Superior optical Kerr effects induced by two-dimensional excitons

Feng Zhou; Cacere Jelah Nieva; Dianyuan Fan; Shunbin Lu; Wei Ji

doi:10.1364/PRJ.447029

Abstract

Materials with strong optical Kerr effects (OKEs) are crucial for a broad range of applications, such as all-optical data processing and quantum information. However, the underlying OKE mechanism is not clear in 2D materials. Here, we reveal key insights of the OKE associated with 2D excitons. An admirably succinct formalism is derived for predicting the spectra and the magnitude of the nonlinear refractive index (

n_{2}

) of 2D materials. The predicted

n_{2}

spectra are consistent with reported experimental data and exhibit pronounced excitonic resonances, which is distinctively different from bulk semiconductors. The

n_{2}

value is predicted to be

3 \times 10^{- 10} {cm}^{2} / W

for a 2D layered perovskite at low temperature as 7 K, which is four orders of magnitude larger than those of bulk semiconductors. The superior OKE induced by 2D excitons would give rise to a narrow refractive index-near-zero region for intense laser light. Furthermore, we demonstrate that the 2D layered perovskite should exhibit the best OKE efficiency (

W_{FOM} = 1.02

T_{FOM} = 0.14

) at 1550 nm, meeting the material requirements for all-optical switching. Our findings deepen the understanding of the OKE of 2D semiconducting materials and pave the way for highly efficient all-optical excitonic devices.

1. INTRODUCTION

The optical Kerr effect (OKE) is one of the nonlinear optical (NLO) phenomena observed in dielectric or semiconducting materials, where the refractive index changes in response to intense laser light. The change of refractive index (or so-called nonlinear refractive index, $n_{2}$ ) leads to Kerr lensing [1] and phase modulation [2] of the laser light. By utilizing these characteristics, materials with the OKE have shown great potential for applications, such as all-optical switching, all-optical modulation, four wave mixing, optical parametric amplification, and laser mode-locking [3 –8]. In the above-mentioned context, an ideal OKE material should possess large nonlinear refractive index at low optical power. In addition, a short response time is also crucial. Prior to this work, $n_{2}$ spectra of various 3D dielectrics and semiconductors have been experimentally measured [9 –11] and theoretically verified via a two-parabolic band (TPB) model [12]. Such a theory provides a general guideline on predicting the $n_{2}$ magnitude of bulk semiconducting materials. However, 3D materials usually show either weak nonlinear refractive change or long response time even under illumination with strong optical fields. Such limitations therefore call for alternative strategies in order to obtain OKE materials for efficient all-optical devices.

One such strategy is to exploit excitonic effects of 2D materials (or so-called 2D excitons). The 2D exciton can be naturally formed by optical coupling excitonic transitions in layered materials where the spatial confinement and the reduced screening effect give rise to enhanced Coulomb interactions. Since the excitons are confined in a plane that is thinner than their Bohr radius in most 2D semiconductors [13 –15], quantum confinement enhances the exciton binding energy [16] and alters materials’ optical properties [17]. In particular, the 2D exciton layered materials manifest strong and ultrafast light–matter coupling on nonlinear optical responses in the visible to the near-infrared spectra [18,19]. For instance, a single crystal of Ruddlesden–Popper perovskite (RPP) in the 2D structure has been reported to possess an $n_{2}$ dispersion with a remarkable resonant feature arising from the Pauli blocking [1] in one-photon-induced transitions to the lowest excitonic state [20], giving rise to considerable OKEs in the visible range. The temporal response for this OKE has been reported to be $\sim 100 ps$ , which is attributed to either an exciton-exciton annihilation (EEA) effect or a radiative recombination process of the $1 s$ excitons [21 –24]. Apart from the one-photon absorption, OKEs along with the two-photon absorption (2PA) have also been reported in 2D semiconducting materials, where the photon energy ( $h υ$ ) of the incident light is far below the bandgap ( $E_{g}$ ), i.e., $E_{g} / 2 < h υ < E_{g}$ . It is noted that $n_{2}$ values of layered crystals of RPP, ${(C_{4} H_{9} {NH}_{3})}_{2} {({CH}_{3} {NH}_{3})}_{n - 1} {Pb}_{n} I_{3 n + 1}$ ( $I_{n = 1, 2, 3, 4}$ ), have shown more than an order of magnitude, as compared with a bulk semiconductor and the TPB model [25] at the excitation wavelength of 2.7 μm. The enhancement of the OKEs was demonstrated to arise from excitonic effects on the 2PA [26,27]. Upon the 2PA associated OKEs, “dark” excitons are created instantaneously and then relax through electron-electron interaction, which exhibits a fast temporal decay $< 60 fs$ in transition metal dichalcogenide (TMD) monolayers and RPPs [23,27]. However, a theoretical guideline and systematic understanding on the OKE of 2D layered materials are still lacking.

