Transverse thermal energy conversion using spin and topological structures

This Tutorial provides conceptual guidance for the development of next-generation, high-efficiency thermoelectric (TE) devices based on transverse geometries, in addition to some practical considerations for the accurate measurement and reporting of transverse transport properties. To provide sufficient context and background information, we begin with a basic introduction to the relevant physics of transport phenomena in solids and briefly discuss the evolution of TE technology. We then shift to our main topic of transverse device geometries, including a brief mention of some potential applications. From there, we introduce mechanisms known to produce perpendicular charge and heat fluxes, with emphasis on the spin Seebeck and Nernst effects, as well as topological effects described by Weyl physics. We explore these topics through specific examples and discuss numerous ways in which they might be further developed for use in thermal energy conversion. We conclude with an overview of best practices and common mistakes to avoid when measuring and reporting transverse TE properties, including a case study of an international round robin test comparing spin Seebeck effect measurement methods.

Electronic, magnetic, and crystalline structures comprised of repeating unit cells can be described in terms of band diagrams. These diagrams depict the quantum-mechanically allowable combinations of energy E and momentum wavevector k for low-lying excitations of the (psuedo)particle populations confined within each type of structure (electrons, magnons, and phonons, respectively). This makes band diagrams particularly useful for predicting and understanding the nature of transport phenomena in such materials. Many other types of solids also exist in which electron, phonon, and magnon transport can occur but where band diagrams are not as useful, such as disordered or amorphous materials, quasi-crystals, and polymers. While these materials may ultimately prove to be relevant for transverse TE devices, they are not featured heavily in the present discussion. Two-dimensional (2D) material surfaces and interfaces can also have properties unique from the bulk, including completely different transport behavior. Some examples of 2D phenomena are described in Secs. III–VII. Regardless, the primary emphasis of the present discussion is on transport phenomena in conventional three-dimensional ordered solids.

We begin our exploration of this topic with the simple picture of 1D band diagrams provided in Fig. 1, where the shapes are determined by the dispersion relationships governing each type of population.

In general, transport phenomena occur when (pseudo)particle populations are pushed out of equilibrium, either by a gradient in chemical potential or other driving force. This non-equilibrium state results in either a flux or accumulation, depending on boundary conditions. We provide here only a brief description of the relevant physics; for more in-depth discussions, we refer readers to detailed review articles such as Refs. 1–4 and references therein.

For the purpose of this Tutorial, we provide in Table I a list of the most relevant (pseudo)particle populations in solids and indicate the corresponding type of flux each can carry. In a general sense, and with limited exceptions, each population is directly sensitive only to driving forces that affect one or more of the fluxes that population carries (e.g., phonons respond to thermal gradients, but not electric fields). That being said, interactions between these populations (e.g., phonon-electron scattering) make it possible, under certain conditions, for any type of driving force to indirectly generate any type of flux. Examples include advective transport processes such as phonon–electron drag.

To provide clarity and consistency throughout the rest of this text, we define a general notation system for incident driving forces/fields and resultant fluxes/accumulations like that of Ref. 1. In this notation, an excitation ζ_a along the a-axis results in a flux j_b along the b-axis, with an optional applied field H_c along the c-axis. Put together, these produce a transport coefficient $ηabc=(jb/ζa)|Hc.$ While this three-subscript notation system may visually resemble tensor notation, we emphasize its use in this context is for describing transport coefficients for which an applied magnetic field is required. We use rectangular coordinates x, y, z to indicate geometrical relationships between the fluxes and fields on a physical sample, as indicated by the subscripts in Table II, and the axes are drawn in Fig. 2.

There are many possible combinations of these parameters for heat, spin, and charge, as apparent from the examination of Fig. 2. We list in Table II the examples most applicable to the present discussion. For longitudinal effects, where no magnetic field is needed to generate the measured flux or field, the defined coefficients include only two subscripts—though we note these effects can still certainly display magnetic field dependence (e.g., magnetoresistance).

The thermoelectric figure of merit zT was derived as a unitless parameter allowing for standardized comparison of TE performance of different materials at different temperatures.⁵ This metric relates the transport properties most relevant for determining the efficiency of thermoelectric energy conversion at a given temperature T, Seebeck coefficient (S_xx), electrical resistivity (ρ_xx), and thermal conductivity (κ_xx),

This equation expresses mathematically what can be stated intuitively: Solid state thermal energy conversion is most efficient when the thermodynamically reversible contribution (the Seebeck effect) is large and the irreversible components (Joule heating and thermal conduction) are small. It is worth noting that zT is definitively not an efficiency, therefore, it does not have a mathematical upper limit. For reference, Bi₂Te₃, the most commercially available thermoelectric material, has a zT of 1 at 300 K.⁶ 

While zT describes the thermoelectric figure of merit for a single material, ZT describes the thermoelectric figure of merit for a thermoelectric couple or n-type and p-type thermoelectric legs connected electrically in series and thermally in parallel, as shown in Fig. 3(a). ZT is defined in terms of each material's independent Seebeck coefficient, thermal conductivity, and electrical resistivity, as indicated by the subscripts “n” and “p,”⁶ 

The efficiency, η, of a thermoelectric energy generation device is dependent on ZT, the absolute temperature of the hot reservoir (T_H), and the temperature of the cold reservoir (T_C),⁶ 

For values of ZT much greater than one, the efficiency η approaches the Carnot efficiency, $(TH−TC)/TH$⁠. For reference, the record high of reported ZT values is ∼3.1 at 783 K in polycrystalline SnSe synthesized using carefully purified reagents to minimize oxidation during processing.⁷ 

A boom in TE research occurred throughout the twentieth century^8–11 and carried over into the twenty first century.^12–15 While the field has historically been amenable to an extremely wide variety of creative ideas for optimizing TE performance,^16–25 the vast majority of approaches center on two fundamental themes: (1) minimizing phonon thermal conductivity without affecting electronic properties (the so-called “phonon glass-electron crystal” idea first articulated by Slack)²⁶ and (2) manipulating electronic densities of states (eDOS) to create large Seebeck coefficients concurrent with favorable electrical resistivity (Table III).^{20, 27–30}

Inspiration for the present Tutorial comes from the fact that while some of these creative TE optimization strategies have yielded more success than others, none of them can overcome the multiple inherent disadvantages of the conventional longitudinal device geometry, which is depicted in Fig. 3(a). These limitations include

1. The need to return the induced electrical current to an isothermal plane, meaning longitudinal TE devices must always be made with both n- and p-type materials connected electrically in series but thermally in parallel.

2. Charge and heat flows are (anti)parallel to each other, so overall performance depends mainly on intrinsic material properties rather than device design parameters.

These limitations are “baked-in” to the longitudinal design architecture and will persist regardless of any future research progress in TE materials optimization. With that in mind, we propose more opportunities for advancement in this field are likely to come from considering devices that operate in the transverse geometry, which is depicted in Fig. 3(b). Recent developments in the physics of topological states and spin-dependent transport have generated abundant inspiration for novel routes toward transverse TE power generation. The following sections provide guidance on where to look for improved performance, as well as how to know with confidence if/when such improvements are actually found.

The same zT equation defined in Eq. (1) can be generalized for a single material by separating the axes along which the heat and charge currents flow into their x and y components,

where the coefficients are as defined previously.

The S_xyz thermopower term here is different from the Seebeck coefficient described previously; in this case, we are interested in the ratio between the net electric field in the y direction (E_y) generated by a temperature gradient in x (∇_xT), which together constitute a transverse thermopower.

The most well-known transverse thermoelectric phenomenon is the Nernst Effect, which occurs in conducting materials when a magnetic field H_z is applied perpendicular to∇_xT. The applied field deflects thermally excited charge carriers in a direction mutually perpendicular to ∇_xT and H_z, creating an electric field E_y to balance the Lorentz force. The corresponding Onsager reciprocal is the Ettingshausen effect.

