The spin-heat coupling and enabling applications

Understanding the interaction of energy carriers (e.g., electrons, phonons, and magnons) and how they impact materials' transport properties (e.g., electrical conductivity, thermal conductivity, and the transfer of magnetic moment of materials) are of technological importance for a broad range of engineering applications. Electrons are fermions with an electron spin 1/2 that transports charge and spin (angular momentum). Magnons, defined as the quantized modes of spin waves, are bosonic quasi-particles that transport both heat and spin.¹ Similar to magnons, phonons are also bosonic quasi-particles but nominally transport only heat through perturbations of the positions of the atoms. Although phonons are not considered to carry charge or spin, they can implicitly impact the transport of charge and spin through scattering or dragging conduction electrons and magnons.^1,2 In Fig. 1, we summarize the interactions between these three elemental energy carriers and also highlight several applications enabled by taking advantage of such energy-carrier interactions. The representative applications include thermoelectrics,^3–5 piezoelectrics,^6,7 spin-transfer torque magnetic random access memory (STT-MRAM),^8–10 spin field-effect transistors (FETs),^11–13 and heat-assisted magnetic recording (HAMR).^14,15

There have been comprehensive literature reviews covering the electron–phonon coupling^2,19 and the coupled electron-spin and charge transport^20,21 in solid materials. Thus, this Perspective will focus on the interaction between spin and heat and its impacts on the thermal and magnetic properties of materials, together with several resulting technological applications. For example, magnons and phonons coexist and propagate in crystalline magnets, where they can couple with each other and exhibit rich physics. The interaction between magnons and phonons has been observed in some temperature- and field-dependent thermal measurements. This interaction is responsible for the suppressed thermal conductivity,²² the abnormal peaks in the temperature-dependent thermal conductivity,^23,24 and the anomalies in the magnetic-field-dependent spin Seebeck effect.²⁵ In Secs. II–IV, we will discuss the details of the magnon contribution to thermal properties, temperature-dependent magnetic properties, and thermal generation of spin currents, all of which involve the coupling between heat and spin (carried by either magnons or electrons). In addition, we will also briefly highlight several enabling applications and share our perspectives about future directions in Sec. V.

In magnetic materials, magnons can also carry heat.²⁶ Like phonons, magnons are also bosons with a dispersion¹ described as $ω=μBgLH+2JSa2k2$⁠, where μ_B is the Bohr magneton, g_L is the Landé factor, J is the exchange coupling energy, S is the spin quantum number, a is the size of the unit cell, k is the magnon wave vector, and H is the applied magnetic field. Since the magnon density of states depends on the applied magnetic field H, the field-dependent thermal measurements of a ferrimagnet or an antiferromagnet can thus capture the magnon contribution to thermal transport.²⁷ To accurately quantify the magnon contribution to thermal conductivity, the possible coupling between heat and charge, as well as between charge and magnetization, should be isolated. Therefore, the preferred materials should remain both electrically insulating and magnetically ordered within the temperature range of interest.²⁸ For example, insulating ferrimagnetic yttrium iron garnet (YIG) has been extensively studied in spin-caloritronic experiments due to its desirable electronic and magnetic properties.^1,27–32 As shown in Fig. 2(a), the field-dependent thermal conductivity of insulating YIG is significantly suppressed when a magnetic field is applied, as this field reduces the magnon contribution to thermal transport.^30–32 In addition, Jin et al.²³ reported a rapid increase in the in-plane thermal conductivity of monocrystalline quasi-two-dimensional (2D) antiferromagnetic Nd₂CuO₄ at low temperatures above a critical field. This critical field is related to an energy gap in an acoustic magnon branch. They attributed the increase in the in-plane thermal conductivity to the closure of the gap when the applied field was higher than the critical field [Fig. 2(b)].^23,33 More recently, Bao et al.³⁴ performed thermodynamic and inelastic neutron scattering measurements to validate the magnon–phonon interaction in a cubic collinear antiferromagnetic Cu₃TeO₆. Magnons manifest themselves by scattering phonons, causing the suppression of thermal conductivity around the Néel temperature (T_N). Besides ferrimagnets and antiferromagnets, magnetic-field-dependent thermal conductivity is also observed in multiferroic materials (e.g., GdFeO₃), indicating a strong magnon–phonon interaction [Fig. 2(c)].³⁵ 

