Machine learning analysis of tunnel magnetoresistance of magnetic tunnel junctions with disordered MgAl2O4

Through Bayesian optimization and the least absolute shrinkage and selection operator (LASSO) technique combined with first-principles calculations, we investigated the tunnel magnetoresistance (TMR) effect of Fe/disordered-MgAl2O4(MAO)/Fe(001) magnetic tunnel junctions (MTJs) to determine structures of disordered-MAO that give large TMR ratios. The optimal structure with the largest TMR ratio was obtained by Bayesian optimization with 1728 structural candidates, where the convergence was reached within 300 structure calculations. Characterization of the obtained structures suggested that the in-plane distance between two Al atoms plays an important role in determining the TMR ratio. Since the Al-Al distance of disordered MAO significantly affects the imaginary part of complex band structures, the majority-spin conductance of the {\Delta}1 state in Fe/disordered-MAO/Fe MTJs increases with increasing in-plane Al-Al distance, leading to larger TMR ratios. Furthermore, we found that the TMR ratio tended to be large when the ratio of the number of Al, Mg, and vacancies in the [001] plane was 2:1:1, indicating that the control of Al atomic positions is essential to enhancing the TMR ratio in MTJs with disordered MAO. The present work reveals the effectiveness and advantage of material informatics combined with first-principles transport calculations in designing high-performance spintronic devices based on MTJs.


I. INTRODUCTION
Magnetic tunnel junctions (MTJs), which exhibit different resistance depending on the relative magnetization directions of the ferromagnetic electrodes, are among the most important devices for spintronic applications, such as magnetic random access memories (MRAMs) and high-sensitivity magnetic sensors [1,2]. MgO-based MTJs have yielded large tunnel magnetoresistance (TMR) ratios of over 600% at room temperature owing to coherent tunneling through the Δ1 evanescent state of MgO and the half-metallic behavior of the Δ1 state in Fe and CoFe [3][4][5][6]. However, epitaxial growth of crystalline MgO is difficult because of its lattice mismatch with the ferromagnetic electrode, which is about 4% for bcc-Fe and over 5% for half-metallic Density functional theory (DFT) calculations yielded a TMR ratio of 160% for Fe/MAO/Fe MTJs [8], which is one order of magnitude smaller than the 1600% obtained for Fe/MgO/Fe MTJs [3,4]. Since the in-plane lattice constant of MAO-based MTJs is twice that of MgO-based MTJs, boundary edge states in the two-dimensional Brillouin zone are folded to the Γ point (kx=ky=0 found that the unit cell size is reduced when the cations Mg 2+ and Al 3+ in spinel MAO randomly occupy octahedral and tetrahedral sites including vacant sites [9]. Cation site disordering changes the space group to either F-m3m or F-43m and halves the unit cell size of MAO, while the anion sites are always occupied by oxygen atoms, and the fcc sublattice of oxygen is not changed by the disordering.
Cation-site-disordered MAO yields higher TMR ratios than ordered MAO owing to suppression of the band folding effect. However, the TMR ratios observed were still lower than those of MgO-based MTJs. In order to enhance the TMR ratio of MAObased MTJs, it is important to identify the local structures of cation-site-disordered MAO, which leads to high TMR ratios. However, at the present time, the detailed local structure of cation-site-disordered MAO cannot be determined experimentally.
Furthermore, determining the optimal structure through the DFT method alone is challenging, because the number of atomic configurations of disordered MAO increases sharply with increasing MTJ unit cell size.
Materials informatics (MI), which combines data science and traditional simulation/experiments, has become a popular tool to accelerate the materials discovery and design process [10,11]. For instance, in the development of thermal functional materials, MI has greatly reduced the computational cost, and the optimal nanostructures for phonon transport could be calculated by using only a few percent of the total candidate structures through machine learning methods such as Bayesian optimization [12] and Monte Carlo tree search [13]. Various regression models including the Gaussian process and support vector regression models have been applied to predict the thermal boundary resistance, and the results indicate that machine learning models have much better predictive accuracy than the commonly used acoustic mismatch model and diffuse mismatch model [14,15]. In the case of thermoelectric materials, Bayesian optimization has been used improve the thermal and electronic properties simultaneously. As a result, the thermoelectric performance of defective graphene nanoribbons was enhanced by 11-times compared with that of pristine ones [16]. Hou et al. [17] also adopted Bayesian optimization to optimize the Al/Si ratio in Al2Fe3Si3 and improved the power factor by 40%, which significantly reduced the time and labor cost of optimizing thermoelectric materials. More recently, Bayesian optimization has been used to design wavelength-selective thermal radiators [18], which can serve as thermophotovoltaics and infrared heaters. On the basis of these studies, we concluded that complementing DFT calculations with Bayesian optimization would allow us to determine the optimal disordered MAO structure. Thus, in the present work, we combine MTJ calculations with Bayesian optimization to design a disordered spinel barrier structure, aiming to maximize the TMR ratio and elucidate the underlying mechanism.

