We review the spin-Seebeck and magnon-electron drag effects in the context of solid-state energy conversion. These phenomena are driven by advective magnon-electron interactions. Heat flow through magnetic materials generates magnetization dynamics, which can strongly affect free electrons within or adjacent to the magnetic material, thereby producing magnetization-dependent (e.g., remnant) electric fields. The relative strength of spin-dependent interactions means that magnon-driven effects can generate significantly larger thermoelectric power factors as compared to classical thermoelectric phenomena. This is a surprising situation in which spin-based effects are larger than purely charge-based effects, potentially enabling new approaches to thermal energy conversion.

## I. INTRODUCTION

Conventional thermoelectrics are based on the Seebeck effect in which an electric field **E** is generated within a material as a direct result of the application of a temperature gradient ∇*T*. The Seebeck effect occurs because free charge carriers exist inside the material within a diffuse cloud that has a temperature-dependent statistical distribution, which is altered by ∇*T*. If there is sufficient difference between the densities of states of high and low energy electrons (or between the scattering times of electrons interacting with phonons, other electrons, defects, etc.), then a local net difference in electron population appears (i.e., an electric field, **E**). The ratio between **E** and ∇*T* is a state function broadly referred to as the Seebeck coefficient *α*. For small perturbations (Δ*T*/*T* ≈ 1%), *α* is constant vis-à-vis ∇*T* (linear transport) and properly characterized by a thermopower tensor $\alpha \u2194;\u2009E=\alpha \u2194\u2207T$. If **E** and ∇*T* are collinear, the thermopower *α* is a scalar quantity. If the density of charge carriers in a material is low, then their spatial distribution is affected by ∇*T* more strongly, and therefore *α* is large. This correspondence is approximated by the Pisarenko relation *α* ∝ − ln(*n*), where *n* is the charge carrier concentration, and the logarithm arises from the distribution function. When this relation is combined with the Wiedemann-Franz law, which quantifies the amount of heat transported by free charge carriers, these relations produce competing dependencies on *n* that inherently limit the thermoelectric figure of merit *zT* of conventional materials.

In contrast to this classical perspective, here we focus on advective transport, i.e., two-fluid (magnons and electrons) flow. In ferromagnets (FMs) and anitferromagnets (AFMs), ∇*T* can affect charge carriers *indirectly* through perturbations of the magnetic polarization. Since localized magnetization is sensitive to temperature, and free electrons are sensitive to localized magnetization, we can utilize heat flux to drive magnetization dynamics that ultimately generate electrical energy.

Thermal fluctuations of local magnetization produce spin-waves. In the quasi-particle picture, these waves are known as magnons, which are bosons that carry both heat and spin.^{1} Their statistical distribution function depends on both an electrochemical potential and a temperature *T _{m}*. The propagation of magnons follows the classical wave equation. In simple Heisenberg-like FMs near the Brillouin zone center, magnon dispersion is quadratic:

*ħ ω*=

*Dk*

^{2}

*a*

^{2}(

*D*is the stiffness and

*a*the spacing between spins);

^{1}in simple Heisenberg-like AFM’s, it is linear,

*ħ ω*=

*Dka*.

^{1}

To understand how magnons are useful for energy conversion, we first briefly outline two of their fundamental thermal properties: how they couple to phonons and how they carry heat. Then, we concentrate on two ways in which magnons can be used for generating electric fields: the spin-Seebeck effect (SSE) and magnon-electron drag (MED).

Experimentally, heat cannot transfer directly from a non-magnetic material into the magnon bath within an adjacent magnet, such as when non-magnetic thermometers and heat reservoirs are attached to the cold and hot side of a magnetic material. Instead, phonons in the magnetic material serve as intermediaries for heat flow from outside the material into the magnons. In the presence of heat flow (i.e., out of thermodynamic equilibrium), there are differences between the local non-equilibrium temperatures of magnons (*T _{m}*), phonons (

*T*), and/or electrons (

_{p}*T*) in different materials when brought in contact with one another. Since electron-phonon dynamics are extremely fast, the thermodynamic temperature

_{e}*T*of the system is

*T*=

*T*≈

_{p}*T*. Beyond that, the Δ

_{e}*T*values between each bath of excitations depend greatly on boundary conditions as well as bulk and interfacial material parameters. Δ

*T*≈ Δ

_{me}*T*is especially important, since this is considered the driving force for the SSE.

