# Finite Element Approach for the Simulation of Modern MRAM Devices

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## Abstract

**:**

## 1. Introduction

**Figure 1.**Four examples of multi-layer MRAM cell design: (

**a**) standard STT-MRAM with single MTJ; (

**b**) double RL STT-MRAM, where the second RL provides additional torque to reduce the critical voltage required for switching [20]; (

**c**) ultra-scaled STT-MRAM, where the FM layers are elongated and additional oxide layers are added to improve scalability and benefit from the shape anisotropy [21]; (

**d**) SOT-assisted STT-MRAM, where the switching process is kick-started by an initial current pulse in the HM [22].

## 2. Micromagnetic Modeling

#### Spin and Charge Transport

## 3. Finite Element Implementation

#### 3.1. Charge Current Solution

**Figure 2.**Representation of a continuous function u and its finite element approximation ${u}_{\mathrm{h}}$ in a one-dimensional setting. The basis functions for all the nodes are reported at the bottom of the graph. The basis function and solution value associated with the node ${x}_{2}$ are labeled ${\phi}_{2}$ and ${u}_{2}$, respectively.

- For each point inside the TB where the conductivity needs to be computed, referred to as an integration point, the solver loops through the integration points of the RL and FL elements closer to the interfaces.
- The RL and FL points near to or at the interface with coordinates closest to the TB point are selected.
- The integration point number and element number associated with the nearest RL and FL points are mapped to the coordinates of the TB points.

#### 3.2. Spin Accumulation Solution

#### 3.2.1. Tunneling Spin Current

#### 3.2.2. Spin Hall Effect

#### 3.2.3. Complete Weak Formulation

- For each integration point on the RL|TB interface requiring the computation of the tunneling spin current, the solver loops through the integration points of the TB|FL interface.
- The TB|FL point with coordinates closest to the RL|TB one is selected.
- The integration point number and the element number associated with the found TB|FL point are mapped to the coordinates of the RL|TB one.
- The mapping procedure is repeated for the TB|FL interface.

#### 3.3. Magnetization Dynamics Solution

#### 3.4. Demagnetizing Field

## 4. Device Simulation

#### 4.1. Double RL STT-MRAM

#### 4.2. Ultra-Scaled STT-MRAM

#### 4.3. SOT Assisted STT-MRAM

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MRAM | Magnetoresistive random access memory |

CMOS | Complementary metal-oxide semiconductor |

SRAM | Static random access memory |

MTJ | Magnetic tunnel junction |

FM | Ferromagnetic |

NM | Non-magnetic |

RL | Reference layer |

FL | Free layer |

TB | Tunnel barrier |

NMS | Non-magnetic spacer |

HM | Heavy metal |

P | Parallel |

AP | Anti-parallel |

TMR | Tunneling magnetoresistance ratio |

STT | Spin-transfer torque |

SOT | Spin-orbit torque |

SHE | Spin Hall effect |

LLG | Landau–Lifshitz–Gilbert |

FE | Finite element |

CRMT | Continuous random matrix theory |

BEM | Boundary element method |

LHS | Left-hand side |

RHS | Right-hand side |

DSMTJ | Double spin torque MTJ |

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**Figure 3.**Magnetic potential (

**left**) and demagnetizing field (

**right**) computed in a structure with three disconnected ferromagnetic layers. The magnetization orientation in each layer is indicated by the arrows. The color coding indicates the value of the magnetic potential.

**Figure 4.**(

**a**) Structure for an MRAM cell with the addition of a second RL (RL2), separated from the FL by a non-magnetic metallic spacer (NMS). The RL${}_{1}$, RL${}_{2}$, and TB are 1 nm thick, the FL is 1.7 nm thick, and the NM contacts are 50 nm thick. (

**b**) Magnetization reversal of the FL from AP to P for an MRAM cell with a single MTJ (SMTJ, dotted line) and the one with a double RL (DSMTJ, solid line).

