# Tunneling Current Variations in Small-Sized Devices Based on a Compact Threshold Voltage Model
## 5 Conclusions  
In this article, we focus on strained Si NMOSFET devices. We investigate the influences of substrate bias, short-channel effect, and drain-induced barrier lowering effect on threshold voltage, based on the gradual channel approximation and a quasi-two-dimensional Poisson equation. The influence of narrow-channel effect is also studied in detail, and a comprehensive physical model for the threshold voltage is established. The threshold voltage-related parameters are extracted through experiments. Subsequently, the influences of short-channel effect, narrow-channel effect, drain-induced barrier lowering effect, and substrate bias on threshold voltage are discussed in detail. When the drain-source voltage is biased at \(V_{ds}=1.2\) V, the short-channel effect and drain-induced barrier lowering effect cause a threshold voltage shift of 18.57%; a change in channel width results in a threshold voltage shift of 2.75% (as shown in Figure 15); and when the substrate bias is set to \(V_{bs}=1.2\) V, the threshold voltage shifts by 27.22%. Based on the threshold voltage model in this article, we explore the tunneling current in off-state, and provide the experimental results. Future research should involve establishing corresponding models for other structures and smaller-sized devices. The developed physical models for the threshold voltage and tunneling current of strained Si NMOSFETs provide important references for the analysis and design of strained Si large-scale integrated circuits. **Author Contributions:** Conceptualization, Z.Z. and C.Z. ; methodology, C.Z. and T.W. ; software, Z.Z. and T.W. ; validation, Z.Z., T.W. and C.Z. ; formal analysis, C.Z. ; investigation, T.W. ; resources, C.Z. and T.W. ; data curation, Z.Z. ; writing--original draft preparation, Z.Z. and T.W. ; writing--review and editing, Z.Z. ; visualization, M.W. and Y.X. ; supervision, Z.Z. ; project administration, Z.Z. ; funding acquisition, Q.F. and T.W. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was funded by the Fujian Provincial Key Laboratory of Higher Education Open Research Projects, grant number Xnz1905. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** [[http://bsim.eecs.berkeley.edu](http://bsim.eecs.berkeley.edu)]([http://bsim.eecs.berkeley.edu](http://bsim.eecs.berkeley.edu)) (accessed on 11 July 2023). **Acknowledgments:** The authors would like to acknowledge financial support provided by the Opening Project of Key Laboratory of Microelectronic Devices & Integrated Technology, Institute of Microelectronics, Chinese Academy of Sciences. **Conflicts of Interest:** The authors declare no conflict of interest. **Abbreviations**  
**Abbreviation**  
**Si**  
**NMOSFET**  
**MOSFET**  
**Metal-oxide-semiconductor field-effect transistor**  
**MOSFET**  
**AC/DC**  
**Alternating current/direct current**  
**DIBL**  
**Drain-induced barrier lowering**  
**SiGe**  
**Silicon germanium**  
**Ge**  
**Germannium**  
**SSI**  
**PMOSFET**  
**P-channel metal-oxide-semiconductor field-effect transistor**  
**BSIM**  
**Berkeley short-channel IGFET model**  
**References**  
* **[PERSON]; [PERSON] Deep submicron CMOS based on silicon germanium technology. ** _IEEE Trans. Electron Devices_ **1996, 43, 911-918. [CrossRef]**
* **[PERSON]; [PERSON]; [PERSON] [PERSON]; [PERSON]; [PERSON] [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] [PERSON]; [PERSON] Dependences of the hole mobility in the strained Si pMOSFET and gated Hall bars formed on SiGe/Si(110) on the channel direction and the strained Si thickness. ** _J. Cryst. Growth_ **2021, 571, 126246. [CrossRef]**
* **[PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Two-dimensional threshold voltage model of sub-100 nm strained-Si/SiGe nMOSFET. ** _Acta Phys. Sin._ **2011, 60, 077106. [CrossRef]**
* **[PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Study of physically modeling for small-scaled strained Si nMOSFET. ** _Acta Phys.