_{mp}^{2}The coupling between acoustic phonons and magnons in FMs has been considered for over a half-century.

^{3}An elegant model quantifies this coupling via the magnetostriction tensor, which, when multiplied by local atomic displacement vectors (phonons), gives the dependence of the magnetic energy on the lattice parameters.

^{4}Magnon-phonon interactions mean that a magnon flux arises whenever a phonon flux is present in FM’s or AFM’s, in particular during heat transport.

*C*and carry heat (

_{m}*k*) in addition to magnetization (∼

_{B}T**μ**) or spin (∼

*ħ*

**S**). Therefore, a thermally driven magnon particle flux through an FM is accompanied by magnon-mediated spin-flux

*j*or magnetization-flux

_{S}*j*, and magnon heat-flux

_{M}*j*. In this very schematic picture, and considering only the generation of a magnon flux in an infinitesimal volume of an infinite sample, we can relate these currents by $jS=\u0127SkBTjQm$ (defining

_{Qm}*S*= |

**S**|). To write this, we must consider the direction of

**S**invariant, as is the case if it is set by an external magnetic field or the local magnetization of the sample. Fourier’s law applies to magnons if we assume that the deviations from thermal equilibrium are small enough. Consequently, one can describe the thermally driven spin-flux as

*κ*is the magnon thermal conductivity. The fate of this spin flux is determined by boundary conditions. Since spin can be carried only by magnons or free electrons, the magnetic material must be connected to a “spin sink” for spin flux to cross the interface between materials. This includes neighboring materials in which spin-flip transitions are strong (e.g., heavy metals), where spin currents decay away from the interface. Conversely, if the neighboring material is not a good spin sink, then spin accumulates at the interface, leading to changes in spin electrochemical potential. This resembles the electromotive force that appears at the boundaries of a solid in which electrons are subject to a driving force but cannot sustain a current flow in steady state because the boundaries are electrically in open circuit, for example, the Seebeck voltage that appears on the edges of a sample in the presence of a thermal gradient. Thus a magnon thermopower arises analogously under open-circuit conditions at the ends of a magnetic sample. In a simple hydrodynamic magnon model, this is

_{m}^{5}

*C*is the magnon specific heat per unit volume, and

_{m}*n*the magnon volume concentration. Note that Eq. (2) differs from the Mott relation for electronic thermopower because the Mott relation results directly from the Fermi-Dirac statistics, which are applicable to electrons (fermions) but not magnons (bosons).

_{m}Since free carriers interact strongly with localized magnetic moments, magnon currents can significantly affect electron transport. The SSE and MED thermopower are two such processes where a thermally driven magnon flux drives charge accumulation, which manifests as either a Nernst-like voltage (SSE) or a direct magnon-drag thermopower. Either effect can be used in solid-state energy conversion, since the end result is the conversion of ∇*T* into **E**. In both cases, advective spin-based effects (e.g., SSE in InSb^{6} or magnon-drag thermopower in metals^{5}) have been observed experimentally to be larger than conventional charge-based effects.

## II. THE SPIN-SEEBECK EFFECT

The SSE was discovered in FM metals,^{7} semiconductors,^{6,8} a ferrimagnetic insulator (the garnet Y_{3}Fe_{5}O_{12} (YIG)),^{9} and in AFMs when spin polarization is induced by an external magnetic field.^{10,11} The earliest SSE experiments were conducted in the transverse geometry,^{12} now abandoned in favor of the simpler longitudinal geometry^{13} (Fig. 1(a)). Though born from the spintronics community, SSE’s potential for thermal energy conversion was demonstrated^{14} early on. Work on semiconductors^{6} utilized the Zeeman effect to spin-polarize high mobility electrons in a quantizing magnetic field, resulting in a giant SSE coefficient of 8 mV/K, larger than conventional thermoelectric effects in InSb. Since the voltage was generated in metallic Pt, the thermoelectric power factor surpassed that of charge-based thermoelectrics by many orders of magnitude. This large effect is not relevant to applications because it occurs only at cryogenic temperatures and high magnetic fields. Nonetheless, it motivates research into how the power of spin-driven phenomena can be harnessed under more practical conditions.

