[40] In addition, the lattice thermal conductivities of the Li_xMn_1−xTe samples are presented in Figure 4c, which show no obvious transition in the vicinity of T_N. Hence, although there is an additional magnetic scattering of carriers that occurs in Li_xMn_1−xTe, it can be concluded that the contribution of spin actually enhances the TE properties of the MnTe system, owing to the huge increase of S.

In short, what is of most interest for the Li_xMn_1−xTe system is the sudden increase in their S near T_N. Two possible mechanisms were proposed to explain this change and the first one is MD. At 0 K, for the ferromagnetic and antiferromagnetic systems, the spins are completely ordered when having the lowest energy. For T > 0 K, assuming that one of the spins in the system flips, it means an excited state with the lowest energy. From a quantum mechanical point of view, the motion of all spins is coupled together, and the spins will propagate as a wave.^[17] The quantization of spin waves is usually defined as magnons. Similar to a phonon, a magnon also belongs to a quasiparticle, a collective excitation mode used to describe the spin structure in the crystal structure.

For magnetic TE materials, the applied temperature gradient leads to the gradient of the magnon density because the magnon population obeys the Bose–Einstein distribution function. Under this circumstance, the magnons will move from the hot side to the cold side (Figure 4d), which can interact with free electrons and transfer their momentum to electrons, leading to a magnon-drag contribution to the Seebeck coefficient.^[42] In 1962, Bailyn proposed that there will be an MD phenomenon in ferromagnetic metals.^[43] Afterward, the 3d ferromagnetic elemental metals Fe, Co, Ni^[19b,20] and some of their alloys^[44] were reported to exhibit the MD-contributed thermopower in a certain temperature range.

Watzman et al. proposed a hydrodynamic and spin-dynamic theory to describe the MD contribution to thermopower in metals.^[19a] The hydrodynamic theory based on a magnon-electron two-fluid model has been proven to be predictive, suggesting that both an electron and a magnon fluid exist for a magnetic conductor. Magnon can also carry heat and C_m is usually used to describe the specific heat of magnons. For ferromagnetic magnons, the C_m is proportional to T^1.5 at low temperatures.^[45] Similarly, for antiferromagnetic magnons, at low temperatures, C_m ∝ T³.^[17]

The magnon momentum relaxation time resulting from the collision between electron and magnon is defined as τ_em. And the total scattering time for a magnon, including collisions with electrons, phonons, defects, and other magnons, is defined as τ_m. Under steady-state conditions and for zero electric current, the magnon-drag thermopower was given by^[46]2 $$$\begin{equation}S \equiv \frac{{|\vec{E}|}}{{|\vec{\nabla }T|}} = {S}_{\rm{d}} + {S}_{{\rm{md}}} = {S}_{\rm{d}} + \frac{2}{3}\frac{{{C}_{\rm{m}}}}{{{n}_{\rm{e}}e}}\frac{1}{{1 + {\tau }_{{\rm{em}}}/{\tau }_{\rm{m}}}}\end{equation}$$$

where S_md represents the thermopower caused by MD and n_e is the carrier concentration. Compared to FM, AFM magnons have higher group velocity, and longer magnon lifetime τ_m, which means the larger MD thermopower.^[40] Due to the temperature dependence of C_m, S_md generally follows the T³ law below T_N, as experimentally demonstrated MnTe (Figure 4a).

It is possible to use MD to explain the enhanced Seebeck coefficients in some magnetic materials, in terms of the interaction of spin waves with conduction electrons.^[47] But for some materials, it might not be reasonable to apply MD to the paramagnetic state, where a large contribution can be observed above T_C or T_N.^[41] Spin entropy theory offers another alternative explanation of the spin-dependent Seebeck effect.

