Spin–orbit torque field-effect transistor (SOTFET): Proposal for a magnetoelectric memory

The understanding of spin transport in heterostructures¹ led to the realization of magnetic memories based on giant magnetoresistance (GMR)^2–4 and spin-transfer torque (STT).^5–8 Current research aims to make the writing process for magnetic memories more efficient using spin–orbit torques (SOTs).^9,10 STT and SOT magnetic random access memories (MRAMs) offer the virtues of non-volatility, infinite endurance, and good write speeds.⁸ Nonetheless, the modest resistance change between the magnetic “0” and “1” states of STT- and SOT-MRAMs necessitates a substantial current to obtain acceptable readout voltages, impairing read energies and speeds.

In contrast, non-magnetic semiconductor field-effect transistors (FETs) achieve several orders of magnitude resistance change in each switching event. The field effect converts a linear change in the gate voltage into an exponential change in the mobile carrier density in the semiconductor and, consequently, modulates its resistance. Thus, a material that can transduce the change in the spin/magnetic state in a SOT structure into the charge of a semiconductor channel could significantly boost the resistance change of a magnetic memory.

This requirement would be met by recently developed magnetoelectric multiferroic (MF) materials, which simultaneously possess magnetic order and ferroelectricity in a manner that these order parameters are coupled due to the magnetoelectric effect.^11–13 Moreover, exchange coupling of spins in a ferromagnetic (FM) layer to the magnetic order of a multiferroic layer across ferromagnet/multiferroic heterointerfaces has been experimentally demonstrated.^14,15

Inspired by these recent advances in SOT and multiferroic materials, we propose a promising magnetoelectric memory device, the spin–orbit-torque field-effect transistor (SOTFET). This device aims to combine the virtues of magnetic memories with the large resistance change of FETs, providing both memory and logic functionalities. Analysis of the memory aspect indicates that the SOTFET can offer several orders of magnitude increase in the on–off resistance ratio compared to existing magnetic memories, which can potentially lower the operation energy significantly. The potential logic aspect of the SOTFET would also enable circuit architectures for efficient logic or search functions.¹⁶ In this paper, we present the physical operation of the SOTFET along with a device model and will mainly focus on the memory aspect.

Figure 1 shows the structure of a SOTFET. It resembles an ordinary metal-oxide-semiconductor FET (MOSFET), but with a unique gate stack. The SOTFET gate stack comprises three layers (from top to bottom): a spin–orbit (SO) layer, a ferromagnetic (FM) layer, and a multiferroic (MF) layer, adjacent to a semiconductor channel to which source and drain contacts are made.

The working principle of the SOTFET is illustrated in Fig. 1. The state of magnetization M of the FM layer is the memory component. When a charge current J_SO flows in the SO layer, transverse spin-polarized currents are generated due to spin-momentum locking.^17–22 Spin absorption at the SO/FM interface exerts a spin–orbit torque that switches M of the FM,^8,23 as illustrated in Fig. 1(a) and qualitatively plotted in Fig. 1(b). Flowing J_SO in the opposite direction switches the magnetization between 1 and 0, identical to the conventional writing mechanism in SOT-MRAMs.

The SOTFET differs from the conventional SOT-MRAMs in the read mechanism. Coupling M of the FM with the semiconductor channel would be achieved by the magnetoelectric multiferroic layer. Due to the exchange coupling between the FM and the MF layer,^14,15 the magnetic dipole of the MF layer is also switched with M in the FM. Within the MF material, the Dzyaloshinskii–Moriya interaction (DMI)^24,25 effectively couples electric and magnetic dipoles since the weak canted magnetic moment $MC$ originates from the DMI.^14,26,27 When $MC$ switches polarity, electric polarization P in the MF layer deterministically switches in tandem, all in response to J_SO, as shown in Fig. 1(c).

The resulting switching of P gates the semiconductor channel by shifting the surface potential, similar to the effect in ferroelectric-gate FETs.^28–31 The current I_D flowing in the semiconductor channel is the readout signal, which changes by several orders of magnitude due to the resistance change. Consider the direction of J_SO in the SO layer in Fig. 1(a) as writing a 1 in the FM, leading to a high conductivity ON state of the semiconductor. When the current J_SO flows in the opposite direction, all the dipoles in the gate stack are flipped. The flipping of P then depletes the semiconductor channel, putting it in the OFF state. The resulting transistor output current I_D in response to J_SO is shown in Fig. 1(d): it is bi-stable and provides the desired large resistance ratio for efficient readout.