In this paper, we develop a simple model on OKEs originated from 2D exciton-associated 2PA. By utilizing the second-order quantum perturbation theory for two-photon transitions among the energy states of 2D excitons and then performing the Kramers–Kronig (K-K) transformation, the nonlinear refractive index ( $n_{2}$ ) is successfully derived over a broad range of light frequency with an assumption that only $2 p$ exciton is taken into consideration. The theoretically calculated results from our model are comparable with those reported experimental data for monolayer RPPs, TMDs, black phosphorus (BP), and hexagonal boron nitrides (h-BNs). Furthermore, our model explicitly shows that the $n_{2}$ dispersions of 2D semiconductors are dominated by exciton-resonant features, in particular, the resonance with the $2 p$ exciton; and the $n_{2}$ magnitudes are closely related to both its linewidth and energy level. The $n_{2}$ value of a 2D RPP is predicted to be $3 \times 10^{- 10} {cm}^{2} / W$ , which is four orders of magnitude more than those of bulk semiconductors ( $\sim 10^{- 14} {cm}^{2} / W$ ) [10], demonstrating the superior OKEs induced by 2D excitons and implying a narrow refractive index-near-zero region for intense laser light. Last, we evaluate the figures-of-merit (FOMs) of the OKEs for these 2D semiconductors, which proves them to be of great potential for all-optical excitonic devices.

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2. THEORY

The nonlinear refractive index ( $n_{2}$ ) intrinsically accompanied by the 2PA coefficient ( $β$ ) is governed by the K-K relation [1]. In the present context, we write it in the form $n_{2} (h υ) = \frac{c h}{π} \int_{0}^{+ \infty} \frac{β (h υ^{'}, h υ) d (h υ^{'})}{{(h υ^{'})}^{2} - {(h υ)}^{2}} .$ (1)

For a novel K-K transformation, two distinct photon energies for 2PA are needed, i.e., $h υ$ the “cause” and $h υ^{'}$ the integration variable in Eq. (1). Here, we consider a special case for the degenerate 2PA, in which the photon energies $h υ$ and $h υ^{'}$ are equivalent. The degenerate $β$ in the case of exciton-resonant 2PA of 2D semiconductors is schematized by Fig. 1 and has been calculated based on a second-order, time-dependent, quantum-mechanical perturbation theory [28,29]. Here, we simplify the expression of degenerate $β$ further, and the detailed simplification can be found in the appendix. The simplified expression is as follows: $β (h υ^{'}) = C_{2} \frac{{(n_{0}^{2} + 2)}^{4}}{E_{2 p}^{2}} F_{2, exc} (h υ^{'}),$ (2)where $C_{2}$ is a material-related parameter (and its definition can be found in Appendix A); $n_{0}$ is the linear refractive index. $F_{2, exc} (h υ^{'}) = \frac{h υ^{'}}{{(2 h υ^{'} - E_{2 p})}^{2} + {(h γ)}^{2}}$ is the dispersion function with $E_{2 p}$ and $h γ$ referring to the energy level of $2 p$ excitonic state and half of its linewidth, respectively. Here, we assume that only $2 p$ excitons make the contribution to the 2PA, and other excitonic states are ignored completely (see Fig. 1).

Schematic of the optical Kerr effect (self-focusing or -defocusing) induced by 2PA resonant with the exciton energy (E2p). Eg is the bandgap.