According to the Onsager relation,^[48] the S of the material can be interpreted as the average entropy flow per carrier. Hence, the spin entropy, if enhanced, can contribute positively to S. The spin entropy-enhanced thermopower was reported in some strongly correlated systems as well as some transition metal compounds.^[23,49] The S of magnetic TE materials comes from two parts: the change of chemical potential due to temperature and the kinetic part. The total S is expressed as^[50]3 $$$\begin{equation} - S = \frac{1}{e}\frac{{{\rm{d}}\mu }}{{{\rm{d}}T}} + \frac{{{\rm{d}}\psi }}{{{\rm{d}}T}}\end{equation}$$$

For the first part, according to Maxwell's relation,^[51] the differentiation of the chemical potential with temperature is equivalent to the variation of the entropy with the number of particles,^[51] dμ/dT = dS/dN, which can be directly understood as the amount of entropy carried by each particle on average (Figure 4e). Part of the total entropy comes from the spin, which changes as the magnetic properties of the system change, thus eventually affecting S. The spin entropy in magnetic conductors can be written as k_B·lng, so the additional S_m caused by the spin entropy can be expressed by the Heikes formula^[51] as $${S}_{\rm{m}} = \left( {{k}_{\rm{B}}/e} \right)\partial \ln g/\partial N$$.^[52] In addition, the external magnetic field applied will modulate the magnetic moment of the material, reducing the spin entropy and suppressing the S_m.

For FMs and AFMs, when undergoing a transition from magnetically ordered to paramagnetic, the spin entropy will increase and reach a maximum at the fully disordered state (Figure 4e). As a result, the S changes accompany the magnetic transition, a phenomenon that has been reported in several material systems, TiCo₂Sn,^[53] MnSi,^[54] CaFe₄Sb₁₂,^[25,55] MnSb, CrSb,^[56] and Fe₂VAl.^[57] Figure 4f shows that the thermopower of the materials with a step-like change in the Seebeck coefficient near the magnetic transition temperature, T_C or T_N. In addition, when an external magnetic field is present, the transformation of the magnet to paramagnetic at high temperatures will be slower, and the spin entropy will not be saturated immediately when the magnetic transition temperature is reached, the change of S will slow down.

Very recently, a large magneto-thermopower was also reported in AFM EuMnSb₂,^[26] When no magnetic field is applied, the S increases continuously as T decreases, reaching a maximum value of −275 μV K⁻¹ at 68 K while below T_N (≈21 K) S decreases rapidly,^[26] also demonstrating that the spin entropy can contribute a large enhancement to the total S. After the external magnetic field is applied, the spin entropy decreases due to the suppression of the spin-up of the Eu moment and disappears rapidly with the formation of the antiferromagnetic state at a lower temperature.

Magnetism and Nernst Effect

Nernst Effect in Nonmagnetic Topological Materials (TMs)

When an orthogonal magnetic field (H) and temperature gradient (∇T) are applied to a conductor, a voltage normal to both H and ∇T can be observed. This phenomenon is known as the Nernst effect, as schematically shown in Figure 1c. Compared to the Seebeck effect, the Nernst effect has some unique advantages. First, in the Seebeck effect, electrons and phonons transport along the same direction, which inevitably leads to their coupling and mutual interaction, making it challenging to independently optimize the electrical and thermal transport properties. In the Nernst device, the charge carriers are accumulated laterally and the phonons propagate longitudinally, which is probably beneficial for independently optimizing the electrical and thermal transport properties.^[58] Second, under a longitudinal temperature gradient, if both electrons and holes exist in a material, their contributions to the thermopower will be offset in the Seebeck effect but can be beneficial in the Nernst effect since they will be deflected to the opposite lateral sides by the Lorentz force. Third, as mentioned earlier, the electric potential and temperature gradient in the Seebeck device are in the same direction. To obtain a larger Seebeck voltage, it is sometimes necessary to increase the height of the device to maintain a larger temperature difference. In Nernst devices, the electric potential and temperature gradient are in the orthogonal directions, and the output voltage can be increased proportionally by widening the size along the potential direction without increasing the height of the device, which is conducive to the application of the flexible devices. Moreover, the Seebeck devices generally compose of both n-type and p-type TE materials while Nernst devices only need one type of material which is technologically simpler for integration. Last but not least, the unavoidable elements diffusion between electrodes and matrix materials, particularly at the high-temperature side, can be a serious problem for the Seebeck device if serving for the long term.^[59] While in the Nernst setup, one can place the electrodes close to the cold junction, and thus significant diffusion between the electrodes and matrix materials can be avoided.

