Purely Electric-Field-Driven Perpendicular Magnetization Reversal

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Purely Electric-Field-Driven Perpendicular Magnetization Reversal Jia-Mian Hu,†,‡ Tiannan Yang,‡ Jianjun Wang,† Houbing Huang,‡ Jinxing Zhang,§ Long-Qing Chen,*,†,‡ and Ce-Wen Nan*,† †

School of Materials Science and Engineering, and State Key Lab of New Ceramics and Fine Processing, Tsinghua University, Beijing 100084, China ‡ Department of Materials Science and Engineering, Pennsylvania State University, University Park, Pennsylvania 16802, United States § Department of Physics, Beijing Normal University, Beijing, 100875, China S Supporting Information *

ABSTRACT: If achieved, magnetization reversal purely with an electric field has the potential to revolutionize the spintronic devices that currently utilize power-dissipating currents. However, all existing proposals involve the use of a magnetic field. Here we use phase-field simulations to study the piezoelectric and magnetoelectric responses in a threedimensional multiferroic nanostructure consisting of a perpendicularly magnetized nanomagnet with an in-plane long axis and a juxtaposed ferroelectric nanoisland. For the first time, we demonstrate a full reversal of perpendicular magnetization via successive precession and damping, driven purely by a perpendicular electric-field pulse of certain pulse duration across the nanoferroelectric. We discuss the materials selection and size dependence of both nanoferroelctrics and nanomagnets for experimental verification. These results offer new inspiration to the design of spintronic devices that simultaneously possess high density, high thermal stability, and high reliability. KEYWORDS: Perpendicular magnetization reversal, magnetoelectric, multiferroic heterostructure, phase-field simulations

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rotates to the downward direction through precession and damping (see the gray arrow). Electric-field-induced magnetic EA reorientation might be achieved by directly applying an electric field across a perpendicularly MTJ, based on electric modulation of magnetic interface anisotropy via changes in spin-polarized charge densities.16,17 Typical materials for the latter include ultrathin (≤2 nm in thickness) Fe (ref 18), Co (ref 19), Fe−Co (ref 11), and Co−Fe−B (ref 13) with perpendicular interface anisotropy that is inversely proportional to film thickness, and L10-order FePt(Pd) with a thickness-independent perpendicular magnetocrystalline anisotropy.20 However, our calculations and simulations indicate that an over 45° magnetic EA reorientation mediated by such an interface charge effect requires much higher operating electric fields than the dielectric breakdown electric fields of MTJ with either Al2O3 or MgO as a tunnel barrier (see Supporting Information S1 and S2). Thus, it would not happen. An up to 90° magnetic EA reorientation can alternatively be achieved by applying a much smaller electric field across a ferroelectric or piezoelectric layer underneath the perpendicular MTJ.5,21 The electric-field-induced in-plane strains in the piezoelectric act on the overlaying magnetic free layer across

verheating is the major bottleneck for spintronic device miniaturization. One tantalizing solution is to reverse magnetization using an electric field1−4 rather than a magnetic field or spin-torques generated by power-dissipating currents. However, this is challenging because the electric field is timeinvariant. A particularly challenging but desirable configuration is the reversal of magnetization along a perpendicular magnetic easy axis (EA) by an electric field, which if realized would allow the operation of a perpendicular magnetic tunnel junction (MTJ, the key component in most spintronic devices, see Figure 1a) at a minimal amount of power dissipation ( 0 or the opposite −x3 axis when m3 < 0. This suggests that the 180° magnetization reversal can be achieved by tuning the duration of the pulse electric field within a wide range, that is, from the first reversal across the line of m3 = 0 (∼1.3 ns) to close to equilibrium (∼31 ns), as discussed later. Such time span of electric fields should ensure a complete ferroelectric domain switching (typically within 1−10 ns, ref 28), and is also much longer than the RC (resistivecapacitive) delay time of a PZT nanoisland (90°) oscillation between the upper and the lower hemispheres in Figure 2a. Such electric-

Figure 3. Map of Fperp/kBT and magnitude of actual critical strain εcr that the magnetic nanoislands are subjected to for a wide variety of materials systems with the size of 48 nm × 24 nm × 60 nm. The dashed line is the eye guide for materials that have similar strength of magnetoelastic coupling. All data points are obtained by numerically searching the global minima of total free energy in different materials over a broad range of strain. C

