We investigate theoretically the impact of Rashba spin-orbit coupling (RSOC) effect to two-photon absorption (TPA) and its dependence on the polarization direction of the incident light in monolayer black phosphorus (BP) starting from an anisotropic two band k·p model. It is found that the TPA is enhanced several times by RSOC effect which is tuned by the external electric field. And the TPA response shows highly anisotropic, changing periodically with the polarization direction of incident linearly polarized light as the function of cos4θ approximatively. The TPA coefficient reaches its maximum when the polarization direction is aligned along the armchair direction (x-direction), while falls into its minimum along the zigzag direction (y-direction).
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Black phosphorus (BP), a van der Waals-bonded two dimensional (2D) material with puckered honeycomb structure (see Fig. 1), is the most thermodynamically stable allotrope of phosphorus [1–3]. It is a layered semiconductor with direct band-gap tuned by the number of layers from 0.3 eV (bulk) to 2 eV (monolayer), which happens to bridge the blank between graphene with zero band gap and transition metal dichalcogenides (TMDCs) with larger band gap [4,5]. Recently, BP has received intensive attention due to its tunable band gap in visible-infrared region [6,7], its high carrier mobility [8–10], its intrinsic layered nature, and its highly anisotropic structures, which give rise to the highly anisotropic physical properties, such as anisotropic excitons , anisotropic optical response [12,13] and anisotropic plasmons etc. . These make BP as a promising candidate for opto-electronics, which can be applied to photodetectors with high sensitivity [15–17], field-effect transistors [18,19], optical switches, optical sensors  and so on. In many applications, laser excitation through two-photon absorption (TPA), which is one of the third-order nonlinear optical effects plays an important role. Compared with one-photon absorption, longer excitation wavelengths may be utilized in TPA-based applications to provide deeper penetration depths in absorptive media and higher spatial resolution .
Up to now, TPA and other nonlinear optical properties in BP and its related low dimensional materials have been studied intensively, in the context of both theory and experiment. Margulis et al. theoretically simulated the optical kerr effect and two-photon absorption in monolayer black phosphorus starting from the calculation of the third-order optical susceptibility χ(3) . In our previous work, we displayed the TPA spectrum in monolayer BP nanoribbons with armchair edge . The TPA and saturable absorption (SA) coefficients in few-layer BP nanosheets or BP quantum dots have been measured by the Z-Scan and pump-probe spectra techniques [24–26]. As referring to the plasmon absorption, electrostatically tunable localized surface plasmons can generate a strong local electromagnetic field that enhances the nonlinear optical response. X. Chen and L. Wang's group has designed a large number of novel opto-electronic devices based on the terahertz technology and Plasmon excitation in 2D materials, such as phototransistor for sensitive broadband detection , highly sensitive and wide-band tunable field effect transistors , high-gain long-wavelength photodetector , high-Q Fano-resonnator for sensing  and so on.
It is known that in crystals which lack an inversion centre, electronic energy bands are split by spin orbit coupling (SOC). In the presence of the SOC, the electric field induced broken inversion symmetry leads to the Rashba spin orbit coupling (RSOC) effect . As one of the relativistic effects, it is usually small and often neglected in solids, but in 2D materials it may play an important role, especially if the material is placed in an external electric field or on a polar substrate . It can be predicted that some optical effects will be strengthened due to the fact that the splitting of energy levels will enhance the density of states. Anisotropic band structure also can cause a highly anisotropic Rashba splitting in monolayer BP .
In these prior studies, the studies on TPA and RSOC in BP have been conducted separately. However, the hybrid impact of RSOC effect on TPA in monolayer BP by tuning the external electric field as well as polarization direction of incident light has not yet been reported. In the present paper, we theoretically study the anisotropic behaviors of the TPA properties associated with the RSOC in monolayer BP by a linearly polarized light. We obtain the energy band structure of monolayer BP on the basement of the two band k·p model in the low-energy theory when the RSOC is present. Further we numerically calculate the TPA coefficient and display the TPA spectra as functions of both the external electric field and the polarization direction of the incident light. The results indicate that the RSOC effect leads to the spin splitting of energy bands, creating energy bands with spin-up and spin-down, respectively. Both the energy bands and the RSOC effect are anisotropic. It is demonstrated a linear like relation between splitting energy and the wave vector k. The TPA coefficient is enhanced by the effect of RSOC in varying extent, which is dependent on the strength of the applied electric field. The TPA also shows anisotropic, changing periodically with the polarization direction of incident linearly polarized light as the function of cos4θ approximatively with the period π. The maximum of TPA coefficient corresponds to the incident polarization aligned along x-axis while the minimum of TPA coefficient occurs aligned along y-axis. These results provide guidance on potential applications of optoelectronic nanodevices based on the anisotropic BP.
