Thermal contribution to current-driven antiferromagnetic-order switching

Novel non-volatile data storage technologies, including magnetic random-access memory, resistive random-access memory, and phase-change memory, are fundamentally based on current-induced resistance changes.^1–3 Despite the different switching mechanisms inherent in each technology, the contribution of heating to the mechanisms is inevitable and has been a central focus of research in various disciplines.^4–7 With the growing interest in current-induced state transitions in electronic devices, a systematic investigation of Joule heating in electronic devices and the development of comprehensive guidelines for the calculation of temperature changes are imperative and timely.

In the field of spintronics, current-driven magnetization switching in antiferromagnets has been studied extensively over the past decade. Antiferromagnets are a class of magnetic materials in which individual magnetic moments order in such a way that there is no net magnetization. Due to their insensitivity to external perturbations and the potential to achieve high storage densities and fast operating speeds, antiferromagnetic materials are considered as promising candidates for next-generation spintronic devices.^8–11 

The use of antiferromagnets for information devices has been challenging due to the difficulty of detecting antiferromagnetic order on a macroscopic scale and the local compensation of spin-related electrical and optical responses by different magnetic sub-lattices. Recent advances, however, have demonstrated electrical manipulation and detection of magnetic order in antiferromagnets due to their characteristic electronic band structures.^12–20 In these studies, Joule heating must be carefully considered as it can play a key role in the switching or create thermal artifacts.^18,21,22

Current-driven magnetization switching has been demonstrated in D0₁₉-type non-collinear antiferromagnets, Mn₃X (X = Sn, Ge, etc.). These non-collinear antiferromagnets, characterized by a kagome atomic structure and a chiral spin configuration with three sub-lattices, give rise to a non-vanishing Berry curvature in momentum space, characterized by Weyl nodes. Consequently, such antiferromagnets exhibit significant macroscopic electrical and optical responses dependent on their magnetic structure, such as anomalous Hall, anomalous Nernst, and magneto-optical Kerr effects at room temperature, despite their negligible net magnetization.^12,23–26

The positions of the Weyl nodes are influenced by the orientation of a cluster magnetic-multipole, i.e., an octupole moment, which is aligned with a weak net magnetic moment in the case of Mn₃Sn or Mn₃Ge.^27,28 Therefore, the direction of the octupole moment can be controlled by magnetic fields and electric currents due to the interplay between Zeeman energy and spin torques.^12,14,29

Since these non-collinear antiferromagnets, such as Mn₃Sn and Mn₃Ge, have relatively low Néel temperatures, ≲430 K, Joule heating can play a critical role. Although it was initially proposed that the electrical octupole switching is primarily driven by the collective spin rotation induced by the spin–orbit torque and an in-plane magnetic field,^14,30,31 other studies have emphasized the pivotal role of heat and found that the switching occurs around the Néel temperature.^32,33 This issue is a germane problem in spintronics, where current densities are often close to device breakdown.^7,34–37 It is therefore imperative to have a general method for evaluating the Joule-heating effect.

In this article, we present a systematic methodology and an analytical model to investigate the thermal contribution to current-driven octupole switching in the non-collinear antiferromagnet, Mn₃Sn. Toward this end, we varied the effective thermal resistance independently from the electrical resistance of the device, allowing us to unambiguously identify the role of thermal vs electrical effects. Using W/Mn₃Sn films on Si/SiO₂ substrates with different SiO₂ layer thicknesses, $hSiO2$⁠, we obtained the threshold current density for octupole switching, j_th, which depends on both the SiO₂ thickness and the base temperature, T₀. We identified the heating temperature at the threshold current density, T_th, and consistently showed that T_th remains in all cases above the Néel temperature, T_N. We also developed an analytical model for calculating the Joule-heating temperature and showed that it quantitatively describes our experimental results. From numerical simulations, we illustrate the reconfiguration of the magnetic octupoles from the demagnetization state. Our results elucidate the significant role of Joule-heating in current-driven octupole switching in Mn₃Sn and provide a general method for measuring and calculating Joule-heating temperatures that is applicable to broader research areas using micro-electronic devices.