To prove the feasibility of the SOTFET, we quantitatively analyze the dynamical coupling across each interface and the entire device. The analysis to follow shows that the SOTFET behavior is achievable, but requires magnetoelectric multiferroics of specific magnetism and polarization, along with an appropriate hierarchy of strengths for the exchange coupling, DMI, and anisotropy energies within the gate stack.

We choose to build a model based on the most studied heterostructure CoFe/BiFeO₃ in this initial modeling effort of SOTFET since the FM/MF heterointerfaces are still poorly understood to date. For a generic FM/MF interface, it is yet difficult to describe the physical phenomena in relatively simple mathematical forms. However, we also note that our results, described in the later sections, show that BiFeO₃ is not a suitable multiferroic for SOTFETs because of its strong ferroelectricity and weak magnetism. For a SOTFET, the desired energy hierarchy demands that the suitable multiferroic material should possess a strong magnetic order and a strong exchange coupling with the ferromagnetic material, while its ferroelectric order should deterministically follow the switching of the magnetic order. Bi₂Se₃ is selected as the example SO layer. Other material candidates are discussed in Ref. 32. The aim of this model is to guide experiments by pointing toward desired heterointerface choices.

The magnetization M of the FM layer is switched by spin–orbit torque (SOT). For simplicity, we assume single-domain macrospin behavior. The switching dynamics of this process are captured by the Landau–Lifshitz–Gilbert–Slonczewski (LLGS) equation,^6,8,33,34

where $m̂$ is the normalized magnetization of the FM, $Heff$ is the effective magnetic field acting on $m̂$⁠, γ is the electron gyromagnetic ratio, μ₀ is the vacuum permeability, α is the Gilbert damping factor, M_S is the saturation magnetization, and $τ→SOT=τ→AD+τ→FL$ is the spin–orbit torque, the sum of the anti-damping (AD) torque $τ→AD$ and field-like (FL) torque $τ→FL$⁠, which are given by^8,33

Here, $ℏ$ is the reduced Planck constant, e is the electron charge, t is the thickness of ferromagnetic (FM) material, $j=JSO$ is the charge current density in the SO layer, $θAD(FL)$ is the spin Hall angle of the anti-damping (AD) or field-like (FL) torque from the SO layer, and $m̂p$ is the normalized spin polarization.

The effective field $Heff=Hext+Ha+Hdemag+HDMI$⁠, where $Hext$ is any external magnetic field and $Ha$ is the anisotropy field with perpendicular magnetic anisotropy (PMA) calculated by $Ha=2Kμ0MSmzẑ≡Hkmzẑ$⁠,³⁴ where K is the anisotropy constant. $Hdemag$ is the demagnetization field as calculated in the study by Beleggia et al.³⁵ The last term $HDMI$ is the effective magnetic field arising from the effective DMI, which is discussed further below.

Switching of M in the FM switches the electric polarization P of the MF layer due to the exchange coupling and DMI. The dynamics of P are captured by the Landau–Khalatnikov (LK) equation,^31,36,37

where γ_FE is the viscosity coefficient and $Pi(i=x,y,z)$ is the $x/y/z$ component of P. F is the total ferroelectric free energy,^37,38

where α₁, α₁₁, and α₁₂ are the phenomenological Landau expansion coefficients, K_strain is the strain energy, u is the axis of substrate strain, F_ext is the external electric field, and F_DMI is the effective electric field from DMI, which is discussed below. The strain term, $Kstrain(P.u)2$⁠, in Eq. (5) arises from the substrate-induced strain,^37,38 which dictates the energy-favorable planes for the equilibrium states of P, thereby reducing the degeneracy of P orientations in the specific case of the MF BiFeO₃, which is shown in previous studies.^14,39,40