Figure 1.Schematic of the optical Kerr effect (self-focusing or -defocusing) induced by 2PA resonant with the exciton energy ( $E_{2 p}$ ). $E_{g}$ is the bandgap.

By substituting Eq. (2) into Eq. (1), the $n_{2}$ dispersion is derived to be $n_{2} (h υ) \approx \frac{c h C_{2}}{π} \frac{{(n_{0}^{2} + 2)}^{4}}{E_{2 p}^{2}} \times \int_{0}^{E_{2 p}} \frac{1}{{(E_{2 p} - 2 h υ^{'})}^{2} + {(h γ)}^{2}} \frac{h υ^{'}}{{(h υ^{'})}^{2} - {(h υ)}^{2}} d (h υ^{'}) .$ (3)

Because the equivalence of $h υ$ and $h υ^{'}$ would lead to the integral of $β (h υ^{'})$ with divergences, a substitution of $υ^{'}$ with $υ^{'} + Δ υ^{'}$ and a numerical summation were made for an approximate integral of $β$ over the 2PA range ( $0 < h υ^{'} < E_{2 p}$ ). With further simplifications, see the details in Appendix B; we obtain a semiempirical expression as $n_{2} (h υ) = \frac{Z_{2}^{'} {(n_{0}^{2} + 2)}^{4}}{E_{2 p}^{2}} G (x),$ (4)where $Z_{2}^{'}$ is another material-related parameter, which is determined by the density ( $D_{e}$ ), the effective Bohr radius ( $a_{B}$ ), and the linewidth ( $2 h γ$ ) of 2D excitons [24]. The definition of $Z_{2}^{'}$ can be found in Appendix A. The 10 2D materials (see Appendix C) have an average $Z_{2}^{'}$ value of $1 \times 10^{- 14}$ in the SI units such that $n_{2}$ is in ${cm}^{2} / W$ , and $h υ$ , $E_{2 p}$ , and $h γ$ are in units of eV. $G (x)$ is given by $G (x) = \frac{0.5 - x}{{(0.5 - x)}^{2} + 0.3 {(h γ / E_{2 p})}^{2}},$ (5)where $x = h υ / E_{2 p}$ .

3. RESULTS AND DISCUSSION

Note that Eq. (5) interprets the normalized $n_{2}$ -dispersion. As shown by the pink area in Fig. 2, the normalized $n_{2}$ values with $h γ / E_{2 p}$ being in the range from 0.05 to 0.15 exhibit a central-symmetrical dispersion with respect to $x = 0.5$ . For $| x - 0.5 | \leq 0.25$ , the normalized $n_{2}$ values vary from the positive to the negative dramatically, indicating that the OKE would be on (or near) resonance with $E_{2 p}$ . Otherwise, the normalized $n_{2}$ values can be regarded as the off-resonant ones and are nearly independent on $x$ or $h γ / E_{2 p}$ .

Figure 2.Comparison of the normalized $n_{2}$ between the prediction (pink area) by Eq. (5) and the measured values (symbols) of monolayer (or few-layer) TMDs, $RPP (I_{n = 1})$ , and h-BN [25,30 –34" target="_self" style="display: inline;">–34]. The envelopes of the pink area are calculated with $h γ / E_{2 p} = 0.05$ and 0.15.

To validate our model, the calculated $n_{2}$ dispersions are normalized and compared with experimental data of monolayer (or few-layer) TMDs, RPPs, h-BN, and BP [25,30 –34], where the values of $E_{2 p}^{2} / [Z_{2}^{'} {(n_{0}^{2} + 2)}^{4}]$ are obtained from Table 3 in Appendix A. In the off-resonance region, our model is in agreement with the experimental data within an order of magnitude for RPP ( $I_{n = 1}$ ) in polycrystalline structures (at 2.7 μm) [25] and h-BN (at 1.064 μm) [33]. The experimental data of RPP ( $I_{n = 2, 3, 4}$ ) powders are less than the prediction of Eq. (4). It is anticipated that surface states and/or surface scattering from the crystallites may hinder its intrinsic NLO properties of the 2D crystals. The normalized $n_{2}$ value of BP [34] is $\sim 26$ at $x = h υ / E_{2 p} = 1.29$ , which is comparable with the magnitude predicted by Eq. (4), but it is not shown because it is beyond the scale of the $x$ axis in Fig. 2.

