# Simulation of the Impact of Ionized Impurity Scattering on the Total Mobility in Si Nanowire Transistors

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## Abstract

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## 1. Introduction

## 2. Simulation Method and Physics

#### 2.1. Simulation Framework

#### 2.2. Subband Details and Electrostatics

#### 2.3. 1D Scattering Rates

#### 2.3.1. Acoustic Phonon Scattering

#### 2.3.2. Optical Phonon Scattering

#### 2.3.3. Surface Roughness Scattering

#### 2.3.4. Ionized Impurity Scattering

#### 2.4. Mobility Calculation

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**A flowchart illustrating the simulation framework and the corresponding steps needed to calculate the total mobility. l is the subband index; ${m}_{l}$, ${\xi}_{l}$, and ${E}_{l}$ are the calibrated effective masses, the wavefunction, and the energy level for the lth subband, respectively; i is the i-th scattering mechanism; ${\tau}_{i}^{l}$, ${\Gamma}_{i}^{l}$, and ${\mu}_{i}^{l}$ are the relaxation time, scattering rate, and mobility, respectively, for the ith mechanism and lth subband; ${\mu}_{NW}$ is the total mobility for a particular NW structure.

**Figure 2.**Coupling the 2D solution of the Schrödinger equation in multiple slices to a 3D Poisson solution in a structure based on a Si gate-all-around nanowire channel FET.

**Figure 3.**(

**a**) The rates for acoustic and optical phonons, as well as surface roughness scattering as a function of the total energy, for a sheet density of $3.25\times {10}^{12}\phantom{\rule{0.222222em}{0ex}}$cm${}^{-2}$. (

**b**) The rates for ionized impurity scattering, as a function of the total energy and different sheet densities, assuming charge neutrality. The results are for a 3 nm diameter square nanowire structure.

**Figure 4.**(

**a**) Ionized impurity limited electron mobility as a function of the sheet density considering different fixed ionized impurity densities as well as charge neutrality. (

**b**) Electron mobility as a function of the sheet density considering acoustic, optical, total phonon, surface roughness, and ionized impurity (assuming charge neutrality) scattering separately, as well as the combined mechanisms. The results are for a 3 nm diameter square nanowire structure.

**Figure 5.**The electron mobility as a function of the sheet density considering the combined total phonon and surface roughness (Ph + SR) scattering rates, as well as the total scattering (Ph + SR + II). The results are for different fixed ionized impurity densities and the charge neutrality case, for a 3 nm diameter square NW with a [100] channel orientation.

**Figure 6.**The electron mobility as a function of the NW cross sectional area, considering the impact of the combined total phonon and surface roughness (Ph + SR) scattering effects (

**a**), as well as all present scattering mechanisms (Ph + SR + II) with a (

**b**) fixed ionized impurity density (${N}_{I}={10}^{18}$ cm${}^{-3}$) and (

**c**) the charge neutrality case. The results are for both square and circular NWs ([100] orientation) and a sheet density of ${10}^{13}\phantom{\rule{0.222222em}{0ex}}$ cm${}^{-2}$.

© 2019 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**

Sadi, T.; Medina-Bailon, C.; Nedjalkov, M.; Lee, J.; Badami, O.; Berrada, S.; Carrillo-Nunez, H.; Georgiev, V.; Selberherr, S.; Asenov, A.
Simulation of the Impact of Ionized Impurity Scattering on the Total Mobility in Si Nanowire Transistors. *Materials* **2019**, *12*, 124.
https://doi.org/10.3390/ma12010124

**AMA Style**

Sadi T, Medina-Bailon C, Nedjalkov M, Lee J, Badami O, Berrada S, Carrillo-Nunez H, Georgiev V, Selberherr S, Asenov A.
Simulation of the Impact of Ionized Impurity Scattering on the Total Mobility in Si Nanowire Transistors. *Materials*. 2019; 12(1):124.
https://doi.org/10.3390/ma12010124

**Chicago/Turabian Style**

Sadi, Toufik, Cristina Medina-Bailon, Mihail Nedjalkov, Jaehyun Lee, Oves Badami, Salim Berrada, Hamilton Carrillo-Nunez, Vihar Georgiev, Siegfried Selberherr, and Asen Asenov.
2019. "Simulation of the Impact of Ionized Impurity Scattering on the Total Mobility in Si Nanowire Transistors" *Materials* 12, no. 1: 124.
https://doi.org/10.3390/ma12010124