Field-free switching of perpendicular magnetization in a ferrimagnetic insulator with spin reorientation transition

Abstract

Writing magnetic bits through spin-orbit torque (SOT) switching is promising for fast and efficient magnetic random-access memory devices. While SOT switching of out-of-plane (OOP) magnetized states requires lateral symmetry breaking, in-plane (IP) magnetized states suffer from low storage density. Here, we demonstrate a field-free switching scheme using a 5-nanometer europium iron garnet film grown with a (110) orientation that shows a spin reorientation transition from OOP to IP above room temperature. This scheme combines the benefits of high-density storage in the OOP states at room temperature and the efficient field-free SOT switching in the IP states at elevated temperatures. While conventional switching of OOP bits faces the dilemma that high OOP anisotropy is required to improve bit stability and low OOP anisotropy is required to lower switching current density, this scheme disentangles this interdependence, allowing for low switching currents to be possible without sacrificing the bit stability, offering opportunities for future memory devices.

A field-free switching mechanism powered by a spin reorientation transition is insensitive to the magnitude of symmetry breaking.

INTRODUCTION

Electrical control of magnetization is key to modern nonvolatile memory technologies such as magnetic random access memory (1). A spin polarization layer, in the form of a spin-Hall layer (2) or a ferromagnet spin filter (3), converts charge currents to spin currents, which exert spin-orbit torques (SOTs) (2, 4, 5) or spin transfer torques (6, 7), respectively, to switch the magnetization in storage bits. SOT-based switching has attracted research interest because of its favorable properties, including lower write current (4), faster switching speed, and improved endurance (8, 9) compared to the spin transfer torque–induced switching counterpart.

SOT switching has been demonstrated for both in-plane (IP) (10) and out-of-plane (OOP) magnetized films (2, 4, 5). While IP switching requires a lower switching current density (10) and can occur in the absence of an external field, it faces the challenge of a lower storage density. On the other hand, higher storage densities (11) and more convenient readout can be achieved with OOP magnetized bits, but switching relies on symmetry breaking with an IP field, which adds device complexity. SOT switching of perpendicularly magnetized ferromagnets without an external IP field has been achieved by introducing lateral symmetry breaking elements (12–22), unconventional spin torques through competing spin currents or low-symmetry materials (23–31), exchange bias or interlayer coupling (32–40), or vertical composition gradients (41–44) that break the symmetry between up and down states. However, a scheme that can exploit the benefits of OOP storage and IP switching while avoiding their drawbacks has not been explored.

The spin reorientation transition (SRT) is a change in the anisotropy easy axis, for example, from OOP to IP in a magnetic film (45). The SRT has been demonstrated to occur through changes in the film thickness, temperature, magnetic field, or external pressure in a wide range of magnetic materials (46–53). We can envision a storage scheme that stores and reads data in the OOP state and writes data in the IP state, where the magnetization directions in the IP and OOP states are correlated as the material passes through its SRT.

In this work, we present a mechanism for field-free switching that combines the merits of high storage density for OOP bits and low write current density for IP SOT switching by using a 5-nm-thick europium iron garnet (Eu₃Fe₅O₁₂; EuIG) ferrimagnetic thin film, where an SRT exists at elevated temperatures. The switching mechanism relies on two key ingredients. First, the uniaxial anisotropy axis changes from OOP (IP) to IP (OOP) as the temperature increases (decreases) across the SRT. Second, the spin currents injected antiparallel to the IP magnetic moment lead to IP switching through the action of antidamping torques. We show that the OOP up (down) and IP left (right) states are correlated across the SRT because of a finite substrate miscut that ensures deterministic switching instead of demagnetization or stochastic switching. We find that the current density for deterministic switching is independent of the miscut angle. Not only does this scheme combine the benefits of high-density storage in the OOP state and low write current in the IP state, it also shows the added advantage of disentangling the dependence of switching current density on the bit stability, which both depend on the OOP anisotropy in conventional OOP switching. This scheme opens up opportunities to explore hybrid switching protocols by engineering systems with an SRT.

RESULTS

To establish a system with an OOP-to-IP uniaxial SRT, we deposit 5-nm EuIG on a Ga₃Ga₅O₁₂ (GGG) substrate with a (110) orientation using pulsed laser deposition. The symmetry of the (110) orientation gives rise to an IP uniaxial anisotropy with [1$\overline{1}$ 0] being the IP easy axis, in addition to an OOP uniaxial anisotropy with [110] being the OOP easy axis. Contributions to the total anisotropy K_tot include magnetostatic (K_MS), magnetoelastic (K_ME), surface (K_surf), and magnetocrystalline (K_MC) anisotropy (54, 55). The theoretical details of the anisotropy landscape and its contributions established to determine the easy and hard axes are discussed in the Supplementary Materials. As illustrated in Fig. 1 (A and B), the SRT represents a transition in the lowest energy anisotropy axis (easy axis, ${\widehat{\mathbf{n}}}_{\mathbf{e}}$ ) from OOP along [110] to IP along [1$\overline{1}$ 0] when the temperature increases across an SRT temperature (T_SRT). We define the plane containing these two easy axes ${\widehat{\mathbf{n}}}_{\mathbf{e}}$ , OOP [110] and IP [1$\overline{1}$0], to be the SRT plane, which is parallel to the crystallographic plane (001) and normal to the film plane (110). Because of the large uniaxial anisotropy contributed by K_ME and K_surf, the net magnetization is confined within the SRT plane. Both K_ME and K_surf lower the anisotropy energy along OOP [110] compared to IP [1$\overline{1}$0], while K_MS raises the anisotropy energy along OOP [110] compared to IP [1$\overline{1}$0]. As the temperature increases, K_ME and K_surf decrease in magnitude more rapidly than K_MS (54, 56, 57). K_ME and K_surf directly correlate with the rare-earth spin-orbit coupling, which decays quickly with temperature (58). K_MS is less strongly temperature-dependent for EuIG because of the exchange-induced magnetism in Eu³⁺, as illustrated experimentally and theoretically in the literature (59). An SRT occurs when K_MS exceeds K_ME and K_surf at T_SRT and the remanent magnetization state transitions from OOP to IP.