Thermally oxidized Si/SiO₂ substrates were prepared with different SiO₂ thicknesses, $hSiO2$⁠, ranging from 100 to 1000 nm. A W(7.1 nm)/Mn₃Sn(34.4 nm)/MgO(2 nm) film was deposited on these substrates at room temperature by dc magnetron sputtering. The samples were annealed at 500 °C for 30 min. We deposited Mn₃Sn on top of a W layer, because the significant surface roughness of Mn₃Sn could affect the uniformity of the top W layer. In our stack, with Mn₃Sn on top of W, an intermixing layer of WMn₂Sn can be formed at the interface between the W and Mn₃Sn during annealing (see the Methods section and Fig. S7 of the supplementary material).³⁸ Despite the possibility of the WMn₂Sn layer, the effect of the intermixing layer on the current-driven octupole switching is negligible, as shown below.

The films are polycrystalline, and the atomic ratio of Mn₃Sn is Mn:Sn ≈ 77:23 (see Fig. S9 of the supplementary material). The excess Mn atoms stabilize the hexagonal structure of Mn₃Sn.³⁹ Note that the thickness of the SiO₂ layer, $hSiO2$⁠, changes the thermal property of the substrates because the thermal conductivity of SiO₂, $ΛSiO2≈$ 1.3 W m⁻¹ K⁻¹, is about two orders of magnitude smaller than that of Si, Λ_Si ≈ 140 W m⁻¹ K⁻¹.

We first measured the anomalous Hall conductivity of W/Mn₃Sn, which depends on the positions of the Weyl nodes in momentum space.²⁸ Since the Weyl points correlate with the spin ordering in Mn₃Sn, the Hall signal can be manipulated by controlling the octupole-moment direction, $ĝ$ [Fig. 1(a)]. The anomalous Hall effect is maximized when $ĝ$ is perpendicular to the film plane, and in the case of polycrystalline Mn₃Sn, the signal amplitude depends on the average of the out-of-plane component of the octupole moment over the sample, $⟨ĝ⟩$ [Fig. 1(b)].

In Fig. 1(c), we plot the Hall conductivity, σ_xy, as a function of the perpendicular magnetic field, H_z, for different $hSiO2$ at T = 300 K. As can be seen in Fig. 1(c), all samples show similar responses to H_z regardless of $hSiO2$⁠. The residual Hall conductivity is σ_xy,0 ≈ 31 Ω⁻¹ cm⁻¹, which is comparable to the previously reported value, ∼20 Ω⁻¹ cm⁻¹, in a single-layer polycrystalline Mn₃Sn film.^14, σ_xy,0 of polycrystalline Mn₃Sn is lower than that of epitaxially grown W/Mn₃Sn, ∼40 Ω⁻¹ cm⁻¹.³⁰ The coercive field strength, μ₀H_c = 0.6 T, is about three times larger than that of epitaxially grown Mn₃Sn films, μ₀H_c ≈ 0.2 T.^30,40

We also obtained the hysteresis curves at different base temperatures, T₀, ranging from 200 to 400 K. Note that Mn₃Sn changes the phase from the non-collinear state to the incommensurate spin-spiral state below 275 K.^41,42 As shown in Fig. 1(d), the shape of the hysteresis loops depends on T₀. σ_xy,0 increases with increasing T₀ from 200 to 260 K and then decreases with increasing T₀. σ_xy,0 becomes zero when T₀ reaches the Néel temperature, T_N ≈ 410 K. The coercivity decreases monotonically with increasing T₀ due to thermal fluctuations, which leads to a decrease in the effective magnetic anisotropy.

Subsequently, we measured the current-driven octupole-moment switching. Similar to the case of heavy-metal/ferromagnet structures, the octupole moments can be controlled deterministically by the spin–orbit torque from the adjacent heavy-metal layer.¹⁴ Note that the contribution of the intergrain spin-transfer torque to the switching is negligible because it is much smaller compared to the spin–orbit torque.^29,43 To generate the spin–orbit torque, we applied a dc electric pulse to W/Mn₃Sn with a current density j [red arrow in Fig. 2(a)] and simultaneously applied a small in-plane dc magnetic field, H_x, in the current direction [green arrow in Fig. 2(a)]. The pulse duration time and magnetic field strength were 105 ms and 100 mT, respectively. The fall time of the pulse is sufficiently long, ∼150 µs, so that we can avoid multi-stable octupole switching³² (see Fig. S1 of the supplementary material). μ₀H_x = 100 mT is below the coercivity, μ₀H_c = 0.6 T, and the value is optimized to maximize the switching efficiency.³³ The final effective octupole direction is determined by the directions of the spin–orbit torque and the magnetic field. After injecting the write current pulse, we measured σ_xy with a sufficiently small read current density, $∼1.2×109$ A m⁻², to detect the octupole state in Mn₃Sn.

