# Exact Relationship between Black Phosphorus Thickness and Behaviors of Field-Effect Transistors
## 3 Results
(**f**) is an enlarged graph of the curve in (**e**) at small thickness. Figure 2d shows that the kink point of the slope of the fitting curve (the maximum value of the first order differential) appears near \(l_{o}\). The value of \(l_{o}\) is 21.7 nm. Above \(l_{o}\), the slope of the curve starts to decrease slowly, but gradually becomes faster. When \(l\) reaches a certain value \(l_{right}\), the speed of the slope decreases reaches the fastest. Thereafter, the speed of decrease gradually becomes slower. If \(l\) continues to increase, when it approaches 50 nm, the speed of the slope decrease of the curve reaches almost 0, and then the fitting curve gradually flattens, which means that the degree of influence of thickness on device characteristics can be ignored. Figure 2e again proves the above conclusion that when the \(l\) is greater than 21.7 nm, the minimum value of the slope of the dlog\(l_{ds}\)/d-\(l\) occurs at the BP thickness of 26.7 nm (negative, but its absolute value is the maximum). Namely, \(l_{right}\) is 26.7 nm. That is, when the thickness is greater than 26.7 nm, the effect of thickness on device characteristics gradually starts to become less pronounced. Below \(l_{o}\), although we did not measure the transport characteristics of FETs with BP thickness up to 6 nm, it seems possible to predict from Figure 2d that when the channel material thickness is less than \(l_{left}\) (kink point), the effect of \(l\) on \(l_{ds}\) seems to become progressively less pronounced as the \(l\) decreases. The region where the degree of the effect of thickness on \(l_{ds}\) is most pronounced should be around \(l_{left}\) (16.3 nm) because the change of slope of the curve is the fastest near \(l_{left}\), as shown in Figure 2e. From 6 nm to 16.3 nm, the speed of the slope increase of log\(l_{ds}\)-\(l\) is gradually increasing with the increase of the thickness, reaching the maximum at 16.3 nm, after which it gradually becomes smaller and is 0 at 21.7 nm. The extent of the effect of BP thickness on device performance at different thickness ranges can be obtained from Figure 2e. The speed of slope varies from the maximum (at \(l_{left}\)) to the minimum (at \(l_{right}\)), but the thickness spans only 10.4 nm. From 6 nm to 16.3 nm, the speed varies from a non-zero value to a maximum value, although the span of \(l\) is only 10.3 nm, but the span of speed is only half of the interval of 16.3 nm to 26.7 nm. From \begin{table}
\begin{tabular}{c c c c c} \hline
**Thickness (nm)** & _A/arb. Unit_ & _B/arb. Unit_ & _C/nm_ & _I\({}_{o}\)/nm_ \\ \hline
6 & 0.77958 & \(-\)3.95910 & 4.35542 & 21.83318 \\
20 & 1.08207 & \(-\)3.99000 & 4.39502 & 21.72937 \\
30 & 1.25882 & \(-\)3.99619 & 4.39977 & 21.71348 \\
40 & 1.38407 & \(-\)3.98594 & 4.39986 & 21.72556 \\
50 & 1.48156 & \(-\)3.98977 & 4.40987 & 21.69807 \\ \hline \end{tabular}
\end{table} 
Table 1: Fitting parameters. Figure 3: (**a**) The change of device total resistance with material thickness, and its fitting curve of log\(R\) vs. \(l\); (**b–d**) are the change of fitting coefficients and the carrier mobility with the material thickness. 26.7 nm to 50 nm, speed varies from the minimum to 0, where \(l\) spans 23.3 nm and speed spans half of the interval 16.3 nm to 26.7 nm. Alternatively, when the thickness of the BP material is between 16.3 and 26.7 nm, the BP thickness should have a significant effect on device performance, even greater than the effect of thickness on device characteristics when the thickness is between 20 nm and 6 nm (or 26.7-50 nm), although the thickness of the material also has a significant effect on device characteristics when the thickness of the material is thin (20 nm to 6 nm). Of course, Figure 2e shows more clearly that when the thickness reaches or exceeds 50 nm, the \(l\) has almost no effect on \(l_{ds}\). In addition, when the thickness is between 6 nm and 7 nm, it seems that the degree of influence of thin layer thickness on the output characteristics of the device is changed more significantly, shown in Figure 2f.