**Figure 5.**(

**a**) Structure for an elongated MRAM cell with FL composed of two sections (FL1 and FL2), separated by a TB. The RL, FL${}_{1}$, and FL${}_{2}$ are 5 nm thick, all the TBs are 0.9 nm thick, and the NM contacts are 50 thick. (

**b**) Magnetization reversal of the FL from AP to P for the elongated cell.

**Figure 6.**(

**a**) Structure reproducing an SOT + STT-based MRAM cell. The MTJ stack is deposited on top of a heavy metal line (HM). The RL and TB are 1 nm thick, the FL is 2 nm thick, the top NM contact is 50 nm thick, and the HM layer is 4 nm thick, 50 nm wide and 100 nm long. (

**b**) Magnetization reversal of the FL from AP to P for the SOT + STT cell.

LLG parameters | Value |

Saturation magnetization (${M}_{s}$) | $0.81\times {10}^{6}$ A/m |

Exchange constant (${A}_{exc}$) | $2.0\times {10}^{-11}$ J/m |

Interface anisotropy (${K}_{int}$) | $1.29\times {10}^{-3}$ J/m${}^{2}$ |

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

Drift-diffusion parameters | Value |

Conductivity polarization, ${\beta}_{\sigma}$ | $0.52$ |

Diffusion polarization, ${\beta}_{D}$ | $0.7$ |

FM diffusion coefficient, ${D}_{e,FM}$ | ${10}^{-3}$ m${}^{2}$/s |

NM diffusion coefficient, ${D}_{e,NM}$ | ${10}^{-2}$ m${}^{2}$/s |

HM diffusion coefficient, ${D}_{e,HM}$ | $1.1\times {10}^{-3}$ m${}^{2}$/s |

TB diffusion coefficient, ${D}_{S}$ | $2.0\times {10}^{-8}$ m${}^{2}$/s |

FM conductivity ${\sigma}_{FM}$ | $4.0\times {10}^{6}$ S/m |

NM conductivity ${\sigma}_{NM}$ | $5.0\times {10}^{6}$ S/m |

HM conductivity ${\sigma}_{HM}$ | $7.0\times {10}^{6}$ S/m |

FM spin-flip length, ${\lambda}_{sf,FM}$ | 10 nm |

NM spin-flip length, ${\lambda}_{sf,NM}$ | 10 nm |

HM spin-flip length, ${\lambda}_{sf,HM}$ | $1.4$ nm |

Spin exchange length, ${\lambda}_{J}$ | $0.8$ nm |

Spin dephasing length, ${\lambda}_{\phi}$ | $0.4$ nm |

Spin Hall angle, ${\theta}_{\mathrm{SHA}}$ | 0.19 |

TB resistance standard and double RL STT-MTJ | Value |

Resistance parallel (${R}_{P}$) | $4.3\times {10}^{3}$ k$\mathsf{\Omega}$ |

Resistance anti-parallel (${R}_{AP}$) | $9.1\times {10}^{3}$ k$\mathsf{\Omega}$ |

TB resistance ultra-scaled STT-MTJ | Value |

Resistance parallel (${R}_{P}$) | $4.1\times {10}^{5}$ k$\mathsf{\Omega}$ |

Resistance anti-parallel (${R}_{AP}$) | $7.5\times {10}^{5}$ k$\mathsf{\Omega}$ |

TB resistance SOT-assisted STT-MTJ | Value |

Resistance parallel (${R}_{P}$) | $1.4\times {10}^{4}$ k$\mathsf{\Omega}$ |

Resistance anti-parallel (${R}_{AP}$) | $4.2\times {10}^{4}$ k$\mathsf{\Omega}$ |

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**MDPI and ACS Style**

Fiorentini, S.; Jørstad, N.P.; Ender, J.; de Orio, R.L.; Selberherr, S.; Bendra, M.; Goes, W.; Sverdlov, V.
Finite Element Approach for the Simulation of Modern MRAM Devices. *Micromachines* **2023**, *14*, 898.
https://doi.org/10.3390/mi14050898

**AMA Style**

Fiorentini S, Jørstad NP, Ender J, de Orio RL, Selberherr S, Bendra M, Goes W, Sverdlov V.
Finite Element Approach for the Simulation of Modern MRAM Devices. *Micromachines*. 2023; 14(5):898.
https://doi.org/10.3390/mi14050898

**Chicago/Turabian Style**

Fiorentini, Simone, Nils Petter Jørstad, Johannes Ender, Roberto Lacerda de Orio, Siegfried Selberherr, Mario Bendra, Wolfgang Goes, and Viktor Sverdlov.
2023. "Finite Element Approach for the Simulation of Modern MRAM Devices" *Micromachines* 14, no. 5: 898.
https://doi.org/10.3390/mi14050898