# Width Dependent Two-Photon Absorption in Monolayer Black Phosphorus Nanoribbons
## 3 Results and Discussion
In graphene that is disk geometrical, the TPA transitions are only dependent on the scalar of momentum. While the TPA transitions in BP are dependent on the vectorial momentum due. If we simply change the polarization direction of the incident light, the TPA transition resonance frequency can be tuned continuously. As such, TPA resonance enhancement occurs a great deal. Also, it is theoretically predicted the \"quasi-particle\" energy band in limited-layer BP, where the quantum confinement is stronger than in other 2D material. Figure 3: _Cont._  
There is another parameter which will affect the magnitude of TPA coefficient, the relaxation energy \(\tau\) appeared in Equation (17). Relaxation energy-dependent TPA spectra are presented in Figure 4. Since it is reported that the excited carrier recovery time is about \(24\pm 2\) fs in BP suspension at the pump-probe wavelength of 1550 nm and the carrier recombination time is between tens of femtoseconds to hundreds of picoseconds [33]. So, we show two-photon absorption spectra for three different relaxation energies \(\tau=0.01\) eV, \(0.03\) eV and \(0.06\) eV, which correspond to relaxation times \(60\), \(20,10\) fs. It can be drawn that the relaxation energy will not determine the position but the magnitude of the absorption peak. These properties can be interpreted from a mathematical perspective: When TPA resonant transition occurs, \(E_{y}-E_{i}=\hbar\omega\) and \(E_{f}-E_{i}=2\hbar\omega\), then \(W^{(2)}\sim\tau^{-2}\), which means the absorption coefficient is inversely proportional to the square of relaxation energy. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Materials**} & \multirow{2}{*}{**Size**} & **Wavelength** & \multicolumn{2}{c}{\(\alpha_{2}\) **(m/W)**} \\ \cline{3-5}  & & **(nm)** & **Experiment** & **Theory** \\ \hline CdS QDs [28] & 6.4 nm & 790 & \(3.9\times 10^{-13}\) & \\ CdS-Ag QDs [28] & 10.1 nm & 790 & \(1.68\times 10^{-10}\) & \\ CdTe QDs [29] & 20 nm & 1060 & \(10^{-10}\) & \\ Graphene [30] & bilayer & 780 & \((1\pm 0.2)\times 10^{-7}\) & \\  & & 1100 & \((2\pm 0.4)\times 10^{-7}\) & \\  & & 3000 & & \(2\times 10^{-3}\) \\ WS\({}_{2}\)[31] & monolayer & 800 & \((3.7\pm 0.28)\times 10^{-6}\) & \\ MoS\({}_{2}\)[32] & monolayer & 1030 & \((7.62\pm 0.15)\times 10^{-8}\) & \\ BP nanosheets [20] & 2–3 nm & 2000 & \((4.16\pm 0.4)\times 10^{-8}\) & \\ BP QDs [23] & 2–3 nm & 500 & \((7.49\pm 0.23)\times 10^{-14}\) & \\  & & 700 & \((1.68\pm 0.078)\times 10^{-14}\) & \\  & & 900 & \((0.81\pm 0.03)\times 10^{-14}\) & \\ acBPNs & 2 nm & 700 & & \(5\times 10^{-6}\) \\  & & 800 & & \(2.3\times 10^{-6}\) \\  & & 900 & & \(5.7\times 10^{-6}\) \\  & & 1100 & & \(9.7\times 10^{-5}\) \\  & & 1300 & & \(1.8\times 10^{-3}\) \\  & & 2000 & & \(0.16\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Calculated and experimental two-photon absorption coefficients for acBPNs and other related materials. Figure 3: (**a**) Absorption spectra for monolayer acBPNs with three different widths only result from intra conduction band transitions. (**b**)Two-photon absorption spectra of monolayer acBPNs for inter band transitions. (**c**) The peak value of absorption coefficient as a function of the atom numbers in unit cell for acBPNs.