Fig. 1. SRT in a EuIG (5 nm)/GGG (110) sample.

(A and B) Schematics of the film stack and the evolution of the lowest energy anisotropy axis (easy axis, ${\widehat{\mathbf{n}}}_{\mathbf{e}}$ ) with temperature (T) plotted with respect to the crystallographic directions. The easy axis ${\widehat{\mathbf{n}}}_{\mathbf{e}}$ changes from (A) OOP [110] at a temperature below the SRT temperature (T < T_SRT) to (B) IP [1$\overline{1}$ 0] when T > T_SRT. The solid arrows represent the remanent magnetization directions, colored (A) in black for T < T_SRT and (B) in blue for T > T_SRT. The plane containing the two easy axes [110] and [1$\overline{1}$0], represented by the vertical plane colored in orange, is defined as the SRT plane. (C and D) Magnetic hysteresis loops measured (C) along OOP [110] at a temperature of 310 K < T_SRT and (D) along [1$\overline{1}$ 0] at 380 K > T_SRT. (E) Remanent magnetization M_r along [110] and [1$\overline{1}$ 0] is plotted in black circles and blue squares, respectively, as a function of temperature. A stepwise function is simultaneously fitted to the two sets of data, which gives a saturation magnetization M_s = 17 ± 1 kA/m and an SRT temperature T_SRT = 372 ± 5 K, where a high IP M_r is observed.

Experimentally, we characterize the SRT behavior by measuring the magnetic hysteresis loops along the OOP [110] and IP [1$\overline{1}$ 0] as a function of temperature using vibrating sample magnetometry. The square hysteresis loops measured along OOP [110] at T = 310 K (Fig. 1C) and IP [1$\overline{1}$ 0] at T = 380 K (Fig. 1D) indicate a transition in ${\widehat{\mathbf{n}}}_{\mathbf{e}}$ from OOP [110] at a temperature below the SRT (T < T_SRT) to IP [1$\overline{1}$ 0] above the SRT (T > T_SRT). Additional hysteresis loops obtained at intermediate temperatures in the Supplementary Materials (fig. S1) verify this transition. We extract the remanent magnetization M_r from the hysteresis loops at each temperature. M_r along OOP [110] (IP [1$\overline{1}$0]) decreases (increases) as the temperature increases from 300 to 390 K, marking a T_SRT = 372 ± 5 K, where the M_r becomes fully IP along [1$\overline{1}$ 0] (Fig. 1E). This temperature is well below the Curie temperature T_c = 560 K, above which the ferrimagnetic order vanishes. A continuous transition in M_r from fully OOP to fully IP is observed, rather than a sharp transition, because of the effect from a substrate miscut (discussed later) and possible spatial variation of the anisotropy landscape across the film. The saturation magnetization is measured to be M_s = 17 kA/m, which is smaller than the bulk value because of the presence of a magnetic dead layer. The bulk saturation magnetization is extracted to be M_s,0 = 77 kA/m from a thickness series of the EuIG film grown under the same condition (54).

Without additional symmetry breaking, the OOP states (up and down) have an equal probability of transition into the two IP states (left and right) across the SRT and vice versa, which will lead to nondeterministic switching with the proposed scheme. Here, in the presence of a finite substrate crystal miscut, which is commonly present (60) and can be well controlled to 0.1° accuracy (61, 62), an IP state is correlated to an OOP state across the SRT, and the selectivity depends on the sign of the miscut. To specify the angles relevant to describing the crystal miscut in the material, we define a coordinate system (Fig. 2A) where $\widehat{\mathbf{z}}$ is the film normal, and $\widehat{\mathbf{x}}$ and $\widehat{\mathbf{y}}$ are direction vectors within the film plane, which are azimuthally aligned to IP [001] and [1$\overline{1}$ 0] crystallographic directions, respectively. θ and ϕ are polar and azimuthal angles defined from the $\widehat{\mathbf{z}}$ and $\widehat{\mathbf{x}}$ axes, respectively. For the EuIG (5 nm)/GGG (110) sample used in this work, the substrate miscut, defined as the angle between the (110) crystal plane and the film plane, is measured to be δθ_max = 1.13 ± 0.05° at ϕ|_θmax = 28 ± 2° by performing an azimuthal (ϕ) scan with high-resolution x-ray diffraction (HRXRD) (Fig. 2B). The included miscut angle on the yz plane, which coincides with the SRT plane, is extrapolated to be θ_yz = 0.53 ± 0.05° from the sine fit function (red curve in Fig. 2B).

