Introduction

Spin-polarized currents flowing in conductive ferromagnets exert a pseudo-torque on the magnetization, which results from spin-momentum transfer1,2. In layered magnetic nanostructures, this spin-transfer torque (STT) effect gives rise to current-driven magnetization precession3 and even to magnetization reversal4,5. Among such nanostructures, magnetic tunnel junctions (MTJs) comprising ferromagnetic electrodes separated by an ultrathin insulating interlayer are of special interest. Indeed, MTJs are currently employed in modern hard-disk drives and represent a promising basic element for the development of an electric-write nonvolatile memory with nondestructive readout6. Importantly, they have two stable magnetic states (e.g., with parallel and antiparallel electrode magnetizations) which may display very different electrical resistance due to spin-dependent tunnelling. The phenomenon of tunnelling magnetoresistance (TMR), with TMR ratios of over 600% achieved at room temperature7, provides reliable readout of the logic state. This feature, being combined with practically unlimited endurance and very high thermal stability of information storage in MTJ-based memory cells, could make MTJs ideally suited for the next generation of non-volatile memories.

However, the current density necessary for the magnetization reversal providing the data writing is typically too high (~106 A cm−2) for industrial applications6,8. Only recently, Wang et al. discovered an easy electric-field-assisted magnetization switching in perpendicular-anisotropy Co40Fe40B20-MgO-Co40Fe40B20 junctions, which requires small current densities ~ 104 A cm−2 (ref. 9). In their experiments, a constant biasing magnetic field was applied to the junction and reversible switching between parallel and antiparallel orientations of electrode magnetizations has been achieved by voltage pulses of the same polarity but different height. The authors attributed this unipolar switching to voltage-induced modification of perpendicular magnetic anisotropy, which can greatly reduce the coercivity. It should be emphasized that, although the electric field penetrates into metals to a distance of the order of atomic dimensions only, significant electric-field-induced changes in the surface/interface magnetic anisotropy energy were observed experimentally and predicted theoretically for various ferromagnetic films and heterostructures10,11,12,13,14,15.

In this study, we develop a theoretical description of the voltage-driven magnetization switching in junctions having electric-field-dependent interfacial anisotropy. The theory focuses on MTJs with electrodes made of cubic ferromagnets, which are distinguished by the presence of size-induced spin reorientation transitions (SRTs) at room temperature. Special attention is given to junctions with perpendicular-to-plane magnetizations of ferromagnetic electrodes (p-MTJs), which can be stabilized by the interfacial magnetic anisotropy16 or substrate-induced lattice strains u (ref. 17), but qualitatively similar results are obtained for in-plane magnetized MTJs (see Supplementary Information). It is shown that the critical current density needed for the STT-driven magnetization reversal becomes small when the following three conditions are satisfied simultaneously: (i) the initial thickness-strain state of the “free” electrode is close to an SRT, (ii) the applied bias voltage enhances proximity to this SRT and (iii) the junction has relatively low electrical conductance. Importantly, the application of external magnetic field is not necessary for easy voltage-driven magnetization switching, as the same role plays the magnetic coupling between two electrodes.

Results

We studied voltage-driven switching for typical MTJs having two stable magnetic states at zero bias, which correspond to parallel (P) and antiparallel (AP) electrode magnetizations. In an MTJ-based memory cell, the magnetization M of one electrode (termed “free” layer) changes its orientation during the writing process, whilst the magnetization Mfixed of another electrode (“fixed” layer) remains constant. For MTJs with nanoscale dimensions, we may assume that the free layer is homogeneously magnetized up to the saturation magnetization Ms and switches through the coherent mode. Hence the magnetization dynamics in a free layer traversed by a spin-polarized current can be described by the Landau-Lifshitz-Gilbert-Slonczewski equation

where γ is the gyromagnetic ratio, μ0 is the permeability of the vacuum, α is the dimensionless damping parameter and Heff = −(1/μ0)∂F/∂M is the effective magnetic field defined by the Helmholtz energy density F of the free layer. The last two terms in Eq. (1) allow for the current-induced spin-transfer torque (STT) and field-like torque (FLT), respectively18. These torques depend on the voltage V applied to the junction2,19,20 and should be divided by the free layer volume when substituted into Eq. (1). Importantly, in MTJs with the interfacial anisotropy sensitive to electric field in the tunnel barrier, Heff becomes a voltage-dependent quantity as well.

