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## Creator

[Fuya Makino](https://orcid.org/0009-0004-4578-124X), [Takamasa Hirai](https://orcid.org/0000-0002-5577-8018), [Takuma Shiga](https://orcid.org/0000-0002-5103-7853), [Hirofumi Suto](https://orcid.org/0000-0003-4387-5862), Hiroshi Fujihisa, [Koichi Oyanagi](https://orcid.org/0000-0001-8784-078X), [Satoru Kobayashi](https://orcid.org/0000-0002-3545-2977), [Taisuke Sasaki](https://orcid.org/0000-0002-5952-7638), Takashi Yagi, [Ken-ichi Uchida](https://orcid.org/0000-0001-7680-3051), [Yuya Sakuraba](https://orcid.org/0000-0003-4618-9550)

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[Magnetothermal resistance effect in a <math>  <mrow>    <mi>C</mi>    <msub>      <mi>o</mi>      <mn>50</mn>    </msub>    <mi>F</mi>    <msub>      <mi>e</mi>      <mn>50</mn>    </msub>  </mrow>  <mtext>/</mtext>  <mi>Cu</mi></math> multilayer studied via analysis of electron and lattice thermal conductivities](https://mdr.nims.go.jp/datasets/404ccdd0-231b-48d9-a328-2dc8ce5ea5e4)

## Fulltext

1 Magneto-thermal resistance effect in a Co50Fe50/Cu multilayer studied via 1 analysis of electron and lattice thermal conductivities  2 Fuya Makino,1,2,3 Takamasa Hirai,2,* Takuma Shiga,4 Hirofumi Suto,2 Hiroshi Fujihisa,4 Koichi Oyanagi,3 Satoru 3 Kobayashi,3 Taisuke Sasaki,3 Takashi Yagi,4 Ken-ichi Uchida,1,2,5 and Yuya Sakuraba1,2,† 4  5 1Graduate School of Science and Technology, University of Tsukuba, Tsukuba, Ibaraki, Japan. 6 2National Institute for Materials Science (NIMS), Tsukuba, Ibaraki, Japan. 7 3 Faculty of Science and Engineering, Iwate University, Morioka, Iwate, Japan. 8 4 National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki, Japan. 9 5 Department of Advanced Materials Science, Graduate School of Frontier Sciences, The University of Tokyo, Kashiwa, 10 Chiba, Japan. 11  *HIRAI.Takamasa@nims.go.jp, †SAKURABA.Yuya@nims.go.jp 12  13  14 ABSTRACT. This study investigates the giant magneto-thermal resistance (GMTR) effect in a fully-bcc epitaxial 15 Co50Fe50/Cu multilayer through both experimental and theoretical approaches. The applied magnetic field results in a 16 giant change of the cross-plane thermal conductivity (Δκ) of 37 W m-1 K-1, which reaches 1.5 times larger than the 17 previously reported value for a magnetic multilayer and record the highest value at room temperature among the other 18 solid-state thermal switching materials working on different principles. We investigated the electron thermal conductivity 19 for exploring the remarkable Δκ by the two-current-series-resistor model combined with the Wiedemann-Franz (WF) law. 20 However, the result shows the electron contribution accounts for only 35% of the Δκ, indicating the presence of additional 21 spin-dependent heat carriers. Further investigation of the lattice thermal conductivity, which is expected to be spin-22 independent, using non-equilibrium molecular dynamics (NEMD) simulations suggests a striking contrast: the additional 23 spin-dependent heat carrier contribution is significantly enhanced in the parallel magnetization configuration but nearly 24 negligible in the antiparallel configuration. These findings provide a fundamental insight into the origin of large GMTR 25 effect and highlight its potential of active thermal management technologies for future electronic devices. 26  27  28  29 I. INTRODUCTION 30 mailto:HIRAI.Takamasa@nims.go.jpmailto:SAKURABA.Yuya@nims.go.jp  2 Thermal switching devices are pivotal for advancing thermal management technologies, which are essential 31 for enhancing the efficiencies in modern electronic devices [1,2]. By controlling heat flow through a dynamic change in 32 thermal conductivity of solids, these devices address critical challenges in managing heat dissipation across nanoscale to 33 macroscale regimes. To realize practical thermal switching, it is necessary to achieve a significant change of thermal 34 conductivity, a wide operating temperature range, and seamlessly integration with existing electronic systems. Various 35 mechanisms yielding the thermal conductivity change in solids have been extensively investigated so far, including metal-36 insulator transition [3,4], electrochemical reaction [5,6], voltage-controlled ferroelectricity [7,8], superconductors [9,10] 37 and the magnetoresistance (MR) effect [11]. Among them, the MR effect in magnetic materials offers distinct advantages; 38 it enables noncontact, high-durability and high-scalable thermal conductivity switching by applying an external magnetic 39 field and/or controlling a magnetization of magnetic materials. However, the magnetic-filed-induced change for the 40 thermal conductivity in single magnetic materials is much smaller than other mechanisms at room temperature, hindering 41 the practical application of MR-based thermal switching [11].  