Abstract
Spintronics is about the coupled electron spin and charge transport in condensedmatter structures and devices. The recently invigorated field of spin caloritronics focuses on the interaction of spins with heat currents, motivated by newly discovered physical effects and strategies to improve existing thermoelectric devices. Here we give an overview of our understanding and the experimental stateoftheart concerning the coupling of spin, charge and heat currents in magnetic thin films and nanostructures. Known phenomena are classified either as independent electron (such as spindependent Seebeck) effects in metals that can be understood by a model of two parallel spintransport channels with different thermoelectric properties, or as collective (such as spin Seebeck) effects, caused by spin waves, that also exist in insulating ferromagnets. The search to find applications — for example heat sensors and waste heat recyclers — is on.
Main
Thermoelectric and thermomagnetic effects have been known for two centuries^{1,2}. They reflect the coupling of heat and charge currents and find applications in thermometers, power generators and coolers. Heat currents also interact with spin currents^{3,4,5}. This has spawned the field of spin caloritronics (from 'calor', the Latin word for heat)^{6}, which is concerned with nonequilibrium phenomena related to spin, charge, entropy and energy transport in (mostly) magnetic structures and devices. Here we will look at theoretical and experimental evidence for spindependent Seebeck/Peltier coefficients and thermal conductance, thermal spintransfer torques, spin and anomalous thermoelectric Hall effects, the recently discovered spin Seebeck effect and so on. We will not discuss equilibrium phenomena such as the temperature dependence of magnetic properties including phase transitions, nor related topics such as magnetocaloric cooling and thermally assisted magnetic recording.
A challenge for condensedmatter and device physics is to develop 'green' information and communication technologies, and efficient devices for scavenging waste heat. Another issue is the imminent breakdown of Moore's law by the thermodynamic bottleneck: further decreases in feature size and transistor speed go in parallel with intolerable levels of ohmic energy dissipation associated with the motion of electrons in conducting circuits. Thermoelectric effects in meso^{7} and nanoscopic^{8} structures might help. The additional degree of freedom provided by the electron spin and magnetic order provides new strategies to increase the thermoelectric figure of merit as well as offering radically new functionalities in all temperature regimes. Spin caloritronic effects are not limited to solidstate structures, but have been predicted to occur in cold atomic gases as well^{9}.
Spin caloritronic phenomena can be roughly classified into (i) independent electron, (ii) collective and (iii) relativistic effects. The first class (i) is the thermoelectric generalization of collinear magnetoelectronics and effects such as giant magnetoresistance and tunnel magnetoresistance. The second class of effects (ii) is generated by the collective dynamics of the magnetic order parameter that couple to single particle spins via the spintransfer torque and spin pumping. Finally, there are (iii) thermoelectric generalizations of relativistic corrections such as anisotropic magnetoresistance, anomalous Hall effects and spin Hall effects. In what follows, we describe and interpret selected recent experiments to provide a snapshot of a field in motion, rather than a comprehensive review. A brief introduction to the conventional thermoelectrics is given in Box 1. We also make an attempt to harmonize the historically grown and somewhat confusing nomenclature for spin caloritronic effects in Box 2.
Independent electron/spin effects in metallic magnets
Transport in magnetic layered structures with collinear magnetizations is well described by the twocurrent model, in which the majority and minority spin carriers form parallel channels with different resistances for ferromagnets, as well as interfaces between ferromagnets and paramagnetic metals, and tunnelling barriers between metals if at least one is magnetic. In this section we discuss the basic theoretical concepts of the twochannel resistor model and review recent observations of associated spin caloritronic effects.
