# Width Dependent Two-Photon Absorption in Monolayer Black Phosphorus Nanoribbons
## 2 Theory and Calculations
with \(\zeta\!=\!-(i/xk_{y})(\varepsilon+\beta f^{2})\) and \(\overline{\zeta}\!=\!-(i/xk_{y})(\varepsilon+\beta\overline{f}^{2})\). Since the width of the ribbon is \(W\), i.e., \(0\leq x\leq W\), a series relation below can be derived from Equations (5) and (6) at the boundaries:  
\[\phi_{+}(0)=t_{1}+t_{2}+\tilde{t}_{1}+\tilde{t}_{2},\] \[\phi_{+}(W)=t_{1}e^{fW}+t_{2}e^{-fW}+\tilde{t}_{1}\tilde{e}^{fW}+ \tilde{t}_{2}e^{-\overline{f}W},\] \[(\overline{\alpha}-\overline{\beta}\partial_{x}^{2}+\overline{\gamma} \partial_{y}^{2})\phi_{-}-i\chi k_{y}\phi_{+}=E\phi_{-} \tag{4}\]0 \[(\overline{\alpha}-\overline{\beta}\partial_{x}^{2}+\overline{\gamma} \partial_{y}^{2})\phi_{-}-i\chi k_{y}\phi_{+}=E\phi_{-} \tag{4}\]1  
Similar to the armchair graphene nanoribbons, the wave functions of both the atoms at the edges and their corresponding symmetry partners will vanish for acBPNs. Thus, the boundary conditions can be expressed as  
\[(\overline{\alpha}-\overline{\beta}\partial_{x}^{2}+\overline{\gamma} \partial_{y}^{2})\phi_{-}-i\chi k_{y}\phi_{+}=E\phi_{-} \tag{4}\]2  
Here it is defined \(F(x)=\sum_{i=A,B^{\prime}}\phi_{i}(x)\) and \(G(x)=\sum_{i=B,A^{\prime}}\phi_{i}(x)\). Applying the above boundary conditions to Equation (10) and eliminating the constants \(t_{1}\) and \(t_{2}\), two algebraic equations can be obtained  
\[\left\{ \begin{array}{l}(\tilde{t}_{1}+\tilde{t}_{2})(\zeta-\overline{\zeta })=0,\\ (\tilde{t}_{1}e^{\overline{f}W}+\tilde{t}_{2}e^{-\overline{f}W})(\zeta- \overline{\zeta})=0.\end{array} \right. \tag{12}\]  
Noted that \(\zeta\
eq\overline{\zeta}\), so we can conclude \(e^{2\overline{f}W}=1\) and consequently \(2\overline{f}W=2 in\pi(n=1,2,3\ldots)\). And then the dispersion relation for acBPNs can be derived as  
\[E_{n}=\mu_{0}+\eta_{x}k_{n}^{2}+\eta_{y}k_{y}^{2}\pm\sqrt{\left(\sigma+\gamma _{y}k_{y}^{2}+\gamma_{x}k_{n}^{2}\right)^{2}+\chi^{2}k_{y}^{2}} \tag{13}\]  
where \(k_{n}\)=\(n\pi/W\) which results from the assumption \(\overline{f}=ik_{n}\) is a pure imaginary number. It is found that when \(W\rightarrow\infty\), the above dispersion relation is similar to that of bulk BP for \(k_{n}\leftrightarrow k_{x}\). The wave functions of an acBPN are the combinations of all possible modes  
\[\Psi=\sum_{n}A_{n}\bigg{(} \begin{array}{c}1\\ \zeta_{n}\end{array} \bigg{)}e^{ik_{y}y}\sin(\frac{n\pi}{W}x) \tag{14}\]  
where \(A_{n}\) is the normalized constant. Based on the energy spectrum expressed by Equation (13), we calculate the width dependent band gap at the \(\Gamma\) point by taking the difference between conduction band and valence band where \(n=1\),  
\[E_{g}=2(\sigma+\frac{4\gamma_{x}\pi^{2}}{(N-1)^{2}{l_{1}}^{2}}). \tag{15}\]  
It is noted that the two-photon absorption coefficient is proportional to the two-photon transition probability rate, which can be furnished by the second order time-dependent perturbation theory as following [26]  
\[W^{(2)}=\frac{2\pi}{\hbar}\sum_{i,f}\big{|}M_{f,i}\big{|}^{2}\delta\Big{(}E_{ f}-E_{i}-2\hbar\omega\Big{)}, \tag{16}\]  
\[M_{f,i}=\sum_{v}\frac{H_{f,v}^{\rm int}H_{v,i}^{\rm int}}{E_{v}-E_{i}-\hbar \omega-i\tau} \tag{17}\]  
where \(E_{i}\), \(E_{f}\) and \(E_{v}\) represent the energy levels of the initial, final and intermediate states of a carrier, respectively. The electron-photon interaction Hamiltonian \(H^{\rm int}=(e/mc)\)**A**-**p** with \(m\) the effective mass of electron in acBPN, **A** = \(A\)**e** is the vector potential of the light wave with the amplitude \(A\) and the polarization vector **e**, **p** is the electron momentum operator. Substituting the energy levels and wave functions for acBPNs we have obtained, the expression of matrix elements for one-photon absorption transition with the quantum numbers \(n_{i}\) and \(n_{v}\) can be written as  
\[