*T*is applied to the YIG in the

*z*-direction, driving a thermal spin current

*j*. As a result, voltage is generated across the Pt strip along

_{Sz}*y*. The directional magnetic polarization vector

**σ**is determined by the magnetization of the FM layer (

**M**), which is swept through a hysteresis loop by an applied magnetic field (

**H**) along

*x*(Figure 1(b)). From this, we define the SSE coefficient,

*T*. We point out that the sign convention used in the thermoelectric community for the Nernst effect is based on the Gerlach system,

^{15}which we have used when plotting the data in Fig. 1(b), though it actually gives a polarity opposite to that adopted for describing the spin Seebeck coefficient. The physical differences between Nernst and SSE are also significant, in that SSE happens in three distinct steps:

*Within the YIG*, ∇*T*generates a heat-flux**j**_{Q}=**j**_{Qp}+**j**_{Qm}consisting of heat carried by phonons (**j**_{Qp}) and magnons (**j**_{Qm}) (*magnon thermal conductivity*). Since YIG is an insulating material, the heat-flux due to electrons (**j**_{Qe}) is zero.^{16}Eq. (3) directly relates**j**_{Qm}to the spin-flux**j**_{S}carried by the magnons.*At the YIG/Pt interface*,**j**_{S}crosses from the FM to the NM via*spin-pumping*. This process polarizes conduction electrons in the metal, and its efficiency is characterized by an effective conductance*g*_{↑↓}(*spin-mixing conductance*).*Within the Pt*, strong spin-orbit interactions cause the spin-polarized electrons to generate a transverse electric field**E**via the_{ISHE}*inverse spin-Hall effect*(*ISHE*).

### A. Inside YIG: Magnon thermal conductivity

As outlined above, application of ∇*T* to a magnetic material results in a phonon flux, which in turn generates a magnon flux via magnon-phonon interactions, and these magnons carry both heat and spin. The nature of the phonon-magnon interactions varies in different materials and depends on factors like temperature, wave-vector, and applied field.^{17–20} In some instances, thermal conduction appears to be suppressed^{21} by magnon-phonon interactions, whereas in other materials, it is enhanced.^{22} In materials like YMnO_{3},^{23} the interactions can be so intense that the mixed magnon-phonon modes arise.

At low temperatures and in materials where these interactions are weaker (e.g., YIG), it is reasonable^{24} but inexact^{25,26} to consider magnons and phonons as separate reservoirs at different temperatures,^{27} corresponding with separate thermal conduction channels. The total thermal conductivity is then *κ _{m}* +

*κ*, where

_{p}*κ*is the phonon contribution. Applying the kinetic theory for thermal conduction in diffusive systems,

_{p}^{28}we can equate $\kappa m=1/3Cmvmlm$, where

*v*is the magnon group velocity and

_{m}*l*is the mode averaged magnon thermal mean free path (TMFP). The TMFP is a well-known parameter in phonon physics useful for characterizing the length scale traveled by (quasi-)particles between inelastic scattering events.

_{m}Since magnons are perturbations of magnetic order, their dispersion relations are sensitive to an external magnetic field *H*; applying *H* increases the threshold energy necessary for generating magnons by the Zeeman energy *gμ _{B}H*. For large ratios of

*gμ*/

_{B}H*k*, magnons can be frozen out, such that only phonons contribute to the thermal properties. By comparing the behavior at high and low fields, information can be extracted about

_{B}T*κ*,

_{m}*C*, and

_{m}*l*.

_{m}Multiple iterations of this approach have been applied to YIG.^{16,29,30} In Ref. 16, we measured *κ*(*H*, *T*) and the isobaric specific heat *C _{p}*(

*H*,

*T*) up to

*H*= 70 kOe and 2 K < T < 300 K, to isolate

*C*and

_{m}*κ*. This information can be combined with values for

_{m}*v*obtained from neutron diffraction data

_{m}^{31}to provide an estimate of

*l*up to

_{m}*T*∼ 20 K and the phonon mean free path

*l*up to 300 K. Both parameters are ∼100-200 μm at 2 K, but drop precipitously with temperature to ∼10 μm at 20 K, and

_{p}*l*≈ 2 nm at 200 K. This approach for measuring

_{p}*l*is not feasible above ∼20 K due to the limited range of experimentally accessible

_{m}*H*and the complexity of the magnon dispersion in YIG. However, since

*l*and

_{m}*l*track each other at low temperatures, and interactions between these populations only increase with temperature, it is reasonable to suggest that this tracking persists to room temperature and that both

_{p}*l*and

_{m}*l*are likely of the order of a few nm at 300 K.