In the near-resonance region, there is a large discrepancy with the experimental data of monolayer (or few-layer) TMDs [30 –32], which is attributed to the spin-split-off excitonic states. Experimental evidence has shown monolayer TMDs with the spin-orbital splitting possess a pair of generated excitons (as either the A-exciton or B-exciton [35]) transitioning at the $K$ and $K^{'}$ valley in the momentum space with broken inversion symmetry. This gives rise to a unique valley degree of freedom in the $n_{2}$ dispersions that can directly couple to the helicity of excitation photons [17]. Under two-photon excitation, it is necessary to consider the two excitons; hence, as detailed in Appendix A, the $n_{2}$ dispersion is modified to be $n_{2} (h υ) = \frac{Z_{2}^{'} {(n_{0}^{2} + 2)}^{4}}{2 E_{2 p}^{2}} [G_{A} (x) + G_{B} (x)] .$ (6)Here, $E_{2 p} = E_{2 p_A}$ , $G_{A} (x) = \frac{0.5 - x}{{(0.5 - x)}^{2} + 0.3 {(h γ / E_{2 p})}^{2}}$ , $G_{B} (x) = \frac{Λ δ^{2} (1 - 0.5 δ - x)}{{(1 - 0.5 δ - x)}^{2} + 0.3 δ^{2} {(Λ h γ / E_{2 p})}^{2}}$ , $δ = E_{2 p_A} / E_{2 p_B}$ , $h γ = h γ_{A}$ , and $Λ = h γ_{B} / h γ_{A}$ .

Figure 3 shows good agreement between Eq. (6) and the experimental data. $G_{A} (x)$ and $G_{B} (x)$ result in two peaks in the near-resonance region, indicating that the spin-split-off energy of excitons should play a significant role in the OKE of monolayer TMD materials. Furthermore, the B-exciton with a higher energy level from the spin-orbital splitting leads to a broadened and blueshift of the resonance of the individual $n_{2}$ dispersion, in comparing the blue curves to the red ones in Fig. 3. This is attributed to higher energy levels of B-excitons, which exhibit larger linewidths of NLO resonance. The spin-splitting effect makes a great variation of overall $n_{2}$ values as displayed by the purple curves.

Figure 3.Normalized $n_{2}$ dispersion of monolayer ${MoS}_{2}$ , ${MoSe}_{2}$ , ${WS}_{2}$ , and ${WSe}_{2}$ calculated by the two excitons (purple), A exciton (red), and B exciton (blue) with $h γ / E_{2 p} = 0.075 eV$ . The symbols are the experimental data from Refs. [30 32" target="_self" style="display: inline;">–32] and scaled with $Z_{2}^{'} = 1 \times 10^{- 14}$ .

Figure 4 compares calculated $n_{2}$ magnitudes from Eq. (4) or Eq. (6) with the TPB model. It is interesting to note that the maximal $n_{2}$ values by Eq. (4) or Eq. (6) are in the range from $1 \times 10^{- 12}$ to $1 \times 10^{- 10} {cm}^{2} / W$ , which are greater than the TPB model results by at least one order of magnitude. This indicates the 2D excitonic effects greatly enhance the OKE in 2D cases when compared with their bulk counterparts. As derived in Eqs. (4) and (6), the calculated $n_{2}$ values are proportional to $D_{e} \cdot a_{B}^{4}$ , implying 2D materials possessing the higher density of excitons and/or the larger effective Bohr radius would exhibit greater OKE along with degenerate 2PA in many ways analogous to plasma screening effects [36]. As a result, RPP ( $I_{n = 4}$ ) is predicted to show the largest peak $n_{2}$ value of $\sim 1.5 \times 10^{- 10} {cm}^{2} / W$ among the 10 materials. The calculated $n_{2}$ values are of the same order of magnitude as the multilayer graphene at an excitation wavelength in the near-infrared [37]. As compared with the experimental data of monolayer (or few-layer) TMDs in thin films, or dispersion [31,38 –41], the measured absolute $n_{2}$ values are greater than from Eq. (6) [38,40,41]; also see Table 4 in Appendix C. One reason for the discrepancy is that the presence of a large number of defects/impurities in these films would give extra transitions in either 2PA or saturable absorption (SA) processes and, thus, enhance the measured nonlinearities. It is also noted that the absolute $n_{2}$ values from our model are one order of magnitude larger than the reported results of ${MoS}_{2}$ dispersion [31] and ${MoSe}_{2}$ nanosheets [39], where light scattering in those amorphous samples may screen a refractive index change arising from the 2D nature of excitons.