The potential for the Nernst and Ettingshausen effects to be utilized for solid state energy conversion has always been recognized, with the caveat that needing an applied magnetic field is not practical for most technological applications. This points naturally to considering ferromagnetic materials with remanent magnetization at zero applied field; historically, however, the “remanent” or “anomalous” Nernst effect at H_z = 0 is not large enough to be useful in most known magnetic materials, as discussed in more detail below.

Regardless, the appeal of transverse energy conversion devices is immediately apparent from a cursory examination of Eq. (2) and Fig. 3: Transverse devices generate electrical energy based on the ratio of vectors that are perpendicular instead of parallel, so device performance scales with extrinsic dimensions instead of being limited to intrinsic material properties alone. This means conversion efficiency can be improved by manipulating device designs, even in the absence of improvements to the constituent materials.

To elaborate on this point, we consider the general case of transverse, or Nernst, thermopower resulting from a temperature gradient ∇_xT = ΔT/Δx (i.e, a change in temperature over a certain change in position along x), which generates an electric field E_y = ∇φ = ΔV/Δy (i.e., a gradient in electric potential φ corresponding with a voltage difference over a certain length along y). In the conventional longitudinal TE geometry, the two lengths are equal (x = y, Δx = Δy), so S_xx = −E_x/∇_xT = −(ΔVΔx)/(ΔTΔx), making the size of the device irrelevant. In transverse devices, however, perpendicular vectors mean for a given ∇_xT, even a small electric field E_y = ΔV/Δy could be integrated over an almost arbitrary length in y to produce a substantial voltage V, i.e., S_xyz = −E_y/∇_xT = −(ΔVΔx)/(ΔTΔy). If the device also has a large cross section in the x–z plane, then it will have low electrical resistance and, therefore, large power output. In addition to this benefit, transverse devices do not inherently require the use of separate n- and p-type materials, since the plane in which the electric field arises is always isothermal. This creates the potential for significantly less complex design and manufacturing than what is required for longitudinal devices.

While separate n- and p-type materials are not strictly necessary to produce isothermal voltages in the transverse geometry, such devices can still benefit from using materials in which both holes and electrons are mobile. This is because carriers of different polarity deflect in opposite directions in a magnetic field, resulting in additive contributions to the Nernst effect. The opposite is true for longitudinal devices, where thermally excited holes and electrons move in the same direction and, therefore, work against each other to reduce the magnitude of the Seebeck effect. In practice, this means single carrier semiconductors are more favorable for longitudinal TE devices, but a wider variety of material classes—such as semimetals and other two carrier systems—are appealing for use in transverse devices.

Figure 4 illustrates this point schematically, while Fig. 5 provides an example of the Nernst and Seebeck data collected from the classic TE material Bi₂Te₃ and the prototypical semimetal Bi.

Beyond the Nernst effect, we can look to additional transverse transport processes such as the spin Seebeck effect (SSE) and anomalous Nernst effect (ANE) for insight into how to best utilize magnetic materials and heterostructures for transverse TE energy conversion. These spin-dependent mechanisms, combined with topological band structures, constitute promising options for generating mutually perpendicular flows of charge and heat without the need for large applied magnetic fields. We begin our discussion of these topics by examining the available mechanisms known to result in transverse transport phenomena.

The notion of transverse charge flows is well established, with the first reported observation dating back to Edwin Hall's 1879 discovery that applying a magnetic field orthogonal to an electric current generates a measurable voltage in the direction mutually perpendicular to the incident flux and applied field. This sideways deflection of charge carriers in solids by the Lorentz force is referred to as the “Ordinary Hall Effect” (OHE).

Hall's observations included work on magnetic materials, in which he discovered the transverse voltage behaved differently. In modern terms, this “Anomalous Hall Effect” (AHE) in ferromagnets has two signatures: (1) the dependence of the transverse voltage on applied magnetic field matches that of the material's magnetic susceptibility (i.e., the AHE voltage displays hysteresis) and (2) unlike OHE, the magnitude of the voltage at the low field cannot be explained simply by considering the Lorentz force alone.

We now understand this low-field behavior originates from the sideways deflection of charge carriers due to three potential spin-dependent processes depicted in Fig. 6, which can be categorized as arising from either band structure effects (intrinsic) or scattering (extrinsic). The two extrinsic processes are skew scattering, in which spin–orbit coupling between free electrons and localized impurities results in asymmetric scattering rates, and side jump, in which the free electron velocity varies due to spin-dependent electric fields experienced while approaching and leaving an impurity. The intrinsic process, on the other hand, arises from what is known as Berry curvature, a gauge field intrinsic to the band structure of a material that acts like a magnetic field—another topic to which we will return later.

The mechanisms driving transverse deflection in the ordinary Nernst effect (ONE) are largely the same as those in the OHE and thus can be described by very similar equations. In fact, these phenomena are connected analytically via the Mott relation, which shows the Nernst coefficient is the energy derivative of the OHE conductivity³² (i.e., ONE is the thermal analog to OHE). This correspondence does not necessarily apply to the anomalous contributions, though it can under certain conditions.³³ 

While the default approach to transverse transport phenomena is to consider the sideways movement of charge, it is worth also briefly examining the case where the perpendicular flux is heat. This group of phenomena is generally referred to as different types of thermal Hall effects (THEs). Specific to electrical conductors, the Righi–Leduc effect describes transverse heat flow resulting simply from hot electrons deflected by an applied magnetic field such that heat and charge are both pushed sideways. (This is essentially equivalent to an adiabatic Nernst measurement, as discussed later.) Rarer are observations of THE in electrically insulating materials, where heat is carried by only phonons or magnons.^34–38 

Hypothetical THE-driven transverse generators can be envisioned by joining any conventional TE material with favorable zT atop any material in which heat flow is deflected perpendicularly. If using an electrical conductor as the substrate, an electrically insulating but thermally conductive layer would need to be placed between the two materials to avoid shunting.

This concept is depicted schematically in Fig. 7, where the conventional TE material is placed adjacent to a second material exhibiting large THE. When a temperature gradient is applied to the device in the cross-plane direction, the base material produces a substantial temperature gradient in the in-plane direction. This temperature gradient results in a corresponding longitudinal Seebeck voltage in the TE material, thereby converting incident heat flux into transverse charge flow.

There are obvious limitations at the current stage to the effectiveness of this idea in practice, namely, the need for materials with sufficiently large thermal Hall effects. To the best of our knowledge, no such material exists among electrical insulators; while there are reports of phonon magnetoresistance³⁹ and phonon/magnon Hall effects,^34,36 the response of non-electronic thermal currents to applied magnetic fields is typically nowhere near significant enough to produce useful transverse thermoelectric responses in adjacent materials under practical operating conditions, even if we relax the goal of avoiding large applied fields.

This concept becomes more plausible if we consider using a conductive material displaying a very large Righi–Leduc effect, especially if the material is ferromagnetic with large remanent magnetization and coercivity. Inserting a buffer layer between the substrate and TE material on top would be required to avoid shunting the TE film. If implemented successfully, this design would create the desired transverse temperature gradient (and associated transverse voltage) while keeping the TE material electrically isolated from the substrate, all while operating the device with no applied field.

To get an idea for how efficient such a device could be, we consider a hypothetical combination of the current state of the art for longitudinal TE materials near room temperature, Bi₂Te₃, grown as a film on top of Mn₃Sn, a material with large reported transverse thermal conductivity,⁴⁰ with some compatible buffer layer sandwiched between. We then make the following assumptions:

1. The Mn₃Sn substrate and Bi₂Te₃ film are completely isolated electrically.

2. The thermal properties of the substrate dominate the device, meaning the transverse temperature gradient within the Bi₂Te₃ layer matches that in the Mn₃Sn substrate.

3. The out-of-plane temperature gradient across the Bi₂Te₃ film and magnetization of Mn₃Sn have no impact on the Bi₂Te₃ longitudinal thermoelectric properties.

