# Advanced Modeling and Simulation of Multilayer Spin–Transfer Torque Magnetoresistive Random Access Memory with Interface Exchange Coupling

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

_{0}-ordered alloys like FePt and FePd, known for their high magnetocrystalline anisotropy, have shown promise in simplifying the stacking structure of recording layers compared to traditional CoFeB/MgO multilayer systems [15].

_{22}-Mn3Ga and D0

_{22}-Mn3Ge have recently received attention as potential materials for the FL in MRAM applications due to their combination of high magnetocrystalline anisotropy, low magnetic damping, and favorable thermal stability [16,17]. These attributes make D0

_{22}-Mn3Ga especially suitable for spin–orbit torque MRAM applications, offering pathways to enhance energy efficiency and reduce the critical current required for switching [16]. Despite these advantages, integrating D0

_{22}-Mn

_{3}Ga into existing MTJ structures poses significant challenges that need to be addressed to fully capitalize on the material’s potential in future MRAM technologies [18].

## 2. Micromagnetics Model

_{1}) (3 $\mathrm{n}\mathrm{m}$) |MgO ($0.9$ $\mathrm{n}\mathrm{m}$) |CoFeB second free layer (FL

_{2}) (3 $\mathrm{n}\mathrm{m}$) and |MgO ($0.9$ $\mathrm{n}\mathrm{m}$) all of which are interconnected to normal metal (NM) contacts (50 $\mathrm{n}\mathrm{m}$). The overall diameter of this configuration is $2.3$ $\mathrm{n}\mathrm{m}$, highlighting the intricate layering and miniaturization achieved in this ultra-scaled design.

_{Ru}($0.85$ $\mathrm{n}\mathrm{m}$) |RL ($1.1$ $\mathrm{n}\mathrm{m}$) |TB ($0.9$ $\mathrm{n}\mathrm{m}$) and |FL ($1.4$ $\mathrm{n}\mathrm{m}$) respectively, linked to NM contacts (50 $\mathrm{n}\mathrm{m}$).

_{Ru}($0.85$ $\mathrm{n}\mathrm{m}$)|RL ($3.2$ $\mathrm{n}\mathrm{m}$)|NMS

_{Ta}($0.4$ $\mathrm{n}\mathrm{m}$)|PL ($1.3$ $\mathrm{n}\mathrm{m}$)|TB ($0.9$ $\mathrm{n}\mathrm{m}$)|FL ($1.4$ $\mathrm{n}\mathrm{m}$) again concluding in NM contacts (50 $\mathrm{n}\mathrm{m}$). The overall diameter for Stack B and Stack C is 70 $\mathrm{n}\mathrm{m}$. The simulation parameters applied across the various MRAM configurations illustrated in Figure 1 are comprehensively detailed in Table 1, with appropriate references provided in [25,26,27,28].This range aligns with the experimental values documented in NMS [29,30]. Remarkably, instances of coupling strengths over $\pm 2\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{J}/{\mathrm{m}}^{2}$ have been recorded [31].

#### Interlayer Exchange Coupling

## 3. Results and Discussion

#### 3.1. Stack A

_{1}and FL

_{2}, oriented positively along the x-axis. The magnetization reversal from P to AP is computed, highlighting the back-hopping phenomenon. Although typically undesirable in a composite FL, we demonstrated cyclic switching between four distinct states of the FL using the same current direction. This finding challenges the traditional binary perspective of the MRAM operation, offering a new multi-level functionality in ultra-scaled MRAM cells [53].

_{1}and FL

_{2}, and is significantly influenced by the crystalline quality, thickness, and stoichiometry of the MgO layers [54,55], it is crucial to note that our findings focused specifically on the IEC between these two layers. The potential for a similar FM coupling between the RL and FL

_{1}, through a MgO layer of the same thickness as that between FL

_{1}and FL

_{2}, requires separate consideration. The presence of FM coupling between FL

_{1}and FL

_{2}does not automatically suggest a comparable interaction between the RL and FL

_{1}.

_{1}would require specific investigation, focusing on the magnetic properties and structural details of the MgO layers involved.

#### 3.2. Stack B

#### 3.3. Stack C

_{Ru}. Additionally, a CoFeB PL is connected to the RL via FM coupling through Ta NMS

_{Ta}. The AFM bond facilitated by NMS

_{Ru}exhibits a notable coupling force of $-1.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{J}/{\mathrm{m}}^{2}$, indicative of a robust AFM connection. Conversely, the initial FM connection through NMS

_{Ta}, proposed at $0.8\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{J}/{\mathrm{m}}^{2}$ by Devolder et al. [27], represents a substantial FM coupling. However, subsequent works by Devolder et al. [58] and Goff et al. [59] revised this FM coupling strength to $0.21\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{J}/{\mathrm{m}}^{2}$, which is weaker compared to the initial FM coupling estimate.

