Abstract
Voltagedriven 180° magnetization switching provides a lowpower alternative to currentdriven magnetization switching widely used in spintronic devices. Here we computationally demonstrate a promising route to achieve voltagedriven inplane 180° magnetization switching in a strainmediated multiferroic heterostructure (e.g., a heterostructure consisting of an amorphous, slightly elliptical Co_{40}Fe_{40}B_{20} nanomagnet on top of a Pb(Zr,Ti)O_{3} film as an example). This 180° switching follows a unique precessional path all in the film plane and is enabled by manipulating magnetization dynamics with fast, local piezostrains (rise/release time <0.1 ns) on the Pb(Zr,Ti)O_{3} film surface. Our analyses predict ultralow area energy consumption per switching (~0.03 J/m^{2}), approximately three orders of magnitude smaller than that dissipated by currentdriven magnetization switching. A fast overall switching time of about 2.3 ns is also demonstrated. Further reduction of energy consumption and switching time can be achieved by optimizing the structure and material selection. The present design provides an additional viable route to realizing lowpower and highspeed spintronics.
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Introduction
Multiferroic magnetoelectric heterostructures enable switching magnetization with a voltage rather than a current, dissipating much less heat^{1}. Such voltagedriven magnetization switching has been achieved through the transfer of piezostrains^{2,3,4,5,6,7,8,9,10,11,12} or/and exchange coupling^{13,14,15,16,17} across the interface of the constituting magnetic and ferroelectric phases. Given that a timeinvariant voltage cannot break the timereversal symmetry of a magnetization, applying voltage alone typically induces an at most 90° magnetization switching. For example, piezostrain mediated^{6,8,10} and exchange coupling mediated^{13} voltagedriven 90° magnetization switching have both been observed experimentally in multiferroic heterostructures at zero magnetic field. Such 90° switching provides basis for the design of lowpower spintronic devices such as magnetoelectric random access memories (MeRAM), which integrate a magnetic tunnel junction (MTJ) on top of a ferroelectric/piezoelectric layer^{18,19,20}. A voltagedriven full 180° magnetization switching in the free layer of the MTJ would result in significantly larger electric resistance change of MTJ and hence higher signaltonoise ratio. Pioneering experimental demonstrations include strainmediated voltagedriven 180° switching of local magnetic domains in multilayer capacitors of Ni electrodes and BaTiO_{3}based dielectric layer^{12} and exchangecouplingmediated voltagedriven switching of inplane net magnetization in BiFeO_{3}based multiferroic heterostructures^{16}. Theoretical proposals of achieving voltagedriven 180° switching of an almost uniform magnetization have also appeared. These are based on voltagecontrolled two consecutive and deterministic 90° switching in nanomagnets with intrinsic fourfold magnetic anisotropy^{21,22}, or precessional magnetization switching^{3,23,24,25,26,27,28,29,30}.
Building on the strainmediated voltagedriven 180° perpendicular magnetization switching suggested in ref. 3, this work computationally demonstrates an inplane 180° magnetization switching driven by a voltage in an slightly ellipseshaped nanomagnet on top of a 400nmthick (as in ref. 31) polycrystalline Pb(Zr,Ti)O_{3} (PZT) thin film as an example. The amorphous Co_{40}Fe_{40}B_{20} (CoFeB), which shows reasonably good magnetoelastic coupling^{32} and has been experimentally integrated with piezoelectric thin films^{33} and substrates^{34}, is selected as the model magnetic material. The dimension of the amorphous CoFeB disk is 150 nm × 135 nm × 4 nm, an optimized dimension simultaneously allowing small addressing piezostrains and stable magnetization when strain is removed, see discussion later.