Under these assumptions, and using the labels in Fig. 7, we see the device's electrical power generation depends on only the properties of the TE film, while the temperature drop is determined by only the properties of the substrate. This means zT_xyz can be calculated in a fairly straightforward way; using data from Ref. 41, where Bi₂Te₃ films were grown on steel with an AlN buffer layer between them, we conservatively estimate a value for S_xx of at least 115 μV/K at 300 K. Combining this with the reported resistivity values, and then using the bulk thermal conductivity value of 7 W/m K from 41, we estimate a hypothetical intrinsic zT_xyz of 0.06. While this value is not especially high in a conventional sense, we reiterate the point that overall transverse device power output can be enhanced through adjustments to the extrinsic size of the device, e.g., by elongating the structure in x to integrate the electric field over an almost arbitrarily long length. In that regard, zT_xyz is useful for assessing the efficiency of hypothetical transverse device performance only in terms of potential power density, not absolute output. Regardless, improvements in zT_xyz might come from making the entire device out of thin film layers to further reduce the cross-plane thermal conductivity (which may improve stability of the zero-field behavior, especially if pinned through exchange bias), and/or focusing on incorporating materials with especially high Seebeck coefficients and low electrical resistivity without regard for thermal conductivity, such as CePd₃.¹⁶ 

The mechanisms responsible for spin-dependent sideways deflection of charge carriers in the anomalous Hall effect (like that in Fig. 6) can act on any electronic spin current, even in the absence of net charge flow. This special case of AHE is better known as the spin Hall effect (SHE), which occurs when an incident charge current results in a transverse pure spin current.⁴² Unlike AHE, no magnetic field is needed for SHE since it is a purely spin-based phenomenon. The inverse SHE (ISHE) corresponds with an incident pure spin current producing a net transverse charge current.⁴³ A diagram of these effects is included in Fig. 8. The magnitudes of longitudinal and transverse spin or charge conductivities are quantitatively related through a parameter known as the spin Hall angle; larger spin Hall angles mean more transverse spin current for a given longitudinal charge current.

In general, materials with large spin Hall angles also tend to have short spin diffusion lengths, the parameter that characterizes the distance over which induced spin polarization decays.⁴⁴ The consequence of this decay for ISHE is that the magnitude of the transverse charge current will vary with extrinsic parameters like film thickness. This is because charge carriers in the regions of material far from the source of spin polarization will simply rearrange themselves to cancel out the ISHE-induced electric potential. Regardless, the ISHE is a powerful tool for converting spin currents into perpendicular electric fields, making it enormously important for exploring new transverse TE energy conversion concepts.

One of the most straightforward ways to generate spin polarized electronic currents is to induce charge flow in materials with spin polarized eDOS. Some examples of such band structures are depicted in Fig. 9, where we now semi-arbitrarily differentiate the left and right sides of the ordinate axis based on whether the electrons are “spin up” or “spin down” (also referred to as “majority” and “minority” spin populations). When present in real materials, these band structures produce charge currents that are inherently spin polarized.

The first panel of Fig. 9 shows the spin-dependent eDOS for a ferromagnetic (FM) metal, in which the two spin channels have unequal occupancy. When driven out of equilibrium, the imbalance in spin up and spin down populations means the charge carriers will have a net spin polarization, with the polarization direction determined by the magnetization. In FM metals, spin up and spin down electrons can contribute to transverse transport in their own independent conduction channels.⁴⁵ Examples include elements like Fe, Ni, and Co.

The second panel corresponds with what is known as a half metal, where one spin channel has sufficient DOS to enable metallic charge conduction, while the other spin channel has a semiconductor-like gap between allowable states. A perfect half metal has 100% spin polarization, but this behavior has only ever been observed in CrO₂.⁴⁶ Other materials with very large net polarization include Fe₃O₄⁴⁷ and numerous Heusler-based compounds such as CoFeCrZ (Z = Al, Si, Ga, Ge).^48,49

The third panel shows what is referred to as a Spin Gapless Semiconductor (SGS), in which one spin channel resembles a narrow gap semiconductor or semimetal, while the other looks more like a conventional semiconductor. This type of material is expected to have transport properties similar to narrow gap semiconductors, where both electrons and holes are mobile charge carriers, with the difference being significant spin polarization of both carrier types.⁵⁰ Candidate SGS materials are found mostly among Heusler compounds such as Mn₂CoAl.⁵¹ 

Magnons are propagating perturbations of local magnetic order. These perturbations carry heat and spin but not charge. They occur in all magnetically ordered materials no matter what type of order or whether the material is electrically conductive. In general, there are three ways to drive magnons out of equilibrium: resonant excitation (FMR), interfacial spin pumping, and heat flux. Of these, thermal excitation is by far the most powerful.⁵² 

The notion of exciting magnon spin currents using heat was first demonstrated experimentally in 2008 by Uchida et al. Their experiments showed that thermally generated spin fluxes in electrically insulating ferrimagnets can spin-polarize free electrons in adjacent normal metals (NMs), resulting in an electronic spin current in the NM detected via ISHE.⁵³ This process is known as the spin Seebeck effect (SSE).

The magnetic material used in prototypical SSE heterostructures is yttrium iron garnet (Y₃Fe₅O₁₂, or YIG), a widely known and well-studied insulating ferrimagnet with a low damping coefficient.⁵⁴ Films or single crystals of YIG are often combined with Pt as the NM phase, the latter offering the benefit of being easily grown in thin films while also having sufficiently large spin Hall angle to facilitate ISHE electrical detection of spin currents injected from the adjacent magnetic material.

Uchida's 2008 discovery led to an enormous surge of research into the physics of thermal spin transport,^55–61 creating the field of study generally known as “Spin Caloritronics.” While much of this research is oriented toward “magnonics” and information technology,⁶² there is also substantial interest in using these insights to further the development of energy conversion devices.^24,63–68 Ideas and methods for pursuing this second goal are the primary topics of the remaining discussion.

Thermoelectric devices built in the conventional longitudinal geometry have been in use for several decades. Transverse devices of comparable efficiency would be similarly effective (or ineffective) in the same situations where longitudinal devices have already proven useful. The unique geometry of transverse devices might also open up some creative new opportunities, as discussed below.

Perhaps, the most routine technological use of the thermoelectric effect is in the form of thermocouple temperature sensors, which have been commonly used in commercial and industrial applications for the last several decades. In the context of power generation and active temperature control, however, TE devices are generally not as efficient as conventional combustion engines or vapor compression systems. This means TE devices are typically relegated to niche applications where their relatively unique properties make them the preferred choice. Being solid state devices, their efficiency scales independent of the device size, unlike combustion engines or vapor compression coolers. For this reason, TE devices are often attractive for situations when size and weight are a primary concern. Their lack of moving parts means they generate no vibrations and can operate continuously without maintenance, which makes them very attractive for active cooling of sensors in telescopes and electron microscopes, as well as for remote or off-grid power generation, especially if solar power is not an option. NASA has famously turned to TE devices to power many of its interplanetary missions, with the Voyager probes still sending signals to Earth over 50 years since their original launch dates. The last few Mars missions have also used TE generators to supplement their solar panels, which can be susceptible to events such as dust storms.

Transverse devices separate the charge and heat flows into perpendicular directions, making it possible to imagine utilizing such devices in unconventional geometries. For example, since the voltage drop scales with the transverse length of the device, one can imagine wrapping a transverse TE device around a radial heat source to make the device almost arbitrarily long. A version of this idea has already been demonstrated using galfenol wire wrapped around a cartridge heater,⁶⁹ providing proof-of-concept and motivation for further study. In addition, their relatively simple geometry makes it easy to imagine using transverse devices in flexible or stretchable platforms such as wearable sensing or energy harvesting applications.