The analysis of Eq. (1) was restricted to free layers made of cubic ferromagnets, which have the typical (001) crystallographic orientation in the paramagnetic state. Using the expansion of the energy density F in terms of direction cosines of M given for such layers in ref. 21, we calculated the field Heff. The presence of magnetic coupling between the free and fixed layers was quantified via the introduction of an effective interaction field Hint allowing for both the magnetostatic interaction22,23 and the exchange coupling1. Assuming small deviations from the equilibrium magnetization orientation existing at zero applied voltage, we reduced Eq. (1) to a system of linearized equations. An analytic relation was then derived for the Jacobian matrix A of this system of equations. Using this relation, we calculated two eigenvalues of A, which determine stability of the initial magnetic state. The analysis of these eigenvalues demonstrated that the P and AP states become unstable when the trace of the Jacobian, tr(A), goes to zero. Hence the critical voltage destabilizing the initial magnetic state can be found by solving the equation tr(A) = 0.

In the case of p-MTJs (Fig. 1), the linearized system of equations describing small deviations of M from the initial [001] or direction (orientation angle θ = 0 or 180°) was found to be independent of the voltage-induced part of τFLT. Accordingly, by setting tr(A) to zero, we obtain an equation for the critical STT destabilizing the initial magnetization orientation. In the coordinate system (x1,x2,x3) with the x3 axis orthogonal to the tunnel barrier, the critical value of τSTT satisfies the relation

Here and below the upper and lower signs correspond to the P and AP states of MTJ, respectively, H3 is the out-of-plane component of the external magnetic field H, Nik (i,k = 1,2,3) are the demagnetizing factors of the free layer and the critical field Hc > 0 is given by the formula

where K1 is the bulk magnetocrystalline anisotropy (MCA) constant of the fourth order at fixed lattice strains u (ref. 24), Ks is the interfacial anisotropy constant defining the total specific energy of two interfaces, tf is the free layer thickness, B1 and B2 are the magnetoelastic coefficients and c11, c12 and c44 are the elastic stiffnesses at constant magnetization. It should be noted that Eqs. (2)–(3) are valid even at nonzero u12 and N12, but the conditions H1 = H2 = = 0 and N13 = N23 = 0 must be satisfied to ensure stability of the initial magnetization orientation. In the following discussion, however, we will restrict our consideration to the usual case of zero shear strain (u12 = 0) and focus on free layers with positive MCA (K1 > 0), assuming that the magnetostatic shape anisotropy favours the [100] magnetization orientation (N11 ≤ N22 ≪ N33, N12 = N13 = N23 = 0) even at u11 ≠ u22. Importantly, under these conditions, Hc given by Eq. (3) coincides with the theoretical coercive field of a free layer calculated from the condition at H = Hint = 0. A similar relationship between the critical value of τSTT and the coercive field can be derived for free layers with negative MCA (see Supplementary Information).

Figure 1
figure 1

Schematic representation of a tunnel junction with ferromagnetic electrodes having perpendicular-to-plane magnetizations stabilized by the interfacial magnetic anisotropy or lattice strains.

The magnetization Mfixed of the bottom electrode has a fixed orientation assumed to be parallel to the x3 axis, whereas the magnetization M of the top electrode switches between the up (θ = 0°) and down (θ = 180°) directions when the voltage applied to the junction reaches critical values.

