# Width Dependent Two-Photon Absorption in Monolayer Black Phosphorus Nanoribbons
## 3 Results and Discussion  
We now turn to the results of our numerical studies following the expressions derived above. Hereinafter, the calculations and discussions for monolayer acBPNs are using the following parameters [27]: \(\mu_{0}=-0.42\) eV, \(\eta_{x}=0.58\) eV\(\cdot\)A\({}^{2}\), \(\eta_{y}=1.01\) eV\(\cdot\)A\({}^{2}\), \(\sigma=0.76\) eV, \(\chi=5.25\) eV\(\cdot\)A, \(\gamma_{x}=3.93\) eV\(\cdot\)A\({}^{2}\), \(\gamma_{y}=3.79\) eV\(\cdot\)A\({}^{2}\), \(m=1.03\)\(m_{0}\), and \(d=1.84\) A. To demonstrate the modulability of the bandgap for acBPNs, in Figure 2 we show the bandgap as a function of nanoribbon's width, i.e., one half of the number of atoms \(N\) in the unit cell. It tells us the bandgap of acBPN can be tuned ideally from 8.5 eV to 1.6 eV by change the width of the nanoribbon. The wider the ribbon, the smaller the bandgap. Furthermore, the bandgap decreases sharply with the increase of ribbon's width for acBPNs with small width (\(N<10\)), while it decreases gentlely for larger values of \(N\). Actually, it can be derived from Equation (15) that \(E_{g}\to 2\sigma\) for \(N\) trends to infinite. This happens to be the situation of two-dimensional BP. For small values of \(N\), the bandgap \(E_{g}\) is in proportion to \(1/(N-1)^{2}\), i.e., \(1/W^{2}\). We also can analyze several characters of energy band structure from Equation (13). Firstly, the dispersion is discrete with the quantum number \(n\) in \(k_{x}\) direction, but parabolic in \(k_{y}\) direction; Secondly, the energy levels are of bilateral symmetry by \(E_{k_{y},n}=E_{-k_{y},n}\), while \(E_{k_{y},n}^{e}\
eq-E_{k_{y},n}^{h}\) for conduction band and valence band. Figure 2: The band gap energies obtained as a function of the atom numbers in unit cell for armchair black phosphorus nanoribbons. There are three transition modes for TPA depending on the initial, intermediate and final states: intra conduction band transition (all of the states are in conduction band), intra valence band transition and inter band transition (transit from valence band to conduction and any one can be the intermediate state). Adopting the TPA model presented in the previous section, the TPA spectra result from intra conduction band transitions and inter band transitions for acBPNs with three different width are respectively shown in Figure 3a,b. We can see that the magnitude of two-photon absorption coefficient \(\alpha_{2}\) resulted from the transitions intra conduction band is about 10\({}^{-1}\) m/W for acBPNs with width \(N\) = 13, 14, 15. The main absorption peak locates at 0.5-0.8 eV. While associated with inter band transitions, the \(\alpha_{2}\) is in the magnitude of 10\({}^{-5}\) m/W, which is only 10\({}^{-4}\) of that originated from the intra conduction band transitions. Thus, the two-photon absorption is mainly contributed from the intra band transitions. Furthermore, it is found that \(\alpha_{2}\) increases and absorption peaks shift to lower energy direction with the increase of ribbon's width. Figure 3c illustrates the calculated peak value of TPA coefficient for \(N\) varied from 5 to 20 (W\(\sim\)0.6-3 nm) involving the intra conduction band transitions. This width dependence can be explained by the quantum size effect. Actually, the wider the nanoribbon, the larger the density of energy states. Thus, more two-photon transitions are involved in. We summarize the measured TPA coefficient of conventional semiconductor QDs, 2D graphene, transition metal dichalcogenides (TMDs) and BP by other group as well as our theoretical calculated values in Table 1 for comparison. We can find that, similar to other 2D materials, acBPNs have large TPA coefficient (\(\sim\)10\({}^{-6}\) m/W) in visible region, which is nearly four or more orders higher than that of conventional semiconductor QDs. According to the theoretical calculation, there is a giant TPA coefficient for acBPNs in infrared band, which can reach up to 10\({}^{-1}\) m/W. The BP nanoribbons possess so large TPA coefficient due to its in-plane anisotropy to large extent [5].