Spindependent thermoelectrics in metallic magnetic heterostructures. Thermoelectricity in metals is governed by the energy dependence of the electron distribution and conductivity in an energy interval of the order k_{B}T around the Fermi energy, as sketched in Box 1. Here we generalize the concepts to the twospin current model. The thermoelectric properties in isotropic and monodomain metallic ferromagnets^{3,5,10,11,12} are affected by the spindependent conductivities σ^{(s)}(ε), s = ↑,↓ being the spinindex, leading to the conventional charge conductance σ = σ^{(↑)} + σ^{(↓)} and Seebeck coefficient S = (σ^{(↑)}S^{(↑}) + σ^{(↓)}S^{(↓)})/(σ^{(↑)} + σ^{(↓)}). In linear response and using the Sommerfeld expansion S^{(s)} = −eL_{0}T∂ ln σ^{(s)}(ε)/∂εεF (Mott's law^{1}), where L_{0} is the Lorenz constant, and
where J_{c(s)} = J^{(↑)} ± J^{(↓)} and Q = Q^{(↑)} + Q^{(↓)} are the charge (c), spin (s) and heat currents, respectively, and P and P' stand for the spinpolarization of the conductivity and its energy derivative
The expression μ_{c} = (μ^{(↑)} + μ^{(↓)})/2 gives the charge electrochemical potential and μ_{s} = μ^{(↑)} – μ^{(↓)} the spin accumulation. The spindependent thermal conductivities obey the Wiedemann–Franz law κ^{(s)} ≈ L_{0}Tσ^{(s)} when S^{(↑)/(↓)} << L_{0}^{1} and the total thermal conductivity is κ = κ^{(↑)} + κ^{(↓)} → L_{0}Tσ. The symmetry of the spindependent thermoelectric matrix in equation (1) reflects Onsager's reciprocity relation^{13,14}. We observe a spin current driven by a temperature gradient (the spindependent Seebeck effect) as well as a heat current driven by spin accumulation (directly related to the spindependent Peltier effect), both proportional to P'ST.
In equation (1), the spin heat current Q_{s} = Q^{(↑)} – Q^{(↓)} does not appear. This is a consequence of the implicit assumption that a spin temperature (gradient) T_{s} = T^{(↑)} – T^{(↓)} is effectively quenched by interspin and electron–phonon scattering^{10}. This approximation does not necessarily hold at the nanoscale and at low temperatures^{15,16}. The presence of spin temperatures can be observed, for example, as a violation of the Wiedemann–Franz Law or a magnetoheat conductance in spin valves^{10}.
According to equation (1), not only an applied voltage but also a temperature gradient drives a spin current in a conducting ferromagnet. Conservation of charge and spin currents at a contact between the ferromagnet (FM) and a normal metal (NM) then implies spin current injection into NM under a voltage as well as a temperature bias^{3,5,12}. A thermally or electrically created spin accumulation can be detected by a switchable second ferromagnet, either electrically by the induced voltage or through temperature changes due to the Peltier effect. In magnetic clusters embedded in a normal metal matrix, the thermally injected spin accumulation contributes to the magnetothermopower^{17,18}.
Spindependent Seebeck and Peltier effects. The thermoelectric generalization of the (giant) magnetoresistance in FMNMFM metallic spin valves — that is, the difference between the properties for parallel and antiparallel magnetic configurations — is referred to as the (giant) magnetoPeltier and magnetoSeebeck effect/magnetothermopower^{10,12,19}. The magnetoSeebeck effect has been observed in multilayered magnetic nanowires^{5}. A large Peltier effect in constantan (CuNi alloy)/Au nanopillars^{19} has been associated with phase separation^{21}.
A spindependent Seebeck effect has been demonstrated in lateral spinvalve structures^{22}. Here a temperature gradient is generated over an interface by resistive heating of a ferromagnet (FM_{1} in Fig. 1a). Figure 1b shows the spindependent chemical potentials μ(↑), μ(↓) across the FM_{1}NM interface when a heat current Q crosses it but charge current is zero. Because the heat current is conserved, ∇T is discontinuous at the interface. The slopes of the electrochemical potentials in FM and NM reflect the charge Seebeck coefficients of the bulk materials. Because the individual Seebeck coefficients for the two spin channels S(↑) and S(↓) are not equal, a spin current proportional to S(↑) – S(↓) flows through FM even in the absence of a charge current, creating a spin accumulation μ(↑) – μ(↓) close to the interface, which relaxes in the FM and NM on the length scale of their respective spinflip diffusion lengths λ_{F} and λ_{N}. A thermoelectric interface potential Δμ = Pμ_{s} also builds up. On the left side no spin current is allowed to leave, and the spindependent Seebeck effect results in a spin accumulation of opposite sign. Owing to the spindependence of the Seebeck coefficient, the heat current Q thereby induces a spin current into the normal metal. Slachter et al.^{22} detected the spins thus accumulated by means of the voltage difference V with respect to an analysing ferromagnetic contact FM_{2}. Because FM_{1} and FM_{2} have different coercive fields, the spin accumulation is revealed by the voltage traces V(H) as a function of magnetic field H.