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Nano Letters comprehensive diagram of actual εcr (i.e., after relaxation) and the room-temperature thermal stability factor Fperp(= f perpVm)/ kBT for various magnetic islands, all with the same size of 48 nm × 24 nm × 60 nm as the Ni. Here the f perp is the perpendicular magnetic anisotropy energy density, Vm is the volume of the magnet, kB is the Boltzmann constant and T is the Kelvin temperature (298 K). The f perp is given as 0.5(N11 − N33)μ0M2s + [(Ki − ΔKiE)/D] for polycrystalline or amorphous magnetic nanoislands with an additional term of K1/4 or K1 + K2 for (001) cubic or (0001) hexagonal magnets, respectively. Here K1 and K2 are the magnetocrystalline anisotropy coefficients; Ki and ΔKi are the interface anisotropy coefficient and the electric-field (E)-induced alteration, respectively.18,32 Note that the Fperp/kBT must be higher than 40 (ref 33) to ensure a reliable data retention, which are applicable to all investigated materials of the present size. Accordingly, magnetic materials on the left of this diagram (low εcr) may in principle be preferable for the experimental demonstration of electrically driven perpendicular magnetization reversal, including some ferrites (Fe3O4 and CoFe2O4) and giant magnetostrictive Terfenol-D (Tb0.7Dy0.3Fe2). It is also worth noting that Fperp/ kBT is linearly correlated to the magnitude of actual εcr, with the slope of such linear correlation being largely dependent on the magnetoelastic coupling coefficient (see detailed discussion in Supporting Information S5). As a result, materials locating near one common line (see the dashed line in Figure 3) should have a similar strength of magnetoelastic coupling while those above the line show stronger coupling. Now turn to the discussion of the size dependence of the magnitudes of applied critical strain εcr and electric field for large-angle magnetization precession and subsequent reversal. Knowledge of size dependence is indispensable for realistic designs of experiments. Assuming constant aspect ratios of D/L = 1.25, W/L = 0.5 as the Ni island of 48 nm × 24 nm × 60 nm, the top panel of Figure 4 shows the applied εcr as a function of length L with the right vertical axis indicating the available electric field range currently investigated (i.e., less than 1.28Ec, marked by the dashed line). The middle panel of Figure 4

correspondingly shows the room-temperature stability factor Fperp/kBT. Here, the observed increase in Fperp/kBT arise completely from the increase in Vm, as f perp remains invariant due to the invariant Hshape under constant aspect ratios of D/L and W/L. Even so, the magnitude of the applied εcr starts to increase when the size of Ni island is larger than 40 nm × 20 nm × 50 nm, and rapidly increases to −3.6% when the size reaches 72 nm × 36 nm × 90 nm. Such nonlinear increase in the applied εcr when scaling up the size is attributed to the enhanced inhomogeneity of local magnetization as discussed later, given that the degree of strain relaxation η is also invariant (see the bottom panel of Figure 4) under constant aspect ratios. The large strain relaxation (>90%) prevents plastic deformation in the Ni nanoisland.34 The shaded area in Figure 4 marks a potentially realistic scaling range of the Ni islands, in which the stability factor is higher than 40 (see the dashed line in the middle panel) and the corresponding electric fields (strains) do not exceed the maximum value of 1.28Ec (−1.5%) in the present simulations. Nevertheless, a wider scaling range can be obtained by using ferroelectric or piezoelectric materials that have higher electromechanical responses than PZT (e.g., Pb(Mg1/3Nb2/3)O3-PbTiO3, ref 35, or maybe the supertetragonal BiFeO3, ref 36); by using magnetic materials with smaller magnitude of actual εcr than polycrystalline Ni (see Figure 3). The left vertical axis of Figure 5a shows the magnitude of average perpendicular magnetization component m3 in Ni islands of various sizes before applying electric fields. As seen, the magnitude of m3 shows a sizable, nonlinear decrease from 0.998 to 0.915 when the size is larger than 40 nm × 20 nm × 50 nm. Consequently, the magnitude of applied εcr shows a nonlinear increase such that stronger effective field Heff can be generated to compensate the loss of precession torque M × Heff. The decrease in average m3 is induced by the decreasing magnitude of the average perpendicular stray field hd3 (= Hd3/ μ0Ms) inside the Ni islands (Methods), as shown in the right vertical axis of Figure 5a. The decreasing hd3 further results from the tendency of reducing long-range magnetostatic energy when the short-range exchange energy cannot lock all spins together as a uniform single-domain37 at larger sizes. For detailed illustration, Figure 5b shows the spatial distribution of local magnetization vectors and stray fields (its orientation against m) within the x1x3 plane at the center transects of the three-phase system of Ni island, air, and substrate (methods), with Ni islands of different sizes. In the uniform single-domain Ni island (e.g., 32 nm × 16 nm × 40 nm), the orientation of stray field is also uniform, but with much stronger magnitudes (see the longer vector length) near the top and bottom surfaces. As the size increases, local magnetization gradually becomes inhomogeneous, for example, the “flower” domain (with tilting magnetization vectors at the corners) and the “buckle” domain in the Ni islands of 56 nm × 28 nm × 70 and 72 nm × 36 nm × 90 nm, respectively, associated with the presence of nonuniform stray fields. Figure 6 shows the phase diagram of the magnitude of applied strain (with corresponding available electric fields) and the pulse duration for perpendicular magnetization reversal in the Ni/PZT nanoislands. Note that the magnetization always remains along the initial upward direction if the amplitude of electric field pulse is smaller than 0.9Ec. Upon larger electric fields, the perpendicular magnetization reversal occurs when the pulse duration falls into desired switching regions (black with downward arrow). Such an alternating occurrence of