Results
According to our finiteelement analyses (see Methods), applying identical voltages to the two top electrodes and grounding the bottom electrode (Fig. 1a) generate nonuniform outofplane electricfields (Fig. 1b) and inplane piezostrains (that is, ε_{p} = ε_{yy} − ε_{xx}, in Fig. 1c) on the PZT film surface, consistent with existing experiments^{35}. According to the simulated threedimensional (3D) electric field distribution (see Supplemental Figure S1), local electric fields are largely parallel to the downward polarization, indicating that polarization switching is unlikely. In addition, even the maximum local electric field (~16.1 MV/m) is well below the dielectric breakdown field of the 400nmPZT film (~25 MV/m from ref. 31). The average uniaxial piezostrain (<ε_{p}>) inside the CoFeB region (the central ellipse in Fig. 1b,c) is approximately 1056 ppm. Our analysis further shows that approximately 90% of such uniaxial piezostrain is transferred to the top CoFeB magnet across the interface (see Supplemental Figure S2), because of the shearlag induced strain relaxation^{36}. However, it will be shown that the remaining uniaxial strain (about 950 ppm on average) along the yaxis is sufficient to overwhelm the inplane magnetic shape anisotropy, switching the initial magnetic easy axis (EA) by 90° from the initial inplane long axis (x) to the short axis (y).
More importantly, we propose a full 180° magnetization switching can be enabled by exploiting the dynamics of such 90° magnetic EA switching. As sketched in Fig. 1d, upon the application of the uniaxial piezostrain along the yaxis, the initial magnetization vector (M) along the +xdirection starts to precess around the yaxis (i.e., the new magnetic EA) in the film plane and eventually will stabilize along the +y or −y direction, completing a 90° magnetization switching. If the voltage is turned off (strain will then be released) when M possesses −x inplane component, the initial EA along the xaxis would reappear and M would eventually stabilize along the −xdirection (the EA) via precession and damping.
Thus a fast rise (release) of the piezostrain when the voltage is turned on (off) is essential to manipulate the magnetization dynamics. In this regard, finiteelement models coupling linear piezoelectricity with elastodynamics (see Methods) are established. As shown in Fig. 2a, local piezostrain reaches its first peak with 0.1 ns followed by quickly attenuated oscillation around the equilibrium value of 1056 ppm. Similar feature is exhibited in the process of piezostrain release when voltage is turned off (see Fig. 2b), where the first peak also appears at about 0.1 ns. The rise and fall time of the voltage is set as 0.07 ns, the same as the experimental setup in ref. 37. We will show below that such minimum time of strain rise/release is much shorter than the time required by straininduced magnetization switching.
Figure 3a shows the precessional 90° inplane switching of average magnetization from +xaxis (<m_{x}> ≈ 1) to −y axis (<m_{y}> ≈ −1) under the static voltage of 0.43 V, calculated from phasefield simulations (see Methods). The relatively small outofplane average magnetization component (that is, <m_{z}><0.1) suggests that the precession occurs largely in the film plane. If the voltage is turned off at 2.17 ns where xcomponent of the average magnetization (<m_{x}>) reaches its negative maximum or any other stages when <m_{x}> is negative (see Supplemental Figure S3), a 180° inplane magnetization switching can occur. As shown in Fig. 3b, applying a squarewave voltage pulse of 2.27 ns (pulse I) can switch the magnetization by 180° from +xaxis (<m_{x}> ≈ 1) to −xaxis (<m_{x}> ≈ −1). The time 2.27 ns, accommodating the time required by both the straininduced magnetization switching (2.07 ns) and the strain rise/release processes (ca. 0.1 × 2 = 0.2 ns), represents the minimum voltage pulse duration for magnetization reversal in the present design. Even shorter pulse durations can be achieved by applying larger voltage (therefore larger piezostrain), or/and by further reducing the rise/release time of piezostrain through structural optimization of both the PZT film and the top electrodes. Furthermore, applying another voltage pulse (pulse II in Fig. 3b) can switch the magnetization back to the +xaxis in a similar manner, where magnetization firstly precesses around the yaxis (see the lower panel of Fig. 3b and more clearly, in Fig. 3c) followed by precessional relaxation. This demonstrates that the 180° inplane magnetization switching is repeatable.
