# Quantum Enhancement of a S/D Tunneling Model in a 2D MS-EMC Nanodevice Simulator: NEGF Comparison and Impact of Effective Mass Variation

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## Abstract

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## 1. Introduction

## 2. Simulation Framework and Device Structures

#### 2.1. Description of the Simulated Devices

#### 2.2. Description of the 2D NEGF Module Inside NESS

#### 2.3. General Overview of the 2D MS-EMC Tool

#### 2.4. Description of the S/D Tunneling Model Inside the 2D MC-EMC Tool

#### 2.5. Description of the Effective Mass Calculation

## 3. Simulation Results and Discussions

#### 3.1. Comparison of MS-EMC with S/D Tunneling Models vs. NEGF

#### 3.2. Impact of the Effective Mass Choice

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Double-Gate Silicon-On-Insulator (DGSOI) and FinFET structures analyzed in this paper with L${}_{G}$ ranging from 5 to 15 nm and T${}_{Si}$ = 3–5 nm. The 1D Schrödinger equation is solved in the confinement direction for each grid point and the Boltzmann Transport Equation (BTE) is solved by the MC method in the transport plane.

**Figure 2.**(

**a**) Longitudinal (${m}_{l}$) and (

**b**) transverse (${m}_{t}$) effective masses calculated using Density Functional Theory (DFT) as well as the bulk effective masses as a function of the silicon thickness (T${}_{Si}$) for DGSOI ((100) Confinement Orientation) and FinFET ((0$\overline{1}$1) Confinement Orientation) devices.

**Figure 3.**Deviation (%) of the longitudinal (${m}_{l}$) and transverse (${m}_{t}$) effective masses and their combinations needed in Table 2 as a function of the silicon thickness (T${}_{Si}$) for DGSOI ((100) confinement orientation) as well as FinFET ((0$\overline{1}$1) confinement orientation) devices.

**Figure 4.**I${}_{D}$ vs. V${}_{GS}$ in the DGSOI and FinFET devices at V${}_{DS}$ = 500 mV with L${}_{G}$ 5 nm (

**a**,

**e**), 7.5 nm (

**b**,

**f**), 10 nm (

**c**,

**g**), and 15 nm (

**d**,

**h**), considering the four types of simulations are: (1) Non-Equilibrium Green’s Function (NEGF) approach in the Nano-Electronic Simulation Software (NESS) tool, (2) Multi-Subband Ensemble Monte Carlo (MS-EMC) tool without any type of tunneling, and MS-EMC tool with the Source-to-Drain tunneling (S/D tunneling) module using (3) T${}_{WKB}$(E${}_{x}$) and (4) T${}_{DT}$(E${}_{x}$).

**Figure 5.**Average number of electrons (in arbitrary units) affected by S/D tunneling as a function of the V${}_{GS}$ in the (

**a**) DGSOI and (

**b**) FinFET devices at V${}_{DS}$ = 500 mV with L${}_{G}$ = 5, 7.5, 10, and 15 nm, for the MS-EMC tool with the S/D tunneling module using T${}_{WKB}$(E${}_{x}$) and T${}_{DT}$(E${}_{x}$).

**Figure 6.**$\Delta $SS as a function of the gate length in the DGSOI and FinFET devices at V${}_{DS}$ = 100 mV (

**a**,

**d**), V${}_{DS}$ = 500 mV (

**b**,

**e**), and V${}_{DS}$ = 1 V (

**c**,

**f**), calculated as the difference between the 2D NEGF-NESS and the 2D MS-EMC tools considering the three combinations: a simulation without any tunneling module and both S/D tunneling modules with T${}_{WKB}$(E${}_{x}$) and T${}_{DT}$(E${}_{x}$) probabilities.

**Figure 7.**$\Delta $V${}_{TH}$ as a function of the gate length in the DGSOI and FinFET devices with silicon thickness T${}_{Si}$ = 3–5 nm at V${}_{DS}$ = 100 mV (

**a**,

**d**), V${}_{DS}$ = 500 mV (

**b**,

**e**), and V${}_{DS}$ = 1 V (

**c**,

**f**), considering the three 2D MS-EMC combinations: without any tunneling module and with both S/D tunneling modules using T${}_{WKB}$(E${}_{x}$) and T${}_{DT}$(E${}_{x}$) probabilities.

**Table 1.**Silicon bulk effective masses (m${}_{bulk}$) for the different crystallographic directions considered in the DGSOI and FinFET devices. Herein, ${m}_{x}$ and ${m}_{z}$ are the transport and confinement masses, respectively; ${m}_{y}$ is the effective mass in the periodic transverse direction; m${}_{0}$ is the free electron mass; and the subindex of $\Delta $ represents the degeneracy factor associated with the conduction band valley.