Figure 4. Applied critical isotropic in-plane strain εcr (with available electric-field range marked by the dashed line in the upper panel), room-temperature thermal stability factor Fperp/kBT, and the degree of strain relaxation η[= 1 − |εeff 11/εIP|] as a function of the length L of the Ni island with aspect ratios of W/L = 0.5 and D/L = 1.25. The shaded area marks the size range of Ni island for potential applications. Influences of the aspect ratios of W/L and D/L on the applied εcr, Fperp/kBT, and η are shown in Supporting Information Figure S5. D

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turning off the electric field pulse, even when the m3 becomes relatively small after applying an electric field pulse of long duration (up to 31 ns). Moreover, the asymptotic behavior of these boundaries indicates an almost constant pulse area (i.e., magnitude multiplied by duration). In such case, electric field pulses of smaller magnitude need longer duration to drive a perpendicular magnetization reversal. In summary, nonvolatile and repeatable electrically driven perpendicular magnetization reversal has been demonstrated by applying an electric field pulse of certain duration. The proposed reversal path by successive precession and damping (Figure 1b) is achieved based on the strain-mediated volatile electrically induced reorientation of the perpendicular EA to the in-plane long axis of a nanomagnet. In contrast to previous reports,6,7,10−14 neither static nor dynamic magnetic fields are required for the reversal. Because only unipolar electric fields are applied across ferroelectrics, there is no full polarization reversal and hence greatly alleviates ferroelectric fatigue.38 The use of both reversible and irreversible electric-field-induced strain is allowed. A wide tunable range (from 1.3 ns up to 31 ns, or even longer under smaller damping coefficients) of the electric field pulse duration is exhibited, providing great flexibility for experimental verification and device designs. Methods. Analytical Calculations. The magnetic EA in a single-domain nanomagnet can be determined by minimizing the total magnetic free energy density f tot with respect to magnetization under various electrostrains ε(E) and/or electric fields E directly. f tot is expressed as the sum of magnetocrystalline anisotropy fanis (biaxial or uniaxial), stray field (demagnetization) energy fstray, elastic energy felast, and magnetic surface energy fs densities, that is, f tot(m,ε,E) = fanis(m) + fstray(m) + felast(m,ε(E)) + fs(m,E). The mathematical expressions of fanis, fstray, and felast (including magnetoelastic energy density) for both cubic and hexagonal magnets in Figure 3 with related materials parameters are summarized in Supporting Information S5. fs is written as fs = −[(Ki − ΔKiE)/D]m23 (ref 32), which leads to perpendicular magnetic EA (i.e., f tot shows minimum at m3 = ±1) in ultrathin nanomagnets as Ki > 0 in most cases (Supporting Information S5). The E-induced magnetic EA reorientation is therefore attributed to the electrostrain-tunable felast and/or E-tunable fs. Phase-Field Simulations. When applying biaxial isotropic electric-field-induced strains onto an isolated magnetic island, spatially variant strain arises due to mechanical relaxation.39 To theoretically study the effect of such nonuniform strain on both the magnetic domain structure and the magnetization dynamics, a phase-field model that incorporates mesoscale theories of elasticity and micromagnetics is employed. A threephase system of magnetic nanoisland, substrate (i.e., the PZT of larger lateral size), and air is used for simulations23 and discretized into a 3D array of cuboid cells of 40Δx × 40Δy × 50Δz. Among them, the bottom and their overlaying cells of 40Δx × 40Δy × 20Δz and 32Δx × 32Δy × 20Δz are designated as the substrate and the magnetic island, respectively, while the rest are the air. To describe a magnetic island of 48 nm × 24 nm × 60 nm, the cell size are taken as Δx × Δy × Δz = 1.5 nm × 0.75 nm × 3 nm in real space. By varying the total number of cells and/or the cell size, magnetic islands of different sizes can be treated. Local magnetization vector M(x) = Ms(m1(x),m2(x),m3(x)) acts as the main physical order parameter, the spatial distributions of which represent magnetic domains. Temporal evolution of the magnetization vector (i.e., the dynamics) is