42 In magnetic multilayers consisting of alternatively stacked ferromagnetic metal (FM) and non-magnetic metal 43 (NM) layers, their electrical conductivity depends on the relative magnetization angle of the adjacent ferromagnetic layers, 44 a phenomenon known as the giant magnetoresistance (GMR) effect [12,13]. The GMR effect is one of the most familiar 45 phenomena in spintronics and has been widely investigated for various applications, such as read heads for hard disk 46 drives and magnetic sensors. Similar to electrical conductivity, thermal conductivity is influenced by the magnetization 47 configuration in these multilayers, leading to a phenomenon called the giant magneto-thermal resistance (GMTR) effect 48 [14-16]. The GMTR effect enables a large change of the thermal conductivity from the parallel (P) and antiparallel (AP) 49 magnetization configurations. The performances of the GMR and GMTR effects are commonly characterized by the MR 50 ratio and magneto-thermal resistance (MTR) ratio, defined as (𝜎!" − 𝜎")/𝜎!" and (𝜅"  − 𝜅!")/𝜅!", respectively. Here, 51 𝜎!"(")  and 𝜅!"(")  represent the electrical conductivity 𝜎 and the thermal conductivity 𝜅 in the AP state (P state), 52 respectively. It is known that 𝜎  and 𝜅  in the metallic materials and systems are often well connected through 53 Wiedemann-Franz (WF) law, i.e., the linear relationship between the electron thermal conductivity and the electrical 54 conductivity. Additionally, thermal conductivity has lattice contribution. Therefore, when only these two contributions 55 are assumed, the MTR ratio should be smaller than the MR ratio because the lattice component should be independent of 56 magnetization direction. Previous studies also suggested that in the presence of inter-spin and spin-conserving inelastic 57 scattering in metallic magnetic multilayers where the whole thermal transport is explained by the contribution of electrons, 58 the WF law does not hold and the MTR ratio can be further reduced [17,18]. However, Nakayama et al. recently reported 59 that the MTR ratio (~150%) is significantly higher than the MR ratio (~60%) in a fully-bcc epitaxial Co50Fe50 (3 nm)/Cu 60 (1.6 nm) multilayer, in which Cu forms a metastable bcc structure at room temperature [19,20]. To explain this behavior, 61 they investigated the applicability of the WF law in the case where the density of state shows a steep change near the 62 Fermi energy [21] and performed the first-principles calculation of the spin-dependent ballistic electron transmittance to 63 analyze the GMTR effect. Although their calculated MTR ratio can be larger than MR ratio at the certain temperature 64 range, this range does not include room temperature and the difference between MTR and MR ratios was too small to 65   3 fully explain the measured MTR ratio. The experimentally observed large MTR ratio holds promise for future thermal 66 management applications and raises fundamental interest in uncovering additional contribution of thermal transport. In 67 this respect, the direct comparison between MTR and MR ratios in the cross-plane direction is essential to understand the 68 electrical contribution. However, and such comparison have so far been limited to the fully-bcc epitaxial Co50Fe50 (3 69 nm)/bcc-Cu (1.6 nm) multilayer system. Furthermore, the quantitative analysis of the contribution of the lattice thermal 70 conductivity arising from phonons, one of essential heat carriers, remains unexplored in such a magnetic multilayer.  71 In this study, we observed the GMTR effect in a Co50Fe50/metastable bcc-Cu multilayer with a thicker Co50Fe50 72 layer (5.1 nm) and compared the behavior to that with a thinner Co50Fe50 layer (3.0 nm) used in the previous study [19]. 