Figure 4.Calculated $n_{2}$ dispersions of (a) monolayer RPP ( $I_{n = 1, 2, 3, 4}$ ) and (b) TMDs by Eq. (4) and Eq. (6). Calculated $n_{2}$ dispersions of (c) monolayer RPP ( $I_{n = 1, 2, 3, 4}$ ) and (d) TMDs by the two-parabolic-band model [12]. Parameters used in the calculation are displayed in Table 3 in Appendix A.

With the effects of 2D excitons, further enhancement of the OKE is predicted at lower temperature. As derived from Eq. (4) or (6), $n_{2}$ values on resonance are inversely proportional to the square of the linewidth ( $2 h γ$ ) of 2D exciton, which would narrow down (as $\sim 35 meV$ ) for RPPs, with temperature decreasing to $\sim 7 K$ [42]. The temperature dependence of the OKE could enlarge $n_{2}$ values at peak (or valley) by one order of magnitude [see an example of $n_{2}$ dispersion of RPP ( $I_{n = 1}$ ) in Fig. 5(a)]. The $n_{2}$ value of RPP ( $I_{n = 1}$ ) is predicted to be $3 \times 10^{- 10} {cm}^{2} / W$ at 7 K, which would lead to a change in the refractive index of 3 (i.e., around 150% of the linear refractive index) in the case of $I = 10 GW / {cm}^{2}$ . This would give rise to a narrow refractive index-near-zero region of intense laser light when it is on resonance with 2D exciton. In the off-resonance region, our calculated $n_{2}$ values are nearly independent of $x = h υ / E_{2 p}$ or the linewidth ( $2 h γ$ ). The scaled $n_{2}$ values explicitly show an $E_{2 p}^{- 2}$ dependence, which gives a variation by two orders of magnitude for 2D semiconducting materials with $E_{2 p}$ ranging from 1 to 10 eV. This variation is clearly illustrated in Fig. 5(b).

Figure 5.(a) Calculated $n_{2}$ values as a function of photon energy ( $x$ axis) and temperature ( $y$ axis) for monolayer RPP ( $I_{n = 1}$ ). (b) A log-log plot of the scaled $n_{2}$ in the off-resonance region versus $E_{2 p}$ . The experimental $n_{2}$ values are scaled by $Z_{2}^{'} {(n_{0}^{2} + 2)}^{4} G (x)$ with $n_{0}$ and $Z_{2}^{'}$ listed in Table 3 in Appendix A. The solid line is the theoretical result of Eq. (4) with no adjustable parameters and a slope of $- 2$ .

Indeed, in our model, the absolute magnitude of the nonlinear refractive index is enhanced through two-photon transitions, when the photon energy approaches $h υ / (2 E_{2 p})$ . Such a resonant enhancement of $n_{2}$ is basically accompanied by absorption losses, commonly given by $α = α_{0} + β I$ , where $α_{0}$ is the linear absorption coefficient. The OKE efficiency/transparency trade-off ( $n_{2} / α$ ) at a specific wavelength $λ$ is therefore of importance to the assessment of a material for all-optical switching devices. The material requirements are as follows: $W_{FOM} = \frac{| n_{2} | I}{α_{0} λ} ≫ 1$ and $T_{FOM} = \frac{β λ}{| n_{2} |} ≪ 1$ [43,44]. Table 1 lists the FOMs for selected materials operated at room temperature, $λ = 1550 nm$ , and $I = 10 GW / {cm}^{2}$ . Among them, RPP ( $I_{n = 4}$ ) is the best: its $n_{2}$ is predicted to be as large as $1 \times 10^{- 10} {cm}^{2} / W$ with both $W_{FOM} = 1.02$ and $T_{FOM} = 0.14$ . Compared with multilayer graphene, Si, and GaAs, RPP ( $I_{n = 4}$ ) performs even better, as its nonlinear refractive index is enhanced by two orders of magnitude, while $T_{FOM}$ is comparable [45,46]. Though RPPs ( $I_{n = 1, 2, 3}$ ) exhibit better $T_{FOM}$ ( $< 0.1$ ), indicating less 2PA losses, the $W_{FOM}$ values are less than 1 [29]. Note that the required light irradiance ( $I = 10 GW / {cm}^{2}$ ) would be high. To reduce it, 2D materials in microcavities or photonic structures should be considered. For example, one may integrate 2D materials on clad nanophotonic cavities, such as ring and disk resonators, to increase the effective path length for light–matter interaction [47]. Arbitrary vertical heterostructures such as intentionally designed sequences of graphene, h-BN, and TMD monolayers (or 2D perovskites) [48,49] would give also rise to higher 2D-exciton density in order to reduce the operating light intensity.Table 1.