In addition, we observed the octupole switching using the magneto-optical Kerr effect. First, we measured the Kerr rotation angle, θ_K, as a function of H_z [Fig. 2(b)]. The coercivity in this hysteresis loop is about 0.6 T, which corresponds to the coercivity obtained from the anomalous Hall effect in Fig. 1(c). Figures 2(c) and 2(d) show the differential magneto-optical Kerr effect microscopy images after applying a positive (j > 0) and a negative (j < 0) pulse with positive H_x, respectively. The brightness in the current path changes when the pulse direction is reversed, while the outside brightness is almost conserved. These Kerr images clearly show the current-driven octupole switching.

We measured the hysteresis loops of the current-driven octupole switching for different $hSiO2$ and obtained the threshold current density, j_th, for switching [see Figs. 2(e) and 2(f)]. We used the total thickness of the film, h_f = 41.5 nm, including both W and Mn₃Sn layers to calculate the current density, j. In contrast to the field-driven switching, j_th largely decreases with increasing $hSiO2$⁠. j_th decreases by ∼40% as the thickness of $hSiO2$ increases from 100 to 1000 nm. This result shows that the substrate choice plays a crucial role in spin–orbit-torque-driven octupole switching. It also implies that the intermixing layer, WMn₂Sn, has a negligible effect on the current-driven octupole switching, which is confirmed by the switching behavior.

Note that we did not perform the current-driven octupole switching without a SiO₂ layer because the resistance of the Si wafer, which is on the order of k℧ or less, could change the threshold current density due to current shunting.

We investigated the temperature excursion due to Joule heating for different $hSiO2$ to identify the switching temperature. At steady state, where the generated heat balances the heat loss to the substrate, the temperature increase, ΔT, in the W/Mn₃Sn microstrip is proportional to the input power, P, i.e., ΔT = R_θP, where R_θ is the effective thermal resistance of a substrate. It is important to note that the temperature approaches a saturated state at 105 ms because the heat diffusion distance in silicon, ∼2 mm, is much larger than any length scale in our device, ≲100 μm (see Fig. S2 of the supplementary material).

To measure R_θ for different $hSiO2$⁠, we first determined the longitudinal resistivity, ρ_xx, of W/Mn₃Sn as a function of the base temperature, T₀. A small electrical measurement current density, $∼1.2×109$ A m⁻², was applied to minimize the temperature increase due to Joule heating, $<$1 K. Figure 3(a) shows a representative ρ_xx–T relationship for $hSiO2$ = 500 nm. ρ_xx of the W/Mn₃Sn film increases monotonically, but the slope gradually decreases because ρ_xx of W and Mn₃Sn has different temperature dependencies. For W, ρ_xx is mostly proportional to T₀, as is typical for conventional metals, whereas Mn₃Sn exhibits a non-linear response. In particular, ρ_xx of Mn₃Sn increases with increasing temperature below 300 K and then decreases moderately [Figs. 3(b) and 3(c)].

We also measured ρ_xx as a function of j at room temperature [Fig. 3(d)]. For this measurement, we applied a 105 ms current pulse, similar to those used in current-driven switching, and observed the maximum resistivity before the end of the pulse. As shown in Fig. 3(d), ρ_xx increases with increasing j due to Joule heating. Note that ρ_xx largely follows a quadratic function of j, but there are deviations because ρ_xx is not strictly proportional to T₀.

Using the relationships of ρ_xx–T₀ and ρ_xx–j in Figs. 3(a) and 3(d), we determined the temperature increase in W/Mn₃Sn due to Joule heating, ΔT. The results are plotted as a function of P for different $hSiO2$ in Fig. 3(e). The plots show that ΔT is proportional to P and the heating temperature depends on $hSiO2$⁠. We obtained the effective thermal resistance, R_θ, from the slope of the linear fits and plotted R_θ as a function of $hSiO2$ in Fig. 3(f) (dot symbols). We observed that R_θ increases monotonically with increasing $hSiO2$⁠.

