#
Variability Predictions for the Next Technology Generations of n-type Si_{x}Ge_{1−x} Nanowire MOSFETs

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Simulation Framework

#### 2.1. Device Structure with the Variability Sources Included

#### 2.2. Quantum Transport Formalism

#### 2.3. Extraction of Effective Masses

## 3. Simulation Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of the elliptical gate-all-around nanowire metal-oxide-semiconductor field-effect transistors (GAA NWFET) (a = 3 nm and b = 5 nm) highlighting variability sources. For the square and circular nanowires (NWs), a = b = 5 nm. ${L}_{\mathrm{S}}$ = ${L}_{\mathrm{D}}$ = 28 nm, ${L}_{\mathrm{G}}$ = 10 nm, and ${L}_{\mathrm{V}}$ = 26 nm. The doping concentrations in source/drain and channel regions are 10

^{20}(n-type) and 10

^{15}(p-type) cm

^{−3}, respectively. RDD–random discrete dopants, LER–line edge roughness and MGG–metal gate granularity.

**Figure 2.**Band structures of (

**a**) Si and (

**b**) ${\mathrm{Si}}_{0.2}{\mathrm{Ge}}_{0.8}5\times 3$ nm

^{2}elliptical NWs. The bulk conduction band edge is set to 0.0 eV. $\Delta {E}_{Q}$ is also remarked.

**Figure 3.**Transfer characteristics of ${\mathrm{Si}}_{0.2}{\mathrm{Ge}}_{0.8}$ elliptical GAA NWFETs associated with (

**a**) random discrete dopants (RDD), (

**b**) RDD and line edge roughness (LER) and (

**c**) RDD, LER and metal gate granularity (MGG. The ideal device refers to a device with continuous and uniform doping profiles in the source and drain and no variability sources. Corresponding standard deviation of ${V}_{\mathrm{th}}\sigma \left({V}_{\mathrm{th}}\right)$ is also indicated. ${V}_{\mathrm{DS}}$ = 0.6 V.

**Figure 4.**Distributions of threshold voltage (${V}_{\mathrm{th}}$) for the elliptical NWFETs with different mole fractions. RDD, LER, and MGG are taken into account.

**Figure 5.**Correlation between important FoMs for the elliptical GAA NWFETs with different Ge mole fraction. The bottom left of the table shows correlation scatter plots and the top right shows correlation coefficients which are also listed in the following order: Si (blue), ${\mathrm{Si}}_{0.8}{\mathrm{Ge}}_{0.2}$ (magenta), ${\mathrm{Si}}_{0.5}{\mathrm{Ge}}_{0.5}$ (red), and ${\mathrm{Si}}_{0.2}{\mathrm{Ge}}_{0.8}$ (black).

**Figure 6.**Dependence of ${V}_{\mathrm{th}}$ of the elliptical GAA NWFETs on the variability sources and the Ge mole fraction.

**Figure 7.**Dependence of (

**a**) ${V}_{\mathrm{th}}$ and (

**b**) drain induced barrier lowering (DIBL) on the Ge mole fraction and cross-sectional shape. RDD, LER, and MGG are considered

**Table 1.**Calculated effective masses of Si and ${\mathrm{Si}}_{x}{\mathrm{Ge}}_{1-x}$ nanowires (NWs) with various cross-sectional shapes. Herein, unit is ${m}_{0}$, the rest electron mass.

Degeneracy | Square | Circle | Ellipse | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{m}}_{\mathit{x}}$ | ${\mathit{m}}_{\mathit{y}}$ | ${\mathit{m}}_{\mathit{z}}$ | ${\mathit{m}}_{\mathit{x}}$ | ${\mathit{m}}_{\mathit{y}}$ | ${\mathit{m}}_{\mathit{z}}$ | ${\mathit{m}}_{\mathit{x}}$ | ${\mathit{m}}_{\mathit{y}}$ | ${\mathit{m}}_{\mathit{z}}$ | |||

Si | ${\Delta}_{x}$ | 2 | 0.918 | 0.240 | 0.240 | 0.915 | 0.224 | 0.224 | 0.927 | 0.464 | 0.146 |

${\Delta}_{y}$ | 2 | 0.233 | 0.953 | 0.237 | 0.236 | 0.887 | 0.215 | 0.241 | 0.839 | 0.220 | |

${\Delta}_{z}$ | 2 | 0.233 | 0.242 | 0.875 | 0.236 | 0.208 | 0.896 | 0.241 | 0.206 | 0.886 | |

${\mathrm{Si}}_{0.8}{\mathrm{Ge}}_{0.2}$ | ${\Delta}_{x}$ | 2 | 0.861 | 0.235 | 0.235 | 0.849 | 0.287 | 0.287 | 0.875 | 0.321 | 0.198 |

${\Delta}_{y}$ | 2 | 0.240 | 0.884 | 0.221 | 0.235 | 1.342 | 0.262 | 0.251 | 0.757 | 0.224 | |