Figure 5. (a) Length L-dependent magnitudes of perpendicular average magnetization m3 and stray field hd3 before applying electric fields. (b) Corresponding distributions of perpendicular local magnetization and local stray fields at the central transects of the simulated system within the x1x3 plane, where (from left to right) L = 32 nm (32 nm × 16 nm × 40 nm), 56 nm (56 nm × 28 nm × 70 nm), and 72 nm (72 nm × 36 nm × 90 nm). The aspect ratios are therefore W/L = 0.5 and D/L = 1.25. Corresponding analysis for the dependence on aspect ratios (D/L, W/L) are shown in Supporting Information Figure S6 and S7.

Figure 6. Vertical axes: electric-field-induced isotropic in-plane strains (ε11 = ε22) with corresponding available electric fields applied across the Ni/PZT nanoislands currently investigated.

switching regions results from the oscillatory behavior of the magnetization precession (see Figure 2b) and is reminiscent of a similar stripy phase diagram for electric-field-driven magnetization reversal in the presence of a static magnetic field where the horizontal axis is damping coefficient14 rather than pulse duration herein. The sharp boundaries between different switching regions suggest there are no intermediate states between the upward and downward magnetization states. Therefore, the magnetization reversal can happen as long as the out-of-plane magnetization component m3 is negative after E

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respectively, with the λs taken as −32.9 ppm (ref 22). The heterogeneous strain εhet with a volume average of zero is calculated as εhet = 0.5(∂ui/∂xj + ∂uj/∂xi), where u is the ij displacement. The spatial distribution of u can be obtained by solving the mechanical equilibrium equation ∇·σ = 0 using a Fourier spectral iterative perturbation method.47 Setting a zero stiffness constant c for the air phase automatically incorporates the stress-free boundary condition along the lateral and top surfaces of the magnetic nanoisland. In all simulations, the upward uniform magnetization is evolved upon zero electric field (strain) until equilibrium is achieved, and the obtained magnetization distribution is further used as the initial magnetic domain structure (e.g., see Figure 5b). This treatment mostly captures the real processing for the magnets, in which almost uniform magnetization distributions are obtained by applying a strong static magnetic field and then removing it. A full transfer of strain is considered in all simulations above. Accordingly, the interface effect on the strain transfer and thereby on the electric-field-driven perpendicular magnetization reversal has not been considered, which is a fairly good approximation in the presence of a well-controlled Ni/ ferroelectric interface (e.g., in ref 48 and 49). The imperfect interface would cause some loss of the strain transferred, weaken the magnetoelectric coupling, and increase the actual critical strain εcr. A good interface bond could ensure an efficient strain transfer across the interface to gain a more efficient control of magnetization reversal with electric fields.

