Thickness dependence of spin torque effect in Fe/MgO/Fe magnetic tunnel junction: Implementation of divide-and-conquer with first-principles calculation

AIP Advances 11, 015036 (2021)

Bao-Huei Huang*, Chia-Chia Chao, Yu-Hui Tang*

Thickness dependence of spin torque effect in Fe/MgO/Fe magnetic tunnel junction: Implementation of divide-and-conquer with first-principles calculation

在第一原理計算材料的領域中，計算複雜度通常會跟原子數成正比，所需要的計算時間與自洽困難度也會隨之增長。在本篇研究中，我們應用分治演算法 (divide-and-conquer, DC) 的概念，來避免直接求解大型系統的哈密頓量。利用原子軌道的局域特性，與符合邊界條件的情況下，我們可以先將大型系統分成若干個小型系統個別求解，最後再組合出大系統的哈密頓量。搭配我們開發的JunPy軟體來計算系統的非平衡格林函數，可以求解系統在加電壓下的自旋電子流與自旋力矩。我們結合這些方法，來研究磁性穿隧接面Fe/MgO/Fe的自旋傳輸特性，結果顯示大部分的自旋力矩會在接面附近被完全吸收，因此如果想要利用電流來有效率地翻轉磁矩，Fe磁性層薄膜的厚度須盡量小於2奈米。

In this study, our newly developed JunPy package is generalized by divide-and-conquer (DC) methodology to calculate the layer-resolved spin torques in Fe/MgO/Fe magnetic tunnel junction (MTJ), efficiently but without losing accuracy. Instead of directly carrying out the first-principles calculation for whole device with large amount of Fe layers, we divide it into smaller components that can be solved individually. The device Hamiltonian is thus recombined by the self-consistent block tridiagonal Hamiltonian matrix of each sub-system. Finally, the JunPy-DC calculation is applied to investigate the oscillatory decay of layer-resolved and cumulative spin torques for the Fe electrode with finite thickness. These results suggest that the Fe layer thickness should be no larger than 2 nm to preserve the current-driven magnetization switching process.

　(a) The atomic structure of Fe/MgO/Fe MTJ in our calculation. The central MgO with 5 atomic layers (about 8.94 Å) is sandwiched by two ferromagnetic Fe electrodes, where the magnetization direction of the left (fix) Fe electrode, Mˆ L, is fixed in z axis, and the right (free), Mˆ R, is rotated by an angle θ around y axis to form a magnetically noncollinear system. (b) The schematic plot of divided sub-devices of MTJ. The region C including Fe (7)/MgO(5)/Fe (6), where sufficient buffer Fe layers is chosen to fit the boundary condition with left and right electrodes. Since the Hamiltonian converge to the bulk value far from the interface, the region R can be simply repeated to increase the right Fe layers.
The transmission spectrum of Nanodcal package (two-probe Hamiltonian) and DC device Hamiltonian for θ = π/2 and Vb = 0.1 V
(a) The layer-resolved spin torques of right Fe electrode for θ = π/2 and Vb = 0.1 V. Here i = 1 denotes the first site of the right Fe electrode next to the right MgO/Fe interface. (b) The cumulative spin torques acting on right Fe electrode with finite length, which are the summation of all layer-resolved spin torques from the right MgO/Fe interface to a given cutoff length.