73 Notably, the change in the cross-plane thermal conductivity of the present sample Δ𝜅 =  𝜅"  − 𝜅!" and MTR ratio were 74 37 W m-1 K-1 and 108% respectively; the obtained Δ𝜅 exhibits the highest value among GMTR reported so far. We 75 compared the observed MTR ratio with the MR ratio evaluated by the two-current-series-resistor (2CSR) model [22] in 76 the cross-plane direction to gain insight into the electrical contribution. In addition, we analyzed the atomic resolution 77 micro-structure of Co50Fe50 and bcc-Cu using high-angle annular dark field scanning transmission electron microscopy 78 (HAADF-STEM) and calculated the phonon contribution using a nonequilibrium molecular dynamics (NEMD) 79 simulation [23,24] based on the observed microstructure. Our quantitative analyses revealed that approximately 42% of 80 the 𝜅"  value and 65% of the Δ𝜅  value were arising from other spin-dependent contributions that are neither the 81 contributions of pure electron nor phonon transports based on the conventional theories. 82 II. METHODS 83 The magnetic multilayer film with the structure of Co50Fe50/[bcc-Cu/Co50Fe50]20 was deposited at room 84 temperature onto a [001]-oriented MgO single crystalline substrate using an automated ultra-high-vacuum magnetron 85 sputtering system (hereafter, Co50Fe50 and bcc-Cu are referred to as CoFe and Cu for simplicity.) The CoFe thickness was 86 designed to be thicker that in the previous study [19], and the Cu thickness was fixed at 2.0 nm to achieve an AP 87 magnetization configuration at zero field via the anti-ferromagnetic interlayer exchange coupling (IEC) between the CoFe 88 layers through the Cu layers. The surface of the multilayer was covered by an Al layer without breaking the chamber 89 vacuum. The Al layer serves as a transducer for the thermal conductivity measurement using the time-domain 90 thermoreflectance (TDTR) method. The actual thickness values of each layer will be shown later. Prior to the deposition 91 of the bottom CoFe layer, the MgO substrate was heated up to 600°C in the sputtering chamber for the surface cleaning. 92 After the deposition of all layers, the sample was annealed at 250°C with applying a constant magnetic field of 300 mT 93 for 1 h [20]. 94 The crystal structure of the multilayer film was characterized by the x-ray diffraction (XRD) with the Cu Ka 95 radiation and HAADF-STEM with the acceleration voltage of 200 kV. The GMR effect in the current-in-plane 96 configuration and the magnetization were measured using a standard dc four probe method and a vibrating sample 97 magnetometer, respectively, by applying a magnetic field parallel to the in-plane direction. To investigate the GMTR 98 effect in the CoFe/[Cu/CoFe]20 multilayer film, we performed the TDTR measurement [25,26], one of the optical pump-99   4 probe methods, in the front-heating and front-detection configuration. In this method, ultrafast pump laser pulses heat the 100 surface of the Al transducer layer on the CoFe/[Cu/CoFe]20 multilayer film, while probe laser pulses irradiated with 101 controlled delay time detect the transient response of the surface temperature via thermoreflectance, i.e., the temperature 102 dependence of the reflectivity, which enables the quantitative determination of thermal transport properties of thin films. 103 Further details on the TDTR experiment are found in Refs [27-31]. All the TDTR measurements were performed at 104 ambient temperature under air atmosphere. 105 III. RESULTS 106 An out-of-plane XRD pattern of the sample is shown in Fig. 1(a). The diffraction peak with the fringe around 107 66○ corresponds to the 002 peaks of bcc-CoFe, confirming the [001]-oriented growth of bcc-CoFe on the MgO substrate 108 and the formation of the smooth interfaces with a small roughness. Since the peaks from a normal fcc-phase of Cu were 109 not observed, Cu layer was expected to grow in bcc-structure with the [001]-orientated growth in between CoFe layers in 110 our multilayer. Because the lattice constants of bcc-CoFe and bcc-Cu are almost the same, the 002 diffraction peaks 111 originated from bcc-CoFe and Cu cannot be distinguished. Moreover, the 011 peak from the Al layer is overlapped with 112 the 002 peak from the MgO substrate. Figure 1(b) shows a schematic of the fabricated multilayer film with the actual 113 thicknesses confirmed by the microstructure analysis using a transmission electron microscopy (TEM): 5.1 nm for CoFe 114 layers, 2.0 nm for Cu layers, and 44 nm for the Al layer. We confirmed each CoFe layer was thicker than that in the 115 previous study (3.0 nm). The TEM analysis also confirmed that the total thickness 𝑡%&%'(  of the CoFe/[Cu/CoFe]20 116 multilayer was 147 nm. Figures 1(c)-1(f) show the cross-sectional HAADF-STEM images captured along the [100] zone 117 axis of the MgO substrate. Figure 1(c) indicates that the 20-period CoFe/Cu multilayer structure maintained the interfacial 118 flatness consistently from the bottom to the top