In a fortuitous coincidence, the spin Seebeck effect occurs in the exact same transverse geometry as the Nernst effect. Since these two phenomena arise from very similar physical processes, this coincidence turns out to be inevitable. For purposes of experimental design in the pursuit of fundamental physics knowledge, this overlap has led to many difficulties with isolating SSE signals from ANE and other possible contributions.^70,71 For example, angular-dependence measurements have been relied on in many cases to separate SSE from the in-plane Nernst effect (S_xyy).⁷² Much work has also gone into exploring the potential for Pt films deposited on FM substrates to cross the Stoner instability and become magnetic,^73,74 in which case they produce their own ANE voltage in addition to the desired SSE signal.^70,75 For the purposes of maximizing energy conversion efficiency, this overlap turns out to be desirable, since it allows us to explore a wide range of ideas for combining ANE and SSE. The primary caveat is that one must understand the relative sign of the two effects in any system under investigation to avoid combining them in a way that they cancel each other out.

A cartoon of generic SSE heterostructure types is provided in Fig. 10. A typical thin film SSE heterostructure (e.g., Pt/YIG) is shown in Fig. 10(a). The earliest reported attempts to add ANE and SSE together manifested as multilayer stacks of NM films sandwiched between conducting FMs such as Fe₃O₄ and CoFeB;^76–79 this geometry is depicted schematically in Fig. 10(b). Included in this category are the “spin Hall thermopile” multilayer concepts demonstrated by Uchida,⁸⁰ as well as flexible thin film devices proposed by Kirihara et al.⁸¹ and Wang et al.⁸² 

Figure 10(c) shows an entirely different approach for a bulk composite material where the different phases are randomly mixed in three dimensions. This design architecture was first explored by Boona et al. in 2016, who demonstrated a net positive SSE contribution to the transverse thermopower in Ni–Pt nanocomposites.⁸³ Evidence of SSE signals was again observed in MnBi–Bi composites of the same architecture.⁸⁴ 

This third option of a bulk composite approach is appealing, in general, because it overcomes two main problems with thin film heterostructure devices: (1) thermal energy throughout the entire volume of the composite is involved in electrical power generation, not just the heat flux near the film interface(s) and (2) the entire composite cross section can be used to extract electrical power, resulting in significantly lower electrical resistance compared to thin film structures. Despite these advantages, care must be taken to ensure adequate interface quality between the different phases, since surface roughness of thin film heterostructures has been shown to significantly affect the resulting SSE signals.⁸⁵ 

Although proof of concept has been demonstrated for SSE + ANE in composite materials, there have so far been few reports of improvements on this approach. This is likely due in part to the challenges involved with practical considerations of working with bulk materials instead of relatively easily-grown thin film structures, such as selecting appropriate constituent materials and optimizing the processing conditions. To assist with further development of this concept, we provide here a set of basic design rules for SSE composites

1. The NM or “spin sink” phase should have a large spin Hall angle and a long spin diffusion length.

2. The FM or “spin source” phase should produce large spin fluxes for a given heat flux.

3. Material processing should be optimized to efficiently transfer spin flux from FM to NM.

4. Strategies should be explored to minimize shunting of transverse voltages.

5. Material combinations should be chosen to enable facile and tunable synthesis of intimately mixed phases with desired composition and microstructures (grain size, texture, orientation, etc.).

To address the first design rule—large spin Hall angles and long spin diffusion lengths—we consider two classes of materials known to have large spin Hall angles, then assess their viability for use in composites.

The obvious initial group of options for the NM phase consists of metallic elements, alloys, and compounds with large spin Hall angles like Pt and W, as discussed previously. While these materials work well in planar heterostructures, their spin diffusion lengths are on the order of only a few nanometers, meaning measurable ISHE voltages come from films no thicker than ∼10 nm. Utilizing such NM materials in composites, therefore, requires dispersing nanogranular particles intimately mixed between and/or coating the FM phase(s). This was the approach used for the first demonstration of SSE in the nanocomposite of Pt-coated Ni.⁸³ While realizing this architecture can be challenging, it is not necessarily uncommon; in fact, Pt-coated Fe₃O₄ nanoparticles are routinely synthesized and studied for potential use in catalysis⁸⁶ and medical applications.⁸⁷ Proper compaction and sintering of these or similar nanostructures into pellets may result in significantly enhanced transverse thermopowers due to combined SSE and ANE contributions. However, this specific combination of materials raises concerns related to establishing that both ANE and SSE contributions are actually present, since the transport properties of oxide materials like Fe₃O₄ can be significantly affected by nanostructuring.⁸⁸ This concern is also true in general, and care should always be taken to make single phase control samples using synthesis and processing steps as similar as possible to the composite. Establishing the validity of the SSE contribution can also be done by conducting single variable experiments, such as adjusting the volume ratio of the constituent phases with otherwise identical synthesis conditions. Detailed microstructural characterization, such as that in Ref. 84, will almost always be required for proper interpretation of any changes in transverse TE performance of SSE composites since there is a wide variety of relevant factors that may change with even slight adjustments in processing conditions.

The second class of materials that may satisfy the first design rule includes high atomic number alloys and compounds with large charge carrier mobility, such as GeSn,⁸⁹ GaAs,⁹⁰ InSb,⁹¹ or other similar semiconductors. For the most part, these materials are all extremely well characterized due to their potential for use in a variety of applications. Many of these alloys and compounds are available for purchase in different forms and can be readily synthesized from pure elements. They can also be incorporated into processing workflows involving wet chemistry, melting, or powder processing, opening a range of options for producing composites with desirable textures and microstructures. Their relatively long spin diffusion lengths also provide far more leeway than elemental metals in terms of the grain sizes required to observe ISHE-based SSE contributions to the transverse thermopower.

A third category of materials may also be worth exploring: topological insulators, in which charge carriers' spin and momentum states are locked at the material surface due to spin–orbit coupling.⁹² One primary challenge preventing the use of this mechanism for spin-to-charge conversion is the occurrence of electrical conductivity through the material bulk, which greatly undermines the utility of the surface states. In practice, overcoming this challenge typically requires very precise control over the composition ratio of alloys such as BiSb to optimize the relative magnitudes of surface vs bulk conduction. Given these challenges, utilizing such materials in bulk composites will likely be successful only after first understanding all of the relevant parameters and considerations through experiments on simpler thin film structures.

The second design rule—ensuring large spin flux for a given heat flux—can be satisfied by utilizing materials with highly spin polarized eDOS and/or materials with moderately strong magnon–phonon coupling. The argument in favor of spin polarized eDOS is obvious, in that any thermally induced charge flow will necessarily be highly spin polarized (the “spin-dependent Seebeck effect”). While it is also desirable for the total carrier density of electrons and holes to be large so as to generate a proportionately large total spin current, materials that are too conductive may run afoul of the fourth rule regarding shunting.

For magnon spin currents, we must consider that heat is typically injected across interfaces from one material to another primarily via phonons. This means the magnon population must first couple with the phonons in order to be brought out of equilibrium and thereby generate a spin current. Sufficiently strong magnon–phonon coupling is, therefore, required to efficiently generate large magnon spin currents for a given heat flux. Too much magnon–phonon scattering may be detrimental. Nuanced consideration should be given to the wave vector-dependence of phonon and magnon coupling strength, spin diffusion length, and energy mean free paths, as has been explored extensively in YIG.^60,61,93,94

Selecting materials that satisfy the third design rule—ensuring efficient interfacial spin transfer—require some insight into whether it is preferable in composites for thermally excited spin currents to be carried primarily by electrons or magnons. The answer to that question is not obvious a priori, though we can make some inferences. Grain boundaries are unavoidable in composite materials, and these defects are significant hindrances to transport properties of all kinds, especially spin injection. Adopting that perspective narrows the question to whether interfacial spin injection is more or less sensitive to interface quality when done by magnons in the FM polarizing free electrons in the neighboring material or by spin-polarized free electrons traveling directly across the interface from the FM to the NM. Since it has already been established that magnon-driven spin injection is highly sensitive to the interface quality,^85,95 electronic spin injection may prove more effective, especially since electron spin diffusion lengths are generally different from their momentum mean free paths.⁹⁶ Magnon and electron spin currents co-exist in ferromagnetic conductors, so it is not strictly necessary (or necessarily possible) to choose only one or the other. Either way, these considerations suggest materials with highly spin polarized band structures may be the ideal choice for the FM phase, such as the half metals and spin gapless semiconductors described previously.