II. METHODOLOGY
We constructed the supercell of a Fe/disordered-MAO/Fe (001) MTJ containing 10 atomic layers of bcc Fe and 3 atomic layers of disordered MAO, as shown in Fig. 1.
The rock-salt structure was assumed for disordered MAO, where the cation sites are randomly occupied by Al, Mg, and a vacancy. This kind of defective rock-salt structure has already been experimentally demonstrated [9]. We prepared two types of supercells with different in-plane unit cell sizes for the Fe/MAO/Fe MTJ. One had cross-sectional size of 5.732 Å × 5.732 Å, which is the smallest in-plane unit cell of ordered MAO. The other was 5.732 2 Å × 5.732 2 Å in cross-section, which corresponds to a 45 (001) in-plane rotation of the former structure. 5.732 Å is twice the lattice constant of bcc Fe (2.86 Å).
The TMR ratio for each MTJ structure is defined as, where RAP(P) is the electrical resistance in the anti-parallel (parallel) state.

A. Performance of Bayesian optimization
First, the performance of Bayesian optimization was tested for a system with 3(a) and 3(b). All Bayesian optimizations reached convergence within 300 structure calculations, while random search required at most around 1500 calculations. Figure   3(c) shows the averaged trends of convergence, which clearly indicate that Bayesian optimization is much more efficient than random search. The total number of calculations needed to obtain the global optimal structure is 17%, which is somewhat larger than in phonon transport optimization [11]. This is mainly due to that the total number of candidates in the current case is relatively small, as the efficiency of machine learning is usually limited for small data. Therefore, the efficiency is expected to increase with increasing number of candidates.
To check the accuracy of Bayesian optimization, we performed a full calculation for all of the 1728 candidates. The global maximum TMR ratio reached as high as 600.18%, and the corresponding structure was confirmed to be exactly the same as that obtained by Bayesian optimization. Figure 3(d) shows the TMR ratio distribution.
There are few candidate structures located in the high-TMR-ratio region, which indicates that Bayesian optimization is suitable to search for MTJs with the highest TMR ratio. for the parallel magnetization shown in Fig. 4(a) has a broad peak around the center of the 2D Brillouin zone, which is much stronger than the minority-spin conductance and the conductance for antiparallel magnetization. This agrees with the typical behavior of the coherent tunneling conductance of Δ1 electrons at the Brillouin zone center.
Furthermore, the in-plane wave-vector dependence in antiparallel magnetization has no peak around (kx,ky)=(0,0) and shows a hot-spot-like peaked structure in the 2D Brillouin zone. This means that the contribution of the Δ1 evanescent state to the tunneling conductance is completely blocked in the MTJ because of the suppression of the band holding effect in disordered MAO.

B. Effect of cation site distance on TMR ratio
To identify local structures of MAO with high TMR ratios, we performed the Least

Absolute Shrinkage and Selection Operator (LASSO) regression of the TMR ratio for
Fe/disordered MAO/Fe with 1728 cases. As a descriptor for the LASSO regression, we plane of disordered MAO, with cross-sectional size of 5.732×5.732 Å, two Al atoms tend to be located in the diagonal positions for larger dXY and they tend to be located in the vertical or horizontal positions for smaller dXY (see Fig. 1(a)). Thus, we can say that the TMR ratio tends to increase when two Al atoms are located in the diagonal positions. To clarify this point, Figs. 6(c) and 6(d) show the correlation between TMR ratio and tunneling conductance in the parallel magnetization configuration. We found that there is a correlation between TMR ratio and majority-spin conductance: TMR ratio increases with increasing majority-spin conductance in parallel magnetization, as expected on the basis of the definition of the TMR ratio. By contrast, the correlation between TMR ratio and minority-spin conductance is very weak. This means that the high TMR ratio in Fe/disordered-MAO/Fe MTJs is due to the large majority-spin conductance in the parallel magnetization configuration.

C. Effect of concentration distribution on TMR ratio
Having demonstrated the effectiveness of Bayesian optimization for MTJ design, we turn our attention to the MTJ with larger cross-section shown in Fig. 1(b). Instead of using a constant Mg:Al:vacancy distribution in the disordered spinel barrier region, we tried the three types of distributions listed in Table I. For each distribution, we performed Bayesian optimization separately, and the highest TMR ratios obtained were 153.8%, 312.5%, and 373.2%. We found that the maximum TMR ratios of the MTJs in Table I were not the same as those of the MTJs with a unit cell size of (a × a) shown in

IV. CONCLUSION
We have optimized the disordered spinel barrier in MTJs to obtain a high TMR ratio using Bayesian optimization. The TMR ratio of Fe/disordered MAO/Fe MTJs is successfully optimized by Bayesian optimization. The maximum TMR ratio obtained is over 600%, which is much larger than the TMR ratio of Fe/ordered MAO/Fe MTJs (160%). We found that the in-plane distance between two Al atoms plays an important role in determining the TMR ratio from LASSO analysis. Since the increase in the Al-