## 2 Preparation, Measurements, and Fitting Algorithm  
### Preparation and Measurement of NFinFET  
The grown 10 \(\upmu\)m epi-silicon layer was ionic dry etched to form a 3-dimensional 120 nm wide and corresponding 9 times high fin channel. With two ends as Source and drain, the whole epi-silicon channel looks like a letter \"I\", which was followed with a grown thin gate oxide. The slim channel of \"I\" was then covered with arsenic heavily doped poly-silicon, called Gate. As the Gate is applied with a positive bias, the channel surrounded with Gate is then almost completely depleted, and right underneath the gate oxide, about 200 angstroms [16] p-type silicon is strongly inversed into effectively 90 nm long n-type silicon. The I\({}_{\text{DS}}\) versus V\({}_{\text{DS}}\) was then measured at various Gate biases on Agilent 4156C (Santa Clara, CA, USA). Fitting I\({}_{\text{DS}}\)-V\({}_{\text{DS}}\) and I\({}_{\text{DS}}\)-V\({}_{\text{GS}}\)  
The two-regime conventional formulas for FinFET and MOSFET transistors have to be modified as follows:  
\[ \begin{array}{ll}I_{\text{DS}}(triode)&=k_{N}[(V_{\text{GS}}-V_{th})V_{\text {DS}}-\frac{V_{\text{DS}}^{2}}{2}](1+\lambda V_{\text{DS}})\\ &-\alpha\exp[-\beta(V_{\text{DS}}-\chi)^{2}]\end{array}  \tag{1}\]  
and  
\[ \begin{array}{ll}I_{\text{DS}}(saturation)=&k_{N}[\frac{(V_{\text{GS}}-V_{th})^ {2}}{2}](1+\lambda V_{\text{DS}})\\ &-\alpha\exp[-\beta(V_{\text{DS}}-\chi)^{2}]\end{array}  \tag{2}\]  
_where \(k_{N}=\frac{C_{\text{cs}}{}^{(1)}W_{eff}\mu}{L_{\rho}}\), \(\lambda=\frac{1}{|V_{A}|}\), and \((\chi,\alpha)\equiv(V_{\text{DS}\_{kink}},I_{\text{DS}\_{kink}})\),_where V\({}_{\rm A}\) is early voltage, C\({}_{\rm ox}\)(1) is gate oxide capacitance per square meter, L\({}_{\rm o}\) is channel length, \(\mu\) is the mobility of carriers, and W\({}_{\rm eff}\) = 19W\({}_{\rm o}\) (W\({}_{\rm o}\), fin width); and where \(\beta\) (V\({}^{-2}\)) is for the kink width adjustment [16; 17; 18; 19; 20; 21]. Charge carriers travel through the strongly inversed layer with the speed variation in triode region which is proportional to the slopes of I\({}_{\rm DS}\)-V\({}_{\rm DS}\). Charge carriers radiate electromagnetic wave, called Bremsstrahlung, with the speed of forward light at 2.997 \(\times\) 10\({}^{8}\)/11.9 m/s, intensively heat up the lattice in advance, and definitely cause the extra vibration of the lattice ahead of carriers (quantized as phonons). The corresponding current corrections (due to heat vibration of lattice) ensure the reduction of I\({}_{\rm DS}\) as presented in Figures 1-4. The subtraction of current-voltage data of substitutions in Equations (1) and (2) from the as-measured data is named \(\Delta\)I\({}_{\rm DS}\), which is always positive in triode region forming a Gaussian-like peak at (\(\chi\),\(\alpha\)) with a \(\beta\)-controlled width as drawn in Figures 2b and 3b. In order to achieve kink effect adjustment, those peaks are expected to be eliminated as \(\delta\)-deviations are minimized. Equation (1) or Equation (2) work as V\({}_{\rm DS}\) is less than or larger than (V\({}_{\rm GS}\)-Vth), respectively. And those parameters (k\({}_{\rm N}\), Vth, \(\lambda\), \(\alpha\), \(\beta\), \(\chi\)) are deliberately determined to minimize the delta (8) in Equation (3) as follows:  
\[\delta\ =\ \sqrt{\frac{\sum\limits_{i=1}^{N}\left(I_{fitting}-I_{measured} \right)_{i}^{2}}{N}} \tag{3}\]  
Figure 1: _Cont._  
Figure 1: **(a)** FinFET fitting without kink effect adjustment on thermal phonon collisions. **(b)** Minimum deviation without kink effect adjustment: \(\delta\) = 1.17 \(\times\) 10\({}^{-6\,\rm A\,\rm(at\,V}_{\rm GS}\) = 1.0 V), \(\delta\) = 9.77 \(\times\) 10\({}^{-7\,\rm A\,\rm(at\,V}_{\rm GS}\) = 0.75 V), \(\delta\) = 9.60 \(\times\) 10\({}^{-7\,\rm A\,\rm(at\,V}_{\rm GS}\) = 0.5 V), and \(\delta\) = 1.51 \(\times\) 10\({}^{-7\,\rm A\,\rm(at\,V}_{\rm GS}\) = 0.25 V). **(c)** FinFET fitting with kink effect adjustment on thermal phonon collisions.