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Equation (2) represents a general relation for the critical voltages VP and VAP of the P → AP and AP → P switching processes in the absence of thermal activation. It should be noted that, in our approximation, the field-like torque influences these voltages via a zero-bias contribution to the interaction field Hint only. Based on the results of first-principles calculations11, a linear variation of the interfacial anisotropy constant Ks with the electric field intensity E in the tunnel barrier can be assumed. This yields the voltage dependence of Ks in the form , where is the sensitivity of interfacial anisotropy constant to the electric field and tb is the barrier thickness. In this paper, the bias voltage V is taken to be positive when the electric field in the barrier is directed from the free layer surface to that of the fixed one (Fig. 1). It should be noted that, as demonstrated by the first-principles calculations11,25, the magnetic moments of surface atoms in metallic ferromagnets also change in the presence of electric field in an adjacent dielectric. Hence the saturation magnetization Ms in the free magnetic layer should be regarded as an electric-field-dependent quantity as well. However, for the CoFeB-MgO-CoFeB junctions with tf ~ 1.5 nm, which are considered in this paper, the voltage-induced variations of Ms appear to be negligible. Indeed, from the data obtained for an ultrathin Fe50Co50 film contacting the MgO layer25 it follows that the change in the free-layer magnetization Ms is less than 1% even at the junction's breakdown field Ebreak ≈ 1.7 V nm−1 (ref. 9).

The dependence of STT on the bias voltage also can be approximated by a linear one (constant torkance)2. Accordingly, we can represent τSTT as , where e is the electron charge, ћ is the Planck constant and GSTT is the normalized torkance having the dimension of conductance per unit area. The theoretical studies demonstrate that GSTT should strongly depend on the height Ub and width tb of potential barrier, electron effective mass mb in the tunnel barrier (which may differ markedly from the free electron mass me) and on physical properties of electrodes2,26. Since both STT and junction's electric conductance G are proportional to the exponential factor (ref. 26), a strong correlation between the torkance and the conductance G is expected. In particular, for elastic tunneling in symmetric MTJs, the theory gives , where P is the spin polarization and GP is the MTJ conductance per unit area in the P state2.

In the linear approximation, the critical voltages defined by Eq. (2) become

where Hc0 = Hc(V = 0). Therefore, the critical current densities for the P → AP and AP → P switching processes can be found as JP = GP(VP)VP and JAP = GAP (VAP)VAP, where GAP is the MTJ conductance per unit area in the AP state. Equation (4) shows that the voltage-dependent interfacial anisotropy facilitates one of the switching processes (P → AP or AP → P) and hinders the other. Since the magnetization of the fixed layer usually can be reversed by sufficiently strong magnetic field, it is possible to magnetize an MTJ in such a way that the current-induced switching hindered by the interlayer magnetic coupling will be facilitated by the interfacial anisotropy.

According to the available experimental data9,15, in the case of CoFeB-MgO-CoFeB junctions the sensitivity ks should be negative in our free-energy formulation, where the out-of-plane magnetization orientation is favoured by negative Ks. Hence the electric-field-dependent interfacial anisotropy facilitates P → AP switching and hinders AP → P one in these MTJs. Using typical values of ks ≈ −50 μJ m−2 (V nm−1)−1, α ~ 0.01 and tb ~ 1 nm, we obtain |2α ks/tb| ~ 10−6 C/m2. At the same time, the measured voltage dependence of STT (ref. 19) demonstrates that, for the Co60Fe20B20-MgO-Co60Fe20B20 junction with tb = 1 nm, the normalized torkance GSTT is about 6 × 1010 Ω−1 m−2 at small voltages. Therefore, (ћ/e)GSTT ≈ 4 × 10−5 C/m2 so that the first term in the denominator of Eq. (4) appears to be much larger than the second one in this particular case. However, the conductance of this MTJ is very high (GP ~ 5 × 1011 Ω−1 m−2)19, which indicates that the STT is much higher here than in junctions with small conductances. For instance, the Co40Fe40B20-MgO-Co40Fe40B20 junctions with tb = 1.2 nm have GP ~ 108 Ω−1 m−2 (ref. 9) so that the second term is expected to be much larger than the first one, which implies strong effect of the electric-field-dependent interfacial anisotropy on critical voltages and currents.

Importantly, the coercive field Hc0 involved in Eq. (4) decreases drastically in the vicinity of size-induced or strain-driven SRT. At K1 > 0, such SRT has the form of an abrupt transition between perpendicular-to-plane and in-plane magnetization states21,24 and the relation can be derived for the coercive field of free layer at H = Hint = 0, where t* is the critical thickness at V = 0 given by Eq. (S4) (see Supplementary Information). We see that Hc0 reduces as tf tends to t* but remains finite at this first-order transition. Nevertheless, the coercive field Hc0 and, therefore, the critical voltages and currents can be strongly reduced by fabricating MTJs with an optimum free layer thickness close to t*.