A spindependent Peltier effect has been discovered recently in a dedicated lateral/perpendicular nanostructure (Fig. 2). The heat current induced by injecting a spin accumulation into the ferromagnet was detected in terms of the associated temperature changes by a thermocouple^{23,24}.
Tunnel junctions. The electrical resistance of FMIFM magnetic tunnel junctions, where I denotes an electric insulator, depends on the magnetic configuration, leading to huge tunnel magnetoresistance ratios. Large tunnelling magnetoSeebeck, magnetoPeltier and magnetoheat resistance effects may be expected^{25,26}. A magnetoSeebeck effect in MgObased magnetic tunnel junctions has been observed under heat gradients created electrically^{27} and optically^{28}. Results can be compared with ab initio calculations^{29,30}. Large Seebeck and magnetoSeebeck effects were found for magnetic tunnel junctions with amorphous Al_{2}O_{3} barriers that were interpreted in terms of the Jullière model^{31}. A spindependent thermopower has been predicted from first principles for molecular spin valves^{32}.
As discussed by Jansen et al.^{33} in this issue, the spins injected into semiconductors can be detected in a threeterminal structure with only one magnetic contact, which acts as an injector as well as a detector of spins. The spinrelated signal can be disentangled by the Hanle effect — that is, the dephasing of the spin accumulation by a transverse magnetic field. Le Breton et al.^{34} showed that (Joule) heating of a ferromagnet contact to Si with alumina barriers leads to Hanle curves that prove the presence of a thermally injected spin accumulation. This effect was measured at room temperature and referred to as Seebeck spin tunnelling.
A lowtemperature tunnelling anisotropic magnetothermopower has been observed in (Ga,Mn)AsiGaAsnGaAs structures^{35}. Here the spin dependence is evident by the uniaxial symmetry of the thermovoltage as a function of the magnetization direction, reflecting the strong spin–orbit interaction in the GaAs valence band.
Collective effects
Ferromagnetism is a quantum coherent ground state with broken rotational and timereversal symmetry. Its extraordinary robustness is reflected in critical temperatures up to 1,400 K. Magnetization is also resilient against spatial deformations. In nanoscale structures, the magnetization dynamics are often well described by a single timedependent vector, the 'macrospin' model. In extended systems the elementary excitations are spin waves or magnons^{36,37}. The magnetization interacts with the electron charge current through spintransfer torques and spin pumping, as discussed by Brataas et al.^{38} in this issue. Here we focus on phenomena such as heatcurrentinduced magnetization dynamics, as well as the related 'spin Seebeck effect'.
The heat current that travels by means of spin waves (magnons) carries spin angular momentum opposite to that of the magnetization^{39,40}, as sketched in Fig. 3b. In metallic ferromagnets, the magnon spin current runs in parallel with the conventional particle current. The two modes interact weakly, causing, for example, a currentinduced Doppler shift of spin waves^{41}, contributions to the spindependent Seebeck coefficients^{42}, and the dissipative part of the currentinduced spintransfer torque^{43}. The magnondrag effect on the thermopower of permalloy has been observed recently in nanostructured lateral thermopiles^{44}. Spin waves can be actuated and detected electrically even in magnetic insulators^{45}.
At elevated temperatures, phonons, electron–hole excitations and magnons coexist and carry heat currents in parallel. Most nonequilibrium states are well explained in terms of a weakly interacting threereservoir model, in which phonons, electrons and magnons are at separate thermal equilibria with possibly different temperatures^{46}. The coupling of different modes can be important for thermoelectric phenomena such as the phonondrag effect on the thermopower at low temperatures.