Fig. 2. Miscut characterization with HRXRD and the resultant correlation between IP and OOP states.

(A) The crystal miscut δθ, defined as the included angle between the (110) crystal plane and the film plane, is a function of the azimuthal angle ϕ in the xy plane. δθ_max represents the maximum miscut angle that occurs at an azimuthal angle denoted as ϕ|_θmax. δθ_yz ≡ δθ (ϕ = 90°) is the crystal miscut projected on the yz plane, which is also the SRT plane. (B) The crystal miscut δθ is measured as a function of the azimuthal angle ϕ using HRXRD. Data are fitted with a sine function, shown as the red curve, from which δθ_max is extrapolated to be 1.13 ± 0.05° and δθ_yz to be 0.53 ± 0.05°. (C) Schematics for the correlation between IP and OOP states across the SRT. A single angular variable, θ_M, represents the net magnetization direction (M). The correlated IP and OOP states are represented by arrows of the same color (up and left in blue and down and right in red). (D) Magnetization components $m_y$ (dashed) and $m_z$ (solid) plotted as a function of the effective OOP anisotropy $K_{\text{OOP}}^{\text{eff}}$ for different miscut angles, $\mathrm{\delta }{\mathrm{\theta }}_{\mathit{yz}}=0.001°$ , $0.01°$ , $0.1°$ , and $1°$ . The inset shows an illustration of the OOP-to-IP transition. The top panel shows the transition between up ( $m_z=1$ ) and right states ( $m_y=1$ ), while the bottom panel shows the transition between down ( $m_z=-1$ ) and left states ( $m_y=-1$).

We illustrate the correlation between IP and OOP states arising from a finite substrate miscut by considering how the anisotropy landscape evolves across the SRT (Fig. 2, C and D). Because the large IP anisotropy confines the net magnetization within the yz plane (SRT plane), the magnetization vector M can be represented by a single angular variable θ_M (Fig. 2C). The total anisotropy energy can be considered as a sum of two uniaxial anisotropies: K_e,1 with the axis aligned along ${\widehat{\mathbf{n}}}_{\mathbf{e}\mathbf{,}\mathbf{1}}=(0,\text{sin}\mathrm{\delta }{\mathrm{\theta }}_{\mathit{yz}},\text{cos}\mathrm{\delta }{\mathrm{\theta }}_{\mathit{yz}})$ and K_e,2 with the axis aligned along ${\widehat{\mathbf{n}}}_{\mathbf{e}\mathbf{,}\mathbf{2}}=(0,1,0)$ . The total anisotropy energy can be written as $\begin{array}{c}\begin{array}{c}\begin{array}{c}{K}_{\text{tot}}=-K_{\mathrm{e},1}{({\widehat{\mathbf{n}}}_{\mathbf{e}\mathbf{,}\mathbf{1}}·\mathbf{m})}^2-K_{\mathrm{e},2}{({\widehat{\mathbf{n}}}_{\mathbf{e}\mathbf{,}\mathbf{2}}·\mathbf{m})}^2=-K_{\mathrm{e},1}{\text{cos}}^2({\mathrm{\theta }}_{\mathrm{M}}+\mathrm{\delta }{\mathrm{\theta }}_{\mathit{yz}})-K_{\mathrm{e},2}{\text{sin}}^2{\mathrm{\theta }}_{\mathrm{M}}.\hfill \end{array}\hfill \end{array}\hfill \end{array}$ We can define an effective OOP anisotropy as a difference between the two uniaxial anisotropies $K_{\text{OOP}}^{\text{eff}}=K_{\mathrm{e},1}-K_{\mathrm{e},2}$ . A transition in the easy axis occurs when one uniaxial anisotropy dominates over the other, i.e., $K_{\text{OOP}}^{\text{eff}}$ changes sign. Contributions to the uniaxial anisotropies change with temperature. Specifically, with an increase in temperature, changes in the magnetoelastic anisotropy and surface anisotropy can lead to a decrease in $K_{\text{OOP}}^{\text{eff}}$ . Given that the 5-nm EuIG film shows an easy axis along ${\widehat{\mathbf{n}}}_{\mathbf{e}\mathbf{,}\mathbf{1}}$ with positive $K_{\text{OOP}}^{\text{eff}}$ at room temperature, a change in the sign of $K_{\text{OOP}}^{\text{eff}}$ is expected at an elevated temperature. As illustrated in Fig. 2C, at T < T_SRT, K_e,1 dominates ( $K_{\text{OOP}}^{\text{eff}}>0$ ), and the easy axis is ${\widehat{\mathbf{n}}}_{\mathbf{e}\mathbf{,}\mathbf{1}}$ , which is tilted from $\widehat{\mathbf{z}}$ by δθ_yz, while at T > T_SRT, K_e,2 dominates ( $K_{\text{OOP}}^{\text{eff}}<0$ ), and the easy axis is ${\widehat{\mathbf{n}}}_{\mathbf{e}\mathbf{,}\mathbf{2}}$ aligned to $\widehat{\mathbf{y}}$.