_{p}The magnon spin diffusion length *L _{S}* is a completely distinct length scale relevant to spin transport. Whereas

*l*is related to interactions of magnons where energy is exchanged,

_{m}*L*is related to the magnon spin lifetime. Generally,

_{S}*L*≫

_{S}*l*; in YIG,

_{m}*L*at 300 K is reportedly 10 μm,

_{S}^{32,33}roughly four orders of magnitude larger than our estimate for

*l*. At 23 K,

_{m}*L*can reach 45–73

_{S}*μ*m,

^{33}comparable to macroscopic sample dimensions and only one order of magnitude larger than

*l*. Like the magnon dispersion,

_{m}*L*is also

_{S}*H*-dependent.

^{34}

A third relevant length scale appears in SSE experiments by Kehlberger *et al.*,^{35} who measured SSE in YIG films of various thicknesses *t*. The SSE signal increased with *t* before saturating at *t* ≈ 200-250 nm at room temperature. This length scale falls in between *l _{m}* (a few nm) and

*L*(tens of

_{S}*μ*m). Its origin presently is unclear, partially because it appears to depend on film quality and growth methods. This may be reflective of a number of recent studies that highlight the significant impact of surface conditions, such as the formation of native or “curing” oxide phases, which can significantly alter the observed spin-dependent signals.

^{36–40}On the theory side of the problem, the influence of a length scale associated with the spin electrochemical potential is documented.

^{41}At the FM/metal interface, the SSE involves the balance between thermal spin pumping from the FM and the spin polarization of the free charge carriers on the metal side.

^{2}The metal does not behave like a perfect spin-sink. The boundary conditions at the interface result in a degree of spin accumulation that is intermediate between that of a spin open-circuit and a perfect spin-sink; this in turn results in a spatial dependence of the spin electrochemical potential with a length scale that must play a role in the SSE.

The SSE dependence on temperature (Fig. 1(c)) and applied magnetic field^{42} in YIG reflects mostly that of the population of magnons with energies lower than ∼40 K, which are partially suppressed in thin films and in large applied fields. This is reasonable since the rest of YIG’s complex magnon dispersion contains branches that contribute little to spin transport, either because they interact too strongly with phonons^{20} or because they have too little propagation velocity.^{31}

### B. YIG/Pt interface: Spin-pumping and spin-mixing conductance

In insulating FMs, *j _{S}* is transported only by perturbations of localized magnetic moments. In NMs, the spin flux

*j*

_{S,NM}is carried only by conduction electrons when spin-polarized by external forces or torques. Transferring

*j*from magnetic insulators to non-magnetic metals is known as spin-pumping,

_{S}^{43}wherein the magnons’ magnetic moment polarizes the NM conduction electrons near the interface, resulting in a spin-flux

*j*

_{S,NM}in the NM. This is akin to

*s*-

*d*scattering in transition metals, except here the relevant

*s*and

*d*electrons are in different materials. Due to spin-flip interactions, the spin-polarization in the NM decays exponentially with distance from the interface with a characteristic spin diffusion length

*L*

_{S,NM}. Measured values of

*L*

_{S,NM}for various materials are tabulated in a recent review.

^{44}They can be several

*μ*m in semiconductors like GaAs, and vary three orders of magnitude in metals, from

*L*

_{S,NM}∼ 1

*μ*m in those with relatively low atomic numbers (Al, Cu, Ag), to

*L*

_{S,NM}∼ 1-10 nm in heavier elements (Pt, Ta, W). Multiple experiments point to a spectral dependence of spin transfer across YIG/Pt interfaces,

^{42,45,46}showing that low energy magnons (<40 K) account for an unexpectedly large fraction of

*j*

_{S,NM}. Theoretical investigations of this behavior are ongoing.

^{20,47}

The efficiency of spin-pumping is characterized by the spin-mixing conductance, *g*_{↑↓}, which quantifies the ratio between spin current excitation energy in the FM and the resulting *j*_{S,NM} in the NM. One study^{48} confirmed this proportional relationship using techniques with excitation energies spanning five orders of magnitude and found *g*_{↑↓} = 10^{19} m^{−2} for YIG/Pt. Their results also confirm that SSE produces larger *j*_{s,NM} than microwave or electrical injection, owing to the larger energy scale associated with thermal excitations.