Thermal spintransfer torques. In the presence of a noncollinear magnetic texture, either in a heterostructure such as a spinvalve and tunnel junction, or a magnetization texture such as a domain wall or magnetic vortex, the magnetic order parameter absorbs a spin current or rotates its polarization. According to the conservation law of angular momentum, this is equivalent to a torque on the magnetization that, if strong enough, leads to coherent magnetization precession and ultimately magnetization reversal^{47}. A heat current can exert a torque on the magnetization as well^{10}, leading to thermally induced magnetization dynamics^{48}. Such a torque acts under closedcircuit conditions, in which part of the torque is exerted by charge currents induced by the conventional thermopower, as well as in an open circuit without charge currents^{10}.
The angular dependence of the thermal torque can be computed by circuit theory^{10,12} (Box 1). Indications for a thermal spintransfer torque have been found in experiments on nanowire spin valves^{49}. Slonczewski^{50} studied the spintransfer torque in spin valves in which the polarizer is a magnetic insulator that exerts a torque on a free magnetic layer in the presence of a temperature gradient. The thermal torque is found to be far more efficient at switching magnetizations than a chargecurrentinduced torque, but the parasitic heat conductance channels may spoil the advantage. Note that the physics of heatcurrentinduced spin injection by magnetic insulators is identical to that of the longitudinal spin Seebeck effect, as discussed below.
Thermal torques have been predicted by firstprinciples calculations for magnetic tunnel junctions with thin barriers^{30}. At ambient conditions the critical temperature difference over the barrier for switching from antiparallel to parallel configurations is estimated to be 6 K, but it must be an order of magnitude larger to switch back. The large torques for the antiparallel configuration can be explained by interface states in the thermal window close to the Fermi energy on one side of the barrier, which allow for multiple scattering processes that lead to very efficient spin transfer close to antiparallel configurations.
Domainwall motion induced by charge currents^{47} can be understood in terms of angular momentum conservation in the adiabatic regime, in which the length scale of magnetization texture such as the domainwall width is much larger than the scattering mean free path or Fermi wavelength, as appropriate for most transition metal ferromagnets. In spite of initial controversies, the importance of dissipation in the adiabatic regime^{51} is now generally appreciated. In analogy to the Gilbert damping factor α, the dissipation under an applied current is governed by a material parameter β_{c} that for itinerant magnetic materials is of the same order as α (ref. 51; for a review see ref. 52). In the presence of electron–hole asymmetry at the Fermi energy, the adiabatic thermal spin transfer torque^{10} is associated with a dissipative β_{T} correction^{53,54}, which has been explicitly calculated for GaMnAs (ref. 55). Nonadiabatic corrections to the thermal spintransfer torque in fastpitch ballistic domain walls have been calculated by firstprinciples^{56}. Laserinduced domainwall pinning might give clues for heat current effects on domainwall motion^{57}.
Spin waves can move domain walls, leading to domainwall motion in the opposite direction to the spinwave propagation^{58,59}. Recently, this topic has been addressed in the modern context of heatcurrentinduced domainwall motion in magnetic insulators that induces motion to the hotter edge of the wire^{60,61,62,63}.
Spin Seebeck effect. The spin Seebeck effect is the transverse electromotive force in a paramagnetic contact to a ferromagnet by a temperature bias, as illustrated in Fig. 3d and e for the two principal sample geometries. This effect is interpreted in terms of a spin current injected into the normal metal by the ferromagnet^{64} that is transformed into an electric voltage by the inverse spin Hall effect (ISHE)^{65,66,67} (Fig. 3c). The ISHE is caused by the bending of electron orbits of up and down spins into opposite directions normal to their group velocity, owing to the spin–orbit interaction. It generates a relatively large voltage for heavy metals such as Pt while being virtually absent for Cu, and it has the advantage of scaling linearly with the wire length (for details see Jungwirth et al. in this issue^{68}).
The spin Seebeck effect was discovered first in permalloy^{64}, and later in electrically insulating yttrium iron garnet (YIG)^{69}, ferromagnetic semiconductors (GaMnAs)^{70} and Heusler alloys^{71}, with very similar phenomenology. Its physics is completely different from the spindependent Seebeck effect discussed above, because the conduction electron contribution is negligible^{72} (see, however, ref. 73). This became obvious only after the observation of the spin Seebeck effect generated by an insulating ferromagnet^{69} (Fig. 3f,g). The spin current is the result of a thermal nonequilibrium at the interface between the ferromagnet and the normal conductor, as explained in the following in terms of an imbalance of the thermally excited spin currents over the interface by spin pumping^{74} and spin torques^{47}.