Figure 2D shows how the equilibrium magnetization components vary as a function of the effective OOP anisotropy $K_{\text{OOP}}^{\text{eff}}$ . The magnetization component transitions from OOP to IP as $K_{\text{OOP}}^{\text{eff}}$ decreases from positive to negative, crossing 45$°$ between the OOP and IP states when $K_{\text{OOP}}^{\text{eff}}=0$ . Because of the finite miscut, which breaks the symmetry in the transition pathways between OOP and IP states, each IP state is correlated to an OOP state across the SRT. With a positive miscut angle, δθ_yz > 0, an initial magnetization with positive m_z (up state) transitions to a final magnetization with positive m_y (right state), while a down state transitions to a left state as the effective OOP anisotropy decreases from positive to negative. Under equilibrium consideration, even a miscut angle as small as 0.001° is sufficient to give rise to the correlated transition. The dynamical analysis for nanosecond timescales is discussed later. An increasing miscut leads to a broader transition, which agrees with the experimental observations shown in Fig. 1E. The cubic symmetric magnetocrystalline anisotropy creates a small hysteresis in the SRT of ~2 K as discussed in the Supplementary Materials (fig. S2). The deterministic transition pathway across the SRT makes deterministic switching feasible with the proposed scheme. This finding is further supported by experimental data included in the Supplementary Materials (fig. S3), which shows that setting the IP state with an IP field above the SRT deterministically sets the resultant OOP state when the sample is cooled through the SRT.

To demonstrate the proposed field-free current-induced switching through the SRT, we deposited 4-nm Pt on the EuIG (5 nm)/GGG (110) sample and patterned Hall cross devices with current along the $\widehat{\mathbf{x}}$ axis (Fig. 3A). The spin-Hall effect in Pt leads to an injection of spin current polarized along $\widehat{\mathbf{y}}$ into EuIG (63), which aligns with ${\widehat{\mathbf{n}}}_{\mathbf{e}}$ above the SRT. The OOP states, up (m_z = 1) and down (m_z = −1), are detected through the transverse Hall resistance, R_H, measured along $\widehat{\mathbf{y}}$ , which gives R_H = ∓ R_AHE for m_z = ±1 through the spin-Hall magnetoresistance (64). At an environmental temperature below the SRT, T_Env = 340 K, where the M_r is fully OOP, we demonstrate conventional SOT switching where we apply an IP field along the current direction with a magnitude of μ₀H_x = ±0.16 T (Fig. 3B). Opposite field polarities give rise to opposite switching chiralities, which is predicted by conventional SOT switching for a perpendicularly magnetized film. At zero field, we observe full switching with the same chirality as is observed under a negative H_x (Fig. 3B). The switching current densities, plotted in open (solid) circles for 25% (75%) switching in Fig. 3C, show a linear dependence on the IP field (H_x), symmetric about the zero field. The switching current densities at the zero field are much lower compared to the extrapolated values from the linear fits. This indicates that the field-free switching mechanism is much more efficient compared to the conventional SOT switching of OOP states. Additional measurements at smaller IP fields than plotted in Fig. 3C (μ₀H_x = 0.02 to 0.12 T) are shown in the Supplementary Materials (fig. S4). Incomplete switching occurs within this field range because conventional SOT switching has a higher threshold current density at a lower IP field, and field-free switching through the SRT process is suppressed by the IP field, which always favors one IP state. The field dependence of threshold current density and full switching regimes is symmetric about zero (positive and negative IP fields equally suppress field-free switching), which distinguishes it from the previously reported field-free switching enabled by a tilted easy axis, where the threshold current density and full switching regimes show a lateral shift about the zero IP field (65).

Fig. 3. Current-induced switching and the effect of IP field and temperature.

(A) Schematics of the film stack and the device geometry. The current is injected along $\widehat{\mathbf{x}}$ , and transverse voltage is measured along $\widehat{\mathbf{y}}$ . (B) Magnetization switching behavior as a function of the injected current density at 340 K with an IP field along $\widehat{\mathbf{x}}$ , μ₀H_x, of 0.16 T (blue), 0 T (green), and −0.16 T (red) in magnitude. In all three cases, deterministic full switching is observed above a critical current density. (C) Dependence of switching current density on H_x. The switching current density corresponding to 25% (75%) switching is plotted in open (solid) circles. Data for the positive field are plotted in blue, the negative field in red, and the zero field in green. Black dashed (solid) lines are linear fits to the 25% (75%) switching current densities as a function of H_x. (D) Magnetization switching using a train of 10-ms-long current pulses, ±1.6 × 10¹¹ A/m² in magnitude, under a zero field measured at T_Env = 340 K. The change in transverse resistance, ∆R_H, normalized by the difference in R_H between the up and down magnetization states determined from the OOP field sweep, 2R_AHE, reflects the ratio between the switched area and the entire device area. (E) Switching ratio ∆R_H/2R_AHE plotted as a function of the environmental temperature T_Env. (F) Dependence of j_thres (onset of switching) on the device temperature T_Device at j_thres extrapolated from Joule heating calibration (see Materials and Methods). The region shaded in red represents conditions that will give rise to field-free switching.