Nonlinear Coefficients and FOMs of Materials at 1550 nm

Material	$n_{2}$ [ $\times 10^{- 12} {cm}^{2} / W$ ]	$β$ [cm/GW]	$W_{FOM}$	$T_{FOM}$	Ref.
2D RPP ( $I_{n = 4}$ )	105.5	96.3	1.02	0.14	This work
2D RPP ( $I_{n = 3}$ )	22.9	9.9	0.27	0.07	This work
2D RPP ( $I_{n = 2}$ )	7.2	1.6	0.12	0.03	This work
2D RPP ( $I_{n = 1}$ )	1.3	0.3	0.03	0.04	This work
2D ${MoS}_{2}$	1.5	5.6	0.008	0.56	This work
2D ${MoSe}_{2}$	19.1	212.5	0.07	1.72	This work
2D ${WS}_{2}$	2.7	7.7	0.02	0.45	This work
2D ${WSe}_{2}$	15.8	79.1	0.06	0.78	This work
Multilayer graphene	−800	900	0.20	1.40	[37]
$Si ⟨ 110 ⟩$	0.045	0.79	–	0.37	[45]
GaAs	0.16	10.2	–	0.10	[45]
GaAs/AlAs superlattice	0.15	1.5	–	0.87	[46]
Conjugated 3,3’-bipyridine derivative	0.0046	$< 0.01$	$> 600$	$< 0.15$	[43]

4. CONCLUSION

In summary, based on the K-K transformation of exciton-associated 2PA, we have successfully developed an admirably succinct model to predict the OKE of 2D semiconductors. In our model, all the parameters are measurable experimentally, except for the linewidth of $2 p$ exciton. On the examining the 10 2D materials mentioned here, we find that our model can provide a general guideline on both the $n_{2}$ magnitude and $n_{2}$ spectra. On our analysis, two-photon resonance with 2D excitons yields a remarkable contribution to the nonlinear refractive index spectra. As compared with the TPB model for 3D semiconductors, the OKE induced by 2D excitons shows a great enhancement. Based on this simple model, we assess RPP ( $I_{n = 4}$ ) crystals to be the best material candidate for all-optical switching at the 1550 nm wavelength. Our findings deepen the understanding of the OKE of 2D semiconducting materials and pave the way for highly efficient all-optical excitonic devices.

APPENDIX A: DERIVATION OF n2 DISPERSION FOR 2D SEMICONDUCTORS WITH TWO EXCITONS

β (h υ^{'}) = C N h υ^{'} {(\frac{E_{loc}}{E})}^{4} \times [\frac{{| μ_{G \to 1 s} |}^{2}}{{(E_{1 s_A} - h υ^{'})}^{2} + {(Γ_{1 s_A} / 2)}^{2}} + \frac{{| μ_{G \to 1 s} |}^{2}}{{(E_{1 s_B} - h υ^{'})}^{2} + {(Γ_{1 s_B} / 2)}^{2}}] \frac{{| W_{n} μ_{1 s \to n p} |}^{2} Γ_{n p} / 2 π}{{(E_{n p} - 2 h υ^{'})}^{2} + {(Γ_{n p} / 2)}^{2}},

