# Width Dependent Two-Photon Absorption in Monolayer Black Phosphorus Nanoribbons
## 2 Theory and Calculations  
The geometry of a monolayer BPN with armchair edges is illustrated in Figure 1, where \(l_{2}=4.19\) A is the period constants for the nanoribbon along the \(y\)-axis. It is convenient to define the width \(W\) of a BPN by one half of the number of atoms \(N\) in the unit cell: \(W=(N-1)\)\(l_{1}/2\). It is can be derived as following: if there is \(s\) (integer or semi-integer) units in the \(x\) direction, the number of the atoms in unit cell \(2N=4s+2\); Since the length of one unit in \(x\) direction is \(l_{1}\) that the ribbon's width \(W=s\times l_{1}=(N-1)\)\(l_{1}/2\). Employing the reduced two-band model, we can write the Hamiltonian of low-energy electrons and holes for monolayer BPNs as following [16; 25],  
\[H=\left( \begin{array}{cc}\alpha+\beta k_{x}^{2}+\gamma k_{y}^{2}&i\chi k_{y }\\ -i\chi k_{y}&\overline{\alpha}-\overline{\beta}k_{x}^{2}-\overline{\gamma}k_{y }^{2}\end{array} \right) \tag{1}\]  
which acts on the eigenstate  
\[\Psi=\left( \begin{array}{c}\phi_{+}\\ \phi_{-}\end{array} \right)=\left( \begin{array}{c}\phi_{A}+\phi_{B^{\prime}}+ \phi_{A^{\prime}}+\phi_{B}\\ \phi_{A}+\phi_{B^{\prime}}-\phi_{A^{\prime}}-\phi_{B}\end{array} \right) \tag{2}\]  
where \(\Psi\) the total wave function of the atomic sites \(A\), \(B\), \(A^{\prime}\) and \(B^{\prime}\). The parameters in the Hamiltonian, which has taken into account the hopping energies between the atoms in BP sheet and its lattice geometry, can be defined as following [16]: \(\alpha(\overline{\alpha})=\mu_{0}+\sigma(\mu_{0}-\sigma)\), \(\beta(\overline{\beta})=\eta_{x}+\gamma_{x}(\eta_{x}-\gamma_{x})\), \(\gamma(\overline{\gamma})=\eta_{y}+\gamma_{y}(\eta_{y}-\gamma_{y})\). Bringing Equations (1) and (2) into the Schrodinger equation \(H\Psi=E\Psi\), we can get two coupled second-order differential equations. For armchair phosphorene nanoribbon (acBPN), considering the symmetries between sublattices owing to the \(D_{2h}\) point group invariance of the BP lattice, these two coupled differential equations must be modified as  
\[(\alpha-\beta\partial_{x}^{2}+\gamma\partial_{y}^{2})\phi_{+}+i\chi k_{y}\phi_ {-}=E\phi_{+} \tag{3}\]  
\[(\overline{\alpha}-\overline{\beta}\partial_{x}^{2}+\overline{\gamma} \partial_{y}^{2})\phi_{-}-i\chi k_{y}\phi_{+}=E\phi_{-} \tag{4}\]  
By solving the two differential equations, the general solution of the component \(\phi_{+}\) or \(\phi_{-}\) has the form  
\[\phi_{+}=t_{1}e^{/x}+t_{2}e^{-/x}+\overline{t}_{1}\overline{e^{f}}x+\overline {t}_{2}e^{-\overline{f}x}, \tag{5}\]  
\[\phi_{-}=p_{1}e^{/x}+p_{2}e^{-/x}+\overline{p}_{1}e^{\overline{f}}x+\overline {p}_{2}e^{-\overline{f}x}. \tag{6}\]  
The coefficients \(f\) and \(\overline{f}\) in the exponentials are defined by  
\[f=\sqrt{\sqrt{\left(\frac{\varepsilon\overline{\beta}+\overline{\varepsilon }\beta}{2\beta\overline{\beta}}\right)^{2}-\frac{\varepsilon\overline{ \varepsilon}-\chi^{2}k_{y}^{2}}{\beta\overline{\beta}}-\left(\frac{ \varepsilon\overline{\beta}+\overline{\varepsilon}\beta}{2\beta\overline{ \beta}}\right)}}, \tag{7}\]  
\[\overline{f}=\sqrt{-\sqrt{\left(\frac{\varepsilon\overline{\beta}+\overline{ \varepsilon}\beta}{2\beta\overline{\beta}}\right)^{2}-\frac{\varepsilon \overline{\varepsilon}-\chi^{2}k_{y}^{2}}{\beta\overline{\beta}}-\left( \frac{\varepsilon\overline{\beta}+\overline{\varepsilon}\beta}{2\beta \overline{\beta}}\right)}}, \tag{8}\]  
where \(\varepsilon=E-\alpha-\gamma k_{y}^{2}\) and \(\overline{\varepsilon}=E-\overline{\alpha}-\overline{\gamma}k_{y}^{2}\). And the constants \(t_{j}\) and \(p_{j}\) (\(j=1,2\)) to be determined are related by  
\[p_{j}=\zeta t_{j},\qquad\overline{p}_{j}=\overline{\zeta}t_{j}, \tag{9}\]  
Figure 1: Crystal structure of a black phosphorus sheet. (**a**) Top view. The sublattices A and B (A\({}^{\prime}\) and B\({}^{\prime}\)) are at the bottom (top) of the monolayer structure. (**b**) Armchair black phosphorus nanoribbon. A dotted rectangular denote the unit cells. (**c**) Side view.