Reproducible full switching at an environmental temperature of T_Env = 340 K under a current density j = 1.6 × 10¹¹ A/m² is shown in Fig. 3D. The fractional resistance change (∆R/2R_AHE), which represents the ratio between the switched area and the entire device area, shows 100% changes upon the reversal of the current pulse polarity. ∆R/2R_AHE increases with the environment temperature T_Env at a fixed current density (Fig. 3E), indicating that the field-free switching process is preferred at higher temperatures. Current injection in Pt creates Joule heating, which leads to a temperature offset between the local device temperature (T_Device) and T_Env. By measuring the threshold current density j_thres at different T_Env values and estimating the corresponding T_Device at j_thres, we produce a phase diagram for field-free switching (Fig. 3F), where the region colored in red indicates the current density and temperature conditions that will give rise to field-free switching. Two regimes are observed: a temperature-limited regime at T_Device = 372 K = T_SRT and a current density–limited regime at j = 0.6 × 10¹¹ A/m². This indicates that switching occurs through two steps: First, T_Device needs to reach the T_SRT where ${\widehat{\mathbf{n}}}_{\mathbf{e}}$ becomes fully IP, and second, j needs to reach a certain threshold to provide sufficient spin torque for the switching of IP states.

On the basis of our observations, we propose that the complete switching process occurs as outlined schematically in Fig. 4A. On the positive current branch, a low-resistance state (up) switches to a high-resistance state (down) at j = 1 × 10¹¹ A/m² (Fig. 4A). The magnetization states before current injection (1), during current injection (2 and 3), and after current injection (4) are shown in Fig. 4B. Once the current pulse is applied, Joule heating locally increases the T_Device to T_SRT with a typical timescale of nano- to microseconds. As a result of the symmetry breaking from a finite substrate miscut, the up state transitions to the right state across the SRT (1 → 2 in Fig. 4C). With a high T_Env close to T_SRT, only a low current density is required to provide sufficient Joule heating for T_SRT to be reached, while at a T_Env much lower than T_SRT, a high current density is required to provide sufficient heating. This step gives rise to the temperature-limited regime. The polarized spin current along $\widehat{\mathbf{p}}$ to the left exerts an antidamping torque τ_SOT = M × $\widehat{\mathbf{p}}$ × M on the magnetization M. This antidamping torque τ_SOT opposes the damping torque τ_α, which acts as a restoring torque to align the magnetization to the effective field (sum of exchange and anisotropy fields). The right state switches to the left (2 → 3 in Fig. 4C) when the SOT τ_SOT overcomes the damping torque τ_α. This step gives rise to the current-limited behavior because a minimum current density is required to provide sufficient SOT. Last, after the current pulse, the T_Device decreases to T_Env below the T_SRT. The left state then transitions to the down state (3 → 4 in Fig. 4C), which completes the process of field-free switching of the OOP state from up to down.

Fig. 4. Mechanism for field-free switching.

(A) Magnetization switching behavior as a function of the injected current density at 340 K without an applied field. (B) Schematics of intermediate states in the switching process occurring on the positive current branch. The blue arrow represents the net magnetization direction. The current pulse injection is represented by the waveform in green. The spin polarization direction $\widehat{\mathbf{p}}\mathbf{\parallel }\mathbf{-}\widehat{\mathbf{y}}$ is represented by the green arrow. (C) Schematics of the intermediate steps for switching. The beginning (end) state of the net magnetization in each step is represented by a light (dark) blue arrow. An additional light blue arrow is shown in step 2 → 3 between the beginning and end states to illustrate the competition of damping-like (τ_α) and antidamping SOT (τ_SOT) from the spin current injection. (D) Simulated magnetization dynamics from the macrospin model. The current density $j$ , effective OOP anisotropy field $H_{\mathrm{k},\text{eff}}^{\text{OOP}}$ , and magnetization components ( $m_x$ , $m_y$ , $m_z$ ) are plotted as a function of time $t_j$ , which shows the switching process through the SRT and IP switching, followed by the SRT. (E) The average and standard deviation of the final OOP magnetization, $⟨m_z^f⟩$ and $\mathrm{\sigma }\left(m_z^f\right)$ for simulation performed over a range of thermal rise times, is plotted as a function of the current density to illustrate switching regimes. The initial OOP magnetization is written as $m_z^i$ . (F) Switching regimes plotted as a function of the current density and miscut. (G) Threshold current density $j_{\text{thres}}$ for the SRT process and IP switching process plotted as a function of the relevant material parameters. (H) $j_{\text{thres}}$ for the SRT process and IP switching process plotted as a function of the EuIG film thickness $t$ . The current density colored in red shows the overall $j_{\text{thres}}$ for field-free switching.

We validate the proposed mechanism and explore the dependence of switching current densities on material parameters by simulating the current-induced dynamics (see Materials and Methods). We assumed a macrospin model, that is, the magnetization direction is described by a single unit vector $\mathbf{m}$ . A miscut angle of θ_yz = 0.5° is introduced to approximate the miscut angle characterized by HRXRD in our film. Figure 4D shows the time evolution of the deterministic field-free switching process at a current density above the threshold for both IP switching and SRT to occur. The magnetization is initialized to the up state. A current pulse is applied at time t_j = 0 and removed at time t_j = 30 ns. The current pulse is modeled to exert a SOT and induce the change in the effective OOP anisotropy field for SRT to occur. The magnetization components show an evolution from OOP (m_z = 1) to IP (m_y = 1) through the SRT, followed by IP precessional switching (m_y = −1) and, lastly, an IP (m_y = −1)–to–OOP (m_z = −1) transition.