There have been studies on the Nernst effect of solid materials since the 1960s.^[27h,60] One common phenomenon about the Nernst effect is that the Nernst thermopower (S_xy) will increase nearly positively with the increased magnetic field. For Bi and its alloys with Sb, benefiting from the high carrier mobility and electron–hole compensation, a large Nernst effect can be generated under the applied external magnetic field. Typically, as for TE refrigeration, the lowest cooling temperature (T = 201 K) was achieved through the Nernst–Ettingshausen effect in Bi single crystals at a field of 11 T.^[60b] The recently discovered TMs provide a new platform for exploring new materials for heat-to-electricity energy conversion using the Nernst effect. For example, a high Nernst z_NT of 0.5 was obtained at 2 T which significantly surmounts its longitudinal counterpart in Dirac semimetal Cd₃As₂.^[61] A maximum Nernst power factor 35 × 10⁻⁴ W m⁻¹ K⁻² was achieved in polycrystalline Weyl semimetal NbP at 9 T and 136 K,^[62] which is ≈4 times higher than the conventional Seebeck power factor. These newly developed TMs with giant anomalous Hall conductivity σ^A and anomalous thermoelectricity α^A provide a new platform for investigating the Nernst effect for possible energy conversion applications.^[27a,63]

The studies of the Nernst effect are less explored compared to the Seebeck effect, which is probably owing to that a large external magnetic field is commonly required, hindering the practical applications. In conventional metals, a magnetic field of a few Teslas is usually needed to obtain a large Nernst signal, which exceeds the magnitude that an electromagnet can achieve. Therefore, finding materials with large Nernst responses under low magnetic fields is of great significance. In the past few decades, the fascinating transport phenomenon caused by the nontrivial band structure in TMs has sparked enormous interest in the community of condensed matter physics.^[64] Topology is a mathematical concept, which refers to continuous deformation and distortion without tearing or breaking. Based on the emerging topological band theory,^[28] when there is band inversion in the system, the inverted bulk band structure can form topological insulators (TIs)^[28a,65] or topological semimetals (TSMs), like Dirac semimetals,^[66] nodal line semimetals^[67] and Weyl semimetals (WSMs),^{[28b,65a,66,68]} depending on the strength of the spin–orbit coupling (SOC). TMs exhibit unique band structures and transport properties, including topologically nontrivial surface states,^[69] Fermi arc,^[70] linear band dispersion,^[71] chiral anomaly,^[72] and giant negative magnetoresistance,^[73] which provide a rich platform for exploring new candidates for energy conversion.^[27a,74]

WSM is a novel quantum state with Weyl fermions in the bulk, characterized by the Weyl equation, which is a simplified version of the Dirac equation and first proposed by Weyl in 1929.^[75] As the solution of the Weyl equation, the chiral massless Weyl fermion has long been undetected in high-energy physics since it was proposed. In recent years, with the increasing understanding of TMs, the low-energy excitation in the vicinity of these band touching can be described by the Weyl equation. These linear intersections are dubbed Weyl points, and the quasiparticles near these points can be regarded as Weyl fermions.^[66] One fascinating feature of WSM is the giant Berry curvature near the Weyl points, a quantity describing the degree of entanglement between the conduction band and valance band that can act as a fictitious magnetic field on the electron wave function in momentum space. It was reported that Berry curvature plays a crucial role in many exotic transport phenomena,^[76] including the anomalous Hall effect, spin Hall effect, topological Hall effect, anomalous Nernst effect, and spin Nernst effect. The integration of Berry curvature over a closed surface within a Weyl point yields a quantized number (Chern number), which defines the chirality of the Weyl point (+ (−) as the source (sink) of the Berry curvature). Weyl points with opposite chirality always appear in pairs and only exist whether time-reversal symmetry or inversion symmetry is broken.