${\Delta}_{z}$ | 2 | 0.240 | 0.220 | 0.885 | 0.235 | 0.259 | 1.366 | 0.251 | 0.192 | 0.905 | |

${\mathrm{Si}}_{0.5}{\mathrm{Ge}}_{0.5}$ | ${\Delta}_{x}$ | 2 | 0.799 | 0.241 | 0.241 | 0.788 | 0.286 | 0.286 | 0.818 | 0.392 | 0.179 |

${\Delta}_{y}$ | 2 | 0.250 | 0.864 | 0.224 | 0.247 | 1.042 | 0.272 | 0.268 | 0.674 | 0.210 | |

${\Delta}_{z}$ | 2 | 0.250 | 0.224 | 0.816 | 0.247 | 0.270 | 1.015 | 0.268 | 0.194 | 0.809 | |

${\mathrm{Si}}_{0.2}{\mathrm{Ge}}_{0.8}$ | ${\Delta}_{x}$ | 2 | 0.759 | 0.237 | 0.237 | 0.739 | 0.285 | 0.285 | 0.788 | 0.448 | 0.174 |

${\Delta}_{y}$ | 2 | 0.266 | 0.788 | 0.217 | 0.258 | 0.952 | 0.272 | 0.286 | 0.657 | 0.206 | |

${\Delta}_{z}$ | 2 | 0.266 | 0.213 | 0.798 | 0.258 | 0.272 | 0.958 | 0.286 | 0.186 | 0.828 | |

L | 4 | 0.350 | 0.134 | 0.297 | 0.500 | 0.147 | 0.449 | 0.600 | 0.327 | 0.152 |

**Table 2.**Medians of ${I}_{\mathrm{ON}}$ and ${I}_{\mathrm{OFF}}$ for the ${\mathrm{Si}}_{x}{\mathrm{Ge}}_{1-x}$ nanowire metal-oxide-semiconductor field-effect transistors (NWFETs). Random discrete dopants (RDD), line edge roughness (LER), and metal gate granularity (MGG) are considered.

${\mathbf{Si}}_{\mathit{x}}{\mathbf{Ge}}_{1-\mathit{x}}$ | ${\mathit{I}}_{\mathit{ON}}$ (mA/μm)/${\mathit{I}}_{\mathit{OFF}}$ (pA/μm) | ||
---|---|---|---|

Square | Circular | Elliptical | |

Si | 1.59/397 | 1.37/98.9 | 0.771/9.26 |

${\mathrm{Si}}_{0.8}{\mathrm{Ge}}_{0.2}$ | 1.71/427 | 1.50/127 | 0.862/11.7 |

${\mathrm{Si}}_{0.5}{\mathrm{Ge}}_{0.5}$ | 1.70/473 | 1.51/151 | 0.861/12.7 |

${\mathrm{Si}}_{0.2}{\mathrm{Ge}}_{0.8}$ | 1.84/668 | 1.63/210 | 0.958/18.1 |

**Table 3.**The comparison of drain induced barrier lowering (DIBL) in ${\mathrm{Si}}_{0.2}{\mathrm{Ge}}_{0.8}$ channel devices obtained from the ideal devices and statistical simulations.

Cross-Sectional Shape (RDD + LER + MGG) | Ideal Device | Median |
---|---|---|

Square | 62.4 mV/V | 64.7 mV/V |

Circle | 42.8 mV/V | 50.2 mV/V |

Ellipse | 20.3 mV/V | 29.2 mV/V |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Lee, J.; Badami, O.; Carrillo-Nuñez, H.; Berrada, S.; Medina-Bailon, C.; Dutta, T.; Adamu-Lema, F.; Georgiev, V.P.; Asenov, A. Variability Predictions for the Next Technology Generations of *n*-type Si_{x}Ge_{1−x} Nanowire MOSFETs. *Micromachines* **2018**, *9*, 643.
https://doi.org/10.3390/mi9120643

**AMA Style**

Lee J, Badami O, Carrillo-Nuñez H, Berrada S, Medina-Bailon C, Dutta T, Adamu-Lema F, Georgiev VP, Asenov A. Variability Predictions for the Next Technology Generations of *n*-type Si_{x}Ge_{1−x} Nanowire MOSFETs. *Micromachines*. 2018; 9(12):643.
https://doi.org/10.3390/mi9120643

**Chicago/Turabian Style**

Lee, Jaehyun, Oves Badami, Hamilton Carrillo-Nuñez, Salim Berrada, Cristina Medina-Bailon, Tapas Dutta, Fikru Adamu-Lema, Vihar P. Georgiev, and Asen Asenov. 2018. "Variability Predictions for the Next Technology Generations of *n*-type Si_{x}Ge_{1−x} Nanowire MOSFETs" *Micromachines* 9, no. 12: 643.
https://doi.org/10.3390/mi9120643