### C. Inside Pt: Spin-Hall and inverse spin-Hall effects

In Pt, the conduction electrons’ spin-polarization decays rapidly over *L*_{S,NM} (∼1-10 nm, with the more recent measurements tending to the short end of this range).^{44} Spin-polarized electrons in Pt are subject to the ISHE,^{44,49} where spin-orbit interactions (SOIs) transfer the spin accumulation into a transverse electric field **E _{ISHE}**. The thickness of the Pt layer (Fig. 1(a)) must therefore be matched to

*L*

_{S,NM}, since any excess Pt in which there are no spin-polarized electrons short-circuits

**E**.

_{ISHE}The spin-Hall effect (SHE) and its Onsager reciprocal, the ISHE, are illustrated in Figs. 2(a) and 2(b). In electrically conducting FMs, the SHE is closely related to the Anomalous Hall Effect (AHE).^{12} AHE has been long studied, but its sensitivity to material impurities made it historically difficult for theories to match experiments. In general, we now understand that AHE occurs in conducting FMs because the conduction electrons are spin-polarized in spin-up and spin-down bands, which can be treated as two conducting channels with electron densities *n*_{↑} and *n*_{↓} and conductivities *σ*_{↑} and *σ*_{↓}, and subject themselves to SHE. SHE by itself does not require materials to be ferromagnetic and can arise in materials with no net spin-polarization (*n*_{↑} = *n*_{↓}). Indeed, SHE requires only that the current transported via these two spin channels must be affected differently by external forces, such as SOI’s or a magnetic field.

For detailed discussions on the microscopic nature of these mechanisms, we refer the reader to specialized reviews.^{44,49} In brief, these effects emerge as consequences of relativistic spin-orbit coupling that occurs in solids containing heavy elements, e.g., those in the last two rows of the periodic table. SOI effects arise when the spin of conduction electrons couples with the effective magnetic fields generated by localized orbitals of bound electrons. The larger these orbitals, the stronger the SOI, which scales roughly with the element’s atomic number to the fourth power.

**j**injected into a sample. Figure 2(a) shows a case where these electrons are deflected solely according to the sign of their spins.

_{C}**j**consists of both

_{C}*n*

_{↑}and

*n*

_{↓}electrons; if

*n*

_{↑}are deflected one way and

*n*

_{↓}the other, they produce a transverse spin-flux

**j**(the SHE). If

_{S}*n*

_{↑}≠

*n*

_{↓}, then there is also a net transverse charge current (the AHE). The ISHE starts with a spin flux

**j**injected into a sample, e.g., through spin-pumping, as in Figure 2(b). In the ideal case of a pure spin-flux,

_{S}**j**consists of

_{S}*n*

_{↑}and

*n*

_{↓}electrons in equal concentration moving in opposite directions, and there is no collinear charge flux. If only spin-dependent scattering occurs, then a net charge current appears normal to the spin-flux (sometimes called the spin-galvanic effect). In electrically open-circuit conditions, this results in the appearance of the ISHE,

**σ**is the direction of spin-polarization (up in Fig. 2) of the spin (

**S**), and

*D*is a proportionality constant. Table I of Ref. 44 gives a recent review of the materials parameters. Several experimental papers provide data for the spin Hall angles and spin mixing conductance of various metals in YIG-based structures.

_{ISHE}^{50,51}

### D. The spin-Peltier effect

The thermodynamic reciprocal of the SSE is the spin-Peltier effect (SPE), which has been measured^{52} in the configuration shown in Fig. 3(a). Here, a layer of Pt is deposited on YIG, an external magnetic field saturates the YIG magnetization, and thermometers measure ΔT across the system. An electric current is sent through the Pt, wherein SHE produces a spin accumulation at the YIG/Pt interface. This spin accumulation couples to the magnons in the YIG through spin transfer torque, which is the Onsager reciprocal of the spin-pumping process described earlier. Since magnons in YIG carry entropy as well as spin, the spin transfer torque results in heating or cooling the YIG, depending on the direction of spin flow. The temperature sensors notice only atomic vibrations and not spin perturbations, so the YIG magnons must exchange entropy with the YIG phonons (like in magnon thermal conductivity) for the heat flow to be detected. Thus, each step of the SPE involves the reciprocal of each step in SSE (see Fig. 4).^{53} The complete Onsager reciprocity between SSE and SPE is confirmed in Ref. 52.