In nonmagnetic Weyl semimetals with broken inversion symmetry, a large external magnetic field is typically required to obtain an appreciable topological response, such as TaX (X = As, P)^[77] and NbX.^[62,78] In the study of the Nernst effect in TaP, an anomalous contribution was reported according to a magnetic field-induced Lifshitz transition. As shown in Figure 5a, the S_xy normalized to temperature as a function of the field depicts unusual phenomena at certain temperatures and resembles the behavior of ferromagnetic materials.^[79] Such an anomalous behavior was also found in the report on Cd₃As₂.^[80] At T = 250 K, the S_xy of TaP proportionally rises with the increased magnetic field while at T below 100 K, S_xy first increases with the field and then reaches a saturated platform. The observed Nernst signal was separated into the conventional and anomalous contributions through the following expressions^[77a,80]4 $$$\begin{equation}{S}_{xy} = S_{xy}^{\rm{N}} + S_{xy}^{\rm{A}}\end{equation}$$$5 $$$\begin{equation}S_{xy}^{\rm{N}} = S_0^{\rm{N}}\frac{{{\mu }_eB}}{{1 + {{({\mu }_eB)}}^2}}\end{equation}$$$6 $$$\begin{equation}S_{xy}^{\rm{A}} = S_0^{\rm{A}}\tanh (B/{B}_{\rm{S}})\end{equation}$$$where μ_e is the mobility,$$S_0^{\rm{N}}$$ and $$S_0^{\rm{A}}$$ are the amplitude of the conventional and anomalous coefficients, respectively, and B_S is the saturation magnetic field above which S_xy attains the saturated plateau.

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The black dashed lines in Figure 5a represent the fitting results through Equation (4) and are in good agreement with the experimental phenomena. The impact of the relative position between the Weyl point and E_F on the σ^A, α^A, and S_xy is presented in Figure 5b. Depending on different weight functions (Fermi–Dirac distribution function for σ^A (blue line) and entropy function for α^A (red line), σ^A and α^A have different sensitivity to the location of E_F. σ^A is a quantity that is not directly proportionally to the magnetization but to the summation of BC over all occupied bands below E_F.^[76b,81] α^A, as the TE counterpart of the σ^A, correlates with each other through the Mott formula at low temperatures,^[82] and is only contributed by the energy bands near E_F. An examination of σ^A and α^A reveals when the Weyl point approaches E_F, both of which will enhance (case (ii) and (iii) in Figure 5b). When the position of the Weyl point is further raised, both the σ^A and α^A will decrease. Consider a Dirac point protected by time-reversal symmetry and inversion symmetry, the Dirac point will split into a pair of Weyl points under the action of SOC when B is nonzero (case (i) in Figure 5c). The double Weyl bands can host more density of states than the single Dirac band, resulting in a lower μ and large S_xy as shown in Figure 5b. When B is further increased (B₂ > B_S), the distance between two Weyl points in the k_z direction is further separated with a further decline of μ, resulting in a Lifshitz transition (case (iii)). In the Lifshitz transition, S_xy approaching maximum, further increasing the magnetic field will only enlarge the relative distance between Weyl points, in this circumstance μ remains constant and will not bring about S_xy improvement (case (iv) B₃ > B_S). According to the above discussions, one can draw inspiration that in a system with Dirac points, the introduction of a magnetic field may bring about a relative change in the chemical potential and Fermi level position, which may lead to large anomalous transport phenomena. Benefiting from unique electronic structure and high mobility, the magneto-thermopower in nonmagnetic TMs can be significantly enhanced, the transverse PF can even be compared with that of the advanced conventional TE materials.^[62,63] However, these TMs are usually semimetals with a high thermal conductivity, resulting in a low zT.