In addition to field-dependent thermal measurements, temperature-dependent measurements of thermal conductivity can also reflect the magnon contribution. Extensive studies have been done to explore the magnon contribution to thermal transport via measurements of the temperature-dependent thermal conductivity with various methods, such as the conventional steady-state method, the four-point steady-state method, and the time-domain thermoreflectance technique.^22,23,34,36 Details of the sample categories and measurement techniques are summarized in Table I. The magnon contribution to the material's thermal conductivity depends on the material systems. For simple insulating antiferromagnetic crystals, such as MnO,³⁷ MnF₂,³⁸ CoF₂,³⁸ and FeCl₂,²² the interactions between magnons and phonons lead to anomalies in the thermal conductivity at the magnetic transition point. In these materials, the scattering of magnons and phonons near T_N increases thermal resistivity and thus reduces the thermal conductivity at the magnetic transition point [see the temperature-dependent thermal conductivity of FeCl₂ and CoF₂ in Fig. 2(d)]. The anomaly of thermal conductivity at the magnetic transition point is caused by the magnon–phonon resonant interaction arising from single-ion magnetostriction and is usually observed near T_N.^22,39

For more-complex magnetic materials, we mainly focus on the following complex insulating material systems: multiferroic manganites^40,41 and spin ladders or 2D quantum magnets with 180° Cu–O–Cu bond configurations.^{26,36,42–44} Multiferroic manganite is an important class of multiferroics, which possess both ferroelectricity and magnetism.^40,41 In this material system, the origin of the strong scattering of acoustic phonons could be attributed to the geometrically frustrated magnetism.⁴⁰ As shown in Fig. 2(e), for temperatures above T_N and up to 300 K, the thermal conductivities of polycrystalline RMnO₃ (R = Sc, Lu, and Y) are widely suppressed. Below T_N, a large increase in thermal conductivity is observed, which is associated with the decrease in acoustic phonon scattering by low-energy spin fluctuations.⁴⁰ 

In spin ladders and 2D quantum magnets, the 180° Cu–O–Cu bond configurations provide strong antiferromagnetic coupling between Cu²⁺ ions and result in magnon excitations. The magnetic excitations in these systems are referred to as magnons in spin ladders and spinons in spin chains. Unlike the typical trend of temperature-dependent thermal conductivity of insulating materials with only phonons carrying heat, the thermal conductivity of Sr₁₄Cu₂₄O₄₁ along the ladder direction shows two peaks as the temperature changes.^44,45 These two peaks are the consequence of the excitation of magnon modes above the spin gap competing with the enhanced magnon–phonon scattering as the temperature increases. Particularly, the second peak is mainly due to the magnon contribution. Similar results have been observed in other transition metal oxide-based magnets, such as La₂CuO₄,⁴³ Sr_14−xCa_xCu₂₄O₄₁,⁴⁴ and Ca₉La₅Cu₂₄O₄₁ [Fig. 2(f)].²⁶ 

In addition to thermal conductivity, magnons also contribute to specific heat. Generally, the magnon specific heat is much smaller than phonon specific heat, influenced by the much higher frequency range of magnons (e.g., ∼105 THz in Fe).^26,28,46 At low temperature, however, the magnon contribution to specific heat can be substantial in comparison to the contribution from other energy carriers. As shown in Fig. 3(a), the heat capacity of manganites at the magnetic phase transition point is anomalous.^47,48 Such a behavior of the anomaly is characteristic of most magnetic materials due to the magnetic phase transitions.^49,50 Besides, anomalous behavior can also be observed even at lower temperatures in magnets. For example, the specific heat of YIG deviates from its characteristic T^1.5-dependence at ultralow temperatures (<0.77 K), attributed to the effect of magnetic dipole–dipole interaction [Fig. 3(b)].³⁰ Further, a change in the specific heat of YIG with and without magnetic fields is observed at low temperatures of less than 15 K, clearly suggesting that the magnon specific heat is responsible for such a change [Figs. 3(c) and 3(d)].²⁸ When an external field of 70 kOe is applied, ∼90% of the magnon contribution can be suppressed. Thus, it is possible to freeze the magnon contribution to the specific heat by applying sufficiently strong magnetic fields.²⁸ 

