# Width Dependent Two-Photon Absorption in Monolayer Black Phosphorus Nanoribbons
## 1 Introduction  
Black phosphorus (BP), which consists of phosphorus atoms arranged to puckered layers with a honeycomb structure, is one of new two-dimensional (2D) materials [1; 2; 3; 4; 5]. BP has a layer-dependent direct bandgap varying from 0.3 eV for bulk BP to 2.0 eV for monolayer BP (also known as phosphorene), which can bridge the energy gap between zero-gap graphene and large-gap transition metal dichalcogenides (TMDs) [6; 7; 8]. So, it is a promising candidate for near- and mid-infrared opto-electronical applications [9; 10; 11]. The instability of the surface due to the chemical degradation in ambient conditions used to be a main problem to its prospective applications. However, the scientists have successfully solved the degradation problem in BP and mitigated the instability issue using many innovative approaches, such as surface functionalization or coatings, capping, and encapsulation [5; 12]. Similar to graphene nanoribbons [13; 14], it is predicted that the electronic structure and the transport properties of confined black phosphorous nanoribbons (BPNs) will be much more abundant compared with the infinite sample, due to its crystal orientation and width dependent electronic states. Accurate knowledge of the nonlinear optical characteristics of BPNs is very important for design and optimizing its optical applications  
Recent theoretical studies of BPNs mainly focus on its electronic energy states on the base of the tight-binding approach [15; 16] or first-principles [10; 17]. The boundary conditions for BPNs have been proposed by [PERSON] et al. in the continuum approach, and they have investigated the energy spectrum of monolayer BPNs using a low-energy expansion of tight-binding theory for energy close to the Fermi energy level [16]. [PERSON] et al. have examined the edge-induced states located in band gap infew-layer phosphorene employing density functional theory [18]. The theoretical treatment of nonlinear optical refraction and absorption in single layer BP has been presented by [PERSON] et al. And they explored the polarization dependent properties [19]. As far as the experimental studies are concerned, although the BP sample is instable in ambient conditions, there still have been a series of advances in experimental investigations into nonlinear optical (NLO) characters in limited layered BP or its different nanoscale forms, which are dispersed in organic solvents that shows no nonlinear properties under laser excitation. It is reported that the few-layer BP nanosheets possess giant saturable absorption (SA) response and the TPA coefficient is measured to be \((4.16\pm 0.4)\times 10^{3}\) cm/GW by the OA Z-Scan and pump-probe spectra techniques [20]. Also, the SA measurements have been conducted by wide-band Z-scan technique. Three different sets of organic solvents (isopropyl alcohol, N-Methylpyrrolidone solvents and ethyl alcohol) were chosen to disperse BP flakes [9]. [PERSON] et al. systematically investigated the excited carrier dynamics and NLO saturable absorption of liquid-exfoliated BP nanosheets over a broad wavelength range from 800 to 2100 nm [21]. The achievement of size-dependent SA and Kerr nonlinearity of BP nanosheets employing Z-scan and pump-probe technique was also reported by [PERSON] and co-workers with pulse duration of about 635 fs centered at wavelength of 1562 nm [22]. In addition, [PERSON]'s group has explored the broadband NLO properties of ultrasmall black phosphorus quantum dots (BPQDs) using OA Z-scan technique and the nonlinear absorption coefficients are found to be \((7.49\pm 0.23)\times 10^{-3}\), \((1.68\pm 0.078)\times 10^{-3}\) and \((0.81\pm 0.03)\times 10^{-3}\) cm/GW for 500, 700, and 900 nm [23]. Nevertheless, all of the experimental data lack the verification from the view of NLO theories. Therefore, in this work we theoretically investigate the two-photon absorption properties of BPNs based on the electronic energy states obtained from the reduced two-band model within the long-wavelength BP Hamiltonian.