Discussion
Switching speed
For the present design, the pulse duration of 2.27 ns (Fig. 3) approximately represents the total time required by such piezostrainmediated voltagedriven inplane magnetization reversal, because both the time required by establishing/removing electricfield (~1.4 ps) and transferring the piezostrain to the top CoFeB magnet (~0.83 ps) are negligible. The former equals the time required by charging/discharging the capacitor, which is typically described by ResistanceCapacitance (RC) delay (that is, t_{RC} ≈ R × C). A typical resistance (R) of about 100 Ω (ref. 38) was used and the capacitance (C) is estimated to be about 13.9 fF according to C ≈ P_{r}S/U, with an area of top electrodes (S ~ 0.04 μm^{2}), a remanent ferroelectric polarization of the 400nmthick PZT film (P_{r}~15 μC/cm^{2} from ref. 31) and the magnitude of voltage pulse (U ~ 0.43 V). The time span for the strain transfer (t_{p}) can be estimated according to t_{p} = d/v from ref. 19, where d is the thickness of CoFeB nanomagnet (d = 4nm) and v is the velocity of elastic wave in the solid, calculated as ≈ 4831 m/s using the Young’s modulus [Y = (c_{11} + 2c_{12})(c_{11} − c_{12})/(c_{11} + c_{12})) ≈ 187 GPa] and the mass density (ρ = 8 × 10^{3} Kg/m^{3}) of the amorphous CoFeB.
Energy consumption
The energy consumption per switching (E_{con}), or the energy stored in the PZT capacitor, was estimated to be about 1.29 fJ according to E_{con} = 0.5P_{r}SU (refs 16, 19, 39, 40, 41. The area energy consumption per switching (that is, E_{con}/S) is about 0.03J/m^{2}, approximately two orders of magnitude smaller than that of BiFeO_{3}filmbased multiferroic heterostructure^{16}.
Energy dissipation
The energy dissipation per swtiching (E_{dis}) was estimated to be about 0.85 fJ according to E_{dis} = 0.25πCtan(δ)U^{2} (ref. 42), where the dielectric loss tan(δ) should depend on the frequeny of applied voltage pulse (f = 1/(2.27 ns + 12.6 ns) ≈ 67.2 MHz) and the corresonding tan(δ) is about 0.42^{43}. Note that we expect a smaller tan(δ) in practice, because the design employs a unipolar shortduration pulse voltage rather than bipolar ac voltage. The area energy dissipation, esimtaed as 0.021 J/m^{2} ( = E_{dis}/S), is approximately three orders of magnitude smaller than that dissipated in currentdriven magnetization switching. For example, an area energy dissipation of approximately 12 J/m^{2} (estimated according to (V^{2}t)/(RA), where V is the voltage, t is the duration of current pulse, R is the junction resistance and A is the junciton (free layer) area, from ref. 44 has been demonstrated in an Orthogonal SpinTransfer Magneic Random Access Memory (OSTMRAM), which integrates an additional spinpolarizing layer with perpendicular magnetic anistropy to apply larger initial spintranfertorque on the magnetization in the free layer^{44,45}. The magnetizaton reversal in such OSTMRAM is also precssional^{44} and can be completed in subnanoseconds^{45}.
Switching energetics
Figure 4a,b show the polar plots of magnetostatic and elastic energy densities of the CoFeB elliptical nanomagnet, respectively, calculated from phasefield simulations. The magnetostatic energy favors a magnetic easy axis along the xaxis (long axis), with a potential barrier of about 3609 J/m^{3} (Fig. 4a) while the elastic energy favors an easy axis along the yaxis due to positive magnetostriction, with a potential barrier of about 3644 J/m^{3}. Thus the magnetostatic anisotropy can be overwhelmed by the elastic anisotropy, triggering a 90° easy axis switching from xaxis to yaxis in the CoFeB. Given the similar magnitudes of the magnetostatic and elastic potential barriers, the present <ε_{p}> of 1056 ppm represents the critical magnitude of the piezostrain pulse (denoted as ε_{cr}) for the easy axis switching and thereby the magnetization reversal.