To demonstrate this effect, we calculated the critical voltage and current for the P → AP switching in a representative CoFeB/MgO/CoFeB junction. The torkance GSTT was evaluated as a function of the junction's conductance GP under the assumption of elastic tunneling2 for the spin polarization P = 0.577 corresponding to TMR = 100%. The relationship between GP and the barrier thickness tb was described by the formula with the height Ub = 1.2 eV and effective mass mb = 0.4 me characteristic of the MgO barrier26 and GP0 providing the conductance GP = 7.96 × 107 Ω−1 m−2 measured at tb = 1.2 nm (ref. 9). Substituting GSTT(GP) and tb(GP) into Eq. (4), we found the critical voltage VP and current density JP as a function of the junction's conductance GP. Figure 2 shows that, irrespective of this conductance, both VP and JP decrease significantly as the free layer thickness tf approaches the critical value t* ≅ 1.67 nm. This decrease results from a thickness-induced reduction of the coercive field , which leads to a term proportional to in Eq. (4). It should be noted that, since t* ~ Ks0, both the minimum critical voltage VP(t*) and the slope ∂VP/∂tf of its thickness dependence are independent of the interfacial anisotropy constant Ks0. Indeed, the same feature is characteristic of the critical current density JP.

Figure 2
figure 2

Theoretical results for spin-transfer-driven magnetization switching in CoFeB/MgO/CoFeB junctions.

(a) Critical voltage and (b) critical current density for the P → AP switching calculated as a function of the junction's conductance per unit area in the P state. Curves 1, 2, 3 and 4 correspond to the free layer thicknesses tf = 1.4, 1.5, 1.6 and 1.65 nm, respectively. The following parameters were employed in the numerical calculations: = 50 Oe, N11 = N22 = 0.01, u11 = u22 = 0, α = 0.01, Ks0 = −1.3 × 10−3 J m−2 (ref. 16), ks = −50 μJ m−2 (V nm−1)−1, Ms = 1.13 × 106 A m−1 (ref. 9), K1 = 1.3 × 104 J m−3, B1 = −29.4 × 106 J m−3, B2 = −3 × 106 J m−3, c11 = 2.8 × 1011 N m−2, c12 = 1.4 × 1011 N m−2 and c44 = 1 × 1011 N m−2 (ref. 21).

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On the other hand, the reduction of conductance is accompanied by a drastic decrease in the current density JP and increase in the voltage VP at all studied thicknesses tf. Remarkably, very low critical current densities JP ≤ 104 A cm−2 are predicted for junctions with GP ≤ 108 Ω−1 m−2. However, such easy current-induced switching can be realized only when the free layer thickness tf is sufficiently close to t* so that the voltage VP does not exceed the junction's breakdown voltage Vbreak.

It should be emphasized that the bias voltage V applied to a ferromagnetic layer with electric-field-dependent interfacial anisotropy can induce an SRT as well11. In the case of negligible interlayer coupling and K1 > 0, the calculation gives the following relation for the voltage V* at which the perpendicular-to-plane and in-plane magnetization states acquire the same energy:

Since at V = V* the energy minima corresponding to these states are separated by a potential barrier, the 90° magnetization switching actually occurs at a higher bias voltage (|V| > |V*|), which makes the initial state unstable (by transforming an energy minimum into a maximum) or reduces the barrier height to a level surmountable by thermal fluctuations. Moreover, expressing V* through the critical thickness t*, we find that . Therefore, the switching voltage is expected to be smaller than the MTJ breakdown voltage only at thicknesses tf close to t*. In the case of CoFeB-MgO-CoFeB junctions, for instance, Vbreak ≈ 2 V (ref. 9) and Ks0 ≈ −1.3 × 10−3 J m−2 (ref. 16) so that the proximity to the SRT must be less than 10%.

