# Conclusive Model-Fit Current-Voltage Characteristic Curves with Kink Effects
## 2 Preparation of as Measured Data and Fitting
### The Kink Effect  
The gate bias strongly inverses the channel layer, whose thickness is about 200 Angstroms [15; 16]. The collective free carriers (electrons) travel through the channel, which is a grossly periodic diamond structure of silicon (double face-center cubic). For simplicity, the electrons are assumed to travel in one dimension from source to drain (1, 0, 0) via center to center or corner to corner. Even if those carriers are collective, the interactions among them are thought to be weak enough to be ignored, such that each of them is treated as individual without losing generality. The variation in the potential energy of an electron is realized as sinusoid due to the periodic diamond structure. A free electron is also supposed to surpass the attraction force from the field of the periodic ion cores. Therefore, instead of V = V\({}_{\rm o}\) [\(1-\cos(2\pi\)x/l\({}_{\rm o}\))], with l\({}_{\rm o}\) as a periodic referencing length, the potential energy density is deliberately written as V(\(\phi_{\rm e}\)) = (a/b)[\(1-\cos(\)b\(\phi_{\rm e}\))], where V(\(\phi_{\rm e}\)) is non-negative. Actually, \(\phi_{\rm e}\) inside the argument is nothing new, e.g., the differential equation of the azimuth angle of a pendulum swinging back and forth. An electron moves from a constant potential (ground/source) to the next constant potential  
Figure 1: \(I_{\rm DS}\)-\(V_{\rm DS}\) characteristic curves of FinFET transistor (fin width = 120 nm and channel length = 90 nm) and the corresponding fitting (**a**) without the term, (1 + \(\lambda V_{\rm DS}\)), in the triode region in Equation (1) and (**b**) with the term, (1 + \(\lambda V_{\rm DS}\)), in the triode region in Equation (1). (\(\rm V_{DS}\)/drain), where a collective \(\rm I_{DS}\) is measured and marked. As the \(\rm V_{DS}\) is increased, the corresponding collective \(\rm I_{DS}\) is measured and recorded again until the current becomes saturated when the applied \(\rm V_{DS}\) is high enough. The whole characteristics are then constructed, just like a series of (\(\rm V_{DS}\) and \(\rm I_{DS}\)) pictures, which are applicable for digital usages, amplifiers, and mixers, etc. Wherever an electron is located in the channel, the electric field varies not only because of different locations, but also because of different \(\rm V_{DS}\). This may be the reason why a nonlinear equation is considered. The carriers and electrons are fermions, but there is no loss of generality to treat the electron wave function as \(\Psi_{e}=\sum\limits_{\alpha}\varphi_{e,\alpha}\psi_{f,\alpha}\), where \(\psi_{f}\) obeys Fermi-Dirac statistics. The non-linear differential equation, addressing a moving electron scalar field, \(\varphi_{e}\), in the strongly inversed layer, refers to the Lagrangian as follows:  
\(L=\frac{1}{2}(\frac{\partial\varphi_{e}}{\partial t})^{2}-\frac{1}{2}(\frac{ \partial\varphi_{e}}{\partial x})^{2}-V(\varphi_{e})\)  
_where_  
\(V(\varphi_{e})=\frac{a}{b}(1-\cos b\varphi_{e})\)  
_and_  
\(\frac{\partial}{\partial t}\frac{\partial L}{\partial(\frac{\partial\varphi_ {e}}{\partial t})}+\frac{\partial}{\partial x}\frac{\partial L}{\partial( \frac{\partial\varphi_{e}}{\partial x})}-\frac{\partial L}{\partial\varphi_ {e}}=0\)  
\(\frac{\partial^{2}\varphi_{e}}{c^{2}\partial t^{2}}-\frac{\partial^{2} \varphi_{e}}{\partial x^{2}}+a\sin b\varphi_{e}=0\)  
which is named as the sine-Gordon equation [18]. The moving electrons accelerated by the electrical field confront phonons with a group velocity c of thousands of meters per second in the lattice [18]. These electrons quickly reach the speeds that are sometimes higher than and sometimes lower than the so-called terminal speed.