In the simulated nanosecond pulse duration, we observe an additional nondeterministic switching regime at a current density between no switching and the deterministic switching regime shown in Fig. 4E. Nondeterministic switching occurs below the threshold current density for IP precessional switching because the dynamical component of the magnetization, which is not fully damped out during the current pulse, affects the transition between the IP and OOP states, causing it to deviate from the equilibrium process. The nondeterministic transition depends on the dynamical phase of the magnetization at the SRT as well as the miscut angle. Figure 4F shows the switching regimes observed in the simulation as a function of current density and miscut angle. The threshold current density for deterministic switching is largely independent of the miscut angle, demonstrating the robustness of this field-free switching mechanism against variations in miscut. The nondeterministic switching regime narrows at larger miscut angles because of the stronger equilibrium effect in the deterministic SRT to overcome the dynamical effect. However, this nondeterministic switching regime is likely relevant for short current pulses of 10 to 100 ns. In the timescale of 10 ms in our experiments, the current-induced oscillations likely have damped out when the SRT is reached.

The two processes involved in the field-free switching process, SRT and IP switching, depend on different material parameters (Fig. 4G). For the SRT, the switching current density j_thres increases with the ratio between the OOP anisotropy at room temperature, $K_{\text{OOP}}^{300\mathrm{K}}$ , and its temperature dependence, $\frac{d{K}_{\text{OOP}}}{\mathit{dT}}$ . While it is ideal for $K_{\text{OOP}}^{300\mathrm{K}}$ to be kept high for bit stability, j_thres can be lowered by increasing $\frac{d{K}_{\text{OOP}}}{\mathit{dT}}$ . For IP switching, $j_{\text{thres}}\propto \frac{\mathrm{\alpha }{H}_{\mathrm{k},\text{IP}}}{\mathrm{\chi }}$ , where $\mathrm{\alpha }$ is the Gilbert damping, $H_{\mathrm{k},\text{IP}}$ is the IP anisotropy field, and $\mathrm{\chi }$ is the spin torque efficiency (expressed in terms of the effective field strength per ${10}^{11}$ A/m² injected current density). Conventionally, both the switching current density and bit stability depend on the OOP anisotropy, which creates a dilemma where if one attempts to decrease the OOP anisotropy to lower the energy barrier for a lower switching current density, the bit stability will inevitably be compromised. However, for the switching process that relies on SRT and IP switching, the switching current density relies on the IP anisotropy and the temperature dependence of the OOP anisotropy. This offers a solution to the dilemma, where the OOP anisotropy can be kept high for a high bit stability and only the temperature dependence should be engineered to be high for a low SRT temperature and the IP anisotropy be engineered to be small for a low IP switching current density.

As we previously demonstrated (54), surface anisotropy contributes to the anisotropy landscape in the Pt/EuIG/GGG(110) system. Using the extracted surface and bulk anisotropies and their temperature dependences, we estimate the corresponding threshold current density for SRT and IP switching. The spin torque efficiency is modeled to be inversely proportional to the EuIG film thickness. While j_thres for SRT decreases with thickness because the OOP anisotropy decreases, j_thres for IP switching increases with thickness as the spin torque efficiency decreases. Given that both processes need to occur for field-free switching, a current density exceeding j_thres for both processes is required. As a result, the j_thres for field-free switching is expected to show a nonmonotonic dependence on the film thickness, as shown in Fig. 4G. Increasing the temperature is expected to lower the threshold current density.

Field-free switching is reproduced in a thicker 14-nm EuIG film with a higher threshold switching current density (Fig. 5A), indicating the generality of this mechanism. The temperature dependence of the threshold current density in Fig. 5B shows that the threshold current density decreases with temperature similar to the 5-nm film. However, the threshold current density saturates at a lower environment temperature compared to the 5-nm film, which agrees with the qualitative trend in Fig. 4H, where the thinner film tends to be temperature-limited and the thicker film tends to be current-limited at a given temperature. For the 14-nm film, we observe that the switching boundary widens at environmental temperatures (T_Env) above 340 K, as evidenced by the larger error bars in the threshold switching current densities in Fig. 5B and more clearly visualized by the repeated current-pulse measurements in Fig. 5 (C to F) (in which the magnetization state is reinitialized between each current pulse), where a clear switching boundary is observed at T_Env = 333 K (Fig. 5, C and D), indicating deterministic switching, and a diffuse boundary is observed at T_Env = 353 K (Fig. 5, E and F), indicating nondeterministic switching. This observation is also consistent with the macrospin simulation showing a nondeterministic switching regime in the current-limited regime, which is not expected in the temperature-limited regime.

Fig. 5. Comparison between 5- and 14-nm EuIG films.