layers. To investigate the interfacial atomic lattice matching at different 119 regions [upper, middle, and lower regions in Fig. 1(c)] of the CoFe/[Cu/CoFe]20 multilayer, high-magnification HAADF-120 STEM images and inverse fast Fourier transform images reconstructed from 110 peaks are separately shown in Figs. 1(d)-121 (f). We directly observed the metastable bcc-Cu that formed in the multilayer from the HAADF-STEM images. Besides, 122 very smooth CoFe/Cu interfaces was observed across all regions. Therefore, we concluded that the nearly complete lattice 123 matching and the atomically flat interfaces were formed in the CoFe/[Cu/CoFe]20 multilayer throughout the entire 124 structure. These microstructure analyses validate that our multilayer can be regarded as the homogenous single medium 125 to evaluate its effective thermal conductivity by the TDTR measurements and analyses. 126 Figures 2(a) and 2(b) respectively show the external magnetic field 𝐻 dependence of the current-in-plane MR 127 ratio and the normalized magnetization 𝑀/𝑀) , where 𝑀)  represents the saturation magnetization. Clear resistance 128 plateaus observed in the range of |𝜇*𝐻| from 2 mT to 15 mT indicate that the AP state was realized by IEC within this 129 magnetic field range, where 𝜇* is the vacuum permeability. At |𝜇*𝐻| increases beyond 15 mT, the resistance notably 130 drops, and both the MR ratio and magnetization showed the saturation behaviors, indicating the formation of the P state. 131 It is worth noting that the gradual changes in the magnetizations in the field range corresponding to the plateau of the MR 132 curves are likely caused by the formation of magnetic domains which affect the magnetization but not MR ratio, as the 133   5 AP state is locally formed in each magnetic domain. 134 TDTR signals were recorded as a function of delay time between pump and probe laser pulses. As shown in 135 Fig. 2(c), the TDTR signal was significantly changed from the AP state (𝜇*𝐻 = 0 mT) to the P state (𝜇*𝐻 = ±75 mT), 136 suggesting the large magnetization-configuration-dependent thermal conduction. By analyzing TDTR signals using the 137 one-dimensional heat diffusion model, we experimentally estimated the effective thermal conductivity of the 138 CoFe/[Cu/CoFe]20 multilayer (𝜅+,-) (see Supplementary Material [27]). The fitting curves [solid curves in Fig. 2(c)] 139 showed good agreement with the experimental results. Figure 2(e) summarizes the 𝐻  dependence of 𝜅+,-  for the 140 CoFe/[Cu/CoFe]20 multilayer. We found that the CoFe/[Cu/CoFe]20 multilayer exhibited a drastic 𝜅+,- change in the 141 range from -25 mT to 25 mT, while 𝜅+,- remains almost constant at |𝜇*𝐻| > 25 mT, following the changes in the 142 magnetization configuration (Fig.2(b)). The field range exhibiting the large change in 𝜅+,- corresponds to the range 143 where the transition from the minimum to the maximum resistance is observed in the MR curve [Fig. 2(a)]. This indicates 144 that 𝜅+,- reaches its minimum and maximum values in the AP and P state, respectively, although the number of data 145 points was limited. Although the observed MTR ratio (108%) was a smaller than the reported one (150%) in the previous 146 study [19], the measured change in the thermal conductivity (∆𝜅+,-= 37 W m-1 K-1) was not only approximately 1.5 times 147 larger than that reported in the previous study (∆𝜅+,-= 25 W m-1 K-1) but also the record-high value among solid-state 148 thermal switching materials. In the previous report [19], the used Al thickness of 5nm may be insufficient for absorbing 149 pump-laser energy, requiring the implementation of a bidirectional heat conduction model in TDTR analyses. In contrast, 150 the present Al thickness (44 nm) was thick enough to assume the one-directional heat diffusion model. Despite different 151 models used, both present and previous multilayers [19] exhibited giant MTR ratios, suggesting the robustness of the 152 GMTR effect.  