Electric-field-driven magnetization switching and nonlinear magnetoelasticity in Au/FeCo/MgO heterostructures

Voltage-induced switching of magnetization, as opposed to current-driven spin transfer torque switching, can lead to a new paradigm enabling ultralow-power and high density instant-on nonvolatile magnetoelectric random access memory (MeRAM). To date, however, a major bottleneck in optimizing the performance of MeRAM devices is the low voltage-controlled magnetic anisotropy (VCMA) efficiency (change of interfacial magnetic anisotropy energy per unit electric field) leading in turn to high switching energy and write voltage. In this work, employing ab initio electronic structure calculations, we show that epitaxial strain, which is ubiquitous in MeRAM heterostructures, gives rise to a rich variety of VCMA behavior with giant VCMA coefficient (~1800 fJ V−1m−1) in Au/FeCo/MgO junction. The heterostructure also exhibits a strain-induced spin-reorientation induced by a nonlinear magnetoelastic coupling. The results demonstrate that the VCMA behavior is universal and robust in magnetic junctions with heavy metal caps across the 5d transition metals and that an electric-field-driven magnetic switching at low voltage is achievable by design. These findings open interesting prospects for exploiting strain engineering to harvest higher efficiency VCMA for the next generation MeRAM devices.

A ubiquitous feature in many HM/FM/I heterostructures is the large lateral strain among the HM, FM, and I components which can in turn tune the SOC and hence modify the magnetic anisotropy (MA) and magnetoelectric interfacial coupling. For example, nanostructures of AuCu/FeCo core-shell has been shown to possess superior strain-induced magnetic properties such as high saturation magnetization and coercivity 15 .
HM cap can considerably modify MA of magnetic layers 16 and a combined strain and capping effect can strongly enhance VCMA of a HM/FM/I junction 17 . However, a direct E-field induced switching of magnetization is still elusive in both computation and experiments 3,[7][8][9][10]13,14,[17][18][19] . This raises further questions that (i) If the synergistic effect of strain and HM on VCMA is robust for cap materials across the 5d transition metals and (ii) Whether there exists a choice of HM which enables a direct E-field induced switching in a magnetic junction.
In this work we present ab initio electronic structure calculations which demonstrate that epitaxial strain has a strong effect on MA of Au/FeCo/MgO trilayer in nonlinear manner, leading to a strain-induced spin-reorientation. Furthermore, strain gives rise to a rich VCMA behavior ranging from a ∨ -shape to inverse-∨ -shape (∧ ) E-field dependence with giant coefficient. This demonstrates the universal and robust VCMA behavior in strained HM/FM/I junction for HM across the 5d transition metals. We also predict that an E-field-driven magnetization switching can be archived at low voltage.
Effect of strain on zero-field MA. Figure 1 shows the variation of the zero-field MA of the Au/FeCo/MgO junction with strain, η FeCo . The inset shows the schematic structure of Au/FeCo/MgO trilayer. The iron atoms at the Fe/MgO and Fe/Au interfaces are denoted by Fe1 and Fe2, respectively. The system shows a nonlinear magnetoelastic (MEL) behavior with a spin-reorientation at ~4% strain, in contrast to that in Ta/FeCo/MgO where MA is linearly dependent on strain with a magnetization switching occurs at ~2.5% 17 . This is a striking example on effect of HM cap on functional properties of a magnetic junction at nanoscale.
To explore the origin of the nonlinearity, we write the general expression MA where t is the FM thickness. The total magnetic energy-density functional of the FM, f M , is defined as where e n (n = 1-6) is strain tensor in Voigt notation, α k (k = 1-3) is direction cosine of the magnetization vector, and K i 2 is the interface magnetocrystalline anisotropy. In a thin film grown epitaxially along the [001] crystallographic direction, there is no shear strain, i.e., e 5 = e 5 = e 6 = 0. The MEL energy density f MEL is expanded to second order 20 2 . For D 11 = 0 or γ = 1, the nonlinear term in equation (3) identically vanishes. A finite D 11 indicates that the presence of magnetization along a given direction induces an additional spontaneous strain on the FM film. Therefore, the combination of phenomenological analysis and ab initio calculation clearly shows that magnetic contribution to elastic behavior is significant and is the origin leading to the nonlinearity in the Au/FeCo/MgO trilayer. By fitting the ab initio results we find K i 2 = − 0.61 erg/cm 2 , B 1 = − 8.0 × 10 8 erg/cm 3 and D 11 = 44.8 × 10 10 erg/cm 3 . For bulk FeCo, the first-order MEL coefficient is B v 1 = − 3 × 10 8 erg/cm 3 22 . From the expression = 1 , the interfacial contribution B i 1 = − 13.4 erg/cm 2 can be inferred. For 100-nm-thick Fe film on MgO experiments reported B 1 = − 0.32 × 10 8 erg/cm 3 and D 11 = 1.1 × 10 10 erg/cm 3 . The value of B 1 is close to that of bulk Fe of − 0.34 × 10 8 erg/cm 3 since the surface and interface effects are small in a thick Fe film 23 . Our values for B 1 and D 11 are an order of magnitude larger than these values and are consistent with the fact that B v 1 of bulk Fe 50 Co 50 is an order of magnitude larger than bulk Fe. Moreover, in the Au/FeCo/MgO nano-juction interface effects are crucial.
