Fully micromagnetic analysis of voltage-controlled magnetization switching in magnetic-topological-insulator-based devices

Voltage-controlled magnetization switching has attracted much attention for next generation spintronic devices such as non-volatile magnetic memory¹ and high-speed logic.² Voltage-controlled magnetic anisotropy (VCMA) is a promising way to realize magnetization switching with high energy efficiency, and to realize spintronics devices with low power consumption.³ The current-induced spin-orbit torque (SOT) on topological insulator (TI) is also another key method of manipulating magnetic moment,^4,5 enabling a deterministic magnetization switching without external magnetic field. Furthermore, engineering the SOT efficiency by means of a voltage is a crucial method for practical device applications.

Recently, Chiba et al. proposed the field-effect-transistor (FET)-like device which consists of TI and magnetic TI (MTI).^6–8 This device realizes magnetization switching with the writing energy of $<0.1fJ/bit$ via SOT and VCMA which originate from 2D-Dirac electronic structure of TI. Moreover, the MTI device operation is extremely robust against electronic circuit delay, noise and temperature on write-error-rate by utilizing the macrospin model.⁹ 

There are three possible ways to magnetic ordering in TI to realize the MTI device as follow: (i) magnetically-doped TIs,^10,11 (ii) intrinsic MTIs(i-MTIs),^12,13 and (iii) magnetic material - TI heterostructures.^14,15 In general, magnetically doped TIs and i-MTIs might have low exchange coupling due to long-range interaction between magnetic moments in TIs and i-MTIs. The low exchange coupling leads to some domains in the MTI device. However, the influence of domains and their treatment in MTI devices have not been clarified yet.

In this study, we demonstrate fully micromagnetic simulation of magnetization switching in the MTI devices, and investigate the influence of domain size, and compare with the switching behavior simulated by the macrospin model.

Since the exchange stiffness constant in MTI, which is needed for fully micromagnetic simulation, has not been directly reported yet, the exchange stiffness constant was estimated as mentioned below. Based on molecular field approximation, the exchange integral J is associated with the Curie temperature T_c as $J=32kBTc$⁠. Since the Curie temperature of Cr-doped $(Bi0.1Sb0.9)2Te3$ (BST) is about 15 K,¹¹ the exchange integral J can be estimated to be about 2 meV, and the exchange stiffness can be estimated from the exchange integral A = nJs²/ℓ. By assuming the averaged distance ℓ between magnetic moments to be about 0.5 nm, the exchange stiffness constant A is estimated to be about A ∼ 2 × 10⁻¹² J m⁻¹. Since the Dirac magnetic anisotropy in this system is $Ku*/d=Ku=1kJ m−3$⁠, the domain wall width, $δ=πA/Ku$⁠, can be estimated to be about 150 nm. The exchange length, $ℓex=2A/μ0Ms2$⁠, is estimated to be about 400 nm which corresponds to the domain size D. The reported domain size D of Cr-doped BST was about 500 nm, and the domain wall width was ranging from 150 to 300 nm.¹⁶ 

Herein, note that the thickness dependence of domain size D. The thickness dependence of domain size has been discussed by some authors.^17–20 The domain size D can be determined by the energy balance between magnetostatic energy and domain wall energy. The dipolar length D₀ can be expressed as $D0=2AKu/(πMs2)$⁠. The dipolar length D₀ in this study is estimated to be about 2.8 µm, much larger than the film thickness due to the small domain wall energy and the small saturation magnetization. Thus, the domain size is sufficiently larger than the MTI device dimensions.

In the VC-MTI device, the VCMA is a trigger to switch the magnetization direction while the SOT induces magnetization switching trajectory, i.e., the SOT due to source-drain electric field takes a role of an external magnetic field in voltage-controlled magnetic tunnel junction (VC-MTJ) devices. After the application of the adequate gate voltage, the perpendicularly magnetic anisotropy is vanished. Then, the magnetization starts precessional motion and the magnetization passes through equator or polar angle θ of π/2. Whether the magnetization orients northern or southern hemisphere can be determined by the periodic oscillation depending on the SOT strength. The periodic oscillations with pulse width in the device is very similar to that of VC-MTJ. Thus, the periodic oscillations in the VC-MTI were also investigated by the fully micromagnetic simulation.

First, we have investigated domains in the VC-MTI device with the size L. In the VC-MTI devices with the size of 5 and 1 µm, the equilibrium magnetization distributions were calculated when random magnetizations were the initial inputs. In case of 5 µm size, two domain structure appears (not shown). On the other hand, in case of 1 µm size, the equilibrium state is an uniform magnetization. These calculation results are in agreement with the experimentally reported domain size of about 500 nm. For practical application of VC-MTI, since the device size L is smaller than 100 nm, the influence of domains can be neglected.