(A) Magnetization switching behavior as a function of the injected current density for 5-nm (top) and 14-nm (bottom) EuIG films measured at an environmental temperature, T_Env = 340 K. (B) Threshold current density j_thres plotted as a function of T_Env for both film thicknesses. (C) Color map of 10 repeats of the normalized change in transverse resistance (∆R/R_AHE) measured as a function of injected current density ( j) at T_Env = 333 K for the 14-nm film. The magnetization is initialized to high (low)–resistance state for negative (positive) inject current. (D) Data points represent the average of the repeats in the color map in (C). Red curves are error function fits to extract the threshold current density from the center of the fitted error function and the error bars from the width of the fitted error function. (E) and (F) are similar to (C) and (D) but measured at T_Env = 353 K for the 14-nm film.

DISCUSSION

We compare the switching current density (j = 0.6 × 10¹¹ A/m² in the current-limiting regime) with previous works on field-free switching in Fig. 6. We note that the switching current density in this work (dashed line) is lower than most previous dozens of reports and is comparable to the current state of the art, where only three other works show a slightly lower current density. This work offers the advantage of a simple two-layer thin film stack, which is easy to fabricate. At the same time, it combines the benefits of thermally stable storage in the OOP state at room temperature and field-free current-induced switching in the IP state at an elevated temperature. The current density in this work is also lower than in four other works on a similar garnet/Pt system (12, 31, 32, 40). An interesting feature unique to this field-free switching mechanism is that the bit stability can be engineered separately from the switching current density. The bit stability can be increased by increasing the room-temperature OOP anisotropy, while the switching current density can be lowered by increasing the temperature dependence of OOP anisotropy and decreasing the IP anisotropy.

Fig. 6. Summary of the literature on field-free switching and comparison with this work.

The switching current densities from the literature of field-free switching are plotted in open symbols. Plotted data points are classified into four categories according to their strategies to achieve field-free switching. Each data point is labeled with the materials stack and the reference number: 1 (13), 2 (14), 3 (15), 4 (16), 5 (17), 6 (18), 7 (19), 8 (20), 9 (21), 10 (22), 11 (12), 12 (24), 13 (25), 14 (26), 15 (27), 16 (28), 17 (29), 18 (30), 19 (23), 20 (31), 21 (33), 22 (34), 23 (35), 24 (36), 25 (37), 26 (38), 27 (39), 28 (40), 29 (32), 30 (42), 31 (43), 32 (44), and 33 (41). The minimum current density for field-free switching in this work is plotted as the data point colored in black.

The switching current density could be further lowered by reducing the EuIG thickness to increase the spin injection efficiency or using a material system with a lower damping constant. The SRT can be brought closer to room temperature by further anisotropy landscape engineering through strain tuning or composition optimization. Notably, a relatively small included miscut angle on the SRT plane (0.5° in this work) is sufficient to break the symmetry between OOP and IP states to allow for the proposed mechanism. Theoretically, an arbitrarily small nonzero miscut angle should break the symmetry across the SRT. Given that the miscut is only required to ensure the correlation of the states across the SRT, the critical current density for deterministic switching is independent of the absolute value of the miscut and, therefore, robust to fabrication imperfections. With a previous miscut-induced field-free switching scheme, a miscut angle of 8° is required for switching to occur in a similar materials system (TmIG/Pt), together with a very small coercive field (12).

This scheme of field-free switching with the SRT can be generalized to other materials systems. Two ingredients are critical: an SRT combined with symmetry breaking between the states across the SRT. The SRT has been a widely studied topic with a wide range of materials demonstrating an SRT with temperature (47, 66, 67). Aside from the effect of substrate miscut, symmetry breaking can be achieved with a tilted OOP easy axis, which could be introduced through off-axis sputtering (65, 68), field application along a tilted angle during sputtering, or fabrication of a wedge-shaped device (18). Our scheme allows for storage in the OOP state and switching in the IP state, which maximizes the benefits of high storage density in the OOP configuration and the low write current density in the absence of an applied field in the IP configuration. Several thermally assisted magnetic storage protocols have readily developed into cutting-edge commercial products including heat-assisted magnetic recording (69) and its next-generation heated-dot magnetic recording (70, 71), with the advantage of high storage density. Similar technology such as thermomagnetic patterning has also been a mainstream technique for creating magnetic patterns (72). Our scheme of field-free switching through the SRT benefits from a similar concept of localized heating and, at the same time, allows for operation in a field-free manner. This provides a pathway to the next generation of hybrid low-energy data storage devices.

MATERIALS AND METHODS

The 5-nm EuIG film is deposited from a 99.9% elemental purity EuIG target onto a GGG substrate with a (110) crystal orientation (MTI Corporation) using pulsed laser deposition. A 248-nm KrF excimer laser (Coherent) at an energy of 350 mJ and a repetition rate of 5 Hz is focused on the target at a fluence of 1 J cm⁻². Oxygen is used as the growth atmosphere at a pressure of 150 mtorr. The film thickness and strain state are characterized by performing HRXRD using a Bruker D8 High-Resolution Diffractometer. The temperature-dependent magnetization hysteresis loops are measured using a MicroSense Vibrating Sample Magnetometer, and the temperature is varied by flowing resistively heated nitrogen gas through the sample.