The fourth design rule—minimizing shunting of the transverse voltage—addresses a major potential source of power loss in bulk composites: If the FM phase is too conductive, then the net voltage generated by the NM phase may be reduced due to short circuiting through the FM. This may be mitigated by tuning the volume ratios and processing conditions to avoid percolated connectivity of the FM phase, and/or by adding a third “inert” phase to the composite that has minimal effect on the magnetic or electronic properties, such as lightly doped silicon. This idea has analogs in heterostructure experiments where buffer layers of different materials are added between the FM and NM, and inspiration on materials selection might be drawn from such studies. If the inert phase is selected appropriately, this approach might also have the added benefit of reducing the total thermal conductivity of the composite.⁹⁷ 

Finally, to address the fifth design rule—facile and tunable synthesis—we consider factors relevant to producing composite materials with microstructures and compositions that can be readily manipulated to produce desirable transport properties. In addition to wet chemistry methods like those used to make Pt-coated Fe₃O₄ nanostructures, there are a range of well-established bulk materials synthesis and processing techniques that may be very useful for making SSE composites, such as arc melting, solid state reactions, and powder processing. An open mind should be kept when considering potential methods for producing composites, however, since there are many techniques common in other research areas that may prove very useful here. These include, for example, solgel⁹⁸ or laser synthesis and processing of colloidal YIG nanoparticles,⁹⁹ as well as porous Ni foams developed for electromagnetic shielding applications.¹⁰⁰ Traditional binary and pseudo-binary phase diagrams are also excellent resources to assess possibilities for one-pot synthesis candidates. For example, some very simple FM and NM materials like elemental Fe and Bi are completely insoluble,¹⁰¹ making it possible to mix the materials in any desired manner without concern over the formation of intermetallic phases with unknown or undesirable properties. Creative manipulation of phase diagrams and use of non-equilibrium processing methods may lead to a wealth of interesting composite microstructures.

While interfaces can sometimes be beneficial, as in the case of superparamagnetic nanoparticles²³ or interfacial doping in semiconductor composites,¹⁰² we can take the SSE composite concept to the extreme by removing the interfaces entirely and imagine single phase materials in which magnons, spin polarized electrons, and a large spin Hall effect is present simultaneously. In contrast to thin film SSE heterostructures, in which a NM film atop an insulating FM converts thermally excited longitudinal magnon spin currents into transverse charge currents via ISHE [as illustrated in Fig. 11(a) below], magnetic materials with large spin Hall angles continuously convert these magnon spin currents into transverse charge currents throughout the bulk of the material. This process is referred to as the “self-spin Seebeck effect,” and it is distinct from the anomalous Nernst effect. The most obvious type of material in which this combination of properties might be found is heavy metal ferromagnets.

This class of materials includes single phase compounds such as FePt, MnGa, and MnBi. Indeed, FePt thin films show large remanent ANE signals at the zero field,¹⁰³ and FePt/MnGa thermopiles have also been studied.¹⁰⁴ Examination of single crystal MnBi reveals evidence for self-SSE producing enormous transverse thermopower values of 8 μV/K at the 0.4 T applied field.¹⁰⁵ A comparable observation has also been made on a ferromagnetic bulk metallic glass.⁵⁵ These materials are similar to heterostructures and composites in which SSE and ANE are combined [Fig. 11(b)], except here the resultant transport properties do not depend on the presence of interfaces between different materials. Instead, the thermally generated spin currents moving within the bulk are continuously deflected sideways by the ISHE within the material itself [Fig. 11(c)]. Further exploration of these materials through methods such as solid solution alloying with other isostructural compounds may enable tuning their properties enough to make them viable for transverse TE energy conversion.

Berry curvature is responsible for the intrinsic mechanism in the AHE and ANE. It can also increase the OHE and ONE, but it contributes to each effect in different ways. In the Hall effect, Berry curvature is averaged over all occupied states, while in the Nernst effect, the anomalous contribution depends on the Berry curvature only at the Fermi level.^117,118 This difference means designing materials with large Nernst coefficients requires different strategies than those for enhancing the Hall effect, i.e., increased emphasis not just on the presence or extent of Berry curvature but also on where the Fermi level sits. This is where Weyl physics comes into play.

The idea of a chiral, massless Weyl fermion was first predicted to exist in high-energy physics,¹⁰⁶ then in solid state physics by considering a band structure that is the 3D equivalent to graphene.^107–109 This prediction was confirmed in 2015 through experiments on TaAs, the first known Weyl semimetal (WSM).^110–114 The unique band structure of WSMs centers around linearly dispersing Dirac bands rotated in momentum space to form a Dirac cone (Fig. 12). The Dirac bands cross and invert due to spin–orbit coupling at a low-symmetry point in momentum space. These crossing points are called Weyl points and always come in pairs. WSMs are characterized as Type I, with symmetric Dirac bands in momentum space and a zero density of states at the Weyl points, or Type II, with tilted Dirac bands. Additionally, WSMs are characterized by breaking either inversion symmetry (IS) or time-reversal symmetry (TRS), where the broken symmetry lifts the double degeneracy of a Dirac point, separating it in momentum space into two Weyl points. Weyl points act as magnetic monopoles in momentum space and thus as sources/sinks of Berry curvature (Table III).

Initial experimental work in WSMs focused on using angle-resolved photoemission spectroscopy (ARPES) and scanning tunneling spectroscopy to search for the existence of Fermi arcs, which are the momentum space signature of WSMs.^110–114 Electromagnetic studies followed to explore the transport properties of these materials, quickly discovering that WSMs display ultra-high mobility and giant magnetoresistance. These characteristic signatures of WSMs are attributed to their unique band structure and topology.^115,116

It has been theoretically predicted that large Berry curvature in the vicinity of Weyl points could give rise to a significantly enhanced Nernst effect if the Fermi level is also located near those points.^117,118 The Nernst thermopower S_xyz in WSMs is a function of the corresponding transverse Onsager transport coefficient L_xyz^ET, which is described at the zero field by the following equation:^117,119

where k_B is the Boltzmann constant, e is the elementary charge, ℏ is the reduced Planck constant, k is the wave vector, Ω_z is the Berry curvature along the z axis, and s(E) is the entropy density function that is Gaussian-like and centered at the Fermi level. We emphasize that since Berry curvature is generally anisotropic, its contribution to the Nernst thermopower is as well, which is why the net Berry curvature direction must be specified in this equation. As defined in Introduction, the xyz subscript notation used here indicates heat flux in the x direction, voltage measured in y, and the net Berry curvature (i.e., the effective internal magnetic field) aligned along z. Since this function describes a phenomenon that occurs in three dimensions, it is necessary to indicate along which directions the other two associated vectors (temperature gradient and electric field) are pointing; these axes are otherwise irrelevant to the equation, as long as the signs and relative orientations satisfy the expected relationships for a right-handed coordinate system.