_{2}in panel (a), the RL in panel (b), and the PL in panel (c) undergo reversal due to back-hopping.

_{1}and FL

_{2}are slightly tilted toward the negative and positive z-axis, respectively. When the applied bias remains constant for an extended period or is increased, the magnetization in FL

_{2}undergoes a magnetization reversal. Torques from RL and FL

_{2}stabilize FL

_{1}. The torque contributions from FL

_{1}initiate the magnetization reversal in FL

_{2}and overcome the interface-induced uniaxial anisotropy contribution. This initiates the so-called back-hopping effect in FL

_{2}. As displayed in Figure 3a, even a weak FM coupling between FL

_{1}and FL

_{2}is sufficient to improve the switching speed, as the magnetization reversal begins more uniformly when FL

_{1}and FL

_{2}are coupled. Moreover, the FM coupling prevents the field-like torque from inverse FL

_{2}magnetization, leading to the higher stability of the structure.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

STT-MRAM | spin–transfer torque magnetoresistive random access memory |

MTJ | magnetic tunnel junction |

SRAM | static random-access memory |

DRAM | dynamic random-access memory |

RL | reference layer |

FL | free layer |

TB | tunnel barrier |

NMS | non-magnetic spacer |

NM | normal metal |

IEC | interlayer exchange coupling |

HL | hard layer |

PL | spin–polarization layer |

LLG | Landau–Lifshitz–Gilbert |

FEM | finite element method |

BEM | boundary element method |

TMR | tunnel magnetoresistance |

RKKY | Ruderman–Kittel–Kasuya–Yosida |

FM | ferromagnetic |

AFM | antiferromagnetic |

SAF | synthetic antiferromagnetic |

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**Figure 1.**Schematic illustration to provide a clear understanding of multi-layered MRAM cell structures, each representing a distinct structural configuration, labeled as (

**a**) Stack A, (

**b**) Stack B, and (

**c**) Stack C. To aid in distinguishing between the different components, color coding is applied. The regions where interfacial engineering is performed are denoted by black zigzag lines.

**Figure 2.**Schematic depiction of a trilayer structure composed of left and right semi-infinite ferromagnetic (FM) regions, separated by an NMS layer. The interface magnetization on the left FM interface points in an arbitrary direction, whereas the interface magnetization in the right FM forms an angle $\Delta \varphi $ relative to the magnetization of the left FM.

**Figure 3.**Magnetization trajectories for P to AP switching, labeled as (

**a**) Stack A, (

**b**) Stack B, and (

**c**) Stack C.

**Figure 4.**The computation of spin torque is based on the spin current boundary condition, as outlined in Equation (9). It is applied to three different configurations: Stack A (illustrated in panel (

**a**)), Stack B (shown in panel (

**b**)), and Stack C (depicted in panel (

**c**)). These figures demonstrate the torque patterns as magnetization in FL

_{2}in panel (

**a**), the RL in panel (

**b**), and the PL in panel (

**c**) approach reversal due to back-hopping. In these diagrams, the direction of magnetization within the ferromagnetic sections is indicated by black arrows. The graphical representation reveals that ${T}_{S,x}$ acts as a field-like component of the spin torque along the central axis of the structure, whereas ${T}_{S,z}$ serves as a damping-like component. The notation ${I}_{current}$ is used to represent the direction of electron flow.

LLG Parameters | Stack A | Stack B | Stack C |
---|---|---|---|

Saturation magnetization (${M}_{S}$, HL) | $0.73\times {10}^{6}\phantom{\rule{3.33333pt}{0ex}}\mathrm{A}/\mathrm{m}$ | $0.85\times {10}^{6}\phantom{\rule{3.33333pt}{0ex}}\mathrm{A}/\mathrm{m}$ | |

Saturation magnetization (${M}_{S}$, RL) | $0.81\times {10}^{6}\phantom{\rule{3.33333pt}{0ex}}\mathrm{A}/\mathrm{m}$ | $1.1\times {10}^{6}\phantom{\rule{3.33333pt}{0ex}}\mathrm{A}/\mathrm{m}$ | $0.8\times {10}^{6}\phantom{\rule{3.33333pt}{0ex}}\mathrm{A}/\mathrm{m}$ |

Saturation magnetization (${M}_{S}$, PL) | $1.1\times {10}^{6}\phantom{\rule{3.33333pt}{0ex}}\mathrm{A}/\mathrm{m}$ | ||

Saturation magnetization (${M}_{S}$, FL) | $0.81\times {10}^{6}\phantom{\rule{3.33333pt}{0ex}}\mathrm{A}/\mathrm{m}$ | $1.1\times {10}^{6}\phantom{\rule{3.33333pt}{0ex}}\mathrm{A}/\mathrm{m}$ | $1.1\times {10}^{6}\phantom{\rule{3.33333pt}{0ex}}\mathrm{A}/\mathrm{m}$ |