From R_θ and P_th, we calculated the threshold temperature, T_th = T₀ + R_θP_th, for various $hSiO2$ [Fig. 4]. Here, P_th represents the threshold input power obtained from j_th and ρ_xx(j). T_th is mostly independent of $hSiO2$⁠, and the average T_th is about 460 K, which is higher than the Néel temperature, T_N ≈ 410 K. This result shows that temperature plays an important role in the octupole switching of Mn₃Sn, and the switching requires temperatures above the Néel temperature. Note that the obtained T_th in Fig. 4 is the threshold temperature in the electrode, and the actual temperature at the center of the Hall bar cross is lower than T_th due to the heat and current distribution at the cross-structure.

We also measured T_th below room temperature. As shown in Fig. 1(d), σ_xy,0 peaks around T₀ ≈ 260 K and then decreases with decreasing temperature due to a phase transition.⁴¹ Therefore, here, we measured T_th between 220 and 300 K. As shown in Fig. 5(a), the hysteresis behavior varies with T₀. In particular, j_th decreases monotonically with increasing T₀ [Fig. 5(b)]. We measured ΔT as a function of P at different T₀ and obtained R_θ ≈ 164.5 K W⁻¹ [Fig. 5(c)], which remains constant in this temperature range. By combining j_th and R_θ, we calculated T_th [Fig. 5(d)]. T_th is independent of T₀; averaging T_th is about 460 K, consistent with the average T_th in Fig. 4. These consistent results confirm the reliability of our temperature measurement method.

In Figs. 4 and 5(d), we show that the switching temperature, T_th, remains always above the Néel temperature over all substrate choices and base temperatures. This result implies that temperature plays an important role in current-driven octupole switching in Mn₃Sn films. Assuming that switching occurs when the temperature reaches a certain threshold, ∼T_N, the dependence of j_th on $hSiO2$ and T₀ can be explained by the current density required to reach the Néel temperature, j_N.

We calculated the current density required to reach T_N. Let us consider a conductive strip placed on a sufficiently wide Si/SiO₂ substrate. The film thickness, h_f, is significantly thinner than the width of the strip, w, i.e., h_f ≪ w. As soon as an electric current is applied, the temperature increases due to Joule heating and stabilizes after a while within 105 ms, because 105 ms is about 10³ times longer than the characteristic thermal time scale of the device, ∼0.1 ms (see Fig. S2 of the supplementary material). In this study, we define this condition as steady state, where the temperature rise can be calculated by ΔT = R_θP, which is independent of time. The effective thermal resistance of a Si/SiO₂ substrate, R_θ, can be obtained from the thermal resistance of the SiO₂ layer, $Rθ,SiO2$⁠, and Si substrate, R_θ,Si.

First, we calculate the effective thermal resistance of the thin SiO₂ layer, $Rθ,SiO2$⁠. When the SiO₂ layer is very thin, $hSiO2≪w$⁠, the generated heat mostly flows perpendicular to the film plane, and the lateral heat flow can be neglected. In this case, $Rθ,SiO2$ can be calculated by $Rθ,SiO2=hSiO2/(wlΛSiO2)$ at steady state, where $ΛSiO2$ is the thermal conductivity of SiO₂ and l is the length of the conductor. Note that the effect of contact pads on the electrode is excluded here because Joule heating in these regions is drastically reduced due to the low current density.

Next, we consider the effective thermal resistance of the bulk Si substrate, R_θ,Si. In a two-dimensional model with a long electrode on a large substrate, R_θ,Si is given by R_θ,Si = ln(ηl_d/w)/(πΛ_Sil), where $ld=Dtp$ is the thermal diffusion length, η is a constant, D = Λ/(ρC) is the thermal diffusivity, ρ is the density, and C is the specific heat.⁴⁴ However, in our system, R_θ,Si is almost saturated within ∼1 ms, which is much shorter than the pulse length, 105 ms (see Fig. S2 of the supplementary material). Therefore, R_θ,Si is considered as a time-independent constant in this study.

