# Exact Relationship between Black Phosphorus Thickness and Behaviors of Field-Effect Transistors
## 3 Results
According to previous literature reports [30; 31], we believe that other factors may be involved in influencing the performance of device with thin BP thickness. By linear fitting Figure 2a,b, from their slopes the total resistance of devices with BP thickness of about 50 nm, 40 nm, and 30 nm are extracted as about 1.52 k\(\Omega\), 2.18 k\(\Omega\), and 5.4 k\(\Omega\), respectively. It seems to decrease linearly with the increase of thickness. However, the resistance of about 20 nm and 6 nm BP devices are nonlinear increased to 420.2 k\(\Omega\) and 12,733.9 k\(\Omega\). That is, when the device material thickness changes from \(\sim\)30 nm to \(\sim\)20 nm, the resistance of the device increases by a factor of nearly 78; while the material thickness changes from \(\sim\)20 nm to \(\sim\)6 nm thickness, its resistance increases by a factor of 30. This corroborates the results in Figure 2 that the thickness of the BP material is more influential on the device performance when the thickness range is between \(\sim\)16.3 nm and \(\sim\)26.7 nm. The specific relationship between the resistance of devices and the thickness of channel materials is shown in Figure 3a. This super nonlinearity may be due to the fact that when the material thickness is reduced to a certain value, the band gap of the material increases, resulting in a sudden increase in resistance. In addition, using 1 stOpt we fitted the log\(R\) vs. BP thickness \(l\) in Figure 3a, and found that the function of the relationship between fitted log\(R\) and \(l\) is also the above Formula (1), and the fitting parameters \(A\), \(B\), \(C\), and \(l_{o}\) are 3.22190, 3.987341, 4.34556, and 21.79109, respectively. The coefficients are almost identical to those in Table 1, except for the \(A\) coefficient in Equation (1). However, after a closer look at Equation (1), it is found that the coefficient \(A\) is a constant determined independent of thickness. Therefore, both log\(I_{ds}\) and log\(R\) vs. \(l\) should be mainly determined by the second term on the right side of Equation (1), so the Equation (1) can also describe the relationship between log\(R\) and \(l\). From this experiment, we can predict that when the band gap starts to change with the thickness, the corresponding value of thickness should be \(l_{o}\), \(\sim\)21.7 nm. Namely, when the BP thickness is less than 21.7 nm, the energy band of BP material will increase significantly with the thickness thinning, and when the BP thickness is greater than it, the modulation relationship between the energy band of BP material and the thickness will gradually weaken. This conclusion is similar to that of Figure 2d. The exact relationship of the four fitted coefficients in Table 1 with thickness is shown in Figure 3b and c, it seems to confirm that only the \(A\) coefficient is determined with thickness, while the other coefficients vary little or not at large material thicknesses (for \(\sim\)20 nm). Figure 4 shows the transfer characteristic (\(I_{ds}\)-\(V_{g}\)) curves of BP FET devices with thickness of about 50 nm, 40 nm, 30 nm, 20 nm, and 6 nm, respectively. The \(V_{ds}\) is 100 mV. It is obvious that the larger the thickness, the less the applied electric field can regulate the device. For the device with BP of \(\sim\)50 nm, the current only changes from a maximum value of 75 mA to a minimum value of 60 mA at \(-\)40 V to 40 V electric control voltage, a change of 1.25 times. The same is true for a 40 nm thick device. This weak dependence on the gate voltage (\(V_{gs}\)) due to strong Thomas-Fermi charge screening in thick BP flakes. In addition, if the FET device has a thick BP material (greater than 20 nm), no \(l_{off}\) is seen in the measured voltage range, namely, the device material exhibits a very clear p-type because no minimum value of current (flatband condition voltage, \(V_{FB}\)) is seen [32]. In the inset of Figure 4b, we found that the dominant carrier type in BP will change from naturally p-type to n-doping in FET with thinner BP flakes at \(V_{FB}\).