While magnetic fields can be used to tailor the thermal transport properties, temperature, in turn, can also affect magnetic properties at thermally equilibrium states. Such temperature-dependent magnetic properties are important for applications that involve large temperature variations (e.g., heat-assisted magnetic recording, HAMR) or require highly sensitive temperature control and detection (e.g., metrology development of novel temperature sensing and thermal characterization).

The most apparent temperature-dependent magnetic property is magnetization (M, the volumetric density of magnetic moments). In ferromagnets, the exchange coupling tends to align the direction of magnetic moments. As a competing effect, thermal energy, on the contrary, drives magnetic moments away from their equilibrium directions and, therefore, decreases the magnetization. The magnetization of a ferromagnet will almost vanish when the temperature is higher than its Curie temperature (T_C). This strong temperature dependence of magnetization is illustrated in Fig. 4(a) for bulk L1₀-phase FePt with a T_C of ≈720 K as a representative.⁵¹ The granular form of this material has been frequently used as the recording medium for HAMR. Apart from magnetization, the magnetic anisotropy (K) also decreases with temperature, following a trend that scales with the temperature dependence of magnetization [see Fig. 4(a)].⁵¹ The decreasing trend of K as temperature increases also plays an important role in HAMR applications, allowing switching of magnetic polarization when the temperature is briefly and locally near T_C, while maintaining stable magnetization of bits at lower temperatures.

For magnetization dynamics, the magnetic damping parameter α describes how fast the magnetization relaxes to its equilibrium direction. This factor is crucial to determine the energy consumption in device applications such as magnetoresistance random access memory (MRAM). α is preferred to be small to reduce the critical switching current and eventually the energy consumption for MRAM devices.⁵² For recording applications, a high α is instead favored to achieve fast switching and improve the signal-to-noise ratio.⁵³ Since the operating temperature ranges from −55 to 150 °C for MRAM⁵⁴ and the writing temperature is close to T_C for HAMR,¹⁵ the temperature dependence of magnetic damping is of technological importance.

Magnetic damping consists of both intrinsic contribution and extrinsic contribution. Theoretically, spin–orbital coupling^55–57 and phonon-drag⁵⁸ have been proposed as the two main mechanisms causing intrinsic damping of thin films. For extrinsic damping, the common sources include two-magnon scattering, spin pumping, and inhomogeneous broadening.^59,60 For L1₀-FePd, a promising material candidate used in MRAM applications, the temperature dependence of magnetic damping within the operating temperature range (−55 to 150 °C) is relatively weak.^54,61 While for L1₀-FePt in HAMR applications, the medium temperature can approach or even exceed T_C during phases of the writing process. At such high temperatures, magnetization decreases sharply. The change in the magnetization amplitude leads to the longitudinal relaxation, which could be modeled by the Landau–Lifshitz–Bloch (LLB) equation.⁶² Figure 4(b) shows the damping (α) of L1₀-FePt derived from the LLB equation. At high temperatures, the large number of excited magnons will be non-linearly scattered and resulting in a significant enhancement on α.⁵¹ Experimental studies of magnetic damping at high temperatures are extremely challenging and related publications are presently rather limited in number.^53,61 Phonon-drag induced intrinsic damping is much weaker compared with the spin–orbit coupling contribution for metallic magnetic materials and thus will not be discussed here.⁶³ The temperature dependence of extrinsic damping has been less often reported in the literature likely due to the complex and diverse contributing factors, which can vary with sample materials, stack configurations, and both the sample fabrication and measurement conditions.

The coupling between heat and spin further allows the generation of spin currents by temperature gradients. This is analogous to the thermoelectric effect through which waste heat can be directly converted to a charge current without any moving parts. Thermal generation of spin current (also called “thermal spin injection” in the magnetic community) provides a manner of producing large spin current densities compared with traditional techniques^1,64 and thus has become a promising technique for spintronic applications.