Since the Nernst effect is a direct probe of the Berry curvature at the Fermi level, thermal measurements are becoming increasingly recognized as powerful tools to probe the transport of solids, especially those with non-trivial topological structures. Beyond the Nernst effect, magneto-thermal conductivity measurements are also expected to show signatures of Fermi arc-mediated transport.¹²⁰ This has been very recently confirmed in SbBi alloys by Vu et al.¹²¹ 

The prediction of a large Nernst effect has been experimentally demonstrated in nonmagnetic WSMs such as NbP,¹²² TaP, and TaAs.^123,124 A maximum Nernst thermopower of ∼800 μV/K was found in NbP at 109 K, 9 T, exceeding the longitudinal thermopower of conventionally good thermoelectric materials by threefold to fourfold.¹²² This peak occurs at the same temperature where the chemical potential reaches the energy of the Weyl points, which is also the location of the minimum DOS and the energy at which holes and electrons are compensated in the Dirac bands. This is rigorously the case in Type I WSMs, while Type II WSMs may have a minimum DOS at another energy due to the tilting of the Dirac bands. Furthermore, it was found that transport is dominated not by conventional parabolic bands as suggested by DFT, but instead by linear Dirac bands.¹²² 

In addition to WSMs, Dirac semimetals such as Cd₃As₂,¹²⁵ HfTe₅,¹²⁶ and ZrTe₅¹²⁷ have also shown an impressive Nernst effect under external magnetic fields. The applied magnetic field induces a transition in the band structure such that the Berry curvature is enhanced near the Fermi level. Very recent work on Fe-based binary ferromagnets Fe₃Ga and Fe₃Al¹²⁸ has shown that even without Weyl points, the formation of a flat “nodal web” structure near the Fermi level can enhance the Berry curvature, leading to a considerable Nernst effect. A colossal ANE signature was recently reported by Asaba et al. in a single crystal of the strongly correlated kagome ferromagnet UCo_0.8Ru_0.2Al.¹²⁹ These results all support the idea that WSMs are a rich hunting ground for new materials with potential use in transverse devices.

The unprecedentedly large Nernst thermopower found in NbP at ∼109 K naturally inspires further inquiry: Is there a WSM in which the chemical potential reaches the energy of the Weyl points at room temperature? The recent progress in the Materials Genome Initiative, coupled with high-throughput DFT, offers potential to determine if such a compound already exists. Band structure engineering could also make this happen, potentially by increasing the residual doping level in a WSM to move the Fermi level further into the Dirac band, slowing its motion toward the energy of the Weyl points as a function of temperature. If these or any other proposed search methods are successful, then WSM-based Nernst generators become significantly more viable.

While these results are certainly encouraging, there are still several factors limiting the immediate practicality of Weyl-based transverse TE generators. These include the need for large magnetic fields applied to single-crystalline samples that are small, challenging to make, and have high thermal conductivity. Inspiration for overcoming these limitations can be drawn from some of the similar efforts made in other areas of TE research referenced previously.

For example, introducing defects to lower thermal conductivity may be a viable option, such as polycrystalline NbP.¹³⁰ Since grain boundaries limit thermal conductivity, polycrystalline NbP may show overall improved zT_xyz. The Nernst thermopower in polycrystalline NbP remains large despite a threefold to fourfold decrease relative to single crystal samples, and the high electrical conductivity contributes to a large Nernst power factor over a broad temperature range. Nernst power factor PF_xyz quantifies the power density of a transverse TE device and is defined as follows:

The conventional longitudinal power factor PF is analogous to the Nernst power factor in the same way zT and zT_xyz are related.

Another way to make WSMs more practical is to consider ways of producing similarly large Nernst thermopower values without requiring large magnetic fields. One proposed mechanism for accomplishing this is to examine WSMs that break TRS, as this is anticipated to correspond with a net Berry curvature intrinsic to the material's band structure that acts like an effective magnetic field in the context of transport properties.¹¹⁷ There are two potential manifestations of this concept currently being explored. The first is ferromagnetic WSMs, which intrinsically break TRS due to their magnetic structure. Examples include Co₂MnGa¹³¹ and Co₃Sn₂S₂,¹³² where studies indicate Berry curvature indeed leads to a large ANE beyond that expected from intrinsic magnetization alone. The second manifestation is nonmagnetic WSMs and Weyl metals that break TRS, of which most candidate materials are non-collinear antiferromagnets such as Mn₃Sn,^40,133 Mn₃Ge,¹³⁴ and YbMnBi₂.¹³⁵ Recent studies in the chiral antiferromagnets Mn₃Sn and Mn₃Ge showed ANE signals of similar magnitude to those seen in ferromagnetic WSMs, and ANE signals in YbMnBi₂ exceed those seen in topological ferromagnets, indicating this mechanism is an equally viable option. Work in this area is ongoing.

Table IV summarizes current results of Nernst thermopower experiments using Berry curvature and/or Weyl physics as intrinsic mechanisms to achieve a large Nernst thermopower. Results in Table IV show the maximum magnitude of Nernst thermopower reported for a given material. Worthy of note is that although the magnetic materials and TRS-breaking WSM exhibit promising Nernst thermopowers at or near zero-field, the Nernst thermopowers of elemental Bi and even some of the non-magnetic WSM are still orders of magnitude larger at relatively small externally applied magnetic fields {e.g., S_xyz of Bi at 6.8 K and 1 T = 30 000 μV/K;¹³⁶ S_xyz of Bi at 102 K and 1 T = 350 μV/K [Fig. 5(b)]; S_xyz of NbP at 109 K and 1 T = 127 μV/K}.¹²¹ 

While the influence of intrinsic Berry curvature on the Nernst effect has been studied in depth, the detailed impacts of extrinsic mechanisms have seen increased interest only more recently. As mentioned previously and illustrated in Fig. 6, there are two known extrinsic mechanisms induced by disorder: skew scattering and side jump. Side jump occurs due to a spin-dependent sideways wave packet displacement during scattering events, while skew scattering arises from a difference in spin-dependent scattering rates between one orthogonal direction and the other.

One way to induce such an imbalance is strong spin–orbit coupling between free electrons and local impurities. This idea has been theoretically predicted and experimentally demonstrated via AHE measurements in AuW,¹³⁸ CuIr,¹³⁹ and CuBi¹⁴⁰ alloys.

For reasons described previously, AHE and ANE are impacted by the intrinsic mechanism in ways that can be very different within the same material. The extrinsic mechanisms, on the other hand, are expected to impact both AHE and ANE in comparable ways. Data demonstrating the influence of extrinsic mechanisms on the Nernst effect are less common than those exploring the Hall effect, with most studies reported so far focusing only on theoretical considerations. Papaj and Fu¹⁴¹ performed calculations using the semiclassical Boltzmann equation formalism and reported that the extrinsic contributions can be more significant than the intrinsic band structure contribution in ferromagnetic Dirac semimetals such as Fe₃Sn₂ and WSMs such as Co₃Sn₂S₂. It is interesting to note this result is contrary to the other studies of Co₃Sn₂S₂ mentioned previously,¹³² wherein the enhanced Nernst effect is attributed to the intrinsic contribution of large Berry curvature near Weyl nodes. This contradiction warrants further experimental studies to understand the role of intrinsic vs extrinsic mechanisms. This could be accomplished by, for example, systematically varying the concentration of impurities or other disorders in a given material system and looking for any corresponding changes in the Nernst effect. The results of such a study may provide additional insight into methods for boosting the Nernst effect through tuning the extrinsic contributions, which could then be applied to other material systems.

Another interesting research direction regarding the extrinsic mechanisms is to explore nonlinear transport physics. Besides the linear Hall effects, the recent discovery of nonlinear Hall effect in WSMs with broken inversion symmetry^142,143 has opened the door toward designing novel transverse TE devices operating with the zero magnetic field. Regarding the extrinsic contributions, Du et al.¹⁴⁴ have recently reported a theoretical work where they show that disorder-induced extrinsic mechanisms can play a more dominant role in the nonlinear Hall effect than they do in the linear regime. Since no nonlinear counterpart has been observed in the Nernst effect yet, this direction of research offers ample opportunities for exciting discoveries. Insights learned from studying the role of defect scattering in the nonlinear Hall effect could be applied to the nonlinear Nernst effect as well, which may lead to significantly larger Nernst signals.