Device | Valley | m${}_{\mathit{bulk}}$ | ||
---|---|---|---|---|

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

DGSOI | ${\Delta}_{2}$ | ${m}_{t}$ = 0.193 m${}_{0}$ | ${m}_{t}$ = 0.193 m${}_{0}$ | ${m}_{l}$ = 0.912 m${}_{0}$ |

(100)<011> | ${\Delta}_{4}$ | $\frac{2{m}_{l}{m}_{t}}{{m}_{l}+{m}_{t}}$ = 0.319 m${}_{0}$ | $\frac{{m}_{l}+{m}_{t}}{2}$ = 0.553 m${}_{0}$ | ${m}_{t}$ = 0.193 m${}_{0}$ |

FinFET | ${\Delta}_{2}$ | ${m}_{t}$ = 0.193 m${}_{0}$ | ${m}_{l}$ = 0.912 m${}_{0}$ | ${m}_{t}$ = 0.193 m${}_{0}$ |

(0$\overline{1}$1)<011> | ${\Delta}_{4}$ | $\frac{{m}_{l}+{m}_{t}}{2}$ = 0.553 m${}_{0}$ | ${m}_{t}$ = 0.193 m${}_{0}$ | $\frac{2{m}_{l}{m}_{t}}{{m}_{l}+{m}_{t}}$ = 0.319 m${}_{0}$ |

**Table 2.**Effective masses (m${}_{eff}$) considering the DGSOI and FinFET devices herein studied with silicon thickness T${}_{Si}$ = 3–5 nm using DFT simulations included in QuantumATK of Synopsys [19]. Notice that ${m}_{x}$ and ${m}_{z}$ are the transport and confinement masses, respectively, ${m}_{y}$ is the mass in the direction normal to transport, m${}_{0}$ is the free electron mass, and the subindex of $\Delta $ represents the degeneracy factor associated with the conduction band valley.

Device | Valley | T${}_{\mathit{Si}}$ = 3 nm | T${}_{\mathit{Si}}$ = 5 nm | ||||
---|---|---|---|---|---|---|---|

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

DGSOI | ${\Delta}_{2}$ | 0.144 m${}_{0}$ | 0.144 m${}_{0}$ | 1.002 m${}_{0}$ | 0.166 m${}_{0}$ | 0.166 m${}_{0}$ | 0.93 m${}_{0}$ |

(100)<011> | ${\Delta}_{4}$ | 0.252 m${}_{0}$ | 0.573 m${}_{0}$ | 0.144 m${}_{0}$ | 0.282 m${}_{0}$ | 0.548 m${}_{0}$ | 0.166 m${}_{0}$ |

FinFET | ${\Delta}_{2}$ | 0.15 m${}_{0}$ | 1.134 m${}_{0}$ | 0.15 m${}_{0}$ | 0.171 m${}_{0}$ | 0.956 m${}_{0}$ | 0.171 m${}_{0}$ |

(0$\overline{1}$1)<011> | ${\Delta}_{4}$ | 0.642 m${}_{0}$ | 0.15 m${}_{0}$ | 0.265 m${}_{0}$ | 0.563 m${}_{0}$ | 0.171 m${}_{0}$ | 0.29 m${}_{0}$ |

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**MDPI and ACS Style**

Medina-Bailon, C.; Carrillo-Nunez, H.; Lee, J.; Sampedro, C.; Padilla, J.L.; Donetti, L.; Georgiev, V.; Gamiz, F.; Asenov, A.
Quantum Enhancement of a S/D Tunneling Model in a 2D MS-EMC Nanodevice Simulator: NEGF Comparison and Impact of Effective Mass Variation. *Micromachines* **2020**, *11*, 204.
https://doi.org/10.3390/mi11020204

**AMA Style**

Medina-Bailon C, Carrillo-Nunez H, Lee J, Sampedro C, Padilla JL, Donetti L, Georgiev V, Gamiz F, Asenov A.
Quantum Enhancement of a S/D Tunneling Model in a 2D MS-EMC Nanodevice Simulator: NEGF Comparison and Impact of Effective Mass Variation. *Micromachines*. 2020; 11(2):204.
https://doi.org/10.3390/mi11020204

**Chicago/Turabian Style**

Medina-Bailon, Cristina, Hamilton Carrillo-Nunez, Jaehyun Lee, Carlos Sampedro, Jose Luis Padilla, Luca Donetti, Vihar Georgiev, Francisco Gamiz, and Asen Asenov.
2020. "Quantum Enhancement of a S/D Tunneling Model in a 2D MS-EMC Nanodevice Simulator: NEGF Comparison and Impact of Effective Mass Variation" *Micromachines* 11, no. 2: 204.
https://doi.org/10.3390/mi11020204