Electric field induced magnetization reversal in magnet/insulator nanoheterostructure

ABSTRACT Electric-field control of magnetization reversal is promising for low-power spintronics. Here in a magnet/insulator nanoheterostructure which is the fundamental unit of magnetic tunneling junction in spintronics, we demonstrate the electric field induced 180 magnetization switching through a multiscale study combining first-principles calculations and finite-temperature magnetization dynamics. In the model nanoheterostructure MgO/Fe/Cu with insulator MgO, soft nanomagnet Fe and capping layer Cu, through first-principles calculations we find its magnetocrystalline anisotropy linearly varying with the electric field. Using finite-temperature magnetization dynamics which is informed by the first-principles results, we disclose that a room-temperature 180 magnetization switching with switching probability higher than 90% is achievable by controlling the electric-field pulse and the nanoheterostructure size. The 180 switching could be fast realized within 5 ns. This study is useful for the design of low-power, fast, and miniaturized nanoscale electric-field-controlled spintronics.


Introduction
Nowadays magnetic storage plays a critical role in the development of fast, high-density, nonvolatile memory technology [1]. The typical example is magnetic random access memory (MRAM) device which relies on the magnetic tunneling junction (MTJ) to storage bit information. MTJ is usually constituted of insulator, free magnetic layer in which the magnetization can be switched by an external field, and fixed magnetic layer in which the magnetization direction is firmly pinned. For example, if the configuration of magnetization in the free layer antiparallel to that in the fixed layer represents bit '0ʹ, switching the magnetization in the free layer during the writing process to achieve a parallel configuration leads to bit '1ʹ. A huge number of MTJ units realize the information storage. The writing in MTJ is intrinsically a magnetization reversal process in the free layer. Therefore, developing new strategies to switch magnetization in the free layer is indispensable for revolutionizing spintronics.
Generally, three methods have been proposed to switch magnetization. Firstly, the switching or writing process can be driven by an external magnetic field. In MRAM, built-in wires in every memory cell are required for the switching of a nanomagnet, i.e., the magnetic field must be generated by passing a current through a wire. The extra wires not only make the device circuit complicated and thus hinder the high density but also generate current to result in energy dissipation and overheating. Secondly, spin-transfer torque and spin-orbit torque can be used to switch magnetization. They allow for high density, but are not a low-power method in terms of the high current density. Thirdly, the electric-field control of magnetization is recently massively explored to switch magnetization. Since this method is free of electric currents, it is very promising for the future extremely low-power next-generation spintronics based memory devices.
Generally, multi-field coupling provides more freedom for the design of nanoscale functional structure [2][3][4][5][6][7][8][9]. In detail, the electric-field control of magnetization can be realized either in multiferroic materials which possess more than one ferroic effects and the coupling between two of them, or in insulator/magnet heterostructure whose interface magnetic property can be well modulated by an external electric field. For example, through the magnetoelectric (ME) coupling between the electrical polarization of a ferroelectric material and the magnetization of a ferromagnet, the magnetization can be controlled by applying an electric field. However, the ME coupling is weak in singlephase systems. Such a control is usually implemented in ferroelectric/ferromagnetic heterostructures through strain-mediated elastic coupling , interface bonding [33][34][35][36], and exchanging coupling [37][38][39][40]. On the contrary, in insulator/magnet heterostructure, the electric field is found to induce charge change of the magnetic atoms at the insulator/magnet interface and thus tune the spin-orbital coupling and magnetocrystalline anisotropy of these interfacial atoms. Although this effect is limited to the interface, it fits well for the high-density devices in which the MTJ thickness is nanoscale and the interfacial effect is strong enough to induce magnetization reversal.
In this work, we study the electric-field control of magnetization reversal in nanoscale magnet/insulator nanoheterostructure MgO/Fe/Cu, with a focus on the multiscale scenario for predicating the electric field induced magnetization reversal behavior in the free layer of MTJ. First-principles calculations are carried out to reveal the dependence of magnetocrystalline anisotropy of Fe on the electric field applied to MgO/Fe/Cu. Informed by the first-principles results, magnetization dynamics simulations without and with temperature-induced thermal fluctuations are performed to identify the thermal energy barrier for the equilibrium state, the necessary conditions for magnetization switching, the conditions for 180 � switching, and the temperaturerelated probabilistic switching events. Figure 1(a) shows a typical MTJ unit which is a Cu/Fe/MgO/Fe/Cu nanoheterostructure. Since the magnetization of the top Fe layer is firmly pinned in the plane, only the bottom Fe layer which is free to switch is of interest. By applying a voltage V to the MTJ system, no electric current will be generated in Cu and Fe, and only electric field will be generated in MgO. The atomic structure for first-principles calculations is illustrated in Figure 1(b). For a good lattice match between MgO and Fe, MgO unit cell is rotated by 45 degrees. At the interface, Fe atom is put on the top of O atom for lowering the total energy of the supercell. The supercell is constructed along the [001] direction (x axis), containing nine-layer Fe, nine-layer MgO, four-layer Cu, and 10-Å-thick vacuum. The electric field is imposed by the dipole layer method [41], with the dipole placed in the middle of the vacuum region.