The thermal generation of spin currents in magnetic materials typically involves two effects, namely, the spin-dependent Seebeck effect (SDSE) and the spin Seebeck effect (SSE). The SDSE refers to the thermal generation of spin currents carried by conduction electrons in a metallic magnet owing to the spin-dependent thermoelectric properties of the magnet. The SDSE process can be described by a model considering two spin-transport channels (majority and minority spin channels) with different thermoelectric properties.⁶⁵ As shown in Fig. 4(c), when a temperature gradient is applied, a net spin current can be induced due to the different thermoelectric properties of these two channels consisting of electron diffusion. Since conduction electrons are the carriers for spin currents, SDSE only occurs in conductive magnets.

Unlike SDSE, magnons are the major carriers for spin currents thermally generated due to the SSE. Therefore, SSE can also occur in (electrically) insulating magnetic materials.^66–69 Figure 3(d) illustrates the SSE experimental schematic of a ferromagnetic (FM) insulator under the longitudinal configuration where the direction of spin current is in parallel with the temperature gradient.^66,69 When a temperature gradient is applied, spin currents are injected to the adjacent normal metal (NM, usually Pt) layer, where the spin currents are converted to detectable charge currents (or voltages) through the inverse spin Hall effect. In SSE, the injection of spin currents from the FM layer to the NM layer is attributed to the s-d coupling between the magnons in the FM layer and the conductive electrons in the NM layer. A model based on the fluctuation-dissipation theorem has been developed to describe this process.^66,68,70 It should be noticed that the measurement configuration of the longitudinal SSE is very similar to that of the Anomalous Nernst effect (ANE),⁷¹ where a transverse charge voltage/current is generated by a longitudinal temperature gradient across a magnetic material. Therefore, the longitudinal SSE is only well defined for magnetic insulators.⁶⁶ Recently, several groups have investigated the SSE on ultrafast timescales.^72,73 It opens the door to apply the SSE in terahertz spintronic devices.

Apart from the SDSE and SSE, the spin Nernst effect (SNE) can also generate spin currents thermally.^74–76 The SNE describes the generation of a transverse pure spin current from a longitudinal temperature gradient applied on a non-magnetic material with high spin–orbit coupling (e.g., W and Pt). It has shown potential for manipulating magnetization dynamics.⁷⁷ 

One of the applications initially proposed for the SSE was waste heat recovery.^66,78,79 Compared with the traditional thermoelectric effect, however, the SSE is generally a weaker effect (<1 μV/K) at room temperature.⁸⁰ In fact, for thermoelectric applications, the “magnon-drag” effect (referring to the magnon transport on the charge Seebeck effect) is reported to have a much higher thermopower and thus serves as a better choice.⁸⁰ Other than thermoelectrics, SSE could find potential applications in two-dimensional position sensing,⁸¹ spin-based detection of light,⁸² and driving micro-mechanical devices.⁸³ 

The spin-heat coupling via interactions between different energy carriers has enabled a number of applications, ranging from metrology advancement to cutting-edge technology innovation. The temperature-dependent magnetization of magnetic materials has inspired the development of highly sensitive measurement techniques for studying thermal properties of materials. One example is the use of magnetic thin films as transducers in the ultrafast pump–probe technique (time-resolved magneto-optical Kerr effect, TR-MOKE) to measure the thermal conductivity of materials^84,85 and interfacial thermal conductance.^86–90 The magnetic transducer can be much thinner and less thermally conductive than the nonmagnetic transducer used in the standard time-domain thermoreflectance technique. In this case, the measurement sensitivity to in-plane thermal properties can be greatly enhanced. Due to its ultrafast nature, TR-MOKE is also advantageous for investigating magnetization dynamics of ferromagnets and spin transport on the picosecond timescale.^91,92 Recently, researchers at NIST invented a novel approach of 3D thermal magnetic imaging and control (Thermal MagIC) to achieve localized and remote temperature sensing and regulation. The technical foundation of this approach is based on the temperature-dependent magnetic response of magnetic nano-objects loaded into a 3D volume. These nano-objects act as distributed nano-sensors and provide magnetic signals depending on the local temperature, which allows for the construction of high-resolution 3D temperature mapping.^93,94