Anomalous Nernst Effect in Magnetic TMs

The aforementioned Nernst effect in TMs generally requires a high magnetic field, which might limit the large-scale application. Therefore, searching for giant Nernst responses under a low magnetic field is important. Nowadays, in some ferromagnetic or antiferromagnetic materials, due to the existence of intrinsic Berry curvature, the materials can obtain giant topological responses at a low magnetic field, which conventional electromagnets could achieve. If the material possesses a net magnetization, a large external magnetic field is not needed to obtain a large S_xy, which is known as ANE (Figure 1d). The ANE is related σ^A and α^A through the equation $$S_{xy}^{\rm{A}} = \frac{{{\sigma }_{xx}}}{{\sigma _{xx}^2 + {{(\sigma _{xy}^{\rm{A}})}}^2}}( {\alpha _{xy}^{\rm{A}} - {S}_{xx}\sigma _{xy}^{\rm{A}}} )$$, and the intrinsic σ^A and α^A can be given by the equation^[81]7 $$$\begin{equation}{\sigma }^{\rm{A}} = - \frac{{{e}^2}}{\hbar }\int_{{{\rm{BZ}}}}^{{}}{{\frac{{{d}^3k}}{{{{\left( {2{{\pi}}} \right)}}^3}}f(k){\Omega }_{\rm{B}}(k)}}\end{equation}$$$8 $$$\begin{equation}{\alpha }^{\rm{A}} = - \frac{{e{k}_{\rm{B}}}}{\hbar }\int_{{{\rm{BZ}}}}^{{}}{{\frac{{{d}^3k}}{{{{\left( {2{{\pi}}} \right)}}^3}}s(k){\Omega }_{\rm{B}}(k)}}\end{equation}$$$where f(k) is the Fermi–Dirac distribution function, s(k) is the entropy function through the equations(k) = −f(k)ln (f(k)) − (1 − f(k))ln (1 − f(k)), and Ω_B(k) represents the Berry curvature. In certain topological FMs or noncollinear AFMs, a giant ANE and AHE have been found even when there is almost no net magnetization.^[27d,79c,83] Actually, the ANE or AHE does not completely dependent on the magnitude of the magnetization but is the nonzero Berry curvature of the electronic band near E_F. In the past several years, many topological materials have been found showing large ANE, including antiferromagnetic Mn₃Sn and Mn₃Ge,^[84] YbMnBi₂,^[85] YMn₆Sn₆,^[86] ferromagnetic Co₂MnGa,^[27d,79c] Co₂MnAl,^[87] Co₃Sn₂S₂,^[79a,88] Fe₃Z (Z = Ga, Al),^[89] Fe_xGa_4-x,^[90] UCo_0.8Ru_0.2Al,^[79b] Fe₂YZ (Y = Co, Ni),^[27b] and Fe₃Sn.^[91] The large ANE of these topological materials was thought to mainly come from the intrinsic Berry curvature. Therefore, searching for magnetic WSM with large Berry curvature is an important prerequisite for generating large ANE and thus for the practical application of ANE. However, hunting for WSM based on first-principles calculations or high-throughput computations is not an easy task. In magnetic Heusler compounds, the influence of mirror symmetry and Fermi level position on σ^A and α^A was revealed through high-throughput calculations,^[92] suggesting that more mirror planes are conducive to larger BC and proper Fermi level position in the band structure is favorable for large σ^A and α^A. In Co₃Sn₂S₂, by comparing samples with different impurity concentrations, the ANE was found inversely proportional to the average carrier mean free path, and the introduction of appropriate impurity concentrations is favorable to enhance the ANE.^[88] Moreover, the local disorder-induced band broadening effect and Fermi level shift caused by Ni atoms doping significantly enhance the σ^A of Co₃Sn₂S₂.^[93] So far, there are still very few works highlighting the enhancement of AHE and ANE by designing the BC distribution. Therefore, it is of great significance to find ways to regulate BC distribution aiming at the enhancement of AHE and ANE.

Magnetic Heusler compounds are a group of multifunctional materials, which have a large composition zone for tuning different properties.^[92,94] Large AHE and ANE were reported in Co₂MnZ due to their coexistence of magnetism and topology.^[27d,j,79c] However, several disorder structures typically exist in the compound, which will significantly deteriorate the anomalous properties.^[87,95] In Fe-based Heusler compounds Fe₂YZ, it was found that there is a strong disorder between Y and Z atoms in the prepared samples,^[27b] as schematically shown in Figure 6a. ground-state Fe₂YZ prefers to crystallize in the inverse Heusler structure (X) rather than the normal Heusler structure (L2₁) according to previous studies.^[96] However, the experimentally obtained σ^A is among the two calculated values using the X structure and L2₁ structure, respectively (Figure 6b). This discrepancy was reckoned to be the disorder between Y and Z atoms, which results in more mirror planes compared to that of the X structure despite still being less than that of the L2₁ structure. After annealing, the experimentally obtained σ^A is reduced because of the diminished mirror planes. These results indicate that Berry curvature in a material can be adjusted by controlling the crystal symmetry and disorder, which helps explore new compounds with large σ^A and α^A.