A novel idea for realizing transverse TE generation was very recently proposed by Zhou et al.¹⁴⁵ The thermopower of ANE (S_ANE) can be expressed by two components as $SANE=ρxxαxy−ρAHEαxx,$ where ρ_xx, ρ_AHE, and α_xx (α_xy) are the longitudinal resistivity, anomalous Hall resistivity, and the diagonal (off-diagonal) component of the Peltier tensor, respectively. While the first term on the right-hand side of the equation originates from the intrinsic ANE, the second term originates from the AHE of the longitudinal carrier flow induced by the Seebeck effect and can be rewritten as $−SSE×ρAHEρxx$⁠, where S_SE is the Seebeck coefficient. Inspired by this second term, the authors artificially engineered the contributions of the Seebeck effect and AHE to drive transverse TE generation using a system shown in Fig. 13. The system consists of thermoelectric and magnetic materials electrically connected at both ends to form a closed circuit. When $∇T$ is applied to the hybrid system, a longitudinal electric field generated by the Seebeck effect of the thermoelectric material induces a charge current in the magnetic material. This charge current is then converted into the transverse electric field by the AHE of the magnetic material, manifested as transverse thermopower. The authors named this phenomenon as Seebeck-driven transverse thermoelectric generation (STTG). A proof-of-concept experiment revealed that the transverse thermopower $Stoty=(EMy−∇T)$ resulting from the combinations of STTG and ANE is one order of magnitude larger than that from ANE only, and the sign of the STTG contribution can be easily engineered by the transport properties of the thermoelectric material. Thus, this newly proposed STTG concept provides a viable way to realize transverse TE generation with a significantly larger transverse thermopower than existing ANEs. The same authors have also published a phenomenological calculation establishing how to characterize the thermopower, power factor, and figure of merit for STTG devices.¹⁴⁶ 

One drawback of this design concept is the need for at least two different materials to produce the working structure, which requires optimizing numerous parameters to achieve peak performance. For example, controlling the thermal conductivities of both thermoelectric and magnetic materials and the heat flow across the interface between them could be quite challenging for different combinations of materials. Nevertheless, the proposed STTG concept offers an attractive option for realizing high-performance transverse TE generation and could potentially also be utilized in a bulk composite architecture similar to the one described previously.

This section discusses various considerations for measuring and reporting transverse thermoelectric properties, providing guidance and best practices with the intent of enabling more accurate “apples-to-apples” comparisons of properties measured by different groups in different types of materials and structures. In addition to many well documented considerations that must be made for thermoelectric transport property measurements in general,¹⁴⁷ transverse measurements introduce additional complicating factors related to off diagonal transport coefficients and unintended consequences of applied magnetic fields.

Strictly speaking, the “Nernst coefficient” is not expressed in the same dimensional units as the “Seebeck coefficient,” the former typically having units of Volts per Kelvin per Oersted, and the latter having units of Volts per Kelvin. In general, V/K are used to describe thermopowers, recognizing that for transverse measurements (i.e., the Nernst thermopower), the resulting quantity is actually a ratio of perpendicular vectors (electric field and temperature gradient), and these vectors have embedded within them two different lengths whose units cancel out. To determine a Nernst coefficient, we must also account for the magnitude of the applied magnetic field necessary to produce that thermopower. For materials with magnetic hysteresis, this definition is straightforward only in the high field regime where the magnetization is saturated. At fields below saturation, the so-called “anomalous” behavior can be treated in a similar way by looking at where the thermopower crosses from positive to negative and/or negative to positive. The exact values of the applied field at which this transition occurs depend primarily on the material's magnetic coercivity, which, in turn, depends on both intrinsic and extrinsic properties such as crystalline anisotropy, grain size, domain size, texturing, etc.

This framework allows us to see how all measurements of transverse thermopower are inherently tied together, regardless of which type of structure or mechanism is involved, whether they be magnetic compounds, thin film heterostructures, or bulk composites driven by SSE, ANE, self-SSE, or STTG (Figs. 10, 11 and 13). Figure 14 shows a progression of how Nernst effect data change as the magnetic coercivity and remanence are increased. Panel (a) corresponds with a non-magnetic material in which there is no anomalous contribution such that the ONE coefficient can be defined by the slope of the thermopower vs field at any point. The transverse thermopower is zero at the zero applied field. Panel (b) shows three examples of an anomalous signal coming from a phase with low remanence and coercivity, all with the same ONE slope but with varying ANE magnitude and sign. Here, the transverse thermopower is again zero or nearly zero at zero applied field. Panel (c) shows a very pronounced and easily distinguishable step-like transition associated with larger remanence and coercivity, as is typical in single domain structures like thin films.

From the plots in Fig. 14, we can think about another way to characterize the magnitude of the SSE and/or ANE contribution, i.e., how different are plots 14(b) and 14(c) from 14(a)? In the general case of 14(c), in particular, the ONE signal is often orders of magnitude smaller than the ANE component, making it essentially negligible. This is extremely common in SSE experiments where the only ONE contribution comes from the NM film, and the SSE signal, therefore, appears in the data like a very sharp step change. This difference in transverse thermopower between saturated magnetization states is often referred to in the literature as the “SSE coefficient.” ANE data are also sometimes discussed this way, especially in studies attempting to combine the two effects. In any case, to avoid confusion, we strongly recommend explicitly defining what is meant by the terms “ANE coefficient” or “SSE coefficient” whenever they are used in research reports.

We note that Nernst signals should almost always be odd functions of the applied magnetic field; the presence of any component with even field dependence suggests potential experimental problems such as contamination from the magneto-Seebeck effect, which may arise from misalignment of electrical probes on the sample. This can be overcome in many cases by symmetrizing the data to separate the even and odd components of the signal.

One long standing and well-known issue in measuring the Nernst effect and related transport phenomena is the debate between adiabatic vs isothermal measurements. By definition, “adiabatic” measurements are those in which no heat is transported to or from the system, while “isothermal” measurements are those in which the only temperature gradient in the sample is the one created by the incident heat flux (i.e., the sample temperature at any point along the direction of heat flow is constant in the transverse direction). In practice, many adiabatic measurements end up being essentially isothermal as well and vice versa. However, the difference between these setups is important to recognize because conventional theory generally assumes isothermal conditions for determining the magnitude of thermoelectric transport coefficients. Pure thermal transport effects, on the other hand, such as thermal conductivity and the thermal Hall effect, must be measured adiabatically in order to observe their true magnitude. This distinction between adiabatic and isothermal measurements becomes especially important in any material with a large thermal Hall effect, and/or where the Nernst effect is similar in magnitude or even larger than the Seebeck effect. Both of these conditions apply to WSMs, making it extremely important in research reports to clarify the type of measurement performed.

Adiabatic measurements assume heat transfer only through conduction from the heater to the heat sink through the sample, with no heat transfer from the sample to the surrounding measurement system via convection or radiation. In an adiabatic mount, heat flow is directed through the sample, but heat flow within the sample is unrestricted [as depicted in Fig. 15(a)]. This means unintentional transverse temperature gradients can be produced via magneto-thermal effects such as the thermal Hall effect when a temperature gradient is applied perpendicular to an external magnetic field. Transverse temperature gradients can also develop when the thermal contact resistance at the interfaces between the heater, sample, and heat sink is not uniform across the contact area. Adiabatic sample mounting techniques are significantly more common than isothermal techniques, as they require simply adding thermal and electrical probes to a sample, then measuring transport properties in an evacuated environment covered by a radiation shield.

Isothermal measurements, on the other hand, require preparing the specimen in such a way that heat flow in-plane is restricted and transverse temperature gradients cannot be generated. One useful method for accomplishing this is depicted in Fig. 15(b) and described below.