Electric field control of magnetocrystalline anisotropy
The first-principles calculations are carried out within the density functional theory and the framework of the projector augmented-wave formalism as implemented in the Vienna ab initio simulation package (VASP) [42]. The Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional in the generalized gradient approximation (GGA) is employed. An energy cutoff of 500 eV and a Monkhorst-Pack k-mesh 21 � 21 � 1, at which a good convergence of magnetocrystalline anisotropy energy is achieved, are utilized. The in-plane lattice parameter of the supercell is set as that of Fe (0.287 nm). The MgO layers except for the three layers close to Fe are fixed. The atomic positions of all other layers in the x direction are relaxed. The convergence criteria for the structure relaxation are set as 10 À 6 eV and 2 meV/Å for the energies and forces, respectively. By using the self-consistent charge density, non-self-consistent calculations with the spin-orbit coupling are performed to get the total energy as a function of the orientation of the quantization axis. The total energy difference between the different quantization axes is used to determine the magnetocrystalline energy. The magnetocrystalline anisotropy constant (K) of the Fe layer is evaluated as the difference of the total energy per unit Fe volume (nominal thickness 1.2 nm) when the magnetization is along (100)/(010) (z=y) and (001) (x) directions. Positive and negative K indicates perpendicular and in-plane magnetocrystalline anisotropy, respectively.  a voltage jump appears there. From the slope of the voltage distribution in MgO, the electric field (E) can be estimated as 0.625 V/nm which is usually related to the real electric field measured in experiments. Repeating the similar calculation procedure for K at different E, the dependence of K on E can be obtained, as shown in Figure 2(b). Without the applied electric field (E ¼ 0), K is around 1.4 MJ/m 3 . If additional electric field is applied, K is found to linearly increase with E. By linearly fitting the data in Figure 2 (b), we obtain the following relationship The large slope of 0:5257 MJ/m 3 /(V/nm) indicates remarkable magnetoelectric coupling in MgO/Fe/Cu system, and means that K can be modulated in a wide range by applying an electric field.