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Although the magnetic Heusler compounds are conductive to achieve performance regulation by doping or alloying,^[94,95d,97] the reported Co-based or Fe-based Heusler compounds have face-centered cubic structures and show isotropic transport measurements.^[27d,67] They are generally soft magnetic materials with saturation magnetization ≈1 T and show negligible anomalous transport signals when the magnetic field is zero. Therefore, finding materials with large transverse thermoelectric responses under zero magnetic field is more beneficial for practical applications. In a recent work on the permanent SmCo₅ magnet, a large S_xy of ≈4 μV K⁻¹ was obtained at 300 K and the anomalous Ettingshausen effect was observed,^[98] benefiting the practical applications of transverse thermoelectric refrigeration. For the topological ferromagnets discovered in recent years, the ternary chalcogenide Co₃Sn₂S₂ exhibits a relatively large coercive field. Combined with the topological band structure, it is expected to realize anomalous responses even without applying the magnetic field. Co₃Sn₂S₂ crystallizes in a hexagonal lattice with a space group of $$R\mathop 3\limits^ - m$$(No. 166). As shown in Figure 6c, in the ab-plane, six Co atoms form a cage structure with Sn(2) atoms wrapped in and two opposite parallel Co₃ atoms surrounding the Sn(1) atoms with S atoms interpenetrating between layers. Through angle-resolved photoemission spectroscopy measurements,^[83b,99] it has been determined that Co₃Sn₂S₂ is a topological ferromagnet with Weyl points located in the Brillouin zone (BZ). If the Co₃Sn₂S₂ sample is first magnetized and the anomalous Nernst thermopower ($$S_{xy}^{\rm{A}}$$) can be detected even under zero field (Figure 6d, red diamond).$$S_{xy}^A$$ first increases with temperature and reaches the maximum value of ≈3 μV K⁻¹ at 80 K, and then gradually decreases with the increase of temperature which is caused by the weakening of the magnetization when it tends to the Curie temperature (≈177 K). By comparing the data measured from the sweep field (Figure 6d, blue diamond), they are almost identical, indicating the remanent state could contribute to$$S_{xy}^{\rm{A}}$$ under zero-field. Although a large zero-field Nernst effect can be achieved in Co₃Sn₂S₂, its application is limited by the relatively low T_C. Therefore, it is very meaningful to find topological materials with high Curie temperature and giant Nernst response under zero field for future anomalous TE applications. It is worth noting that the calculation of z_NT in Nernst depends on whether the transverse thermal boundary conditions are isothermal or adiabatic. In the isothermal condition, precautions have been taken to make the applied temperature gradient and heat flow parallel: the two sides of the sample where the voltage is measured are at the same temperature. In the adiabatic condition, no heat flux leaves the sample by the sides where the voltage is measured.^[27g]

Device and Application

The anomalous transport phenomena associated with the unique electronic structure in TMs have aroused extensive research interest. In nonmagnetic TSMs, Weyl points that are generated by the broken inversion symmetry due to non-centrosymmetric crystal structure have no contribution to AHE and ANE in the absence of an external magnetic field.^[62,72b,77a] In AFMs or FMs^{[27b-d,79a,b,83a,86,91]} with Weyl points near the E_F, the spontaneous magnetization results in the broken time-reversal symmetry, contributing to giant intrinsic AHE and ANE under a small, or even zero, external magnetic field, which facilitates the practical applications.^[100]

One of the advantages of the Nernst device is the output voltage can be increased proportionally by increasing the length in the electric potential direction without increasing the height of the device as shown in Figure 7a. The Nernst voltage can be expressed as 9 $$$\begin{equation}{V}_{xy} = N{S}_{xy}\Delta T\frac{c}{a}\end{equation}$$$where N represents the number of Nernst TE legs in series, c and a are the lengths in the voltage and temperature directions, respectively. Through Equation (9), the output voltage can be increased proportionally by increasing the length of the electric voltage direction without thickening the device, facilitating the flexible application. It is worth noting that the increase of the device length will also lead to the larger internal resistance. By optimizing the structure design of the device and selecting good electrode interfacial materials, the internal resistance can be significantly reduced.