153 From the perspective of practical applications of thermal switching devices, achieving both low and high 154 thermal conductivity states under zero external magnetic field (non-volatile bistability) is crucial. In the present multilayer, 155 a clear hysteresis appears in both MR and magnetization curves [Figs. 2(a) and 2(b)], whereas no such hysteresis was 156 observed in previous study [19]. The observed hysteresis probably originates from the increased crystalline magnetic 157 anisotropy energy of CoFe due to the thicker CoFe layer thickness compared to the previous study. By utilizing this 158 hysteretic behavior, we demonstrated the bistable control of the thermal conduction. As shown in Figs. 2(d) and 2(e), by 159 adjusting 𝜇*𝐻  to 10 mT with changing the sweeping direction of 𝐻 , the TDTR curve and estimated 𝜅+,-  value 160 reproduced both data at the AP and P states at the same 𝜇*𝐻 value. This bistability was also realized in negative field 161 region at – 6 mT. Although detailed mechanisms of non-volatile bistability are to be investigated, this result suggests that 162 non-volatile bistability could be also obtained by stabilizing P and AP states at zero magnetic field, potentially through 163 the exchange bias effect instead of IEC, exhibiting the feasibility of a nonvolatile thermal switching device based on the 164 GMTR effect in the future. 165  166 IV. DISCUSSION 167   6 Next, we analyzed the component of the electron thermal conductivity by simulating the RA (electrical 168 resistance-area product) and MR ratio in the current-perpendicular-to-plane (CPP)-configuration using the generalized 169 2CSR model [Fig. 3(a)]. The 2CSR model [32-34] is the well-established theoretical approach that predicts the resistance 170 change in the CPP-GMR structure by considering the individual series resistors for majority- and minority-spin channels. 171 In the simulations, the CPP-GMR structure was modelled as Cu (1 nm)/[CoFe (5.1 nm)/Cu (2.0 nm)]20/CoFe (5.1 nm)/Cu 172 (1 nm), where the 1-nm-thick bottom and upper Cu layers considered as the electrodes. The simulation requires several 173 parameters including the bulk resistivity (𝜌), bulk spin-scattering asymmetry coefficient (𝛽), spin diffusion length (𝜆), 174 thickness of each layer (𝑡), interfacial resistance area product (𝑟), and interfacial spin-scattering asymmetry coefficient 175 (𝛾) [35,36]. Reasonable parameters, such as 𝜌 of CoFe and Cu; 𝛽 of CoFe; 𝜆 of CoFe and Cu, were adopted from 176 previous studies [35,36], which were summarized in Table I. We used the 𝜌 and 𝜆 reported for fcc-Cu in place of those 177 for bcc-Cu as there is no reported values for bcc-Cu. It should be noted that these parameters in bulk region of Cu have 178 negligibly weak effect on the simulated results because the thickness of Cu (2.0 nm) is much shorter than 𝜆. However, it 179 is necessary to obtain values of 𝑟 (𝑟.&/+/.1) and 𝛾 at the metastable CoFe/Cu interface (𝛾.&/+/.1) since the previous 180 study has reported that the metastable bcc-Cu has a significantly different interfacial electronic band matching with CoFe 181 compared to fcc-Cu [20]. In this study, we quantitively analyzed 𝑟.&/+/.1 and 𝛾.&/+/.1 based on the experimental data 182 of the [CoFe(3 nm)/bcc-Cu]N multilayer CPP-GMR devices [19]. By analyzing the measured 𝑅𝐴!" for N = 3, 5, and 7 183 [19] using the series circuit model consisting of the bulk and interfacial resistance area products, we determined 𝑟.&/+/.1= 184 0.27 mΩ μm2. We also attempted to obtain the 𝛾.&/+/.1 by comparing the experimentally measured resistance change-185 area products (ΔRA) for N = 3, 5, and 7 [19] with the simulated ΔRA as a function of 𝛾.&/+/.1 [Fig. 3(b)]. As a result, 186 we found 𝛾.&/+/.1 to be 0.76±0.03. By adopting these 𝑟.&/+/.1 and 𝛾.&/+/.1 in the analysis based on the 2CSR model, 187 we finally estimated 𝑅𝐴!" = 47.2 mΩ μm2, 𝑅𝐴" = 29.4 mΩ μm2, and obtained CPP-MR ratio of 60% for the present 188 CoFe/[Cu/CoFe]20 multilayer with the 5.1-nm-thick CoFe layers. The obtained MR ratio was much smaller than the 189 experimentally observed MTR ratio, approximately half of the MTR ratio of 108%, whose tendency is consistent with 190 the previous report [19]. Based on the WF law, the electron thermal conductivity of conventional metals is described as 191 𝜎𝐿𝑇, where 𝜎 (= 𝑡%&%'(/𝑅𝐴) is the electrical conductivity, L is the Lorenz number, and T is the absolute temperature. 192 Assuming that the WF law and L value of free electron (𝐿* = 2.44 × 1023 W Ω K24) can be applied for the present 193 multilayer structure, the electron thermal conductivity values for the AP and P states (𝜅eWF) were estimated to be 22.6 and 194 36.2 W m-1 K-1 at T = 297 K, respectively, using the simulated 𝑅𝐴!" and 𝑅𝐴" (see Table II). The calculated their 195 difference ∆𝜅eWF of 13.6±0.5 W m-1 K-1 is only about 35% of the experimentally observed ∆𝜅+,- (37±10 W m-1 K-1). 