The substrate miscut is measured using HRXRD on the (440) substrate reflection as a function of the IP angle ϕ following a procedure described in (73). The included angle is calculated as δθ = 2θ₍₄₄₀₎/2 − θ₍₄₄₀₎ − θ_tilt, where θ₍₄₄₀₎ and 2θ₍₄₄₀₎ are the angles for the incident and diffracted beams, respectively, of the (440) reflection, and θ_tilt is the sample tilt angle.

For switching measurements, 4-nm Pt is deposited on the EuIG (5 nm)/GGG (110) using dc magnetron sputtering. Six-probe Hall cross devices with a 25-μm current channel width are fabricated using electron beam lithography. The environment temperature T_Env is controlled via a DynaCool PPMS system. The current is applied from a Keithley 6221 current source, and the transverse voltage is read out using a Keithley 2182A nanovoltmeter. A read current density of 1 × 10¹⁰ A/m² is used to determine the state of magnetization without disturbing the state.

Joule heating from current injection in Pt is calibrated by measuring the longitudinal resistance R_xx of the six-probe Hall cross device as a function of temperature and separately as a function of current density. Data for R_xx measured as a function of the temperature show a linear dependence and are fitted with $R_{\mathit{xx}}=a+\mathit{bT}$ . Data for $R_{\mathit{xx}}$ measured as a function of the current density show a quadratic dependence and are fitted with $R_{\mathit{xx}}=c+d{j}^2$ . Therefore, the temperature rise can be correlated with the current density through $\mathrm{\Delta }T=\frac{d}{b}{j}^2\equiv e{j}^2$ . The device temperature is then estimated from the environment temperature at a specific current density through $T_{\text{Device}}=T_{\text{Env}}+\mathrm{\Delta }T=T_{\text{Env}}+e{j}^2$ . The data for Joule heating calibration are shown in the Supplementary Materials (fig. S5).

Switching dynamics are simulated through custom code to solve the Landau-Lifshitz-Gilbert equations with a time step of 1 fs to ensure accuracy, assuming a macrospin model. The Landau-Lifshitz-Gilbert equations used to describe the system are shown below

$$\begin{array}{c}\frac{d\mathbf{m}}{\mathit{dt}}=\mathrm{\gamma }{\mathrm{\mu }}_0\mathbf{m}\times {\mathbf{H}}_{\mathbf{\text{eff}}}+\mathrm{\alpha }\mathbf{m}\times \frac{d\mathbf{m}}{\mathit{dt}}+\mathrm{\gamma }{\mathrm{\mu }}_0\mathrm{\chi }j\mathbf{m}\times (\widehat{\mathbf{p}}\times \mathbf{m})\hfill \end{array}$$ (1)

$$\begin{array}{c}{\mathbf{H}}_{\mathbf{\text{eff}}}=-H_{\mathrm{h}}(\mathbf{m}·{\widehat{\mathbf{n}}}_{\mathbf{h}}){\widehat{\mathbf{n}}}_{\mathbf{h}}+H_{\mathrm{e},1}(\mathbf{m}·{\widehat{\mathbf{n}}}_{\mathbf{e},\mathbf{1}}){\widehat{\mathbf{n}}}_{\mathbf{e},\mathbf{1}}+H_{\mathrm{e},2}(\mathbf{m}·{\widehat{\mathbf{n}}}_{\mathbf{e},\mathbf{2}}){\widehat{\mathbf{n}}}_{\mathbf{e},\mathbf{2}}\end{array}$$ (2)

where ${\widehat{\mathbf{n}}}_{\mathbf{h}}$ is the anisotropy hard axis, and ${\widehat{\mathbf{n}}}_{\mathbf{e}\mathbf{,}\mathbf{1}}$ and ${\widehat{\mathbf{n}}}_{\mathbf{e}\mathbf{,}\mathbf{2}}$ are directions normal to the easy axes below or above the SRT, respectively. H_h is the anisotropy field along the hard axis, and H_e,1 and H_e,2 are the anisotropy fields along the easy axes, respectively. A miscut is introduced into the definition of ${\widehat{\mathbf{n}}}_{\mathbf{e}\mathbf{,}\mathbf{1}}$.

$$\begin{array}{c}{\widehat{\mathbf{n}}}_{\mathbf{e}\mathbf{,}\mathbf{1}}=(0,\text{sin}\mathrm{\delta }{\mathrm{\theta }}_{\mathit{yz}},\text{cos}\mathrm{\delta }{\mathrm{\theta }}_{\mathit{yz}})\end{array}$$ (3)

$$\begin{array}{c}{\widehat{\mathbf{n}}}_{\mathbf{e},\mathbf{2}}=(0,1,0)\end{array}$$ (4)

$$\begin{array}{c}{\widehat{\mathbf{n}}}_{\mathbf{h}}=(1,0,0)\end{array}$$ (5)

The current is introduced as a square function, which turns on at t_j = 0 for a duration of 30 ns. The Joule heating effect is considered to follow an exponential profile that starts to increase the temperature at t_j = 0 and decrease the temperature at the end of the current pulse t_j = 30 ns, asymptotically approaching the equilibrium temperature. H_e,2 is set to approximate the shape anisotropy field and H_e,1 is set to vary with the Joule heating profile. Additional dynamics simulation of different switching regimes is shown in the Supplementary Materials (fig. S6).

Supplementary Materials

This PDF file includes:

Supplementary Text

Figs. S1 to S8