1 s

Normalized Transition Dipole Moments for 2PA Transitions^a

2PA Transitions	$\| μ_{G \to 1 s} \| \| μ_{1 s \to n p} \|$ [a.u.]
$G \to 1 s \to 2 p$	1
$G \to 1 s \to 2 s$	0
$G \to 1 s \to 3 p$	0.41
$G \to 1 s \to 3 s$	0
$G \to 1 s \to 4 p$	0.24
$G \to 1 s \to 5 p$	0.16
$G \to 1 s \to 6 p$	0.11

Note: Transition dipole moments are normalized by $| μ_{G \to 1 s} | | μ_{1 s \to 2 p} |$ amplitude [28].

β (h υ^{'}) = \frac{C N (h γ)}{81 π} {| W_{2} |}^{2} {| μ_{G \to 1 s} |}^{2} {| μ_{1 s \to 2 p} |}^{2} {(n_{0}^{2} + 2)}^{4} \times [\frac{1}{{(E_{1 s_A} - h υ^{'})}^{2} + {(h γ_{A})}^{2}} \frac{h υ^{'}}{{(2 h υ^{'} - E_{2 p_A})}^{2} + {(h γ_{A})}^{2}} + \frac{1}{{(E_{1 s_B} - h υ^{'})}^{2} + {(h γ_{B})}^{2}} \frac{h υ^{'}}{{(2 h υ^{'} - E_{2 p_B})}^{2} + {(h γ_{B})}^{2}}],

n_{2}

h υ

x = \frac{h υ}{E_{2 p}}

Parameters Used in the Calculation of $n_{2}$ Dispersions^a

	$n_{0}$	$a_{B}$ [Å]	$D_{e}$	$E_{g}$ [eV]	$E_{b}$ [eV]	$E_{1 s}$ [eV]	$E_{2 p}$ [eV]	$Z_{2}^{'}$ [ $\times 10^{- 15}$ ]	$h γ$ [eV]	$δ$	$Λ$
RPP ( $I_{n = 1}$ )	2.11	15.3	$0.60 \times 10^{20}$	2.74	0.35	2.39	2.59	0.979	0.15	N.A.	N.A.
RPP ( $I_{n = 2}$ )	2.21	17.1	$1.20 \times 10^{20}$	2.51	0.26	2.25	2.37	3.05	0.15	N.A.	N.A.
RPP ( $I_{n = 3}$ )	2.27	17.9	$1.80 \times 10^{20}$	2.27	0.16	2.11	2.14	5.50	0.15	N.A.	N.A.
RPP ( $I_{n = 4}$ )	2.32	22.0	$2.40 \times 10^{20}$	2.14	0.15	1.99	2.00	16.7	0.15	N.A.	N.A.
${MoS}_{2}$	1.84	9.3	$5.00 \times 10^{20}$	2.70	0.80	1.90	2.50	1.67	0.075	0.939	2
${MoSe}_{2}$	2.10	10.1	$1.81 \times 10^{21}$	2.43	0.66	1.77	2.21	8.40	0.075	0.909	2
${WS}_{2}$	1.82	10.3	$6.28 \times 10^{20}$	2.92	0.82	2.10	2.73	3.17	0.075	0.867	4
${WSe}_{2}$	1.84	10.5	$2.28 \times 10^{21}$	2.57	0.79	1.78	2.35	12.4	0.075	0.848	4
h-BN	2.00	6.1	$2.40 \times 10^{21}$	5.81	0.92	5.50	5.69	1.00	0.15	N.A.	N.A.
BP	2.50	20.2	$3.80 \times 10^{20}$	1.60	0.50	1.10	1.20	20.0	0.15	N.A.	N.A.

Note that no fitting parameter was used in plotting the theoretical curves, except for the linewidth, and the experimental data of $n_{2}$ values [14,15,28,29,53–55] are scaled with the average value of $Z_{2}^{'} = 1 \times 10^{- 14}$ .