On the technology innovation aspect, one cutting-edge application that is heavily relying on the temperature-dependent magnetic properties is the aforementioned HAMR technology. The booming data-storage market stimulates the industry to continuously increase the hard-disk areal density by using smaller grains. With the decrease in grain size, the recording medium becomes less thermally stable and eventually hits the superparamagnetic limit, where the fluctuation of ambient temperature and the polarizations of neighboring bits can randomly switch the magnetization of a single bit and thus lose information. Therefore, materials with a large magnetic anisotropy have been used as recording media to improve thermal stability; however, this also requires a strong magnetic field for the data writing process, which is not yet available in the current industry. These competing requirements of high areal density, thermal stability, and writeability lead to the “Magnetic Recording Trilemma” faced by the conventional data storage technologies. As a proposed solution, HAMR addresses this trilemma by heating a very small area of the recording medium where writing is to occur, then polarizing the magnetic material to write the data bit, and cooling it down to freeze the magnetization and thus lock the information.^14,15,95 The heating is induced by a laser via a near-field transducer integrated into the HAMR head, which momentarily (∼1 ns) heats up the recording medium close to its T_C and thus reduces its magnetic resistance below the available magnetic field provided by the write pole.¹⁵ Compared with the conventional magnetic recording technology, it is envisioned that HAMR holds great potential to extend the storage areal density by at least an order of magnitude.

Besides the media engineering, head thermal management is also of critical importance to the success of HAMR technology. In HAMR operation, the use of the laser, focused down to sub-diffraction limited thermal spots to assist switching the magnetization of the medium, poses many thermal challenges at the nanoscale. The heat dissipation in HAMR devices requires aggressive heat sinking and intense exploration of effective thermal management strategies. In addition, the writing process occurs at close-to-T_C temperatures, which makes it extremely challenging to experimentally investigate the magnetization dynamics at such elevated temperatures, partially due to the pronounced inhomogeneous broadening and more-complex scattering events between magnons and phonons. To date, experimental studies of the intrinsic damping of recording materials remain elusive, despite the importance of damping which is directly related to the switching speed in the recording process. Ongoing effects from both industrial and academic researchers have been devoted to the high-temperature switching process seeking for a deeper fundamental understanding and further optimizing the device performance.

In parallel with this, new ideas for applications have been stimulated by the interaction between spin and heat. For example, built upon the spin Seebeck effect and inverse spin Hall effect, magneto-thermal microscopy has been proposed to detect time-resolved magnetization dynamics in magnetic materials.⁹⁶ In addition, spin-caloritronic heat engines and motors have been envisioned, where the domain-wall motion (one type of magnetization dynamics) is driven by thermally induced currents through spin-transfer torques.^67,97,98 Finally, we would also like to briefly highlight the spin–strain coupling, which is another coupling form involving the interactions between magnons and phonons. It has demonstrated great potential for manipulating spins in quantum systems (e.g., diamond nitrogen-vacancy),⁹⁹ cooling mechanical resonators to a sub-thermal state,¹⁰⁰ and achieving ultrahigh-speed strain-assisted switching in magnetic storage or computing systems.¹⁰¹ 

Overall, the coupling between heat and spin (carried by free electrons or magnons) has become a topic of intense research in view of a wide range of potential applications. Looking toward the future, with more results unveiling the rich physics of thermal spin transport, new strategies to further optimize existing applications and innovate new designs and technologies. It is anticipated to inspire more fundamental research to understand the coupling mechanisms and how they would impact material properties and also lead to the development of remarkable improvements for the next generation of device applications in spintronics, data storage, spin-caloritronics, and quantum computing.

The authors of this work appreciate the support from the National Science Foundation (NSF) under Award No. 1804840, the University of Minnesota MRSEC (under the NSF Award No. DMR-2011401), the Advanced Storage Research Consortium, and the Minnesota Futures Award.

The authors have no conflicts to disclose.

Y.Z. and D.H. contributed equally to this work.

Data supporting the results and conclusions of this study are available within the article.