# Exact Relationship between Black Phosphorus Thickness and Behaviors of Field-Effect Transistors
## 3 Results
The \(V_{FB}\) are about \(-\)4.5 V and 2.2 V for FET with the BP thicknesses of 6 nm and 20 nm, respectively. It seems that for  
natural p-type BP materials, the greater the thickness, the greater the \(V_{FB}\), i.e., the greater the \(V_{g}\) required to shift the polar type of the material. Furthermore, the \(I_{off}\) was observed in devices with 6 nm and 20 nm thickness material, the \(I_{on}/I_{off}\) is \(\sim\)10\({}^{3}\). At different gate voltages (\(V_{g}\)) the \(I_{ds}\)-\(V_{ds}\) curves of BP FET devices with about 50 nm, 40 nm, 30 nm 20 nm, and 6 nm BP are shown in Figure 5, where the source drain voltage (\(V_{ds}\)) scanning range is from \(-\)100 mV to 100 mV. It can be seen that the \(I_{ds}\) of the BP-FET device is directly related to the thickness of the material. The thicker the material is, the greater the source drain current under the same \(V_{g}\). However, the thicker the material is, the weaker the modulation effect of the electric field will be, which can be seen from Figure 5a, at different \(V_{g}\), several curves basically overlap. In addition, when the thickness of the material is reduced to about 20 nm, under the \(V_{g}\) modulation of 40 V, the source leakage current has been reduced by 5000 times compared to BP-FET with the 30 nm material. However, the modulation effect of the electric field is enhanced, and the on/off current ratio (the ratio of the on-state current to the off-state current) of the device reaches \(\sim\)10\({}^{3}\). In addition, from the transfer characteristic curve, it can be seen that when the negative gate voltage is greater, its conductivity is better, so majority carriers are holes. When the BP thickness is about 20 nm and 6 nm, the transfer characteristic curve of the device indicates that the type of carriers involved in conducting in the BP crystal has shown a certain bipolar type (as shown in the Figure 4b), which is similar to the BP FET of the thin layer materials (below 10 nm) [8]. From the inset of Figure 4b, the thinner the material is, the smaller the modulation voltage required for the carrier polar transition. According to the transfer characteristic curve of FET, the carrier mobility can be calculated according to the following formula [31]. \[\mu_{FE}=\frac{\mathrm{d}I_{ds}}{\mathrm{d}V_{g}}\frac{L}{\mathrm{W}C_{l}V_{ds}} \tag{2}\]  
where \(L/W\) is the ratio of length and width of the device channel. In our prepared BP-FET, the ratio is 1.2, \(C_{i}=\varepsilon_{o}\varepsilon_{i}/d\) is the capacitance per unit area, \(\varepsilon_{o}\) = 8.854 \(\times\) 10\({}^{-12}\) fm\({}^{-1}\) is the dielectric constant in vacuum, \(\varepsilon_{i}\) = 3.9 is the relative dielectric constant of silicon dioxide,  
Figure 4: (**a**,**b**) are the transfer characteristic curves of BP-FET devices with thickness of 50 nm, 40 nm, 30 nm, 20 nm, and 6 nm, respectively. \(V_{ds}\) is set to 100 mV. Except for FET with 50 nm BP, the \(I_{ds}\) of FET with other thicknesses have been amplified by 1.5 times, 3 times, 50 times, and 200 times, respectively, for FETs with thickness of 40 nm, 30 nm, 20 nm, and 6 nm. The inset of (**b**) is a magnified view of the \(V_{g}\) between \(-\)20 V and 40 V.  
and \(d\) is the thickness of the insulating layer, that is, the thickness of silicon dioxide. It is 285 nm. We can calculate that the values of the device mobility with BP thickness of about 50 nm, 40 nm, 30 nm, 20 nm, and 6 nm are about 336.4 cm\({}^{2}\)/Vs, 252.5 cm\({}^{2}\)/Vs, 214.4 cm\({}^{2}\)/Vs, 187.5 cm\({}^{2}\)/Vs, and 52.5 cm\({}^{2}\)/Vs. The exact relationship between mobility and BP thickness is shown in Figure 3d. The carrier mobility increases with the increase of material thickness, and from 20 nm to 50 nm, it can be approximately regarded as proportional to the thickness. However, the most significant increase in device mobility was observed when the BP thickness is increased from 6 nm to 20 nm.