Exchange constant (${A}_{exc}$, HL) | $1.0\times {10}^{-11}\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}/\mathrm{m}$ | $1.0\times {10}^{-11}\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}/\mathrm{m}$ | |

Exchange constant (${A}_{exc}$, RL) | $2.0\times {10}^{-11}\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}/\mathrm{m}$ | $2.0\times {10}^{-11}\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}/\mathrm{m}$ | $1.0\times {10}^{-11}\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}/\mathrm{m}$ |

Exchange constant (${A}_{exc}$, PL) | $2.0\times {10}^{-11}\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}/\mathrm{m}$ | ||

Exchange constant (${A}_{exc}$, FL) | $2.0\times {10}^{-11}\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}/\mathrm{m}$ | $2.0\times {10}^{-11}\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}/\mathrm{m}$ | $2.0\times {10}^{-11}\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}/\mathrm{m}$ |

Shape anisotropy (${K}_{u}$, HL) | $7.843\times {10}^{5}\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}/{\mathrm{m}}^{3}$ | $7.843\times {10}^{5}\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}/{\mathrm{m}}^{3}$ | |

Shape anisotropy (${K}_{u}$, RL) | $2.593\times {10}^{5}\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}/{\mathrm{m}}^{3}$ | $8.501\times {10}^{5}\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}/{\mathrm{m}}^{3}$ | $8.318\times {10}^{5}\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}/{\mathrm{m}}^{3}$ |

Shape anisotropy (${K}_{u}$, PL) | $9.974\times {10}^{5}\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}/{\mathrm{m}}^{3}$ | ||

Shape anisotropy (${K}_{u}$, FL) | $4.322\times {10}^{5}\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}/{\mathrm{m}}^{3}$ | $9.261\times {10}^{5}\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}/{\mathrm{m}}^{3}$ | $9.261\times {10}^{5}\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}/{\mathrm{m}}^{3}$ |

Gilbert damping constant ($\alpha $, HL) | $0.02$ | $0.02$ | |

Gilbert damping constant ($\alpha $, RL) | $0.02$ | $0.01$ | $0.02$ |

Gilbert damping constant ($\alpha $, PL) | $0.015$ | ||

Gilbert damping constant ($\alpha $, FL) | $0.015$ | $0.005$ | $0.01$ |

IEC (${J}_{iec}$, Ru) | $-1.32\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{J}/{\mathrm{m}}^{2}$ | $-1.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{J}/{\mathrm{m}}^{2}$ | |

IEC (${J}_{iec}$, Ta) | $0.8\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{J}/{\mathrm{m}}^{2}$ | ||

Resistance parallel (${R}_{P}$) | $4.1\times {10}^{3}\phantom{\rule{3.33333pt}{0ex}}\mathrm{k}\Omega $ | $1.68\phantom{\rule{3.33333pt}{0ex}}\mathrm{k}\Omega $ | $1.68\phantom{\rule{3.33333pt}{0ex}}\mathrm{k}\Omega $ |

Resistance parallel (${R}_{AP}$) | $7.5\times {10}^{3}\phantom{\rule{3.33333pt}{0ex}}\mathrm{k}\Omega $ | $4.22\phantom{\rule{3.33333pt}{0ex}}\mathrm{k}\Omega $ | $4.22\phantom{\rule{3.33333pt}{0ex}}\mathrm{k}\Omega $ |

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## Share and Cite

**MDPI and ACS Style**

Bendra, M.; Orio, R.L.d.; Selberherr, S.; Goes, W.; Sverdlov, V.
Advanced Modeling and Simulation of Multilayer Spin–Transfer Torque Magnetoresistive Random Access Memory with Interface Exchange Coupling. *Micromachines* **2024**, *15*, 568.
https://doi.org/10.3390/mi15050568

**AMA Style**

Bendra M, Orio RLd, Selberherr S, Goes W, Sverdlov V.
Advanced Modeling and Simulation of Multilayer Spin–Transfer Torque Magnetoresistive Random Access Memory with Interface Exchange Coupling. *Micromachines*. 2024; 15(5):568.
https://doi.org/10.3390/mi15050568

**Chicago/Turabian Style**

Bendra, Mario, Roberto Lacerda de Orio, Siegfried Selberherr, Wolfgang Goes, and Viktor Sverdlov.
2024. "Advanced Modeling and Simulation of Multilayer Spin–Transfer Torque Magnetoresistive Random Access Memory with Interface Exchange Coupling" *Micromachines* 15, no. 5: 568.
https://doi.org/10.3390/mi15050568