# Conclusive Model-Fit Current-Voltage Characteristic Curves with Kink Effects
## 1 Introduction  
Transistors fabricated in the semiconductor industry successfully achieve various desired functions, including signal processing, data calculation, and decision making. All the functions are associated with memory transfer at a comparable speed. So, the size of the transistor continues to shrink, not mainly for the benefits of increasing the number of integrated circuits (IC). The speed of the ICs somehow becomes the pursued goal. Three concerns appear manifested in a sense. They are the outrageous loss of controllability, unavoidable heat, and unexpected limitations of photolithographs. The controllability, closely related to the leakage current, is due to proportional dimension shrinkage, even if some adopted prevention, including halo implant or pocket implant, is taken into account at the planar device level. In addition, the promotion of electrical performance at the same scale using the compressive or tensile stress technique is seriously considered as well. Unfortunately, channel lengths below tens of nanometers seem not to work out any more as desired, no matter what has been adopted. Instead of a planar bulk silicon substrate, there comes a 3D structural fin-like field effect transistor (FinFET) and gates all around the field effect transistor (GAFET), which make use of a slim strip of epitaxial silicon as a body substrate wrapped by insulator-separated gate poly-silicon. The strip of channel substrate is depleted as the gate poly-silicon is applied with a bias. This bias causes the depletion region of the substrate strip to impressively and effectively block the leakage current in between the source and drain. As for the generated heat, it is basically linked to the equivalent resistors, whose resistance is actually proportional to the length of the resistor and inversely dependent on the area of the cross-section. The resistance of the decreasing dimension soars, making the increasing Ohm's heat tremendously degrade its electrical performance and thus competitively lose leading status. Therefore, the conduction of heat dominates the whole process flow and material choice. Copper is thus preferred for its higher conductivity. The exposure of ultra-violet rays suggests diffraction immunity,which is identified as the use of a shorter wavelength, improvement in the refraction angle (refractive index, n), and subtle design and combination of masks. Traditional optics and modern optics are both utilized or designed to resolve diffraction issues. Furthermore, the electrical performances for any transistor have to be reliable and repeatable. Models are thus proposed to address these electrical performances. They are able to fit the measured characteristic curves. All the circuit designs, including analog and digital, rely solely on the established model. Research has to be undertaken to understand the electrical data. The most commonly used formulas have been posed for over many decades, and they surely reduce the tremendous work on the design level. Therefore, current-versus-voltage characteristic curves showing the electrical performances of transistors are necessarily parameter-extracted in the model. Nevertheless, researchers are still intrigued to know if the \"modified\" conventional formula is applicable for fitting repeated characteristic curves [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14]. Useful parameters are supposed to be constants in the model, even though they need interpreting. For example, mobility is a constant at a certain fixed V\({}_{\text{GS}}\). But the mobility may be different at different V\({}_{\text{GS}}\)[15]. Moreover, one thing that causes attention is carriers traveling in crystallized silicon, which is a diamond structure. The crystal thermally vibrates at a certain temperature, causing trouble for the carriers. As the carriers speed up, they confront more obstacles. More obstacles generate more heat and more friction, which may slow down the carriers, especially in the triode region.