APPENDIX B: SIMPLIFICATION OF n2 DISPERSION OF 2D SEMICONDUCTORS WITH ONE EXCITON

β (h υ^{'}) = \frac{C N (c h^{2} γ)}{81 π^{2}} {| W_{2} |}^{2} {| μ_{G \to 1 s} |}^{2} {| μ_{1 s \to 2 p} |}^{2} \times \frac{{(n_{0}^{2} + 2)}^{4}}{E_{2 p}^{2}} \frac{h υ^{'}}{{(2 h υ^{'} - E_{2 p})}^{2} + {(h γ)}^{2}} .

C_{2} = \frac{C N (h γ)}{81 π} {| W_{2} |}^{2} {| μ_{G \to 1 s} |}^{2} \cdot {| μ_{1 s \to 2 p} |}^{2}

n_{2}

APPENDIX C: EXPERIMENTAL DATA OF n2 VALUES

Table 4.

Extracted $n_{2}$ Values from Experimental Data

	$n_{2}$ [ $\times 10^{- 12} {cm}^{2} / W$ ]	$\frac{n_{2} E_{2 p}^{2}}{Z_{2}^{'} {(n_{0}^{2} + 2)}^{4}}$	$λ$ [nm]	$n_{0}$	Ref.
RPP ( $I_{n = 1}$ )	0.41	0.16	2700	2.11	[25]
RPP ( $I_{n = 2}$ )	0.45	0.11	2700	2.21	[25]
RPP ( $I_{n = 3}$ )	0.39	0.048	2700	2.41	[25]
RPP ( $I_{n = 4}$ )	0.40	0.053	2700	2.32	[25]
${MoS}_{2}$	−1.96	−1.45	800	1.84	[30]
	350	260.02	800	1.84	[38]
	1.88	1.40	1064	1.84	[38]
	−0.21	−0.15	1064	1.84	[31]
${MoSe}_{2}$	−0.12	−0.034	1064	2.10	[31]
${MoSe}_{2}$	0.20	0.058	1000	2.10	[39]
${WS}_{2}$	−1.10	−1.02	800	1.82.	[30]
	0.81	0.76	800	1.82	[32]
	58.30	54.55	1064	1.82	[40]
	128	119.77	1040	1.82	[41]
${WSe}_{2}$	−18.70	−12.27	1040	1.84	[41]
h-BN	0.12	0.30	1064	2.0	[33]
BP	860	26.73	800	2.5	[34]

APPENDIX D: TEMPERATURE DEPENDENCE OF NONLINEAR REFRACTIVE INDEX

n_{2}

$Nonlinear refractive index, n2, as a function of photon energy (x axis) and temperature (y axis) for monolayer RPP: (a) In=2, (b) In=3, and (c) In=4. These n2 values are calculated with the averaged Z2′=1×10−14.$

Figure 6.Nonlinear refractive index, $n_{2}$ , as a function of photon energy ( $x$ axis) and temperature ( $y$ axis) for monolayer RPP: (a) $I_{n = 2}$ , (b) $I_{n = 3}$ , and (c) $I_{n = 4}$ . These $n_{2}$ values are calculated with the averaged $Z_{2}^{'} = 1 \times 10^{- 14}$ .

$Nonlinear refractive index, n2, as a function of photon energy (x axis) and temperature (y axis) for TMD monolayers: (a) MoS2, (b) MoSe2, (c) WS2, and (d) WSe2. These n2 values are calculated with the averaged Z2′=1×10−14.$

Figure 7.Nonlinear refractive index, $n_{2}$ , as a function of photon energy ( $x$ axis) and temperature ( $y$ axis) for TMD monolayers: (a) ${MoS}_{2}$ , (b) ${MoSe}_{2}$ , (c) ${WS}_{2}$ , and (d) ${WSe}_{2}$ . These $n_{2}$ values are calculated with the averaged $Z_{2}^{'} = 1 \times 10^{- 14}$ .

APPENDIX E: FIGURES OF MERIT OF MONOLAYER RPPS AND TMDS

W_{FOM}

Figure 8.(a) $W_{FOM}$ of monolayer TMDs, (b) $W_{FOM}$ of monolayer RPPs, (c) $T_{FOM}$ of monolayer TMDs, and (d) $T_{FOM}$ of monolayer RPPs. The gray area corresponds to the wavelength range for optical communications.

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