To elucidate the microscopic mechanism of the strain effect on MA, we calculate k-resolved MA according to the force theorem 24 Here, ε(n, k) [ Figure 2(a,c,e) display MA(k) for η FeCo = 0, 2 and 4%, respectively. Figure 2(b,d,f) show the corresponding energyand k-resolved distribution of the orbital character of the minority-spin bands of the Fe1-derived d xy , d xz,yz , − d x y 2 2 , and d z 2 states along ΓM for η FeCo = 0, 2, and 4%, respectively. We find that the strain-induced modification of the zero-field MA is mainly due to changes of the band structure of the interfacial Fe1 atom.
The MA can be expressed approximately in terms of the in-and out-of plane components of the orbital angu- Here u are occupied minority, occupied majority, and unoccupied minority-spin states (energies), and ξ is the SOC constant. Unless otherwise stated throughout the remainder of this manuscript the d-states refer to minority-spin states. At there is a large downward shift of the occupied Fe1 d xy around Γ leading to an increase of its energy separation to the unoccupied Fe1 d xz,yz states and hence an increase of MA(BZP1). At BZP2, the unoccupied d xy and d xz,yz slightly above the Fermi energy shift down and become occupied. This eliminates in turn the SOC of the occupied Fe1 − d x y 2 2 -and d xz(yz) -derived states with the unoccupied d xy and d yz(xz) through L z , respectively. Consequently, the MA(BZP2) is reduced. Because the MA(k || ) increases at BZP1 and decreases at BZP2, the total MA hardly changes upon increasing the strain from 0 to 2% (Fig. 1).
Under 4% strain the k-resolved MA, shown in Fig. 2(e), exhibits a sharp positive peak at BZP1 which is responsible for the magnetization vector switching from an in-to out-of-plane direction. The underlying origin in the electronic structure is that one of the Fe1 d xz(yz) states at BZP1 slightly above the Fermi energy shifts downward and becomes occupied [ Fig. 2(f)]. This in turn induces SOC of the occupied Fe1 d xz(yz) with the unoccupied Fe1 d yz(xz) around BZP1 through the out-of-plane orbital angular momentum operator L z .
Effect of strain on VCMA. The variation of MA as a function of the E-field in MgO is shown in Fig. 3(a-c) for η FeCo = 0, 2 and 4%, respectively. The E-field in the insulator is inversely proportional to the strain-dependent out-of-plane component, ε ⊥ , of the dielectric tensor of the insulator. We find that ε ⊥ increases exponentially with increasing compressive strain on the insulator (i.e., decreasing expansive strain on the FM). The calculated values of the relative ε ⊥ /ε 0 are 10.7, 17.0, and 27.0 for for η FeCo = 4, 2, and 0%, respectively. The results in Fig. 3(a-c) demonstrate that epitaxial strain gives rise to a wide range of intriguing VCMA behavior where the MA changes from (i) asymmetric ∨ -shape field behavior under 0% strain with β values of 1871 (− 101) fJ V −1 m −1 for positive (negative) E-field; to (ii) asymmetric ∧ -shape under 2% strain with β values of − 246 (482) fJ V −1 m −1 for E-field larger (smaller) than the critical field E c = − 0.58 V/nm where the MA reaches its maximum; and to (iii) asymmetric ∧ -shape under 4% strain with β values of − 1061 (393) fJ V −1 m −1 for E ≥ (≤ ) E c = 0.70 V/nm. Note that the range of E I is below the breakdown field of MgO (~1 V/nm). In most experiments E I is below 0.7 V/nm 4,7 , which is the value of E c at 4%. Therefore, experimentally the VCMA appears linear at 4%. These VCMA coefficient values are the highest reported today and are larger by one to two order of magnitude compared to those reported in most experiments, except for those in refs 8,9 where charged defects may play a role.
More importantly, we predict an E-field-driven switching of the magnetic easy axis from in-plane to out-of-plane direction at 0.30 (− 0.80) V/nm for η FeCo = 0 (4)%. These findings have two important implications for magnetoelectric spintronics. First, the predicted VCMA coefficient values are very close to or larger than the critical value of ~200 fJ V −1 m −1 required to achieve a switching bit energy below 1fJ in the next-generation of MeRAMs. Second, the results reveal the feasibility of tailoring the VCMA behavior via strain engineering to achieve desired MeRAM devices. Figure 3(d-f ) show the difference between the out-of-and in-plane orbital moments, [100] , of the Fe1 and Fe2 interfacial atoms as a function of E-field for η FeCo = 0, 2 and 4%, respectively. The E-field variation of Δ m o for Co is much weaker and is not shown here. For single atomic species FMs with large exchange splitting the MA is related to the orbital magnetic moment anisotropy via the Bruno expression MA = ξΔ m o /(4μ B ) 26 . However, for structures consisting of multiple atomic species (as in the case of trilayers) with strong hybridization it has been shown that the expression is not satisfied and needs to be modified 27 . Overall the E-field dependence of Δ m o for Fe1 and to a lesser degree of Fe2 correlates with that of the MA.