Consider first a ferromagnet at thermal equilibrium with an attached normal metal contact (Fig. 4a). When the ferromagnet is thermally excited, by its time dependence the magnetization m(t) 'pumps' a net spin current into the normal metal^{74}
where g_{r} is the real part of the (dimensionless) spinmixing conductance of the FMNM interface. On the other hand, at finite temperatures the normal metal generates thermal (Johnson–Nyquist) noise in the form of current fluctuations that are partially spinpolarized^{75}. These lead to random spintransfer torques that, vice versa, generate magnetization dynamics. At thermal equilibrium the sum of the timeaveraged currents vanishes, by the second law of thermodynamics.
Let us now proceed to a simple model of a ferromagnet FM sandwiched between two reservoirs NM with a temperature difference applied (Fig. 4b). When FM is sufficiently smaller than the magnetic domain wall width, all spins move in unison and the ferromagnet is characterized by a single macrospin temperature that determines the uniform fluctuations of the magnetization around the equilibrium direction.
We now have to consider the other degrees of freedom of the system — the phonons and, in conductors, the electrons. Electrons and phonons are relatively strongly coupled among themselves, but much less to the spins. We assume that electrons/phonons are thermalized, meaning that their distribution function can be represented by a temperature profile that interpolates between the hot and cold terminals, as indicated in Fig. 4, disregarding the thermal resistance of the interfaces. Because on the left side the magnet is now colder than the contact, the pumped spin current governed by the fluctuations corresponding to T_{F} is smaller than the spin current induced by the Johnson–Nyquist noise, which scales with electron temperature T_{L}. On the righthand side the situation is the opposite. When the contacts are identical and in the steady state, the total spin (and heat currents) entering from the left and leaving on the right have to be the same, so T_{F} = (T_{L}+ T_{R})/2. We may conclude that a spin and heat current can be transported by the fluctuations of the magnetization. This mechanism works for either conducting or insulating ferromagnets.
The (transverse) spin Seebeck effect can now be explained by considering the situation for a Pt contact on top of a ferromagnet subject to a temperature bias. Even in Pt the spin–orbit interaction is considered a weak perturbation, so the ISHE generates a voltage for a spin current that may be computed as if injected into a simple metal. If the underlying ferromagnet is macroscopically large, a macrospin approximation is not appropriate. Instead of a single magnetic temperature T_{F} we have to consider now a magnon temperature distribution T_{F}(x) which, as argued above, can differ from that of the electron–phonon system. The latter is assumed to be identical to T_{N}, the electron temperature in the normal metal, by effective thermalization. In the transverse configuration (Fig. 3e), the temperature difference T_{F} − T_{N}, and therefore the spin current and the associated ISHE signal, has to change sign between the hot and cold edges, as observed. Scattering theory leads to a predicted magnitude of the spin current in the transverse configuration of^{76}
where γ is the gyromagnetic ratio, M_{s} the saturation magnetization, V_{coh} a magnetic coherence volume and k_{B} the Boltzmann constant. Adachi et al. subsequently arrived at a similar expression by linear response theory^{77}. Equation (4) summarizes our qualitative understanding of the spin Seebeck effect but raises a few issues that have not all been resolved.
In the macrospin model the spin current is inversely proportional to the total magnetization volume, because the spintorque pumping is a surface effect on the total magnetization. Without corrections this would lead to a very small signal in large samples. This issue can be resolved by the Landau–Lifshitz–Gilbert equation^{78}: only those spins contribute that are close to interfaces within a magnetic coherence volume V_{coh}, which is a material parameter of the order of (10 nm)^{3}, scaling as √(T_{F}D^{3}) where D is the spinwave stiffness^{76}.
The magnitude of the mixing conductance is well established for intermetallic interfaces^{79}, but not for interfaces including magnetic insulators such as YIG. Although initial modelling with a simple Stoner model predicted a small mixing conductance that agreed with experiments, a local moment picture^{50} and band structure calculations^{80} found much larger values, comparable to those of intermetallic junctions. Recent dedicated experiments^{81} indicate that by careful interface preparation the mixing conductance can be greatly increased to agree with theory.