obtained by numerically solving the Landau-Lifshitz-Gilbert (LLG) equation, that is ∂M α ⎛⎜ ∂M ⎞⎟ M× = −γ0(M × Heff ) + ⎝ MS ∂t ∂t ⎠

(1)

On the basis of eq 1, the magnetization should precess around the total effective field Heff driven by the torque M × Heff, and achieve equilibrium along the direction of Heff driven by the damping torque (the second term). Here γ0 and α are the gyromagnetic ratio and the Gilbert damping coefficient, respectively, taken as −2.42 × 105 m·A−1·s−1 (using a g-factor of 2.21 from ref 40) and 0.01 for polycrystalline Ni. The influence of the damping coefficient α on magnetization precession is shown in Supporting Information S8. Our simulation results indicate that the α may not exceed 0.1, otherwise the spatial magnetization precession trajectories would be limited within the upper hemisphere due to the enhanced damping torque, thus the magnetization would not precess across the film plane. Nondimensionalization is performed when numerically solving eq 1 using a Guass-Seidel projection method,41 in which the dimensionless time step Δt* = 0.002 corresponds to a real time step Δt [= Δt*(1 + α2)/ (γ0Ms)] of 0.017 ps (ref 42). Such a small time step ensures a high numerical accuracy. The Heff is determined as −(1/ μ0)(δF/δM), where F is the magnetic free energy of the threephase system of the polycrystalline Ni island, substrate, and air, and is expressed as F=

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∫v [A |∇m|2 − 0.5μ0Ms(Hd·m) + 0.5cijkleijekl]dV

ASSOCIATED CONTENT

S Supporting Information *

(2)

Additional results and discussions, Figures S1−S8, Table S1, and references. This material is available free of charge via the Internet at http://pubs.acs.org.

The first contribution in eq 2 is the isotropic exchange energy depending on the gradient of local magnetization vectors with A denoting the room-temperature exchange constant of 8.6 × 10−12 J/m for Ni (ref 43). This term favors a uniform magnetization m with zero gradients inside the magnet (single-domain). The second term is the energy of stray field Hd (also known as demagnetization energy or magnetic shape anisotropy energy). The spatial distribution of Hd is obtained by solving the magnetostatic equilibrium equation, that is, ∇·(μ0Hd + μ0Msm) = 0, based on numerical algorithm established for a 3D array of ferromagnetic cubes.44 The stray field energy competes with the exchange energy by favoring a zero net magnetization with head-to-tail magnetic domain walls (multidomain). For a single-domain nanomagnet with uniform magnetization, the average Hd inside the magnet can be estimated as Hd = −NMsm, with N the demagnetization tensor that can be analytically calculated based on theory established for rectangular prisms.45 It can be seen that the Hd within a single-domain magnet is unidirectional (i.e., against m), different from the uniaxial effective perpendicular shape anisotropy field (N11 − N33)Ms. The last term in eq 2 describes the elastic energy in which c is the four-rank elastic stiffness tensor, and e is the elastic strain tensor determined as e = εhom + εhet − ε0 following Khachaturyan’s elasticity theory.46 Here εhom (including ε11, ε22, and ε12) is the homogeneous strain corresponding to deformation of the magnetic island, arising from the interface transfer of the electric-field-induced biaxial in-plane isotropic strains (ε11 = ε22, ε12 = 0) in the bottom PZT island. The spontaneous strain ε0 of the magnetic island relates to the local magnetization vector and magnetostriction coefficient as ε0ij = 1.5λs(mimj − 1/3) and ε0ij = 1.5λsmimj for i = j and i ≠ j,

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: (L.-Q.C.) [email protected]. *E-mail: (C.W.N.) [email protected]. Author Contributions

J.-M.H. and T.Y. contribute equally to this work. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported by the NSF of China (Grants 51332001, 51472140, 11234005, and 51221291), and the NSF (Grants DMR-1410714, DMR-0820404, and DMR1210588). The computer simulations were performed on the LION and Cyberstar Computing Systems at the Pennsylvania State University supported in part by NSF Major Research Instrumentation Program through Grant OCI-0821527 and in part by the Materials Simulation Center and the Graduated Education and Research Services at the Pennsylvania State University.

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