196 Thus, ∆𝜅+,- should include the additional spin-dependent contribution ∆𝜅'88(=∆𝜅+,-  − ∆𝜅eWF), which is calculated to 197 be very large 24±10 W m-1 K-1. It should be also mentioned that the L value in ferromagnetic materials can be smaller 198 than 𝐿* due to several factors (e.g., spin-dependent scattering mediated by exchange interaction and high localized 199 density of states near Fermi energy). For instance. the experimentally reported L values in Ni and Co at room temperature 200 are slightly smaller than 𝐿* [11,37]. It was also reported that L value for magnetic Co/Cu multilayer is smaller than 𝐿* 201 due to the predominant electron-phonon scattering [15]. By considering the reduction of L from 𝐿* in present CoFe/Cu 202 magnetic multilayer suggested from these earlier studies, intrinsic electron thermal conductivity may be smaller than 𝜅eWF, 203   7 respectively. Therefore, it is expected that the deviation of L more likely result in an underestimation of the ∆𝜅'88 204 component, rather than overestimation. Another potential factor in correcting electron thermal conductivity is the Seebeck 205 effect, expressed as 𝜎𝑆4𝑇, where S is the Seebeck coefficient. However, assuming the S values of our multilayer in the 206 AP and P states are the same as those measured in the previous study [38], the correction on the electron thermal 207 conductivity is estimated to be less than 1 W m-1 K-1 in both the AP and P states, which cannot explain the observed large 208 ∆𝜅'88. 209 Here, we explore the lattice component of thermal conductivity in CoFe/Cu multilayers. In actual CoFe layers, 210 Co and Fe atoms are expected to be randomly alloyed, as annealing at 250°C is insufficient to form the ordered B2 211 structure of CoFe [39]. Even though the layer thicknesses are on the nanometer scale, phonons are expected to experience 212 significant alloy scattering. Furthermore, considering the Debye temperatures of the constituent elements [40], the 213 quantum effect should have a minimal effect. Therefore, we employed NEMD simulations to evaluate the lattice thermal 214 conductivity. As illustrated in Fig. 4(a), the CoFe/Cu multilayer was modeled by stacking a bcc-based CoFe/Cu slab along 215 the z direction. The coordinates of Co and Fe atoms in each layer were randomly distributed while maintaining an atomic 216 fraction of 1:1. Furthermore, their coordinates in different slabs were set to be uncorrelated [Fig. 4(b)]. The lattice 217 constants of both CoFe and Cu slabs were set to 2.86 Å, as determined from the lattice spacing analysis of TEM images. 218 The interatomic interactions among the Co, Fe, and Cu atoms were described by the embedded atom method potential 219 with the parameterization of Zhou et al. [41]. In the NEMD simulation, a system with a length (𝑙) and a cross-sectional 220 area (𝐴) was sandwiched by hot and cold thermostats with the length of 𝑙/2. The periodic boundary condition was applied 221 to the cross-sectional direction. The temperatures of the thermostats were controlled by the Langevin thermostat. Denoting 222 inflow and outflow of the energies in the thermostats at a time interval 𝛥𝑡 as Δ𝑄:(Δ𝑡) and Δ𝑄;(Δ𝑡), the heat flux in 223 the system was given by the average; 𝑞(𝑡 = 𝑘𝛥𝑡) = ∑ [Δ𝑄:(𝑗Δ𝑡) − Δ𝑄;(𝑗Δ𝑡)]<= /2𝐴𝑘Δ𝑡, where 𝑘 is the total time steps 224 of the NEMD simulation. We first performed a 5 ns-long NEMD simulation to ensure the system reached a non-225 equilibrium steady state more smoothly. Subsequently, we conducted an additional 5 ns-long NEMD simulation to sample 226 𝑞(𝑡) and evaluated its mean value 𝑞M was evaluated after a nonequilibrium steady state was established. An effective 227 lattice thermal conductivity of the CoFe/Cu multilayer system was calculated by 𝜅LNEMD = 𝑞M𝑙/Δ𝑇, where Δ𝑇 denotes a 228 temperature difference between the hot and cold thermostats. 𝜅LNEMD was finally evaluated through the ensemble average 229 for five different atomic configurations. For the comparison with measurements, 𝑙 was set to approximately 143 nm. In 230 addition, 𝐴 was set to 2.29×2.29 nm2, which ensures the convergence of 𝜅LNEMD. To fix the translational and rotational 231 motions of the entire system, the CoFe/Cu slabs with frozen internal degrees of freedom were attached to both edges of 232 the system. All NEMD simulations were performed by the LAMMPS package [42] with a time step of 0.5 fs. As shown 233 in the Fig. 4(c), a steady state was achieved at 2.5 ns to 5.0 ns, 𝜅LNEMD was evaluated from the heat flux in the time 234 range. Although small fluctuations appeared in the obtained temperature profilefor Δ𝑇 = 50 K [Fig. 4(d)], a nearly 235 linear temperature profile was established, confirming the homogeneity of the CoFe/Cu multilayer. The resulting 236 𝜅LNEMD of the CoFe/Cu multilayer was 5.7 W m-1 K-1 at room temperature. The 𝜅LNEMD value remained independent 237 of the magnitude of Δ𝑇 (see Supplementary Material [27]]. Note that the embedded atom method-type potential 238   8 used in this calculation is not fully optimized for modeling heat conduction in the present multilayer system. 