In order to investigate how to deal with cases where domains exist, the semi-stable state of two domains was forcibly formed in the VC-MTI device with 1 µm as shown in Fig. 2. The calculated domain wall width in the initial state was about 120 nm, which is in good agreement with the above-mentioned domain wall width. The gate pulse of 0.39 V in which the Dirac magnetic anisotropy goes to zero was induced to erase the domain wall as shown in Fig. 2(b). Moreover, after turning off the gate voltage V_g, the source-drain electric field E_sd of 40 kV/m was immediately applied. Then, the magnetization gradually orients along the perpendicular direction due to Dirac magnetic anisotropy as shown in Fig. 2. Figure 2(c) shows the time evolution of each component of averaged magnetization ⟨m⟩. It was confirmed that the domain wall was swept out in about 20 ns after applying the gate voltage. When the gate voltage turns off, that is when the magnetic anisotropy is restored, the magnetization orients along the perpendicular direction, and is stabilized. This operation corresponds to the refresh operation in the VC-MTI device for practical use. Thus, if the domain exists in the device, the domain wall can be swept through the refresh operation. If there are polycrystals or defects in the MTI device, a multi-domain structure is likely to be formed. However, since some magnetic domains can not exist when the device size is smaller than domain wall of 120 nm, the influence of magnetic domains can be neglected for practical application.

The magnetization reversal in the VC-MTI device with the size L of 100 nm by combination of Dirac-VCMA and Dirac-SOT was simulated by the fully micromagnetic simulation. The switching behaviors with the macrospin model are in good agreements with those of the fully micromagnetic simulation as shown in Fig. 3. In the case of domain wall motion mode for the magnetization reversal, imperfections in the thin film such as polycrystallinity and defects result in domain wall pinning. However, magnetization reversal in MTI devices is a rotation mode, and MTI devices are expected to be less susceptible to defects. Thus, for the VC-MTI device with less-than-100 nm size, the macrospin model is a good approach for analysis of device operation and write-error-rate.

As previously reported, the magnetization reversal can be determined by the gate pulse width depending the source-drain electric field.⁹  Figure 4 shows the final state of magnetization after applying gate pulse with square shape and the duration τ_g for various source-drain electric field strength E_sd in the VC-MTI device with the size of 100 nm. For comparison, the macrospin calculation results are also shown in Fig. 4. The calculated pulse width in which magnetization reversal occurs becomes wider as E_sd decreases. This trend in the fully micromagnetic analysis is consistent with the macrospin analysis. However, the gate pulse width obtained by the fully micromagnetic analysis does not quantitatively agree with the macrospin results. Because the demagnetizing field is spatially distributed in the device, the magnetization near the device edges is slightly different from the magnetization in the center region. Moreover, the in-plane component of demagnetizing field during magnetization reversal is not considered in the macrospin analysis. Since the in-plane component of demagnetizing field modulates the Dirac SOT field, the gate pulse duration in which magnetization reverses is modulated. Thus, the gate pulse duration in which magnetization reverses shifts to wider pulse width. The macrospin analysis is convenient for analyzing write-error-rate and device operation deviation. The gate pulse modulation in the macrospin analysis can be incorporated by modifying the demagnetizing field as $Bd=Ñm$⁠, where $Ñ$ is the demagnetizing tensor corresponding to the device shape. It was confirmed that magnetization reversal can be controlled by gate pulse application through the fully micromagnetic simulation. As a result, if the device size is smaller than 100 nm, the magnetization uniformly rotates under the application of the gate voltage and the source-drain electric field. Thus, the macrospin model analysis is valid for analysis of magnetization switching behavior and write-error-rate.

We numerically investigate magnetization switching behavior in voltage-controlled magnetic-topological-insulator-based (VC-MTI) devices by means of the fully micromagnetic simulation. First, the influence of domain in VC-MTI devices was investigated. When the device is less than 100 nm, the macrospin model is adequate for investigation of switching behavior in VC-MTI devices and the magnetization switching occurs in rotation mode. Even if the exchange stiffness is low, although magnetic domains appear in the VC-MTI device, domains disappear by applying gate voltage and source-drain electric field, which is the refresh operation in the VC-MTI device. The switching behaviors in the fully micromagnetic simulation are in good agreements with those of the macrospin model. Therefore, for the VC-MTI device with less-than-100 nm size, the macrospin model is a good approach for analysis of device operation and write-error-rate.

This work was partly supported by Grants-in-Aid for Scientific Research (Grants Nos. 20H02196, 22K14591, 22H01805, 20K03814, 18KK0132) from the Japan Society for the Promotion of Science, by the Spintronics Research Network of Japan (Spin-RNJ). This work was partially performed under the Research Program of “Dynamic Alliance for Open Innovation Bridging Human, Environment and Materials” in “Network Joint Research Center for Materials and Devices.”

The authors have no conflicts to disclose.

Takashi Komine: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Takahiro Chiba: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.