# A Comprehensive Study of Temperature and Its Effects in SOT-MRAM Devices

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## Abstract

**:**

## 1. Introduction

## 2. Method

#### 2.1. Implementation

#### 2.2. Simulated Structures

#### 2.3. Simulation Parameters

## 3. Results

#### 3.1. Temperature of the Structure

#### 3.2. Effect of Temperature on the Initial Switching Dynamics

#### 3.3. Field-Free Switching—Combined STT-SOT-MRAM

#### 3.4. Switching with External Fields

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Parameter | Value |
---|---|

MTJ Parameters | |

Tunnel magnetoresistance ratio (TMR) | 200% |

Current spin polarization, ${\beta}_{\sigma}$ | $0.7$ |

Diffusion spin polarization, ${\beta}_{\mathrm{D}}$ | $1.0$ |

Resistance parallel | $1.4\xb7{10}^{4}$ $\Omega $ |

Resistance antiparallel | $4.2\xb7{10}^{4}$ $\Omega $ |

Magnetic Parameters of FeCoB | |

Gilbert damping, $\alpha $ | $0.02$ |

Gyromagnetic ratio, $\gamma $ | $1.76\xb7{10}^{-11}\phantom{\rule{0.166667em}{0ex}}\mathrm{rad}\xb7{\mathrm{s}}^{-1}{\mathrm{T}}^{-1}$ |

Saturation magnetization (300 K), ${M}_{\mathrm{S}}$ | $0.81\xb7{10}^{6}\phantom{\rule{0.166667em}{0ex}}{\mathrm{Am}}^{-1}$ |

Exchange stiffness, ${A}_{exch}$ | $2\xb7{10}^{-11}\phantom{\rule{0.166667em}{0ex}}{\mathrm{Jm}}^{-1}$ |

Anisotropy energy density, ${K}_{a}$ | $0.539\xb7{10}^{6}\phantom{\rule{0.166667em}{0ex}}{\mathrm{Jm}}^{-3}$ |

Material ↓ / Parameter → | ${\mathit{D}}_{\mathit{e}}\phantom{\rule{4pt}{0ex}}({10}^{-3}{\mathbf{m}}^{2}/\mathbf{s})$ | ${\mathsf{\lambda}}_{\mathbf{sf}}\phantom{\rule{4pt}{0ex}}\left(\mathbf{nm}\right)$ | ${\mathsf{\lambda}}_{\mathit{\phi}}\phantom{\rule{4pt}{0ex}}\left(\mathbf{nm}\right)$ | ${\mathsf{\lambda}}_{\mathit{J}}\phantom{\rule{4pt}{0ex}}\left(\mathbf{nm}\right)$ | ${\mathsf{\theta}}_{\mathbf{SHA}}$ |
---|---|---|---|---|---|

FeCoB | 1 | 10 | 0.4 | 0.8 | - |

MgO | - | - | - | - | - |

$\beta $-W | 0.2 | 2.4 | - | - | −0.3 |

Contacts, Vias | 1.1 | 1.4 | - | - | - |

SiO | 0.1 | 1.4 | - | - | - |

Substrate | 0.2 | 1.4 | - | - | - |

Material ↓ / Parameter → | $\mathsf{\sigma}$ ($\mathbf{\Omega}\mathbf{m}$) | ${\mathsf{\rho}}_{\mathbf{m}}\left(\mathbf{kg}\phantom{\rule{0.166667em}{0ex}}{\mathbf{m}}^{-3}\right)$ | ${\mathit{c}}_{\mathbf{V}}\phantom{\rule{0.166667em}{0ex}}\left(\mathbf{J}\phantom{\rule{0.166667em}{0ex}}{\mathbf{kg}}^{-1}{\mathbf{K}}^{-1}\right)$ | $\mathsf{\kappa}\left(\mathbf{W}\phantom{\rule{0.166667em}{0ex}}{\mathbf{m}}^{-1}{\mathbf{K}}^{-1}\right)$ |
---|---|---|---|---|

FeCoB | 4$\xb7{10}^{6}$ | 8800 | 612 | 36 |

MgO | - | 3580 | 877 | 0.4 [49] |

$\beta $-W | 0.6$\xb7{10}^{6}$ | 19,300 | 134 | 173 |

Contacts, Vias | 7$\xb7{10}^{6}$ | 8800 | 420 | 122 |

SiO | 0 | 2200 | 730 | 1.4 |

Substrate | 1$\xb7{10}^{6}$ | 2330 | 710 | 150 |

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**Figure 1.**(

**a**) A schematic illustration of STT- and (

**b**) SOT-MRAM cells. Two separate current paths are present for the SOT-MRAM, where an HM layer is placed underneath the magnetic FL.