2. Theory and calculation
When we consider the effect of RSOC induced by the applied electric field, the low-energy dispersion of the monolayer BP can be well described by an anisotropic two band k·p model 
When the monolayer BP sheet is irradiated by a linearly polarized light with frequency ω, the Perierls substitution, i.e. $\hbar$ki→Pi=$\hbar$ki+eAi should be used here. Then the electron- photon interaction Hamiltonian Hint can be separated by Hint = H′-H, where H′ is the total Hamiltonian including the light field. If the angle between the polarization direction and the armchair edge (x-direction) is θ, we can rewrite the electron-photon interaction Hamiltonian as
3. Result and discussion
In pristine monolayer BP, parabolic energy bands reflect anisotropy and the massive 2DEG with a direct bandgap about 1.52 eV at the Γ point. The presence of applied electric field leads to the splitting of the energy levels with the effect of RSOC. Then distinct electronic and optical properties emerge rather than being observed in hexagonal symmetric graphene. We start by demonstrating the energy dispersions of monolayer BP with and without external electric field respectively in Figs. 2(a)–2(c). The different slopes along kx and ky directions imply both the anisotropic energy dispersion relations and anisotropic effective masses of carriers in monolayer BP without external electric field [see Fig. 2(a)]. While for the situation that the external electric field is applied, it is shown in Fig. 2(b) that the conduction band and valence band split to two bands separately. In order to get a better view, we plot the energy dispersions along the wave vector kx at ky = 0 in Fig. 2(c). It is noted that the Rashba term results in the spin splitting of energy bands, creating an energy band of spin-up and spin-down, respectively. The energy dispersion along the wave vector ky at kx = 0 shows similar phenomena, which is not shown here. We can see that the spin splitting energy between the spin-up and spin-down bands is in the magnitude of µeV, which is also dependent on the wave vector k. It is obvious in Fig. 2(c) that the larger the kx, the larger the spin splitting energy. Figure 2(d) demonstrates clearly the linear like relationship between splitting energy and the wave vector k. The different slopes of kx and ky for electron and hole indicate the anisotropy and asymmetry of RSOC effect.
In Fig. 3, we plot the energy contour of the conduction Rashba splitting bands in the kx- ky plane. We can see that the energy band along the wave vector kx direction increases faster than that along the wave vector ky direction. This phenomenon will cause anisotropic nonlinear optical effects because of the more optical transitions in the x-direction than that in the y-direction.
Figures 4(a) and 4(b) present 2D plots of the TPA as a function of the incident light wavelength for the different polarization directions both in the presence and in the absence of the RSOC in monolayer BP. We demonstrate the TPA spectra with θ=0°, 30°, 45°, 60°, 90°. It is observed that the TPA spectra have two peaks, one of which is very small at the wavelength of about 825 nm and the other one is much larger at the wavelength nearly 1641 nm. We can clearly see that the TPA coefficient decreases as the polarization angle of the linearly polarized light increase from 0° to 90°. The maximum TPA occurs when the polarization direction is aligned with the x-direction. This can be explained that the density of the energy states is larger along x-direction than that along the y-direction (see the Fig. 3). So more TPA transitions occur along x-direction. This phenomenon is consistent with the results of previous studies [13,22]. It indicates that the strong anisotropic nonlinear absorption can be realized in the monolayer BP. Comparing Fig. 4(a) with 4(b), we also can find that the TPA coefficient is enhanced nearly doubly by the effect of RSOC when the external electric field is 2 V/Å, which will be enhanced much more with a stronger external electric field. Figure 4(c) shows the TPA spectra for different values of the external electric field in the presence of the RSOC when the polarization direction is along the x-direction. It can be seen that the resonant peaks of the TPA increases with the increasing of the electric field. Due to the RSOC effect caused by the external electric field, the stronger electric field leads to larger spacing spin splitting energy levels, which may make the probability of two-photon transition resonance. Then the TPA coefficient increases. Nevertheless, the Rashba splitting of the energy bands are in magnitude of µeV so that the shift of the absorption peaks is too tiny that they are hard to be observed in Fig. 4(c). In brief, we can control the strength of TPA by adjusting polarization direction as well as the external electric field. What we must clarify is that once the direction of external electric field is not perfectly perpendicular to the atom plane, the situation will become more complicated. For example, the electric field effect must be considered . And then these results will not be applicable for this situation.
It is already mentioned that the nonlinear optical response of monolayer BP shows strong anisotropy with the incident light’s polarization in Figs. 4(a) and 4(b). Now in Fig. 5 we display the angular dependence of the TPA as a function of polarization direction relative to the x-axis (armchair direction) of BP varying from 0 to 2π when the wavelength is fixed at 825 nm and 1641 nm, at which there are TPA peaks. The TPA coefficient changes periodically with the polarization angle of linearly polarized light and the period is π. The maximum of TPA coefficient corresponds to an incident polarization aligned along x-axis while the minimum of TPA coefficient occurs aligned along y-axis. This periodical function relationship is consistent with the third harmonic generation (THG) in Ref. . The functional relations can be fitted well by cos4θ approximatively since HAx>> HAy resulting from m′cx <<m′cy.
In conclusion, we have studied the anisotropic behaviors of TPA properties of monolayer BP associated with the RSOC effect by the linearly polarized light. We obtain the spin-orbit coupling splitting energy bands of monolayer BP in the low-energy theory when the RSOC effect is taken part in under an external electric field. The TPA is enhanced by the RSOC effect in different extent which is dependent on the strength of applied electric field. The TPA coefficient increases with the increase of the strength of the external electric field. Both the RSOC effect and the TPA properties are highly anisotropic. The TPA coefficient changes periodically with the polarization angle of linearly polarized light which appeared as the function cos4θ. The TPA coefficient enjoys its maximum when the polarization direction is along the x-direction (armchair direction), while falls into its lowest point when along the y-direction (zigzag direction). So we can control the strength of TPA by adjusting polarization direction as well as the external electric field. For multilayer BP, which is relatively easier to exfoliated than monolayer BP, both the RSOC effect and the electric field effect between layers will make the energy states much more complicated and many new physical properties will emerge subsequently. In our future work, we can systematically study the layer-dependence of nonlinear optical effect in multilayer BP.
National Natural Science Foundation of China (11764046, 11764047); Applied Basic Research Foundation of Yunnan Province (2017FB009).
The authors declare no conflicts of interest.
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