To reveal peculiarities of the magnetization switching occurring in the presence of interlayer coupling or external magnetic field, we analyze the voltage-induced evolution of the free layer energy density ΔF as a function of the angle θ between the magnetization M and the x3 axis. Figure 3 shows representative energy profiles ΔF(θ) calculated for a CoFeB-MgO-CoFeB junction with the interlayer coupling characterized by the interaction field = 50 Oe. It can be seen that, at V = 0 and at negative voltages not exceeding the threshold Vth ≅ −1.02 V, the energy minima exist only at θ = 0° and θ = 180°, which correspond to the perpendicular-to-plane magnetization orientations parallel and antiparallel to the magnetization of the fixed layer, respectively. At higher negative voltages, however, an intermediate minimum corresponding to almost in-plane magnetization direction appears at 70° < θ < 90°. Furthermore, at V = Vc ≅ −1.49 V, the energy minimum at θ = 180°, which is related to the AP state disadvantaged by the interlayer coupling, transforms into a maximum. Therefore, the voltage Vc can induce a switching of the free layer magnetization in an MTJ having initially the AP state. In this case, the free layer magnetization M will rotate by about 96° and acquire almost in-plane orientation. Remarkably, the reduction of voltage back to Vth will give rise to further magnetization rotation towards θ = 0° because the intermediate minimum disappears at this voltage. Thus, a voltage pulse of sufficient height (|V| > |Vc|) can induce the AP → P switching facilitated by interlayer coupling. It should be noted that the influence of STT on the AP → P switching in a CoFeB-MgO-CoFeB junction with = 50 Oe and a low conductance GP < 3 × 108 Ω−1 m−2 can be neglected because the ratio τSTT/(γ μ0 Ms) is less than 1% even at the bias voltage V = −2 V close to the breakdown voltage. A similar voltage-assisted 180° magnetization rotation was observed in the Co40Fe40B20-MgO-Co40Fe40B20 junctions subjected to an external magnetic field9.

Figure 3
figure 3

Effect of bias voltage on the energetics of a free magnetic layer in a CoFeB-MgO junction with nonzero interlayer magnetic coupling.

The free layer energy density ΔF is plotted as a function of the magnetization orientation angle θ shown in Fig. 1. Panel (a) corresponds to the initial state of this junction (V = 0), whilst panels (b) and (c) demonstrate the energy profiles at bias voltages of −1.02 V and −1.49 V, respectively. The calculations were performed for a junction with = 50 Oe, tf = 1.6 nm and tb = 1 nm using the set of parameters listed in the caption of Fig. 2.

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At the same time, the appearance of an intermediate energy minimum between θ = 0° and θ = 180° might interrupt the P → AP switching initiated by the spin-polarized current. Indeed, the STT and FLT torques destabilizing the initial P state may be insufficient for the escape from this energy minimum. Then, at the critical voltage VP given by Eq. (4), the free layer magnetization M will acquire almost in-plane orientation instead of being rotated by 180°. Moreover, owing to the interlayer magnetic coupling, M will rotate back to the initial direction after completion of the voltage pulse. In this situation, which is beyond the scope of the present paper, the actual voltage needed for the P → AP switching may be higher than VP. Hence Eq. (4) can be safely used only when it gives the critical voltage not exceeding significantly the voltage Vth (≈ −1.02 V for the considered MTJ). At |Hint| ≪ Ms, Vth can be approximated by the voltage V* defined by Eq. (5).