Size scaling
Figure 5a shows how the scaling of inplane dimension influences the ε_{cr} (see the upper panel) and in the bottom panel, the roomtemperature (T = 300 K) thermal stablity factor (F_{barrier}/k_{B}T). Here the energy barrier F_{barrier} equals the magnetostatic potential barrier (e.g., the barrier between x and y axes shown in Fig. 4a) multiplied by the volume of the magnet and k_{B} is the Boltzmann constant. A thermally stable magnetic state typically requires a minimum stability factor of 40 (ref. 19), as marked by the dashed line. For a fixed dimension of the inplane long axis, reducing the length of the short axis decreases the aspect ratio (r). This leads to stronger anisotropy in magnetostatic energy density (that is, deeper potential barrier) and thereby enhanced thermal stability. The ε_{cr} for overcoming the energy barrier rises accordingly. On the other hand, for a fixed aspect ratio, scaling up the elliptical nanomagnet (represented by the increasing length of longaxis) also enhances the thermal stability. However, this is purely due to the increase in volume and the ε_{cr} remains virtually unchanged as long as the nanomagnet remains as a uniform domain (e.g. see Fig. 5b). The increased heterogeneity of inplane magnetization appears at large dimensions, for example, the leaflike domain shown in Fig. 5c. This will also increase the ε_{cr}, similarly to our previous report of pulsevoltagedriven perpendicular 180° magnetization switching^{3}.
Summary
By combining finiteelement analysis and phasefield modeling, we have demonstrated a nonvolatile 180° inplane magnetization switching driven by a unipolar pulse voltage in a multiferroic heterostructure with amorphous Co_{40}Fe_{40}B_{20} elliptical nanomagnet on top of PZT film. We have shown that the magnetization vector, initially along the inplane long axis, would precess across the inplane short axis (>90° precession) when pulse voltage is on. As the voltage is turned off, the magnetization will relax to the other direction of the long axis via damped precession, completing an 180° switching. Compared to previous reports utilizing outofplane magnetization precession^{3,12,23,24,25,26}, the magnetization precession trajectory is virtually within the horizontal plane mainly due to the strong outofplane demagnetization. Compared to the previous experimental demonstrations^{26,27,28,29,30} of voltagecontrolled precessional magnetization reversal (via voltagecontrolled magnetic anisotropy) in the presence of a static magnetic field, the present design does not involve the use of any magnetic fields. Furthermore, we have shown that such inplane 180° magnetization switching is repeatable, fast (switching time ~2.3 ns) and requires ultralow energy consumption (~1.3 fJ) and ultralow energy dissipation (~0.9 fJ). Further enhancement of switching speed and reduction of energy consumption/dissipation are possible, for example, by using a thinner piezoelectric layer or/and a piezoelectric layer with larger piezoelectricity than PZT (e.g., Pb(Mg_{2/3}Nb_{1/3})O_{3}PbTiO_{3}, ref. 46), or using magnetic materials with stronger magnetoelastic coupling than amorphous CoFeB (e.g., Fe_{81}Ga_{19}, ref. 47). We believe that these promising features will make the present design a competing alternative to spintorquemediated currentdriven magnetization switching that underpins current spintronic device technologies.