239 Furthermore, this potential may not accurately capture phonon transport mechanisms in certain pure metals [43]. 240 However, this obtained value exhibits magnitude similar to previous superlattice calculations [44,45]. Since it is 241 reasonable to assume the lattice thermal conductivity is independent of the magnetization configurations, we put the same 242 𝜅LNEMD for AP and P states and ∆𝜅LNEMD = 0 in Table II. 243 Finally, we quantitatively compared the value of 𝜅+,-, 𝜅eWF, 𝜅LNEMD, and explored the additional contribution 244 of the thermal conductivity (𝜅'88 = 𝜅+,- − 𝜅eWF − 𝜅LNEMD) as summarized in Table II. The quantitative analysis of 𝜅LNEMD 245 has led to an important finding: 𝜅+,- in the AP state is nearly equal to the summation of 𝜅eWF and 𝜅LNEMD, i.e., 𝜅'88 = 246 5±4 W m-1 K-1, whereas in the P state, it is significantly larger than their summation, i.e., 𝜅'88 = 30±9 W m-1 K-1. This 247 result suggests that the contribution of additional heat carriers in AP state could be negligibly small, although our analysis 248 still contains a certain level of the error bar. Notably, the present quantitative analysis shows that almost 42% of 𝜅+,- in 249 the P state and 65% of ∆𝜅+,- originate from 𝜅'88 that is neither the contributions of electron nor phonon calculated 250 based on the WF law and the NEMD simulation, respectively. A possible additional carriers contribution to the GMTR 251 effect is magnon, collective dynamics of localized magnetic moment in magnetic materials. The magnetization-252 configuration-dependent magnon transport, known as the magnon valve effect, has been demonstrated in magnetic 253 multilayer structures, which highlights the importance of magnon–magnon interactions between adjacent magnetic layers 254 [46,47]. Although the magnon contribution in the thermal conduction is usually discussed at the low temperature [48-50], 255 recent theoretical calculation has shown that the magnon thermal conductivity also depends on the magnetization 256 configuration even at room temperature [51]; in the trilayer consisting of two magnetic insulator layers sandwiching a Cu 257 interlayer, the large magnon-driven MTR ratios of up to 40% have been predicted. However, even if magnons contribute 258 to 𝜅+,-  in the P state, their thermal conductivity in typical ferromagnetic metals is around ~10 W m-1 K-1 at room 259 temperature [31], which is much smaller than the estimated 𝜅'88. Therefore, the conventional sole magnon contribution 260 is unlikely to be a major factor in our observations. Thus, we should also consider the possibility of magnon-electron and 261 magnon-phonon interactions [52-55]. An example of a phenomenon to which such interactions contribute is the magnon-262 drag effect in the thermoelectric conversion [52,53]. However, the reported modulation of thermopower by the magnon 263 origin was very small in ferromagnetic metals, such as CoFe- and NiFe-based system, at room temperature [38,53]. If the 264 magnon-drag-induced change in the thermal conductivity in our CoFe/Cu multilayer is comparable to that in the 265 thermopower, its contribution is negligible. Regarding phonons, there is a possibility that they are influenced by the 266 external magnetic field through spin-lattice interaction [55,56]. However, changes in phonon thermal conductivity have 267 been observed only at the low temperature and under the strong magnetic field, which are completely different from the 268 conditions in this study. Moreover, these changes were on the order of 0.1 W m-1 K-1, making their impact relatively small. 269 We cannot entirely rule out the possibility of that some of these magnon-related interactions may still be significant, as 270 we have not explicitly evaluated their effects in our sample. To clarify the microscopic mechanism of giant MTR ratio 271 and 𝜅+,-, further experiments and calculations from a series of multifaceted perspectives are necessary.  