## Abstract

By applying quantum perturbation theory to two-dimensional excitons in monolayer transition metal dichalcogenides (TMDCs), we develop a theoretical model for two-photon absorption in the near infrared spectral region. By assuming the bandwidth of the final excitonic state to be 0.15 eV, the two-photon absorption coefficients are as high as 50 cm/MW and selenium-based, monolayer TMDCs exhibit greater 2PA coefficients than sulfur-based, monolayer TMDCs. Our model is also compared to the experimental data obtained by Z-scans or nonlinear transmission measurements.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

An exciton is a bound state of an electron-hole pair which is formed by Coulomb attraction. Such a bound state plays a significant role in light-matter interaction especially in semiconducting materials, where an exciton can result in spectrally narrow linewidth, large oscillator strength and efficient radiative recombination [1–4]. The nature of an exciton is a quasi-particle, which makes it great potential for exciton-carriers devices in quantum computation and excitonic circuits [5–8]; and also provides a platform for investigating its rich fundamental physics [9, 10]. The recent advances in the emerging field of two-dimensional (2D) semiconductors facilitate stronger excitonic effects resulting from their 2D spatial confinement and reduced screening effect, as compared to their bulk counterparts [11–15]. For monolayer transition metal dichalcogenides (TMDCs), 2D excitons have been theoretically calculated and experimentally measured to be of considerably large binding energies ${E}_{b}$ (0.5–1 eV, corresponding to an exciton Bohr radius ${a}_{b}$ ≈1 nm) [16–20]. Due to the large binding energy, a series of excitonic states has been revealed by single-photon or two-photon excitation spectroscopies for monolayer TMDCs [21–24]. Furthermore, the 2D excitonic effects make a significant enhancement on the oscillator strengths for single-photon absorption, in particular, transferring oscillator strengths from the direct band-to-band transitions to the excitonic transitions (*1s*-excitons) [13]. However, excitonic effects on both magnitude and spectra of two-photon absorption (2PA) in monolayer TMDCs are little known. Previously, we studied the 2PA spectrum of monolayer MoS_{2} [25]. Here, we report a systematical theoretical study on the wavelength-dependent magnitudes of 2PA in monolayer MoS_{2}, WS_{2}, WSe_{2}, and MoSe_{2}.

As illustrated in Fig. 1, 2D excitonic transitions dominate the transitions of optical absorption whereby there are mainly three peaks in the spectra of monolayer MoS_{2}, WS_{2}, WSe_{2} and MoSe_{2}. Peak A or B corresponds to the lowest energy excitonic state (*1s*-exciton) formed by an electron excited from one of the two splitting valence bands. Peak C is composed of a series of higher-energy excitonic states (or the Rydberg series) for the electron-hole pair. Because of these excitonic states within the bandgap, they can be utilized for intermediate or final states in the 2PA process, in which the *1s*-exciton (A- or B-exciton) act as a real intermediate state leading to the primary transition with an incoming photon in short pulses. Following the first electronic transition, the 1*s*-exciton makes an intra-excitonic transition by absorbing another photon simultaneously to a *p*-exciton as the final state, fulfilling the odd-parity requirement of two-photon transitions. As such, excitonic effects make a significant transferring of oscillator strength for the 2PA process in 2D TMDC, leading to an enhanced 2PA.

## 2. Theoretical calculation

Based on a second-order, time-dependent, quantum-mechanical perturbation theory [26], we can calculate the 2PA cross-section $({\sigma}^{(2)})$ by:

*g*is the line shape function. The 2PA coefficient is given by [27]:where ℎ𝜈 (or $\hslash \omega $) is the photon energy and

*N*is the density of active unit cells. By taking the line shape function to be a Lorentzian function, 2PA coefficient (${\alpha}_{2}$) of monolayer TMDCs can be calculated by:

^{−3}, $h\nu $, ${E}_{ij}$ and ${\Gamma}_{i}$ are in units of eV, and ${\mu}_{ij}$ is in units of esu. We neglect all anti-resonant contributions to the 2PA process, resulting from negative frequency components in the applied laser field [28]. In addition, band-to-band transitions are ignored, because the ${\alpha}_{2}$-value based on band-to-band transition is considerably less than excitonic counterparts [25].

In the calculation of ${\mu}_{qg}$for the generated 1*s*-exciton (either A-exciton or B-exciton), we have${\mu}_{qg}=\u3008{\psi}_{exc,1s}\left({r}_{e}\right)|\left(-e{r}_{e}\right)|{\psi}_{G}\left({r}_{e}\right)\u3009$, where *e* is the electron charge; ${\psi}_{exc,1s}\left({r}_{e}\right)$is the wave function of the excited state; and ${\psi}_{G}\left({r}_{e}\right)$corresponds to the ground state which describes the *d*-orbital electron in the Mo or W atom and the *p*-orbital electron in the S or Se atom.

As for the excited state, we apply the 2D hydrogen model for *1s*-excitons, and hence, its wave function is given by [29]:

In the calculation of ${\mu}_{fq}$, we have

*p*-excitons. For such a

*p*-exciton, its 2D wave function is given by

*n*is the principle quantum number;

*m*is the magnetic quantum number that is taken as ± 1 for

*p*-excitons; $a$ is the Bohr radius for a

*p*-exciton; ${L}_{n-2}^{2}$ is the generalized Laguerre polynomial; and $\phi $ is the rotational angle in the orbital. As such, we can have

*W*is the weight value with ${W}_{1}^{2}+{W}_{2}^{2}+{W}_{3}^{2}+\dots +{W}_{MAX}^{2}=1$, (${W}_{n}^{2}$ = 1/

_{n}*MAX*), and ${E}_{(np)g}={E}_{g}-{E}_{b(n)}$ is the energy level for

*np*-excitonic state.