Static analysis of energy
The magnetization state in a single-domain thin film with a geometry of elliptic cylinder can be described by two angles θ and ϕ, as illustrated in Figure 1(c). The associated total energy density is composed of magnetocrystalline anisotropy energy density and demagnetization energy density [14], i.e.
in which the electric field dependent KðEÞ is obtained from first-principles calculations Figure 1(b) and the saturation magnetization of Fe is taken as μ 0 M s ¼ 2:15 T. The demagnetization factors (i.e. N x , N y , and N z in Eq. 2) of an elliptical cylinder can be calculated as [43] N z ¼ in which the second aspect ratio � ¼ t=2b and the eccentricity 2¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 1 À b=a ð Þ 2 q , with a, b, and t (1.2 nm) as the semi-major axis, semi-minor axis, and thickness, respectively. The coefficients v n and u n can be calculated as functions of � and its complete elliptic integrals of the first and second kind. For the detailed formulations of v n and u n , one is referred to the literature [43]. It should be noted that in principle the stray field and interlayer exchange can also be included in Eq. 2, but are ignored here.
From the viewpoint of information storage such as binary logic and memory applications, the energy barrier at the equilibrium state (E ¼ 0) is usually required to exceed 40 k B T to make a sufficiently stable magnetization. The energy barrier can be calculated as the energy difference between the magnetically easy (z) and hard (x and y) axes, i.e. On the one hand, ΔG should be high enough to stabilize the magnetization state at T ¼ 300 K (i.e. ΔG > 40 k B T). On the other hand, ΔG should be not too high so that applying en electric field could switch the magnetization state. According to Figure 3(a), the geometry a ¼ 2:5b is chosen as an example to analyze the necessary condition for a possible magnetization reversal, and the result is presented in Figure 3(b). In general, an electric field is applied to increase K and thus makes x as the easy axis, indicating a 90 � switching. For example, when there is no electric field, the energy density landscape in Figure 4(a) shows an easy axis along z. If an electric field of E ¼ 0:7 V/nm is applied, the easy axis is changed to x Figure 4(b). The shadow region in Figure 3(b) points out the condition (electric field and film size) under which the magnetization reversal is possible. It is clear that for a ¼ 2:5b, an electric field higher than 0.6 V/nm is required for a magnetization reversal, which is still lower than the typical dielectric breakdown field strength of MgO (2.4 V/nm [44]).
From the above static analysis of energy, it can be deduced from Figures 3(b) and 4(b) that an electric field can change the easy axis from z axis to x axis and thus switch the magnetization by 90 � . However, a 180 � switching is highly desired for the magnetic storage. Especially, a 180 � switching purely by an electric field is critical for the design of low-power spintronics. In following, switching dynamics will be explored to realize the 180 � switching, and a film with a ¼ 50 nm and b ¼ 20 nm will be taken as an example to elucidate the basic idea.