From the equation (4), E-field induced MA can originate from one or both of the following mechanisms: (i) charge screening or band filling effect, which is described by changes of the numerators and (ii) changes in separation of spin-orbit coupled pairs, which are described by the denominators. Thus far most of works assume either the band filling effect 13,[28][29][30] or the change in energy separation 14,31 to be the predominant contribution. In the present work, we do not exclude possible contributions from the band filling effect. However, from our analysis the E-field induced changes in band structures can provide a consistent explanation of the E-field induced MA behavior, indicating that this is a plausible mechanism.
VCMA at zero strain. In order to understand the VCMA behavior under zero strain we show in Fig. 4(a,b) the E-field induced change of MA, Δ MA(k) = MA(k, E) -MA(k, E c = 0), in the 2D BZ for − E I = − 0.37 V/nm and + E I = 0.37 V/nm, respectively. Integration of the Δ MA(k) over the 2D BZ for negative and positive fields yields induced MA consistent with the asymmetric ∨ -shape VCMA in Fig. 3(a). We also show in Fig. 4(c) the E-field induced Δ MA(k) along symmetry lines for positive and negative E-fields. Figure 4(d,e) display the E-field induced shifts of the energy levels (horizontal lines) of the Fe1-derived d states and the non-vanishing SOC matrix elements (vertical lines) between occupied and unoccupied d states at the Γ and I BZ points, where there are large changes of the MA. The decrease of MA at Γ under negative E-field is due to the fact that the occupied minority-spin Fe1 d xy shifts upward while the unoccupied minority-spin Fe1 d xz,yz states shift downward. Consequently the energy separation of this pair of states, coupled via the in-plane orbital angular momentum, L x , decreases resulting in a Δ MA(Γ) < 0. On the other hand, the increase of MA at Γ under positive E-field is associated with a large down- ward shift of the unoccupied majority-spin Fe1 d z 2 state to ~0.1 eV below the Fermi energy ( Fig. 4(e)). This in turn induces a spin-mixed SOC between the majority-spin state and the unoccupied minority-spin d xz,yz states via the in-plane orbital angular momentun L x [second term in equation (4) VCMA at 2% strain. Figure 5(a,b) show the E-field induced change of k-resolved MA, Δ MA(k) = MA(k, E)- Fig. 3(b)]. The contour values in Fig. 5(a) are magnified 4 times for the sake of clarity. Integration of the Δ MA(k) over the 2D BZ for − E I and + E I E-field yields induced MA consistent with the asymmetric ∨ -shape VCMA in Fig. 3(b). The Δ MA(k) is plotted along symmetry lines in Fig. 5(c). In Fig. 5(d,e) 5(d,e)]. In summary, we have demonstrated that epitaxial strain, which is ubiquitous in many HM/FM/I trilayers, has a dramatic effect on the VCMA. It can change the VCMA from a ∨ -to a ∧ -shape E-field dependence with giant VCMA coefficients and tunable critical E-field. Furthermore, we have predicted that tuning of epitaxial strain can give rise to an E-field induced magnetization switching at low voltage. These result demonstrate that the universality and robustness of the VCMA behavior in strained HM/FM/I trilayers and that efficient E-field-driven magnetic switching can be attained by design. These findings open interesting prospects for exploiting strain engineering to harvest higher efficiency VCMA for the next generation MeRAM devices.

Methods
The ab initio calculations have been carried out within the framework of the projector augmented-wave formalism 32 , as implemented in the Vienna ab initio simulation package (VASP) 33,34 . The generalized gradient approximation 35   FeCo/MgO interface are placed atop of Fe atoms. The iron atoms at the Fe/MgO and Fe/Au interfaces are denoted by Fe1 and Fe2, respectively [ Fig. 1 inset]. Due to the large lattice constant mismatch between MgO and FeCo, the FeCo (MgO) is under expansive (compressive) strain, η FeCo (η MgO ), of ~+ 4% (− 5.6%) compared with the lattice of bulk FeCo (MgO). Depending on the experimental conditions η FeCo can vary from zero to 4% 36 . At each strain, the magnetic and electronic degrees of freedom and atomic z positions are relaxed in the presence of the E-field until the forces acting on the ions become less than × 10 −3 eV/Å and the change in the total energy between two ionic relaxation steps is smaller than 10 −6 eV. The plane-wave cutoff energy was set to 500 eV and a 15 × 15 × 1 Monkhorst-Pack k-mesh was used for the relaxation calculations. The SOC of the valence electrons is in turn included using the second-variation method 37