Remarkably, our theory explains easy magnetization switching in the Co40Fe40B20-MgO-Co40Fe40B20 junctions, which was recently discovered by Wang et al9. In this work, p-MTJs with the free layer thickness tf = 1.6 nm and the barrier thickness tb = 1.2 nm were subjected to voltage pulses in the presence of external magnetic field H3 = 55 Oe. It was found that the sequence of unipolar pulses with the duration δp = 200 ms and magnitude alternating between −0.9 V and −1.5 V reversibly switches this MTJ between P and AP states. The P → AP reversal was attributed to an STT created by the spin-polarized current, whilst the AP → P switching was explained by a voltage-induced reduction of coercive field9. At the same time, the substitution of Ks0 = −1.3 × 10−3 J m−2 (ref. 16), Ms = 1.13 × 106 A m−1 (ref. 9), um = 0, N11 = 0.008 and N33 = 0.984 (theoretical values for the free layer with diameter D = 400 nm) into Eq. (S4) shows that, for the discussed MTJ, the critical thickness t* in the absence of magnetic field and interlayer coupling is equal to t* ≈ 1.66 nm. Accordingly, the free layer with tf = 1.6 nm is very close to a thickness-driven SRT already at V = 0. By analyzing the influence of bias voltage on the energy profile ΔF(θ) at H3 = 55 Oe and the measured sensitivity ks = −50 μJ m−2 (V nm−1)−1 (ref. 9) we further find that the threshold voltage Vth is about −1.08 V, whilst the critical voltage Vc at which the AP state becomes unstable at temperature T = 0 K equals Vc ≅ −1.59 V. Hence the AP → P switching observed at V = −1.5 V may be attributed to the two-stage magnetization rotation via an intermediate orientation which becomes unstable when the bias voltage reduces below Vth ≅ −1.08 V during the voltage downswing. The difference between the observed switching voltage and the theoretical critical voltage Vc(T = 0) may be explained by the influence of thermal fluctuations. Indeed, the Néel-Brown formula27 for the switching probability Π(T) shows that Vc(T) can be estimated using the equality Ub(V) = kBT ln(δp f0/ξ), where Ub is the height of the barrier hindering the escape of magnetization from the minimum at θ = 180°, f0 ~ 109 Hz is the attempt frequency and the factor ξ ≅ 5 ensures Π ≅ 1. By studying the voltage-induced evolution of the energy profile ΔF(θ) we found that Vc ≅ −1.4 V at T = 300 K and δp = 200 ms, which agrees well with the experiment.

The critical current needed for the STT-driven magnetization switching reduces significantly in the presence of thermal activation as well6. From the general theory of thermally assisted magnetization reversal28 it follows that the critical STT strength at a finite temperature is given by the formula τSTT(T) = τSTT(T = 0) [1 − kBT ln(δp f0/ξ)/Ub], where τSTT(T = 0) is defined by Eq. (2). Accordingly, the critical voltage VP(T) of the P → AP switching calculated in the linear approximation differs from VP(T = 0) by the factor 1 − kBT ln(δp f0/ξ)/Ub. Since the discussed MTJ has the conductance GP ≈ 1.3 × 108 Ω−1 m−2 and the TMR = 100% corresponding to spin polarization P ≈ 0.577 (ref. 9), the normalized torkance GSTT calculated for elastic tunneling20 is about 2.9 × 107 Ω−1 m−2. Hence, using the measured barrier height Ub = 42 kBT (ref. 9) and damping constant α = 0.013 (ref. 16), we obtain VP ≈ −1.05 V and JP ≈ −1.4 × 104 A cm−2. Remarkably, the predicted ultralow critical current density is very close to the measured value JP = −1.2 × 104 A cm−2 (ref. 9), which is two orders of magnitude smaller than usual densities ~ 106 A cm−2 (ref. 16). It should be noted that the critical current density calculated without the account of voltage-controlled interfacial anisotropy (ks = 0) is about −8 × 105 A cm−2.

Discussion

In this work, we have shown that the electric-field-dependent interfacial magnetic anisotropy has a strong effect on the STT-driven magnetization reversal in MTJs with a low conductance. Depending on the sign of its sensitivity ks to electric field, the varying interfacial anisotropy facilitates either the P → AP or AP → P switching, reducing the critical current density down to ultralow values ≤ 104 A cm−2. Surprisingly, this easy magnetization reversal occurs when the spin-transfer torque is rather small. Although the reverse process may be strongly hindered and even prohibited by the change of interfacial anisotropy, it can be realized via the voltage-induced two-stage magnetization rotation facilitated by the interlayer magnetic coupling. Such switching, however, is an exclusive feature of tunnel junctions distinguished by the proximity of free layer to the size-induced SRT. Thus, the theory provides a recipe for the fabrication of MTJs suitable for commercial electric-write nonvolatile magnetic random access memories.