# An Electron Waveguide Model for FDSOI Transistors

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Quantum Transport Model

_{2}-layer (BOX). Below lies the BP consisting either of a heavily p- or a heavily n-doped silicon layer. The back contact with voltage ${U}_{BG}$ is connected to the BP via a p-doped region (’p-well’). Further abbreviations are HK-MG for the high-k/metal gate stack and STI for shallow trench insulation. Outside the active Si-layer, the wave functions of the current carrying electrons vanish, leading to the boundary condition

#### 2.2. Potential in Abrupt Transition Approximation

## 3. Results

- n-BP with ${U}_{BG}=0$: standard-VT (SVT (i)), identical with Figure 2b;
- n-BP with ${U}_{BG}={U}_{DD}$: low-VT (LVT);
- p-BP with ${U}_{BG}=0$: high-VT (HVT);
- p-BP with ${U}_{BG}={U}_{DD}$ standard-VT (SVT (ii)).

## 4. Discussion

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

MDPI | Multidisciplinary Digital Publishing Institute |

DOAJ | Directory of open access journals |

TLA | Three letter acronym |

LD | Linear dichroism |

## Appendix A. Adoption of a Previous Transistor model for the Calculation of the Drain Current

## Appendix B. The Chemical Potential in Source and Drain

## Appendix C. Solution of the Poisson Equation

- The matching condition ${V}_{T}\left({0}^{+}\right)={V}_{T}\left({0}^{-}\right)$ leads to$${c}_{1}{D}_{G}-e{U}_{G}^{\prime}=\beta .$$
- The matching condition ${V}_{T}^{\prime}\left({0}^{-}\right)/{V}_{T}^{\prime}\left({0}^{+}\right)={\kappa}_{1}$ leads to$${c}_{1}={\kappa}_{1}\alpha .$$
- The matching condition ${V}_{T}\left({D}^{-}\right)={V}_{T}\left({D}^{+}\right)$ leads to$$-\frac{C}{2}{D}^{2}+\alpha D+\beta ={c}_{2}(D-{D}_{BG})-e{U}_{BG}^{\prime}=-{c}_{2}{D}_{Box}-e{U}_{BG}^{\prime}.$$
- The matching condition ${V}_{T}^{\prime}\left({D}^{+}\right)/{V}_{T}^{\prime}\left({D}^{-}\right)={\kappa}_{2}$ leads to$${c}_{2}={\kappa}_{2}(-CD+\alpha )=-{\kappa}_{2}CD+{\kappa}_{2}\alpha .$$

## Appendix D. Analytical Approximations for the Barrier Haight and the Threshold Voltage

## References

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**Figure 1.**(

**a**) N-channel multiple VT, fully depleted SOI-transistor with a back plane (BP) (see text). (

**b**) Active Si-layer for the conduction channel taking the form of a waveguide. (

**c**) One-dimensional effective potential $\overline{V}$ after Equation (22). (

**d**) The (100)-orientation of the active Si-layer with the six constant energy ellipsoids $\alpha =1\dots 6$.

**Figure 2.**For the theoretical parameter set listed at the beginning of Section 3: (

**a**) Energy diagrams of the isolated contacts. (

**b**) Transverse confinement potential ${V}_{T}\left(y\right)$ for an n-BP at ${U}_{BG}=0$ (SVT(i)) for ${U}_{G}$ = 0.0 (black), 0.1 V (red), 0.2 V (green), 0.3 V (blue), 0.4 V (orange), 0.5 V (brown), 0.6 V (grey), 0.7 V (violet), 0.8 V (cyan), and 0.9 V (magenta). (

**c**) In circles: Effective barrier height ${V}_{0}$ resulting from Equation (33), color coding as in part (

**b**). Marked by an arrow: Threshold gate voltage ${U}_{G}^{th}$ determined by condition (39).

**Figure 3.**The threshold voltage versus thickness of the box: n-BP with ${U}_{BG}=0$ (orange), n-BP with ${U}_{BG}={U}_{DD}$ (red), p-BP with ${U}_{BG}=0$ (blue), and p-BP with ${U}_{BG}={U}_{DD}$ (green). Solid lines according to Equation (41) and dashed lines according to Equation (42). The circles mark the values for ${D}_{Box}=10$ nm which enter Table 1.

**Figure 4.**For the system parameters, the barrier height ${V}_{0}\left(e{U}_{G}\right)$ from (33): n-BP with ${U}_{BG}=0$ (SVT(i), orange), n-BP with ${U}_{BG}={U}_{DD}$ (LVT, red), p-BP with ${U}_{BG}={U}_{DD}$ (SVT(ii), green), and p-BP with ${U}_{BG}=0$ (HVT, blue). In dotted black lines: Linear approximation to ${V}_{0}\left(e{U}_{G}\right)$ in Equation (36). The horizontal black solid line marks the chemical potential $\mu $ where the intersection points with ${V}_{0}\left(e{U}_{G}\right)$ give the threshold voltage ${U}_{G}^{th}$, according to (39).

**Figure 5.**For theoretical parameters the transfer characteristics for ${U}_{D}=0.05$ V (dashed lines) and ${U}_{D}=0.9$ V (solid lines). (

**a**) for a n-BP and (

**b**) for a p-BP. Color coding as in Figure 3. In black: Traces taken to extract the SS.

Type | ${\mathit{U}}_{\mathit{G}}^{\mathbf{th}}$ [mV] | ${\mathit{I}}_{\mathbf{ON}}\phantom{\rule{4pt}{0ex}}[\mathsf{\mu}\mathbf{A}/\mathsf{\mu}\mathbf{m}]$ | ${\mathit{I}}_{\mathbf{OFF}}\phantom{\rule{4pt}{0ex}}[\mathbf{pA}/\mathsf{\mu}\mathbf{m}]$ | $\mathbf{SS}$ [mV/dec] |
---|---|---|---|---|

LVT | 440 | 45,900 | 1474 | 70 |

SVT(i) | 563 | 49,452 | 32 | 70 |

SVT(ii) | 563 | 44,952 | 32 | 70 |

HVT(i) | 686 | 39,385 | 0.7 | 70 |

LVT | 278 | 480 | 2139 | 77 |

SVT(i) | 429 | 332 | 16 | 76 |

SVT(ii) | 430 | 333 | 17 | 77 |

HVT(i) | 567 | 208 | 0.1 | 73 |

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**MDPI and ACS Style**

Wulf, U.
An Electron Waveguide Model for FDSOI Transistors. *Solids* **2022**, *3*, 203-218.
https://doi.org/10.3390/solids3020014

**AMA Style**

Wulf U.
An Electron Waveguide Model for FDSOI Transistors. *Solids*. 2022; 3(2):203-218.
https://doi.org/10.3390/solids3020014

**Chicago/Turabian Style**

Wulf, Ulrich.
2022. "An Electron Waveguide Model for FDSOI Transistors" *Solids* 3, no. 2: 203-218.
https://doi.org/10.3390/solids3020014