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Another configuration of Nernst thermopile is realized in a cylindrical heat source through radial temperature gradient and axial magnetic field, where the wires are coiled around the hot cylinder to generate electric voltage through ANE (Figure 7b).^[102] The voltage can be given by 10 $$$\begin{equation}{V}_{xy} = L{S}_{xy}\nabla T\cos \varphi \end{equation}$$$where L is the wire length, and φ is the angle between the wire and the axial direction of the cylinder. Clearly, the voltage can be larger by lengthening the coiled wire.

In the above Nernst thermopile, the voltage of each TE leg is in the same direction, therefore, conducting wires must be used to achieve voltage accumulation, which will reduce the power density of the device. Recently, an intriguing V-shape configuration was put forward to realize voltage accumulation without conducting wires (Figure 7c).^[101] This structure makes use of the large lag angle between the external magnetic field and the magnetization which originates from the interruption of magnetic octupoles at the planar edge of Mn₃Sn. When the magnetic field is applied, the magnetization orientation between adjacent legs is opposite, so the Nernst voltage generated can be accumulated within each leg. This tilted construction possesses a simple structure that facilitates device fabrication and provides new schemes for TE applications.

Summary and Outlook

The introduction of magnetism, or the spin degree of freedom, into TE materials might significantly modulate the electrical and thermal transport properties, providing a possible means to enhance the zT. For conventional nonmagnetic TE compound Ba_0.3In_0.3Co₄Sb₁₂, the introduction of superparamagnetic nanoparticles results in enhanced scattering of phonons and induces multiple scattering of carriers, degrading the carrier mobility while largely improving the Seebeck coefficient. Further understanding these multiple scattering effects would be important for applying this strategy to other nonmagnetic TE materials. Good TE materials usually belong to semiconductors with a narrow bandgap. For currently developed FM semiconductors, the T_C is actually below 80 K,^[103] improving the T_C of the FM semiconductors is still necessary to enable the utilization of magnon-induced thermopower. Whereas, in the AFM semiconductors, for instance, MnTe, the magnon drag and the spin entropy have been proposed to explain the step-like increase of its Seebeck coefficient near T_N. Although the magnetic scattering also worsens the carrier mobility, the largely increased Seebeck coefficient contributed to the enhanced zT. Hence, for inherent magnetic materials, especially antiferromagnetic semiconductors with higher T_N, the spin wave/entropy might contribute to improved TE performance at elevated temperatures, if the magnetic scattering of carriers is not serious.

In contrast to the Seebeck effect-based TE materials, the externally applied magnetic fields to nonmagnetic (topological) materials, typically high-mobility semimetals, might induce a significant Nernst response, providing a platform for studying transverse TE transport. In addition, the magnetic topological materials with a large Berry curvature near the Fermi level could generate an ANE, in which spontaneous magnetization enables the realization of a large anomalous Nernst response under low, or even zero, magnetic fields. Thus, the topological materials provide a rich platform for exploring new candidates for transverse heat-to-electricity conversion.

In summary, the research on the interplay between magnetism and TE transport in solid materials is still in its infancy. Understanding the underlying interaction mechanism and exploring new applications in both conventional TE semiconductors and topological materials by utilizing magnetism as a new degree of freedom will provide new opportunities for developing heat-to-electricity energy conversion technology.

Acknowledgements

S.L. and M.C. contributed equally to this work. This work was supported by the National Key Research and Development Program of China (No. 2019YFA0704900), and the National Natural Science Foundation of China (No. 92163203).

Conflict of Interest

The authors declare no conflict of interest.

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