In the model presented in this study, we take a highly simplified mathematical approach to phenomenologically count for the complex interactions between the FM and MF layers. We merge the exchange coupling, which couples M_C in BiFeO₃ and M in CoFe, and the DMI, which couples P and M_C in BiFeO₃, into one effective DMI term that directly captures the interaction between M in CoFe and P in BiFeO₃, with an effective Hamiltonian,^14,27

where $EDMI,0$ is the energy coefficient of DMI, $P̂$ is the polarization of BiFeO₃, $N̂$ is the Neel vector, and $M̂$ is the magnetic moment in CoFe. All vectors in the equation are normalized vectors. The effective magnetic field (⁠$HDMI$⁠) and electric field (⁠$FDMI$⁠) that enter the equations of motions are then

and

where $HDMI,0$ is the effective DMI magnetic field magnitude and $FDMI,0$ is the effective DMI electric field magnitude. Both fields are assumed to have constant magnitudes for specific material combinations because they originate from the energy and material parameters,

With the direction of N defined as $N̂=−P̂×M̂$⁠, all vectors in the CoFe/BiFeO₃ FM/MF system (P, N, and M) are connected by the DMI. The method of implementing the dynamic evolution of M and P described above is shown schematically in Fig. 2. The initial state of the SOTFET is defined by a set of vectors: M in the FM and $MC$⁠, P, and N in the MF layer. When a current J_SO flows in the SO layer, all four vectors (M, $MC$⁠, P, and N) can switch to different states, with dynamics dictated by the LLGS and the LK equations in each loop. Finally, a set of four vectors in a different equilibrium state will be reached by iteration. The switching behavior of P and M is assumed to be purely rotational, consistent with experimental studies of BiFeO₃.¹⁴ 

Key parameters used in the numerical evaluation of the SOTFET are provided in supplementary material S1. The model is validated by comparing with the micromagnetic simulation tools OOMMF⁴¹ and MuMax3⁴² and other theoretical calculations and experimental results, shown in supplementary material S2. For the SOTFET gate stack to controllably gate the semiconductor channel, a deterministic switching of polarization P in the z-direction in the MF layer is desired.

For the CoFe/BiFeO₃ FM/MF heterostructure and taking P_S = 100 μC/cm² of BiFeO₃⁴³ and M_S = 1.6 × 10⁶ A/m of CoFe,¹⁵ switching behavior for a range of assumed DMI energies is shown in Fig. 3(a). Upon applying a current of $JSO=−30$ MA/cm², different switching behavior of the $x−$ component of the magnetization (M_x) is observed for different DMI energies. For these values, however, the $z−$ component of polarization P_z in the MF layer does not follow the motions of M. This is because, given the large P_S, a moderate DMI energy is not sufficient to overcome the anisotropy energy in P to switch it. For a high DMI energy, with P held in place, M also does not switch because H_DMI then functions as an effective unidirectional anisotropy acting back on M. This is, therefore, a situation when the SOTFET does not achieve the desired functionality.

The natural next step is to explore reduced P_S in the MF layer. Reducing P_S in BiFeO₃ is experimentally feasible, for example, by La substitution of Bi in BiFeO₃.^44,45 Recent experiments by Lin et al.⁴⁶ also show that the exchange interaction between CoFe and La-substituted BiFeO₃ remains strong even with reduced P_S. Qualitatively, this implies that the multiferroic layer should have a relatively weak ferroelectricity, a strong magnetization, and strong coupling between the two order parameters. The calculated results with a reduced P_S = 10 μC/cm² and other parameters unchanged are shown in Fig. 3(b) for a range of $EDMI,0$ values. For the same current J_SO, a critical $EDMI,0$ is observed. Above the critical $EDMI,0$⁠, M_x and P_z concomitantly switch, signaling the required materials parameters for desired SOTFET operation.

Reducing P_S of the multiferroic could help the switching of P for two reasons. First, as shown in Eq. (9), for a fixed $HDMI,0$ and M_S, lowering P_S for the same $EDMI,0$ implies an enhanced $FDMI,0$ to switch the polarization. Second, a reduced P_S leads to a weaker polarization anisotropy as described in the free energy equation, Eq. (5). This lowers the energy barrier between polarization equilibrium states, making the switching easier.