Computing the spatial distribution of the nonequilibrium between electrons or phonons and magnetization is a difficult problem. In the simplest approximation the electrons or phonons, as well as elementary excitations of the magnet (the spin waves or magnons), are fully thermalized on a local scale. The problem then reduces to a simple diffusion picture of weakly coupled subsystems with proper boundary conditions^{82}, which seems to work well for YIG at room temperature^{76}. This picture breaks down at low temperatures at which the phonondrag effect on the magnons kicks in^{83}, explaining the strong enhancement of the spin Seebeck signal found in that regime for GaMnAs^{84}.
In ferromagnetic metals the mean free path length of the magnons is too small to explain the length scale observed in the spin Seebeck effect in the lateral configuration^{64}. Coherent phonons in sample and substrate are therefore likely to be essential for the very observability of the effect in these materials. An important hint was the observation that the spin Seebeck effect is robust against 'scratches' in the ferromagnetic film, proving that the substrate plays an important role^{84}. Further evidence comes from the amplitude of the spinSeebeck effect in GaMnAs, which was found to scale with the thermal conductivity of the GaAs substrate and the phonondrag contribution to the thermoelectric power of the GaMnAs, demonstrating that phonons drive the spin redistribution^{84}. The presence of the coupling of magnons to coherent phonons in sample and substrate explains the spin Seebeck effect even for a single magnetic wire rather than an extended film^{85}. This phenomenon is clearly beyond the driftdiffusion model for the magnon–phonon system, and can be explained for YIG by the linear response formalism^{77,83}. Some progress has been made in understanding how the interaction between magnons and conductionelectron spins affects the spindependent Seebeck effect^{42}, and in understanding the magnon contribution to the dissipative spintransfer torque parameter β_{c} (ref. 43), but a microscopic theory for the spin Seebeck effect in ferromagnetic metals is still lacking.
Initially, experiments were carried out in the 'transverse configuration' of Fig. 3e. However, the 'longitudinal configuration' in Fig. 3d is the most basic setup for the study and application of the spin Seebeck effect in insulators^{86,87}. For the former configuration, precise temperaturedistribution control and careful choice of substrates are important^{71}. Otherwise, signals may be contaminated by artefacts such as the anomalous Nernst^{88} effect (see below). An experiment on ferromagnetic metals would in principle display both the spin Seebeck and spindependent Seebeck effect observed by Slachter et al.^{22}, but detection by the ISHE is very difficult. Weiler et al.^{89} carried out a spatially resolved study on PtF bilayers for conducting and insulating ferromagnets. Whereas for PtYIG the longitudinal spin Seebeck effect was detected, the Hall voltages of PtCo_{2}FeAl turned out to be dominated by the anomalous Nernst effect^{76,78} (see below).
The longitudinal spin Seebeck effect in cooperation with the ISHE converts heat flows into electric voltages that increase linearly with size and do not require complicated thermopile structuring. Therefore, it could be used as a largearea electric power generator driven by heat (A. Kirihara et al., manuscript in preparation). The standard thermoelectric figure of merit must be reconsidered, as the heat conductivity is no longer a relevant parameter, opening the way to new strategies to increase the efficiency of thermopower generation. Applications of the spin Seebeck effect in the longitudinal configuration to positionsensitive detectors have been proposed^{90}. In the longitudinal configuration the spin pumping can be driven by ultrasound excitation as well^{85}.
As mentioned above, the physics of the thermal torque induced by heat currents in spin valves with an insulator as polarizing magnet as proposed by Slonczewski^{50} is identical to that of the longitudinal spin Seebeck effect. The 'loose' magnetic monolayer model hypothesized by Slonczewski seems to mimic the coherence volume V_{coh} that follows from the Landau–Lifshitz–Gilbert equation.
The interaction between charge currents and magnetization dynamics can be cast into a linear response matrix that obeys Onsager reciprocity relations^{38}, which can be extended to include heat currents and temperature differences. Therefore, the spin Peltier effect — the cooling of the magnetization of a ferromagnetic insulator by a proximity spin accumulation — must exist as the Onsager equivalent of the spin Seebeck effect.