### E. Thermoelectric efficiency of SSE/SPE systems

SSE converts heat into electricity, so, in principle, it can be used for thermoelectric energy harvesting. This notion offers mechanical flexibility in thin film devices, as well as remnant voltages that track the device’s hysteretic magnetization, thus alleviating the need for applied external magnetic fields. In reality, single thin-film YIG/Pt-based structures have poor conversion efficiency,^{54,55} primarily due to low electrical power output and inefficient use of thermal energy. The first problem results from typically small ISHE voltages extracted through films approximately as thick as their spin diffusion length (*L*_{S,NM} < 10 nm), leading to high source impedance and low power density. The second problem stems from the spin diffusion of magnons in YIG. Although ∇*T* extends throughout the entire YIG/Pt structure, only thermal energy near the interface drives spin current into the Pt. Kehlberger’s experiment^{35} sets this length scale at 250 nm (see Section II A). While we anticipate further improvements in our understanding of this length scale will come in time, the current picture of SSE implies that less than 1% of the YIG volume actually contributes to the ISHE signal, and the remaining thermal energy goes unused. To make SSE viable for large scale energy harvesting, we consider alternate modalities that address these limitations.

One approach to improve power density is to make the FM substrate out of conducting FMs like CoFeB^{56} or Fe_{3}O_{4},^{57} which couples ISHE in the NM with the anomalous Nernst effect (ANE) in the FM. This approach works well in multilayer structures^{58} to enhance the voltage output and decrease the sheet resistance, although state-of-the-art devices still produce relatively small power factors (∼0.1 pW/K^{2}).^{54} The electrical voltage output can be increased by creating spin-Hall thermopiles,^{59} but this approach only trades-off an increased internal resistance for an increased voltage and does not increase the power output. These two approaches are discussed in another article in this volume.^{60}

A third approach we have developed is to use SSE to enhance the transverse thermopower of bulk composites by embedding NM nanoparticles in conducting FMs.^{61} An ISHE field arises within the nanoparticles due to thermal injection of spin currents from the FM matrix. Since the FM is conducting, ISHE adds to the ANE of the matrix phase, enhancing the voltage and (in principle) offering the same potential advantage of generating remnant transverse voltages in the absence of an applied field. Electrical current can be extracted through the bulk, resulting in lower resistivity. Together, in the prototype systems studied so far, the power factor at ∼300 K and 90 kOe is increased by an order of magnitude over thin film structures. By sizing the FM particles to *L _{S}* (∼

*μ*m) and controlling the microstructure, all of ∇T can be utilized. This addresses both electrical and thermal parts of the efficiency problem and has the additional advantage of creating bulk samples capable of handling much higher power levels than thin-film devices.

## III. MAGNON-ELECTRON DRAG

MED can occur in metallic FMs when magnons directly interact with electrons. This additional contribution to the Seebeck coefficient often completely overpowers the diffusive contribution, even in metals. Utilizing MED circumvents all the issues described above related to heterogeneous SSE materials, since MED occurs in a single material, and therefore makes bulk FM’s and possibly AFM’s viable for conventional thermoelectrics. Unlike semiconductors, which are the traditional material of choice for thermoelectric devices, metallic FM’s offer high mechanical strength, low cost, and ease of processing.

Total thermal conductivity *κ* is often the primary factor limiting the thermoelectric performance of semiconductors, wherein most heat is carried by phonons (*κ _{p}*). In contrast,

*κ*is dominated in metals by electrons (

*κ*). The consequences of this are apparent if we combine the Wiedemann-Franz law with the thermoelectric figure of merit

_{e}*zT*equation. In that case, $zT=\alpha 2L/1+\kappa p\kappa e$, where

*L*is the Lorenz ratio, which is between 0.6 and 1.3 times the free electron value (

*L*

_{0}= 2.5 × 10

^{−8}V

^{2}K

^{2}) in most solids. This expression shows that a high ratio of

*κ*/

_{e}*κ*actually increases

_{p}*zT*, and in the limiting case of

*κ*≫

_{e}*κ*,

_{p}*zT*depends exclusively on

*α*. Except in largely impractical rare earth intermetallic alloys,

^{62,63}no good thermoelectric metals exist today, mostly because diffusion thermopowers are limited to a few

*μ*V/K.

^{64}

*E*being the Fermi energy. This paradigm practically eliminates the possibility of achieving high

_{F}*α*(and therefore

*zT*) in solids where the electron density is comparable to the atomic density, and where

*E*≫

_{F}*k*at reasonable temperatures (i.e., metals).