Electric field induced 180 � switching
The static analysis results in Figure 3(b) only depict the necessary (not the sufficient) condition for a 180 � switching. In contrast, the electric field-induced magnetization dynamics provides more freedom to control the reversal process. With the temperature effect considered as thermal fluctuations, the magnetization dynamics of a single-domain object at finite temperature is governed by [14,45] in which γ 0 is the gyromagnetic ratio constant, α ¼ 0:01 is the damping coefficient of Fe, Δt ¼ 0:2 ps is the time step, and P i (i ¼ 1, 2) is a stochastic process with Gaussian distribution, zero mean value, and completely uncorrelated property in time. P i is generated by the Box-Muller method [46]. The characteristic time τ N is related to volume V and temperature T as τ À 1 N ¼ 2αγ 0 k B T=½M s ð1 þ α 2 ÞV�. It is obvious from the random terms in Eq. 5 that higher temperature and smaller volume will result in more intensive thermal fluctuations. A miniaturized MTJ unit with small volume is good for high density, but possibly suffers from thermal fluctuations induced randomness. It should be mentioned that all the analysis is based on the single-domain assumption. To verify this assumption, 3D micromagnetic simulation [47] has been carried out on the elliptical cylinder with a ¼ 2b ¼ 50 nm and t ¼ 1:2 nm. The micromagnetic simulation results confirm that this cylinder is small enough to maintain the single-domain configuration when an electric field is applied to stimulate the magnetization dynamics.
In contrast to the static analysis, here we apply an electric field pulse to trigger the 180 � switching by utilizing the precessional magnetization dynamics which is described by Eq. 5. As a first step, we investigate the case without thermal fluctuations (T ¼ 0). Figure 5 (a) presents the typical switching trajectory triggered by an electric field with a magnitude of 0.8 V/nm and a pulse duration (t s ) of 0.6 ns. After the removal of electric field at t s ¼ 0:6 ns, the magnetization component m z further reaches À 1 and a 180 � switching is realized. The switching time is around 3 ns and the switching is deterministic at T ¼ 0. It should be noted that in order to achieve a 180 � switching, the pulse duration has to be precisely controlled. For the successful 180 � switching, the switching time (t switch ) as functions of E and t s is shown in Figure 5(b). It can be found that a precise control of E and t s at a reasonable range could achieve a fast 180 � switching within 2 ns. It should be mentioned that a long pulse dose not ensure a fast switching here. The main reason is the precession dynamics of Fe with a low damping coefficient of 0.01. If a long pulse is applied, the magnetization will do precession for a long time and so the switching time will not decrease as expected.
However, if finite temperature is considered (i.e. T > 0 in Eq. 5), the magnetization dynamics will be intrinsically altered. Even in the equilibrium states with E ¼ 0, the magnetization is not exactly aligned along the easy axis z. As shown in the histogram in Figure 6, the finite-temperature effect makes the magnetization fluctuate within several degrees around the easy axis. When the temperature is increased from 300 K in Figure 6(a) to 400 K in Figure 6(b), the magnetization fluctuation around the easy axis is more intensive and the distribution of magnetization components m i becomes wider.
In addition, the finite temperature makes the electric field induced switching as probability events. The deterministic switching at T ¼ 0 will be undeterministic at T > 0. For example, under the same condition E ¼ 0:8 V/nm and t s ¼ 0:6 ns as in Figure 5(a) with T ¼ 0, 180 � switching can either succeed (Figure 7(a)) or fail (Figure 7(b)) at T ¼ 300 K. This kind of probabilistic behavior makes the previous studies of the 180 � at 0 K to be reexamined.
For the switching at finite temperatures, we calculate the switching probability (the percentage of successful 180 � switching) as functions of E and t s at 300 K, as shown in Figure 7(c). It can be seen that even though room temperature (300 K) makes the 180 � switching probabilistic, it is still possible to achieve the switching probability above 90% by carefully designing the magnitude and the pulse duration of the electric field. The wide region with high switching probability ( > 90%) in Figure 7(c) indicates the design flexibility of electric field-induced 180 � switching at room temperature. Unquestionably, increasing the switching probability as much as possible is desired. However, for memory applications where different on-chip error detection and correction schemes exist, the achieved switching probability above 90% here is still practicable. For an estimation of switching time at room temperature, as an example in Figure 7(d) we present 1,000 switching trajectories with a switching probability of ,92:6% at E ¼ 0:76 V/nm and t s ¼ 0:61. The switching time is found to be approximately 5 ns at room temperature. It should be noted in Figure 7(c) that a long pulse does not ensure a high switching probability. The main reason may be related to the slow precession dynamics of Fe. Under a long pulse, the magnetization will slowly precess for a long time during which the accumulated effects of thermal fluctuations will be strengthened to reduce the switching probability.

Conclusions
The electric field-induced 180 � magnetization switching in magnet/insulator nanoheterostructure has been demonstrated by combining first-principles calculations and finitetemperature magnetization dynamics simulations. In a model nanoheterostructure system MgO/Fe/Cu, with the electric field dependent magnetocrystalline anisotropy (K) from first principles as input, the static analysis of total energy density of an elliptic Fe nanomagnet is performed to identify the conditions: (1) the energy barrier at the equilibrium state (E ¼ 0) should be larger than 40 k B T to overcome the temperature-induced thermal fluctuations for practical device applications; (2) the voltage induced K change should exceed the energy barrier and make the magnetization reversal possible. Magnetization switching dynamics at zero temperature indicates that precisely controlling the electric field pulse and the nanoheterostructure size could achieve the 180 � switching within several nanoseconds. Moreover, considering the thermal fluctuations at room temperature as random fields, we find the magnetization reversal as probabilistic events and calculate the switching possibility for 180 � switching. The minimum switching time is found to be around 5 ns, which is less than that in the traditional STT-MRAM, MRAM, and DRAM. The present study provides valuable insight into the rational design of electric field controlled and miniaturized nanoscale spintronic devices where temperature-induced thermal fluctuation has a great impact.

Disclosure statement
No potential conflict of interest was reported by the authors.