Spin caloritronic heat engines and motors. Onsager's reciprocal relations^{13} reveal that seemingly unrelated phenomena can be expressions of identical microscopic correlations between thermodynamic variables of a given system^{14}. The archetypal example is the Onsager–Kelvin identity of thermopower and Peltier cooling (Box 1). The Onsager equivalency between particlecurrentinduced spintransfer torque and spin pumping, on the other hand, is a recent insight^{38}. The reciprocity of heatcurrentinduced spin transfer torque and spin pumping by thermal fluctuations follows from an analogous treatment by scattering theory. The results in linear response for a magnetic wire incorporating a tailtotail magnetic domain wall (Fig. 5) lead to the proposal of spin caloritronic heat engines^{53,54,63,91}. Mechanical and magnetic motions are coupled by the Barnett and Einstein–de Haas effects^{92,93,94}, which are again each other's Onsager reciprocals^{54}. The thermoelectric response matrix including all these variables can be formulated for a simple model system consisting of a rotatable magnetic wire including a rigid domain wall parameterized by its width and position r_{w} (see Fig. 5). The mechanical torque induced by temperature differences may be interpreted in terms of Feynman's ratchet and pawl in the continuum limit. The pawl providing mechanical chirality in the latter is replaced in the former by the magnetic order.
Such a machine has multiple functionalities: it can be operated as an electric generator and dynamo (for metallic ferromagnets only), a thermally driven Brownian motor or a mechanically driven cooler (also for insulating magnets).
Relativistic effects
Thermal Hall effects exist in normal metals in the presence of external magnetic fields and can be classified into three groups^{95}. The Nernst effect represents the Hall voltage induced by a heat current. The Nettingshausen effect describes the heat current that is induced transverse to an applied charge current. The Hall heat current induced by a temperature gradient goes by the name of the Righi–Leduc effect. The spin degree of freedom opens a family of spin caloritronic Hall effects in the absence of an external field, and these are not yet fully explored. We may add the label 'spin' in order to describe effects in normal metals (spin Hall effect, spin Nernst effect and so on). In ferromagnets we may distinguish the configuration in which the magnetization is normal to both currents (anomalous Hall effect, anomalous Nernst effect and so forth) from the configuration with inplane magnetization (planar Hall effect, anisotropic magnetoresistance, planar Nernst effect and so on) as sketched in Fig. 6. Theoretical work has been carried out with emphasis on the intrinsic spin–orbit interaction^{96,97,98}. The thermoelectric figure of merit could possibly be improved by making use of the conducting edge and surface states of topological insulators^{99}.
Seki et al.^{100,101} found experimental evidence for a thermal Hall effect in Au/FePt structures, which could be due either to an anomalous Nernst effect in FePt or to a spin Nernst effect in Au. In GaMnAs, planar^{102} and anomalous^{103} Nernst effects have been observed, with intriguing temperature dependences. Slachter et al.^{104} identified the anomalous Nernst effect and an anisotropic magnetoheating effect in a multiterminal permalloy/Cu spin valve. The anomalous Nernst effect is rather ubiquitous and may interfere with other spin caloritronics effects^{71,84,88,89}.
The heat is on
Spin caloritronics has gained momentum in recent years with the entry of several new groups and even research consortia into the field. Much has yet to be done. Many effects predicted by theory have not yet been observed, and unexpected phenomena such as the spin Seebeck effect might still wait for their discovery. If spin caloritronics is to become more than a scientific curiosity, the thermoelectric figures of merit should be increased. The tunnel (magneto) Seebeck effect is already fairly large, and carries the promise of useful applications, as does the extreme simplicity of spin Seebeck devices. More materials research and device engineering, experimental and theoretical, however, is clearly needed.
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Acknowledgements
We are grateful for collaboration with F. Bakker, A. Brataas, X. Jia, M. Hatami, T. Heikillä, P. Kelly, S. Maekawa, B. Slachter, S. Takahashi, K. Takanashi, Y. Tserkovnyak, K. Uchida, K. Xia, J. Xiao and many others. This work was supported in part by the FOM Foundation, EUICT7 'MACALO', and DFG Priority Programme 1538 'SpinCaloric Transport'.
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Bauer, G., Saitoh, E. & van Wees, B. Spin caloritronics. Nature Mater 11, 391–399 (2012). https://doi.org/10.1038/nmat3301
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