Methods
Finiteelement analyses
We consider polycrystalline PZT film as a model piezoelectric material system. Polycrystalline PZT film has proven by experiments^{35} to be capable of generating inplane uniaxial (biaxial anisotropic) local piezostrains through an electrode design similarly to what has been presented here (i.e., Fig. 1a). The finiteelement model for the PZT piezoelectric material is implemented in the commercial software package COMSOL Multiphysics. The static electric field (E) and piezostrain (ε) are calculated by solving the corresponding electrostatic equilibrium equation () and mechanical equilibrium equation () under the shortcircuit boundary and mechanical clamped boundary conditions, respectively. The piezoelectric constitutive equations are,
where D is electric displacement tensor, σ is the stress tensor, e^{T} is the transpose of piezoelectric stress coefficient tensor, is the vacuum permittivity, is the relative dielectric constant tensor under the mechanical clamped boundary condition and c^{E} is elastic constant tensor under constant electric field. A PZT thin film of 3000 nm×3000 nm×400 nm is considered which is fixed at the bottom (by the substrate) and is free on top. To mimic a large sample, periodic boundary conditions are applied on the four surrounding side surfaces. A free tetrahedral mesh with quadratic shape functions is used for discretizing the domain. The MUltifrontal Massively Parallel Sparse direct Solver (MUMPS) is utilized to solve the governing partial differential equations. The voltage is applied to a pair of trapezoidlike top electrodes and the bottom electrode is grounded. The electromechanical properties of PZT are presented in the Supplemental Materials S5. Shape of two top electrodes is optimized to minimize the local electric field concentration and to obtain an almost uniform strain distribution in the central ellipse region.
Timedependent simulations are performed to calculate the minimum rise (relaxation) time of generated piezostrain upon applying (removing) the voltage. A structural Rayleigh damping is considered in the PZT thin film, which is the source of energy dissipation. The governing elastodynamics equation with damping is
where ρ is density (=7.5 × 10^{3} Kg/m^{3} for PZT), u is the mechanical displacement, t is time and α and β are the mass damping coefficient and the stiffness damping coefficient, respectively. A PZT film typically shows a viscous damping (that is, α = 0 s^{−1})^{48}. The stiffness damping coefficient β is taken as 6 × 10^{−12} s. From local piezostrain vs. time plot, the minimum rise/fall time of strain can be determined at the first strain peak after applying (removing) the voltage. Influence of stiffness damping coefficient β on the rise of piezostrain is discussed in the Supplemental Figure S4 for an applied voltage of 0.43 V.
The same tetrahedral mesh elements with quadratic shape functions are utilized for discretizing the space domain. The timedependent equation (2) is solved using MUMPS direct linear solver with row preordering and pivoting in combination with the implicit timedependent BDF solver where the time steps are chosen automatically by the solver to get best convergence. A scaled absolute tolerance of 0.001 with backward Euler initialization is utilized.
Phasefield model
Phasefield method typically involves the use of an order parameter that changes continuously across the interface of two phases, or across a domain wall separating two domains. In the present phasefield model, local magnetization vector M = M_{s} (m_{x}, m_{y}, m_{z}) is used as the main ‘phasefield’ order parameter, where m_{i} (i = x, y, z) represents the direction cosine (m = M/M_{s}) and M_{s} is the saturation magnetization. Magnetic domain structures are denoted by the spatial distributions of the M. The temporal evolution of the magnetization can be described by the LandauLifshitzGilbert (LLG) equation,
where α and γ_{0} are the Gilbert damping coefficient and the gyromagnetic ratio, respectively. Equation (3) is solved by a semiimplicit Fourier spectral method (see details in refs 49 and 50). The real time span (Δt) corresponding to each normalized numerical time step (Δτ) is about 0.09 ps according to , with Δτ = 0.02. The effective magnetic field is expressed as , with μ_{0} representing the vacuum permeability and F_{tot} the total free energy of CoFeB nanomagnet. For the CoFeB amorphous nanomagnet grown on a ferroelectric layer, F_{tot} can be written as,
where f_{mc}, f_{exch}, f_{elastic} and f_{ms} are the magnetocrystalline anisotropy, exchange, elastic and magnetostatic energy densities, respectively. The magnetocrystalline anisotropy energy is ignored due to the isotropic nature of an amorphous CoFeB nanomagnet. f_{ms} is relevant to the shape of the magnet and is calculated using a finitesizemagnet magnetostatic boundary condition^{51} in the present model. The magnetic exchange energy density (f_{ex}) is calculated as , where A_{ex} indicates the exchange constant. The elastic energy density (f_{elastic}) is calculated as,
where ε^{0} denotes the stressfree strain (see derivation in ref. 52) the total strain ε contains a homogeneous part ε^{hom} and a heterogenous part ε^{het} following Khachaturyan’s theory of microelasticity^{53}. The ε^{hom} describes the macroscopic deformation. It can arise from the lattice/thermal mismatch between the magnet and the ferroelectric layer underneath (assuming to be zero herein), or/and the applied piezostrain calculated from finiteelement simulations (i.e., the average uniaxial piezostrain <ε_{p}> in Fig. 2). The ε^{het}, showing a zero volumetric integral over the entire magnet, can be calculated according to ε^{het} = , where u indicates the local mechanical displacement and is obtained by solving mechanical equilibrium equation .