WSMs, in particular, are expected to have a thermal Hall effect due to a significant electronic contribution to their thermal conductivity. In this case, the measurement of the adiabatic Nernst effect is actually a mixture of the isothermal Nernst effect [given in Eq. (4)] and the longitudinal thermopower along the y-axis, parasitically produced because the condition $∇yT=0$ is not imposed in the adiabatic mount¹²² 

Importantly, we emphasize here the measurement of the adiabatic Seebeck effect contains a contribution of both the isothermal Seebeck [given in Eq. (7)] and the isothermal Nernst effect, where the Nernst effect is parasitically produced due to $∇yT≠0$⁠,¹⁴⁸ 

This means attempting to measure the adiabatic magneto-Seebeck effect actually results in measuring a parasitic Nernst effect produced via the thermal Hall effect. This is especially pronounced in WSMs, where the Nernst effect is significantly larger than the Seebeck effect, indicating that the contamination of the Nernst thermopower by the Seebeck effect is small, but the contamination of the magneto-Seebeck effect by the Nernst effect is large. This has been seen in single-crystalline NbP, where adiabatic magneto-Seebeck measurements indicated a significant contribution, quantitatively similar to the isothermal Nernst effect at the same temperatures and applied magnetic fields,¹⁴⁹ but isothermal magneto-Seebeck measurements indicated no magneto-Seebeck effect.¹²² The supplement of Ref. 122 gives significantly more details on adiabatic vs isothermal sample mounting techniques.

We note a similar concern over unintended temperature gradients can also potentially affect SSE measurements of thin films, as recently reported by Lee et al. in Pt/WSe₂/YIG hybrid structures.¹⁵⁰ In that regard, indicating adiabatic vs isothermal mounting should be a standard consideration when reporting any type of transverse thermopower measurements.

A wide variety of interesting phenomena can be exploited in thin film structures, where interfacial and thickness-dependent effects become comparable to or larger than the corresponding bulk properties. However, the very same features that make thin film structures interesting can also make them very challenging to characterize. This is especially true for transport property measurements, where accurate accounting for cross-sectional areas and interfacial effects is required. For example, measurements of temperature differences and/or voltages along the growth direction of thin films are substantially more difficult to perform than they are on bulk specimens, since the electrodes and/or thermometers used in conventional measurements typically have dimensions larger than the thickness of anything qualifying as a “film.”

Instead, for practical reasons, measurements are often taken of the temperature drop across an entire heterostructure, typically by measuring the temperature of large heat sinks attached to the top and bottom surfaces of the film and substrate. From an engineering and applications perspective, this approach may provide enough information to know what electrical power output can be generated for a given temperature drop across the entire structure since that is essentially how such a structure would function if used as-is in an energy conversion device. From a scientific perspective, however, measuring the temperature drop this way creates significant ambiguity as to the actual temperature gradient present inside the thin film region of interest. That information is critically important for understanding the physics and determining the “true” magnitude of phenomena like SSE, since only a small portion of the total temperature drop occurs within the magnetic thin film where the spin current is generated. Even when the dimensions of the substrate and thin film are well characterized and their bulk thermal properties known, it is not trivial to account for interfacial thermal conductance or for any potential thickness-dependent reduction of thermal conductivity in the film.

One option for improving the measurement accuracy of thin film heterostructure properties is to incorporate techniques like time domain thermal reflectance (TDTR). When properly deployed, TDTR can be useful for assessing the thermal conductivity of thin films as well as the interfacial thermal resistance between them.^151–153 

In certain circumstances, differential measurements can also be used to compare the thermal properties of a heterostructure with and without the active thin film layers present. This can be done, for example, via simultaneous measurements of two nearly identical samples in parallel, or by removing the film from the substrate after the initial data are collected.

These options for improvement come with limitations and may not be straightforward to apply to every experiment. Instead, a better universally applicable method is the one revealed through discussion of the round robin experiment highlighted in Sec. VII D; transverse thermopower measurement data can always be reported in terms of measured electric field per unit heat flux.

A paper published in 2019 reported results from a blind international five-member round robin study of the SSE coefficient measured on a single YIG/Pt heterostructure comprised of a 10 nm Pt film deposited on a 4 μm YIG layer grown atop a bulk Gd₃Ga₅O₁₂ (GGG) substrate. The same sample was passed around to different laboratories active in Spin Caloritronics research, and each lab used its customary setup to determine the structure's SSE coefficient based on measurements of the temperature drop across the entire structure. The results were compiled and published in Ref. 154, with details of the different setups described.

As shown in Fig. 16, the magnitudes of the SSE coefficients, which are described in the figure legend as S_LSSE, were very consistent and repeatable within each group, but varied by almost a factor of 10 between the different groups. This indicates there are many reasonable choices experimenters can make with their setups that will have large impacts on the apparent magnitude of transverse transport properties such as SSE. This was observed despite each group measuring the exact same sample, which eliminated potential sources of variation such as YIG and Pt films and interface quality or YIG and Pt film thickness variations.

These results can be explained only through systematic differences in how each lab measured the voltage and temperature drops. Representative images of each lab's setup are included in Fig. 17. Although each group used a nominally identical method to measure the spin Seebeck coefficient, there are many choices to make regarding the types of voltage leads and thermometers, as well as how to attach them. Heat sinks can have varying thermal properties, and thermal fluxes can be created using a variety of methods. The measurement apparatus itself can also have varying levels of vacuum and quality of radiation shields. The enormous number of possible combinations of these details is what inspired the round robin test to be completed in the first place since the “true” SSE coefficient of any particular heterostructure should not depend on when, where, or by whom it is measured.

In follow-up experiments exploring ways to improve the consistency of these measurements, Sola et al. first deposited fixed Au contacts on the Pt film surface, then attached voltage leads to the Au pads. This approach dramatically increased repeatability of the measured SSE coefficient between different measurement apparatuses.¹⁵⁵ Further improvements on the measurement approach were found by considering not the temperature drop across the entire structure but instead the incident heat flux through the film cross section.¹⁵⁶ These results strongly suggest that the optimal way to report transverse thermopower measurements from any type of structure is in terms of electric field per unit heat flux, as mentioned in the general discussion on thin films. That parameter is the single best way to directly compare transverse thermopower measurements performed on any type of material or structure.

Transverse TE devices have inherent benefits over traditional longitudinal devices, but they have historically been disregarded due to the challenge of generating useful transverse thermopowers at the low or zero applied magnetic field. Recent advances in both theoretical and experimental studies of spin-dependent transport and topological states provide exciting new options that may make transverse TE viable for more widespread applications.

Here, we provided an introduction to these ideas along with numerous suggestions for how they might be further developed from design rules for improved composite materials to unexplored regions of the rapidly growing field of Weyl semimetal physics. We also discussed important considerations for making, interpreting, and reporting results of transverse transport effects in different types of materials and structures.

While much work remains to be done, we hope the information in this Tutorial will be useful to those who also see the potential for transverse TE devices.

S.W. acknowledges support from the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences Early Career Research Program Award No. DE-SC0020154. H.J. acknowledges support from the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2020R1C1C1004291) and the Ministry of Education, Science and Technology (No. NRF-2020K1A4A7A02095438). We also thank Eleanor F. Scott and Abhishek Saini for obtaining transport data on Bi₂Te₃ and Bi shown in Fig. 5 and Min Young Kim and Sang Jun Park for their help with figure artworks.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

The authors have no conflicts to disclose.

2D

Two dimensional

AHE

Anomalous Hall effect

ANE

Anomalous Nernst effect

ARPES

Angle resolved photo-emission spectroscopy

DFT

Density functional thoery

DOS

Density of states

eDOS

Electronic density of states

FM

Ferromagnet or ferrimagnet

FMR

Ferromagnetic resonance

GGG

Gadolinium gallium garnet, Gd₃Ga₅O₁₂

IS

Inversion symmetry

ISHE

Inverse spin Hall effect

NM

Normal metal

OHE

Ordinary Hall effect

ONE

Ordinary Nernst effect

PF

Power factor

SHE

Spin Hall effect

SGS

Spin gapless semiconductor

SSE

Spin Seebeck effect

STTG

Seebeck-driven transverse thermoelectric generation

TE

Thermoelectric

TDTR

Time domain thermal reflectance

THE

Thermal Hall effect

TRS

Time reversal symmetry

WSM

Weyl semimetal

YIG

Yttrium iron garnet, Y₃Fe₅O₁₂

zT

Thermoelectric figure of merit

ZT

Device thermoelectric figure of merit