**Figure 2.**Simulated SOT-MRAM structures. (

**a**) Simple structure from [8]. (

**b**) Realistic structure with contacts, current lines, and a Si buffer beneath.

**Figure 3.**Temperature increase of Structure I [8]. Comparison between the data extracted from [8] (black dotted) and the presented model with two different conductivity of the surrounding oxide (solid). The slow exponential temperature increases are extracted (dashed). A constant current density of $1.1\times {10}^{12}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{Am}}^{-2}$ in the HM is considered.

**Figure 4.**(

**a**) Structure temperature at 0.2 ns after a voltage pulse ${U}_{\mathrm{SOT}}$ = 0.4 V was applied between the contacts. In the beginning, the temperature increase is centralized around the MTJ stack. (

**b**) Maximum temperature increase of the FL for different voltages. The inset shows the maximum temperature increase with respect to ${U}_{\mathrm{SOT}}^{2}$.

**Figure 5.**(

**a**) Simulations of the FL magnetization in-plane flip with SOTs only. Temperature scaling is not included. (

**b**) Simulations of the FL magnetization in-plane flip with SOTs only, with temperature scaling included. An incubation time due to the slow temperature rise can be observed. The critical SOT voltage that flips the FL magnetization in-plane is significantly reduced in comparison to the constant temperature model.

**Figure 6.**Steady-state solutions (gray, black) and real trajectories of the final magnetization state (orange, red, green). When the first oscillation amplitude (orange, green, pink) reaches the unstable solution (gray), an instant jump into the in-plane state appears.

**Figure 7.**(

**a**) Simulations of the combined STT-SOT switching at constant 300 K, and (

**b**) with the full temperature model. The different paths (colors) represent different ${U}_{\mathrm{SOT}}$, ${U}_{\mathrm{STT}}$ = 0.75 V. The SOT current is only present during the first 2 ns, while the STT is kept for the whole simulation. Due to the additional presence of the STT field, the initial oscillatory behavior is amplified and acts as an additional field that amplifies the oscillations. Both of the switching simulations with a constant temperature and with the full temperature model look similar; however, the oscillations are modulated for the latter. The critical switching voltage is also significantly reduced.

**Figure 8.**(

**a**) Parameter change due to the increased FL temperature, and (

**b**) the corresponding FL temperature increase.

**Figure 9.**Switching simulations with an external field for different ${U}_{\mathrm{SOT}}$ and (

**a**) 1.5 ns, (

**b**) 3.0 ns pulse durations. The external field is 50 mT in the ${I}_{\mathrm{SOT}}$ direction. We let the system relax for 1 ns before the pulse is applied. The color coding is identical for both plots.

**Figure 10.**Dependence of the critical SOT switching voltage ${U}_{SOT}^{C}$ on the applied STT heating voltage for the SOT switching with an external field. A parabolic reduction in ${U}_{SOT}^{C}$ is observed due to the increased temperature of the FL. The parabola is shifted due to the additional STT torque.

Structure | ${\mathsf{\tau}}_{1}$ (ns) | ${\mathsf{\tau}}_{2}$ (ns) | ${\mathsf{\tau}}_{3}$ (ns) |
---|---|---|---|

Structure I, 2.4 ${\mathrm{Wm}}^{-1}{\mathrm{K}}^{-1}$ | 0.073 | 0.746 | 5.013 |

Structure I, 2.6 ${\mathrm{Wm}}^{-1}{\mathrm{K}}^{-1}$ | 0.072 | 0.733 | 4.896 |

Structure I, [8] | 0.152 | 1.216 | 5.796 |

Structure II | 0.035 | 0.439 | 2.539 |

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## Share and Cite

**MDPI and ACS Style**

Hadámek, T.; Jørstad, N.P.; de Orio, R.L.; Goes, W.; Selberherr, S.; Sverdlov, V.
A Comprehensive Study of Temperature and Its Effects in SOT-MRAM Devices. *Micromachines* **2023**, *14*, 1581.
https://doi.org/10.3390/mi14081581

**AMA Style**

Hadámek T, Jørstad NP, de Orio RL, Goes W, Selberherr S, Sverdlov V.
A Comprehensive Study of Temperature and Its Effects in SOT-MRAM Devices. *Micromachines*. 2023; 14(8):1581.
https://doi.org/10.3390/mi14081581

**Chicago/Turabian Style**

Hadámek, Tomáš, Nils Petter Jørstad, Roberto Lacerda de Orio, Wolfgang Goes, Siegfried Selberherr, and Viktor Sverdlov.
2023. "A Comprehensive Study of Temperature and Its Effects in SOT-MRAM Devices" *Micromachines* 14, no. 8: 1581.
https://doi.org/10.3390/mi14081581