## 3. Results and discussion

By taking parameters in Table 2, we can calculate the transition dipole moments and 2PA coefficients in the near-infrared spectrum for monolayer MoS_{2}, WS_{2}, WSe_{2} and MoSe_{2}. The only unknown parameter is the linewidth, ${\text{\Gamma}}_{f}$ . In Fig. 2, we plot the 2PA coefficient ${\alpha}_{2}$ as functions of ${\text{\Gamma}}_{f}$ on the y-axis and laser wavelength on the x-axis.

As ${\text{\Gamma}}_{f}$ is increased from 0.1 to 0.5 eV (corresponding to the dephasing times from 3 to 0.2 fs), the following four features are observed: (i) ${\alpha}_{2}$-values vary in the range from 0.05 to 55 cm/MW in the spectral region of interest; (ii) maximal ${\alpha}_{2}$-values decrease and 2PA spectra become broadening; (iii) maximal ${\alpha}_{2}$-values are blue-shifted; and (iv) selenium-based TMDCs exhibit greater 2PA coefficients than sulfur-based TMDCs.

With the increase in the bandwidth ${\text{\Gamma}}_{f}$, higher order *np*-excitons make more contributions towards the 2PA process and hence, decrease and blue-shift the maximal ${\alpha}_{2}$-values. The fact that the 2PA coefficients of monolayer WSe_{2} and MoSe_{2} are larger than those of MoS_{2} and WS_{2} is attributed to the smaller binding energies of monolayer WSe_{2}, and MoSe_{2}, which result in stronger intra-excitonic transitions. Therefore, monolayer WSe_{2} and MoSe_{2} should have greater potential for 2PA-involved photonic devices.

In our previous study [25], we determined the wavelength-dependent 2PA coefficients of monolayer MoS_{2} via two-photon-induced photoconduction measurements. We found that the model is in agreement with the measurement, within one order of magnitude. It was noted that there is a discrepancy around $hv\u2215Eg$ ∼0.5, where the measured data are less than the model. It is due to the fact that the experimental analysis completely ignores the band-to-band recombination. Due to the resonance with the bandgap, some excitons recombine radiatively. As a result, the photocurrent is reduced, thus leading to the smaller values in the measured *α*_{2}.

Here, we compare our model with experiments reported by other research groups [38–42], as shown in Fig. 3, where the 2PA spectra of monolayer MoS_{2}, WS_{2}, WSe_{2} and MoSe_{2} are calculated by taking ${\text{\Gamma}}_{f}$ = 0.15 eV. The 2PA coefficients shown by the symbols are the data measured all-optically (such as Z-scans and nonlinear transmission measurements at specific wavelengths) [38–42]. The ${\alpha}_{2}$-value predicted by our model at 780 nm for monolayer WS_{2} is in excellent agreement with the result by Zheng *at al* [38]. Within one order of magnitude, our model also agrees with the measurement at 1030 nm for monolayer MoS_{2} [39], and with the data at 1040 nm for monolayer WS_{2} [42].

In the case of 1-3 layers of WS_{2} at 1030 nm [40] and films of multilayer WS_{2} at 1040 nm [42], the calculated ${\alpha}_{2}$-values are much less than the experimental results. One reason for the discrepancy is that the presence of a large number of defects/impurities in these samples [40, 42] would give extra transitions in 2PA process and thus, enhance the 2PA coefficients, as compared to the intrinsic nature of 2PA in pure monolayer TMDCs.

It is also noted that the ${\alpha}_{2}$-values by our model are considerably greater than those measured by Zhang *et al.* at 780 nm and 1030 nm [40], by Dong *et al.* at 1030 nm [42], and by Bikorimana *et al.* at 1060 nm [41] on thicker samples. It is anticipated because the 2D nature of excitons diminishes as samples become thicker.

Lastly, it should be pointed out that one possible deviation from our hydrogen-like model arises from the dielectric constant (or relative permittivity). In our calculation, it is taken to be independent of the radii (or energies) of excitonic states for each monolayer TMDC. However, Chernikov *et al.* reported that the model deviates from their experiments, due to the inhomogeneous screening-effects in *ns*-excitonic states resulting in the variable dielectric constants [21].

## 4. Conclusion

By applying quantum perturbation theory to two-dimensional excitons in monolayer transition metal dichalcogenides (TMDCs), we develop a theoretical model for two-photon absorption in the near infrared spectral region. By assuming the bandwidth of the final excitonic state to be 0.15 eV, the two-photon absorption coefficients are as high as 50 cm/MW, and selenium-based, monolayer TMDCs exhibit greater 2PA coefficients than sulfur-based, monolayer TMDCs. Our model is also compared to the experimental data obtained by Z-scans or nonlinear transmission measurements.

## Funding

R144-000-327-112 and R144-000-401-114 by National University of Singapore; 61505117 by National Natural Science Foundation of China (NSFC); JCYJ20170302153323978 by Science and Technology Innovation Commission of Shenzhen; and 2017021 by Natural Science Foundation of Shenzhen University.

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