Figures 3(c) and 3(d) show that the desired stable switching behavior of the SOTFET is achieved by choosing the CoFe/BiFeO₃ heterostructure with a reduced P_S = 10 μC/cm² of the MF layer (BiFeO₃) and an above-critical $EDMI,0$ = 0.8 pJ/μm³, which corresponds to DMI fields of $HDMI,0$ = 5 kOe and $FDMI,0$ = 80 kV/cm. It is seen that switching the current direction in the SO layer switches P_z. The current density used, 30 MA/cm², is about one order of magnitude lower than that of heavy metal-based SOT-MRAMs^47,48 due to the assumed large spin Hall angle of Bi₂Se₃ (⁠$θAD=θFL=3.5$²³) and can be further reduced by using larger spin Hall angle materials such as BiSb.⁴⁹ The current necessary to switch a SOTFET is still rather high, which partly stems from the coupling between the MF and FM layers, which increases the energy barrier to switch M by current. More discussions on the dependence of the critical current on the DMI energy can be found in supplementary material S3. With a suitable current applied, M is observed to switch within the x–y plane and P is switched out-of-plane. The switching trajectories of $M$ and $P$ are shown in the spherical plot in Fig. 3(d). Clear set and reset processes between State 0 and 1 are observed, proving the feasibility of the SOTFET operation for the chosen material parameters.

The switching of P with a reduced P_S = 10 μC/cm² shown in Figs. 3(c) and 3(d) results in a charge difference of $ΔQ=2Pz≈$ 12 μC/cm² in the semiconductor channel, assuming the absence of traps at the interface between the MF layer and the semiconductor channel. For example, for the choice of a silicon channel, this will lead to a surface potential change approximately at $Δψ≈$ 1.3 V by a simple calculation,⁵⁰ accessing the entire operating regime of a MOSFET from strong inversion to accumulation. Thus, by estimation, at least an on/off ratio of 10⁸ in I_D can be achieved due to the resistance change of the channel, which, in practice, will be limited by gate leakage and interfacial trap states rather than the intrinsic capability of a SOTFET. The simulation integrated with a Si-MOSFET model in SPICE verifies that an on/off ratio $>107$ can be achieved,¹⁶ which indicates that a $PS ≈0.1 μC/cm2$ is sufficient to fully control the semiconductor channel for the high on/off ratio when assuming no defect. The high on/off ratio in I_D as the readout component brings the read energy of a SOTFET down to the same level as a conventional semiconductor transistor. As a result, the write energy for a SOTFET should be comparable to that of a SOT MRAM, while the read energy is comparable to that of a FeFET; these features, together with the logic functionalities discussed later, make SOTFET a potentially competitive technology.

An electrically insulating magnetic (FM) layer is more desirable for SOTFET application in order to reduce the shunting current from the SO layer and boost the spin torque efficiency.⁵¹ Besides, the insulating FM layer could reduce the charge injection in the MF layer, thus potentially alleviating the fatigue that is often confronted by ferroelectric materials. The fatigue issue could also be addressed by the fact that the polarization switching is driven by coupling to the magnetic layer rather than an external electric field, which could reduce the tendency for long-distance atom motion.

Another challenge in the development of ferroelectric memory devices has been the presence of the depolarization field that can destabilize the ferroelectric polarization over time.⁵² If the hierarchy of coupling energies within the SOTFET is designed correctly, P cannot switch unless M is switched by J_SO. We envision that, as a result, the exchange coupling of the multiferroic to the magnetic layer could improve the ferroelectric retention.

In addition, by the virtue of simultaneously being a FET, the SOTFET can also provide logic functionality by a gate voltage controlling the channel. As a merger of memory and logic, the SOTFET is capable of performing process-in-memory (PiM) functionalities that significantly lower the energy consumption and physical size of computation, compared to a von Neumann architecture where logic and memory are separated. Some examples are explored in Ref. 16. The experimental realization and various modes of operation of the SOTFET are currently being investigated.

In summary, the SOTFET, a magnetoelectric memory device, is proposed, in which a change in magnetization of a SO/FM layer is transduced to control the semiconductor channel by using a magnetoelectric multiferroic layer, so that a read out with several orders of magnitude change in resistance can be achieved. We establish a quantitative model of the dynamics of the magnetization and polarization of the layers of the SOTFET. From the model, the material needs for the desired operation are identified and the feasibility of the SOTFET is proved in a properly designed CoFe/BiFeO₃ gate stack.

See the supplementary material for the key parameters for numerical simulations, validation of the model, and discussions on the dependence of the critical current on the DMI energy.

This work was supported in part by the Semiconductor Research Corporation (SRC) as nCORE task 2758.001 and NSF under the E2CDA program (No. ECCS 1740286). The authors wish to thank Professor Alyosha Molnar, Professor Christopher Batten, Yu-Ching Liao, Hyunjea Lee, and Yongjian (Helffor) Tang for helpful discussions.