# High Performance GAA SNWT with a Triangular Cross Section: Simulation and Experiments
## 5 Mobility Model  
In this section, we plan to characterize the carrier mobility in the fabricated GAA SNWTs by Bias-Temperature experiments so that surface roughness mobility can be obtained at a low temperature, which can verify the benefit to mobility resulting from the stable and smooth triangular cross-sectional shape.  
The Y-function method can be used to extract the mobility of the fabricated triangular cross-sectional GAA SNWTs [16]. The Y function can be calculated as follows:  
\[\mathrm{Y}\equiv\frac{\mathrm{I_{ds}}}{\sqrt{\mathrm{g_{m}}}}=\sqrt{\mathrm{ k_{0}V_{ds}}}(\mathrm{V_{gs}}-\mathrm{V_{th}}), \tag{2}\]  
Here, k\({}_{0}\) is given by  
\[\mathrm{k_{0}}=\frac{\mathrm{W_{eff}u}\mathrm{C_{ox}}}{\mathrm{L_{eff}}}, \tag{3}\]  
Firstly, the driving current \(\mathrm{I_{ds}}\) and the transconductance \(\mathrm{g_{m}}\) of the GAA SNWTs in the linear region were measured to obtain the Y function, and then the intercept of this straight line at the x-axis was the threshold voltage \(\mathrm{V_{th}}\) and the mobility of the device was extracted by the slope \(\mathrm{k_{0}}\). This method can be adopted to extract the trends of mobility varying with temperature, as shown in Figure 11a.  
It is noticed that the mobility is mainly determined by phonon scattering at room temperature, and the surface roughness scattering dominantly contributed to mobility when the temperature dropped below 70 K. Additionally, the Coulomb scattering had little effect on the mobility over the entire  
Figure 11: (**a**) The trend of mobility with temperature in GAA SNWTs; (**b**) comparison of current simulation results with experimental results of GAA SNWT at T = 4.3 K and 300 K, respectively; (**c**) effective electron mobility as a function of temperature for triangular GAA SNWT achieved by TMAH wet etching and quadratic SNWT with/without H\({}_{2}\) annealing.  
Figure 10: The trends of SS, DIBL, and \(\mathrm{V_{th}}\) with \(\mathrm{L_{g}}\) for triangular cross-sectional GAA SNWTs.  
temperature range because of the lower channel doping of the GAA SNWTs. Here, three kinds of mobility can be obtained by the following equations:  
\[\frac{1}{\mu}=\frac{1}{\mu_{\rm sr}}+\frac{1}{\mu_{\rm ph}}+\frac{1}{\mu_{\rm Coulomb }}, \tag{4}\]  
\[\mu_{\rm ph}=\rm AT^{-\alpha}, \tag{5}\]  
\[\mu_{\rm Coulomb}=\rm BT^{\alpha}, \tag{6}\]  
Here, \(\mu_{\rm sr}\), \(\mu_{\rm ph}\), and \(\mu_{\rm Coulomb}\) denote the surface roughness mobility, the phonon scattering mobility, and the Coulomb scattering mobility, respectively. A and B are fitting coefficients, and \(\alpha\) is the temperature coefficient. According to the scattering theory [17], \(\alpha\) is set to 1.5. From Figure 11a, it can be seen that the modelling results can fit the experimental data quite well. The three fitting coefficients are \(\mu_{\rm sr}=629\) cm\({}^{2}\)/Vs, A = \(2.38\times 10^{6}\) cm\({}^{2}\)K\({}^{1.5}\)/Vs, and B = \(2.42\times 10^{6}\) cm\({}^{2}\)/K\({}^{1.5}\)Vs. As shown in Figure 11b, the device current in the linear region at two different temperatures was calculated using the mobility obtained by this model, and the modelling results are also in good agreement with the experimental results. It is worth mentioning that the mobility in our work is higher than that in GAA SNWTs with or without H\({}_{2}\) annealing [18] in Figure 11c, indicating that the inverted triangle cross section achieved by TMAH wet etching can effectively reduce the influence of surface roughness scattering.