Since the two-dimensional model is not suitable for our system, we numerically calculated the effective thermal resistance of a Si substrate at steady state, R_θ,Si, considering different values of w, D, and l (see Fig. S4 of the supplementary material). From the simulations, we extrinsically obtained an analytical form, R_θ,Si = ln(η′l/w)/(πΛ_Sil), where η′ is about 5 when l ≪ h_Si, where h_Si is the thickness of the substrate. In our experiments, l = 120 μm, w = 20 μm, and h_Si = 500 μm.

To confirm the universality of our model, we calculated the threshold current density using Eq. (2) in the case of 100% octupole configuration switching in epitaxial W/Mn₃Sn heterostructures .³⁰ In this study, the threshold current density is ∼7.5 × 10¹⁰ A m⁻². Using appropriate parameters, Λ_MgO = 40 W m⁻¹ K⁻¹, T_th = 460 K, ρ_xx = 81 µΩ cm⁻¹, h_f = 37 nm, w = 32 μm, and l = 200 μm, we obtain j_N ≈ 7.9 × 10¹⁰ A m⁻², which is in good agreement with their experimental results. This calculation result implies that temperature also plays a crucial role in octupole switching in epitaxially grown W/Mn₃Sn films.

So far, we have demonstrated the importance of temperature in current-driven octupole switching in Mn₃Sn through systematic experiments and an analytical model to evaluate the Joule heating effects.^32,33 When an electric pulse with j > j_N is applied, the temperature rises above T_N, leading to the disappearance of the octupole, while the individual magnetic moments exhibit random dynamics in the paramagnetic state [Fig. 6(a)]. As the current decreases after the pulse, the temperature falls below T_N, and the magnetic moments begin to rearrange from a random orientation. Temperature alone does not provide a preference for the octupole orientation. Instead, the averaged octupole orientation is influenced by the residual spin torques below T_N. Octupole reconfiguration is initiated near the interface between W and Mn₃Sn due to the short spin-diffusion length in Mn₃Sn, ∼1 nm, which seeds the spin texture for the entire layer.³³ 

To investigate the control of the octupole state from the demagnetization state, we performed numerical calculations (see the Methods section and Fig. S6 of the supplementary material). We computed the spin dynamics of a thousand unit cells with random crystal orientations and random initial spin configurations to consider the polycrystalline structure and the demagnetization state just below T_N, respectively. For simplicity, we have not considered the self-induced spin-transfer torque in the polycrystalline structure because its effect is much smaller than that of the spin–orbit torque.^29,43

Once the simulation starts, the octupole is rapidly formed from a demagnetized state in ∼0.1 ns, followed by the rotational dynamics of the octupole moment (see Fig. S6 of the supplementary material). The octupole motion mostly stops at the energy minimum state in a few nanoseconds. The final octupole orientation is determined by the spin–orbit torque, the magnetic field, and the crystal orientation. In these calculations, we assumed that the octupole dynamics occur under nearly constant current and temperature just below T_N, because the fall time of the current, ∼150 µs, is much longer than the time for the octupole dynamics, ≲10 ns.

The distributions of the final octupole orientations are shown in Fig. 6(b) for different current densities, j_hm = −6 × 10¹¹, 0, and 6 × 10¹¹ A m⁻², where j_hm represents the current density in the heavy-metal layer. These orientations are obtained in the ground state after both current and field are turned off. When no current is applied, i.e., j_hm = 0, the octupole directions are mainly oriented in the +x direction due to the magnetic field and are uniformly distributed in the z-direction, resulting in an almost zero average out-of-plane octupole moment, ⟨g_z⟩. In the presence of an electric current, however, the octupoles show a preference for certain orientations depending on the current direction. Positive and negative currents induce more negative and more positive ⟨g_z⟩, respectively. Figure 6(b) shows that octupole orientations can be determined stochastically by the current direction in the polycrystalline Mn₃Sn.

Figure 6(c) shows that ⟨g_z⟩ is proportional to j, and this implies that the switching efficiency is proportional to j_th and j_N, because the octupole formation occurs just below T_N. The threshold current density, j_N, can be obtained from Eq. (2) by assuming a quasi-static state. To confirm the relation between j_th and switching efficiency, we plotted the switching efficiency, j_N, and j_th as a function of $hSiO2$ in Fig. 6(d). The switching efficiency is obtained from $σxy,0sot/σxy,0field$⁠, where $σxy,0field$ and $σxy,0sot$ are the residual anomalous Hall conductivity obtained from field- and current-driven octupole switching, respectively [see Figs. 1(c) and 2(e)]. These results confirm that the switching efficiency is proportional to j_N and j_th, which is consistent with the numerical simulation results in Fig. 6(c).