_{B}T*T*drives magnons, which then drive electrons, and (like phonon-electron drag) the MED thermopower adds to the diffusion thermopower,

*PF*=

*α*

^{2}/

*ρ*), which measure the power density of a thermoelectric generator or cooling capacity of a Peltier cooler. To emphasize this, Figure 4 compares the PFs of elemental iron and cobalt, in which magnon-electron drag is dominant, to those of commercially available thermoelectric materials.

^{65}The metals clearly exceed the semiconductors in peak PF.

This behavior in pure elemental metals is certainly not optimized, so we have recently begun exploring its physical origins in order to engineer materials with enhanced MED. Lucassen *et al.*^{66} first pointed to the relation between the SSE and MED. We have since developed^{5} two quantitative theories for MED in FM metals, as well as a spin-mixing theory for their Nernst effect.

*hydrodynamic theory*for MED,

^{5}which is based on an ideal gas assumption where the FM metal is modeled as two fluids, one being electrons with density

*n*and the other magnons with density

_{e}*n*. These fluids interact through momentum-conserving collisions, i.e., we ignore relativistic effects like SOI and neglect processes where magnons are not conserved. Next, we consider how ∇

_{m}*T*affects these fluids and determine the intensity of the electric field produced by the resulting change in electron density. We derive

^{5}the relation between pressure

*P*and internal energy density

*U*in the presence of ∇

*T*($P=23U$), resulting in an expression for the MED thermopower,

*τ*and

_{me}*τ*represent, respectively, the time scales for magnon-electron and for all other magnon collisions.

_{m}^{5}In this theory, a flux of thermal magnons produces magnetization dynamics that pump an electronic spin current, like the spin-pumping across interfaces described in Section II B, but now in a single homogeneous medium. Using an electric current density with Fourier’s law applied to magnons, and assuming an electrically open circuit in the sample as a boundary condition, the following expression is obtained

^{5}for a spin-pumping thermopower:

*β*, typically of the order of 0.01-0.1;

*s*∼

*a*

^{−3}(

*s*in units of

*ħ*) is the saturation spin density; and

*p*is the spin-polarization of the electric current (typically of order 1).

_{s}^{5}

Despite different physical origins, the two models give identical results at low temperatures in pure metals.^{5} We show this by first assuming that *n _{e}* is the density of electrons in bands of

*s*- or

*p*-orbital character, and that electrons in the

*d*-bands do not conduct, then we insert calculated values for the band structure parameters of elemental Co and Fe into Eqs. (5) and (6). Then, using either (7) or (8), the experimentally measured thermopowers of these elements, which have opposite polarity, can be reproduced with no adjustable parameters (dashed line in the thermopower in Fig. 4).

These models are also useful for determining the anomalous Nernst coefficient of FM metals. MED alone does not give rise to a skew force, so additional mechanisms must be added to produce transverse effects like Nernst fields. To fill this requirement, we consider first SOI, which can generate a skew force via the ISHE, and may actually have been previously observed in SSE-like measurements on amorphous metals.^{67} Alternatively, spin-mixing conduction generates a skew force when the spin-up and spin-down bands in FM metals have different thermopowers, similar to how mixed-carrier effects produce^{5} a net transverse thermopower in semimetals.

## IV. CONCLUSIONS

The main limitation in optimizing *zT* in classical thermoelectrics arises from counter-indicated transport properties in the same solid. Here, we showed that this paradigm can be disrupted by adding spin as an independent parameter; magnon-driven advective transport means that the spin-Seebeck and magnon-drag effects utilize different physics to generate thermoelectric effects. By using heat to excite magnetization dynamics, spin-dependent interactions can then be exploited in solids with high electron density to achieve considerably higher transverse and/or longitudinal thermopower than is possible from direct thermal diffusion of charge carriers. We propose this general approach as a promising new way forward in thermoelectrics.

## ACKNOWLEDGMENTS

This work is supported by the U. S. National Science Foundation., S.R.B. and J.P.H. by its MRSEC program under Grant No. DMR-1420451, and S.J.W. by its Graduate Research Fellowship program under Grant No. DGE-0822215.

## REFERENCES

*Introduction to Solid State Physics*

*Magnetoacoustic Polarization Phenomena in Solids*

_{2}Te

_{3}related materials

*CRC Handbook on Thermoelectricity*