Threedimensional discrete grids of 32Δx × 32Δy × 20Δz with a real grid space Δz = 1 nm and Δx = Δy = 5.0 nm, are used. The influence of grid space on the simulated magnetization dynamics is discussed in Supplemental Materials S6. The bottom grids of 32Δx × 32Δy × 11Δz are utilized to describe the piezoelectric layer (PZT), the top grids of 32Δx × 32Δy × 5Δz represent the air phase and the remaining grids of 32Δx × 32Δy × 4Δz in the middle are used to describe the ellipse CoFeB nanomaget with a thickness d ( = 4Δz) of 4 nm and the air surrounding the magnet. Note that the 4nmthickness is smaller than the exchange length (l_{ex} = , from ref. 54) that is calculated as about 4.9 nm. Thus the local magnetization vectors should very likely be parallel to each other along the thickness direction. The shape of the ellipse nanomagnet is described by a shape function with r the aspect ratio. The top (air) and bottom (PZT) grids allow us to accommodate the 3D magnetic stray field surrounding the CoFeB nanomagnet in simulations. The air phase permits creating stressfree top and lateral surfaces in the CoFeB, leading to nonuniform strain distribution in the magnet (Figure S2). Simulations for materials with different sizes can be achieved by changing the grid number or grid size (Δx, Δy, Δz), which are used to investigate effects of the lateral size and aspect ratio on the magnetization reversal. The material parameters of CoFeB used for simulations are obtained from literatures and the saturated magnetization, elastic constants, magnetostrictive coefficients, Gilbert damping constant, gyromagnetic ratio and exchange constant of CoFeB nanomagnet are listed as follows: M_{s} = 1.0 × 10^{6} A/m (ref. 55); c_{11} = 2.8 × 10^{11} N/m^{2}, c_{12} = 1.4 × 10^{11} N/m^{2} and c_{44} = 0.7 × 10^{11} N/m^{2} (ref. 56); λ_{s} = 3.1 × 10^{−5} (ref. 57); α = 0.005 (ref. 58); γ_{0} = 1.76 × 10^{11} Hz/T (ref. 59); A_{ex} = 1.5 × 10^{−11} J/m (ref. 60).
Additional Information
How to cite this article: Peng, R.C. et al. Fast 180^{o} magnetization switching in a strainmediated multiferroic heterostructure driven by a voltage. Sci. Rep. 6, 27561; doi: 10.1038/srep27561 (2016).
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Acknowledgements
This work was supported by the NSF of China (Grant Nos 51332001, 51472140 and 11234005) and the NSF (Grant No: DMR1410714 and DMR1210588) and was also partially sponsored by the ProjectBased Personnel Exchange Program by the China Scholarship Council.
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J.M.H. conceived the design and supervised the simulations. R.C.P. performed the simulations with feedback from K.M. L.Q.C. and C.W.N. lead the projects. J.M.H. and R.C.P. wrote the paper with feedback from K.M., J.J.W., L.Q.C. and C.W.N. All contributed analyses.
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Peng, RC., Hu, JM., Momeni, K. et al. Fast 180° magnetization switching in a strainmediated multiferroic heterostructure driven by a voltage. Sci Rep 6, 27561 (2016). https://doi.org/10.1038/srep27561
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DOI: https://doi.org/10.1038/srep27561
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