272 V. CONCLUSIONS 273   9 In summary, we investigated the GMTR effect in the CoFe/Cu magnetic multilayer with 5.1 nm-thick CoFe 274 layers. We achieved the giant thermal conductivity change of 37 W m-1 K-1, not only surpassing the previous study [19] 275 but also marking the highest value in the solid-state thermal switching materials. The MTR ratio of 108% is found to be 276 greater than the MR ratio for the CPP configuration, reproducing the tendency of previous study for 3 nm-thick CoFe. By 277 comparing the value of 𝜅+,-, 𝜅eWF, and 𝜅LNEMD, we identified a significant contribution from additional components of 278 the thermal conductivity to the GMTR effect although this contribution has not been uncovered yet in this study. 279 Interestingly, the quantitative analysis of the lattice thermal conductivity suggests that the contribution of this additional 280 heat carriers is remarkable in P state but negligibly small in AP state. This could provide an important clue to elucidate 281 the origin of the unconventional thermal transport in the magnetic multilayers. Furthermore, this study demonstrated the 282 bistability of low and high thermal conductivity states at the same magnetic field due to the hysteretic behavior of the 283 magnetization. These findings highlight the potential of active thermal management technologies using the GMTR effect 284 for electronic devices. 285  286  287 ACKNOWLEDGEMENTS 288 The authors thank G. E. W. Bauer, P. Tang, Y. Yamashita, Y. Miura, R. Iguchi, and W. Zhou for valuation discussion and 289 S. Kuramochi and N. Kojima for the technical support. This work was supported by JST-ERATO “Magnetic Thermal 290 Management Materials Project” (No. JPMJER2201); JST-CREST “Creation of Innovative Core Technology for Nano-291 enabled Thermal Management” (No. JPMJCR17I1); JST-FOREST (JPMJFR222G), JSPS KAKENHI (Nos. 22K20495 292 and 22H04965); and NIMS Joint Research Hub Program. T.H. acknowledged support from the Thermal and Electric 293 Energy Technology Foundation. A part of the present calculations was performed on the TSUBAME4.0 supercomputer 294 at Institute of Science Tokyo. 295  296 DATA AVAILABILITY 297 The raw data are available in the Zenodo repository [57] 298  299   300   10  301 FIG.1. (a) Out-of-plane x-ray diffraction curve of the multilayer film. (b) Schematic structure of the CoFe/[Cu/CoFe]20 302 multilayer film deposited on the MgO (001) single crystalline substrate. (c) Low-magnification high-angle annular dark 303 field scanning transmission electron microscopy (HAADF-STEM) image of the multilayer film. (d)-(f) High-304 magnification HAADF-STEM images and inverse fast Fourier transform (IFFT) images reconstructed from 110 peaks of 305 (d) Upper, (e) Middle, and (f) Lower parts indicated by the rectangular regions in (c), respectively. 306   307   11  308 FIG.2. (a) Magnetic field H dependence of current-in-plane magnetoresistance (CIP-MR) and (b) normalized 309 magnetization for the CoFe/[Cu/CoFe]20 multilayer film with antiferromagnetic coupling via Cu spacer. (c) Temporal 310 response of thermoreflectance (TR) signals, i.e., TDTR signals, for the CoFe/[Cu/CoFe]20 multilayer film with applying 311 the magnetic field 𝜇*𝐻 of 0 and ±75 mT. The black solid curves represent the best fits for each measurement. (d) TDTR 312 signals for the CoFe/[Cu/CoFe]20 multilayer film at 𝜇*𝐻 of 10 mT applied with adjusting the field sweep direction. (e) 313 𝐻 dependence of 𝜅+,- of the CoFe/[Cu/CoFe]20 multilayer film. Black and green arrows in (a) and (e) indicate the sweep 314 direction of the magnetic field 315   316   12  317 FIG.3. (a) Schematic of two-current series-resistor (2CSR) model. (b) Simulated spin interfacial asymmetry  𝛾 318 dependence of resistance change-area product 𝛥𝑅𝐴 (solid curves). Dashed lines show experimentally measured 𝛥𝑅𝐴 of 319 the CoFe/[Cu/CoFe]N multilayer with repeating number N of 3, 5, and 7 [15]. Open circles show the 𝛾 values obtained 320 from the agreement between experimental and simulated 𝛥𝑅𝐴. 321   322   13 TABLE I. Used parameter for the generalized two-current series-resistor (2CSR) model 323  𝜌 [μΩ cm] 𝛽 𝜆 [nm] 𝑡 [nm] 𝑟 [mΩ μm2] 𝛾 CoFe 19.1 [22] 0.62 [23] 15 [22] 5.1 0.27 0.76±0.03 Cu 7.0 [22] 0 100 [22] 2.0 0.27 0.76±0.03  324   325   14  326 FIG.4. (a) Schematic of the model for the nonequilibrium molecular dynamics (NEMD) simulations. (b) Stacking 327 structure for NEMD. The atomic position is determined so as to be different from the adjacent CoFe layers. (c) Time 328 dependence of the heat flux for temperature difference, 𝛥𝑇 of 50 K. (d) The obtained temperature profile for 𝛥𝑇 of 50 329 K and the liner temperature approximation used to calculate the temperature gradient. 330   331   15 TABLE II. 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