# Exact Relationship between Black Phosphorus Thickness and Behaviors of Field-Effect Transistors
## 3 Results  
Figure 2a shows the output characteristics (\(I_{ds}\)-\(V_{ds}\)) of about 50 nm, 40 nm, 30 nm, and 20 nm BP FET devices. The source drain voltage scanning range is from \(-\)100 mV to 100 mV, and the gate voltage (\(V_{g}\)) is 0 V. Figure 2b shows the \(I_{ds}\)-\(V_{ds}\) characteristics of about 6 nm BP FET, but the \(V_{ds}\) scanning range is from \(-\)50 mV to 50 mV. This range is mainly due to the thin material and the need to prevent the device from being damaged. The good linear relationship between \(I_{ds}\) and \(V_{ds}\) (all fit lines are very good straight lines) proves that a good contact is formed between the BP and the electrode. It can be clearly seen that at a fixed \(V_{ds}\), the value of \(I_{ds}\) increases gradually with the increase of the material thickness. Namely, the greater the thickness of the material, the greater the \(I_{ds}\). However, the \(I_{ds}\) changes only a few times for a thickness changing from 20 nm to 50 nm, but from 6 nm to 20 nm, the \(I_{ds}\) does change by several orders of magnitude. Figure 2c shows the exact change rule of \(I_{ds}\) with BP material thickness under different \(V_{ds}\). Interestingly, the laws of several curves are completely consistent, so it is certain that there is a certain relationship between the \(I_{ds}\) of the BP FET device and the BP thickness. Using 1 stOpt (First Optimization, 7D-Soft High Technology Inc.), a simulation software that can easily fit the experimental data, we fitted the experimental data (log\(I_{ds}\)) shown in Figure 2c with Boltzmann model, and the fitted curves can be obtained. The change rule of log\(I_{ds}\) and BP thickness \(l\) can be described by the following semi empirical formula given by software,  
\[\log(I_{ds})=A+\frac{B}{1+e^{\frac{l-l_{ds}}{\mathrm{C}}}} \tag{1}\]  
Figure 1: (**a**) Schematic diagram of device preparation process, (**b**) device schematic diagram, (**c**) the real device diagram, the BP thickness on the right side is about 20 nm, (**d**) its thickness measurement of one material by atomic force microscope. where \(A\), \(B\), \(I_{o}\), and \(C\) are fitting coefficients, their values have very weak changes with BP thickness, as shown in Table 1. The fitting results show that the experimental data of log\(I_{ds}\)-\(l\) in Figure 2c fitted by Formula (1) and the coefficients in Table 1 give an \(R_{adj}\)\({}^{2}\) of over 0.998, which reflects a very perfect fit (usually \(R_{adj}\)\({}^{2}\) = 1 indicates complete agreement and \(R_{adj}\)\({}^{2}\) = 0 indicates complete failure) [29]. That is, Equation (1) can completely describe log\(I_{ds}\)-\(l\). As the material thickness increases, \(A\) and \(C\) slightly increase, while \(B\) and \(I_{o}\) slightly decrease. However, only the coefficient \(A\) exhibits a pattern of increasing with the thickness of the material, for \(B\), \(C\), and \(I_{o}\), the influence of thickness can be almost ignored when the BP thickness is greater than about 20 nm. For example, when BP thickness from 20 nm to 50 nm, \(B\) simply fluctuates between \(-3.990\) and \(-3.989\); \(I_{o}\) also fluctuates within a range (shown in Table 1 and Figure 3b,c). Next, we analyze the impact of \(l\) on \(I_{ds}\) in detail. Figure 2: (**a**) Output characteristics (\(I_{ds}\)-\(V_{ds}\)) of BP FET devices with thickness of 50 nm, 40 nm, 30 nm, and 20 nm, and (**b**) of 6 nm at \(V_{g}\) = 0 V, and their fitting curves; (**c**) the dependence of log\(I_{ds}\) on thickness at different \(V_{ds}\), and their fitting curves. In (**a–c**) the scattered geometry represents the experimental data points and the solid line is the fitted curves. (**d**) First order differential curve of log\(I_{ds}\) vs. \(l\), all curves are completely coincident; (**e**) the second differential curve of log\(I_{ds}\) vs. \(l\) of 10 mV. The left side of the shaded part corresponds to \(I_{left}\) and the right side corresponds to \(I_{right}\), which are approximately 16.3 nm and 26.7 nm, respectively.