Finally, we discuss the switching efficiency in polycrystalline Mn₃Sn films. As shown in Fig. 6(d), the maximum switching efficiency, ∼0.3, was obtained at $hSiO2$ = 100 nm. Achieving 100% switching efficiency is challenging in polycrystalline samples, due to the presence of crystal orientations where the torque required to rotate the octupole moment is negligible. Note that the torque is proportional to the out-of-kagome-plane component of the spin polarization generated by the adjacent heavy metal. In addition, in polycrystalline Mn₃Sn films, the ground state shows sixfold symmetry, resulting in possible film-plane easy axes for the octupole vector. The film-plane easy axes can be eliminated by introducing twofold perpendicular magnetic anisotropy of the octupole using an epitaxial in-plane tensile strain.³⁰ Nonetheless, the switching efficiency can be increased more than 0.3 by using a substrate with high thermal conductivity, Λ, because j_N is approximately proportional to Λ. A higher j_N can induce a larger torque on the octupole moments, resulting in higher switching efficiency. In addition, reducing the magnetic anisotropy can increase the switching efficiency by lowering the energy barrier for the octupole rotation.

In this study, we studied a systematic methodology and an analytical model to evaluate the temperature excursion due to Joule heating in spintronic devices based on current-driven octupole switching in the non-collinear antiferromagnet, Mn₃Sn. The results show that the threshold current density for switching depends on both the substrate choice and the base temperature, while the switching temperature remains essentially constant above the Néel temperature. The calculated current density for reaching the Néel temperature from an analytical model quantitatively describes the dependence of the switching current density. From numerical calculations, we showed the octupole reconfiguration during cooling in polycrystalline Mn₃Sn from the demagnetization state. This work elucidates the role of temperature in octupole switching in Mn₃Sn and offers fundamental insights into the demagnetization-mediated octupole reconfiguration. In addition, it provides a general guideline for characterizing the Joule-heating temperature in micro-electronic devices, thus allowing a careful assessment of thermal effects, which is often a concern in spintronic devices.

It is important to note that, in this study, we focused only on spin–orbit-torque-assisted octupole switching with an adjacent heavy-metal layer. However, it has recently been reported that switching can also be achieved by other spin torques, such as the self-induced spin-transfer torque and the orbital Hall effect, where the role of temperature in octupole switching may be different.^29,43 Therefore, further studies with different spin-torque sources are needed.

See the supplementary material for detailed methods and simulation conditions for temperature distributions and spin dynamics calculations. In addition, the spectra obtained from the x-ray diffraction, the x-ray reflectivity, and the Rutherford backscattering and the magnetization of a W/Mn3Sn film are provided.

This research was supported by the NSF through the University of Illinois Urbana-Champaign Materials Research Science and Engineering Center under Grant No. DMR-1720633 and is carried out in part in the Materials Research Laboratory Central Research Facilities, University of Illinois.

The authors have no conflicts to disclose.

M.-W.Y., A.H., and D.G.C. designed the experiments and developed the concept of the study. M.-W.Y. performed the sample fabrication, measurements, numerical calculations, and data analysis. M.-W.Y. and V.O.L. developed the MOKE magnetometer and microscopy systems. D.G.C. developed an analytical model of the effective thermal resistance. A.H. and D.G.C. jointly supervised this project. All authors contributed to scientific discussions and revised the manuscript.

Myoung-Woo Yoo: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (lead); Methodology (equal); Writing – original draft (lead); Writing – review & editing (equal). Virginia O. Lorenz: Funding acquisition (equal); Methodology (supporting); Supervision (equal); Writing – review & editing (supporting). Axel Hoffmann: Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (supporting); Supervision (equal); Writing – original draft (supporting); Writing – review & editing (equal). David G. Cahill: Conceptualization (lead); Data curation (supporting); Formal analysis (equal); Funding acquisition (equal); Investigation (supporting); Methodology (lead); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article and its supplementary material.