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 Eb (0.5–1 eV, corresponding to an exciton Bohr radius ab ≈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 MoS2 [25]. Here, we report a systematical theoretical study on the wavelength-dependent magnitudes of 2PA in monolayer MoS2, WS2, WSe2, and MoSe2.

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 MoS2, WS2, WSe2 and MoSe2. 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 1s-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.

 figure: Fig. 1

Fig. 1 (a) Energy vs. optical absorption and (b) energy diagram in the momentum space near K point for a monolayer TMDC. The dashed lines represent the energy levels of excitons, and the blue curves show the valence bands (VBs) and conduction band (CB). Δso is the energy split due to the spin-orbital interaction. The auburn arrows show the 2PA process.

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2. Theoretical calculation

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

σ(2)=(2πc)22ω23ε02h2|qμfqμqgωqgω|2g(2ω),
where 𝜔 is the angular frequency of an incident photon,  μij  is the transition dipole moment and g is the line shape function. The 2PA coefficient is given by [27]:
α2=σ(2)Nhν,
where ℎ𝜈 (or  ω) 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 (α2) of monolayer TMDCs can be calculated by:
 α2(hν)=CNhν[ElocE]4[|μqg|2(EAhν)2+(ΓA/2)2+ |μqg|2(EBhν)2+(ΓB/2)2]×[|μfq|2Γf/2π(Efg2hν)2+(Γf/2)2]
where (ElocE)=13(n02+2) is the local-field correction factor;  Ei  and Γi refer to a certain energy level and its linewidth, respectively. C is a material-independent constant which has a value of 3.47×1045 in units such that α2 is in cm/MW, N is in cm−3, hν, Eij and Γi are in units of eV, and μij 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 α2-value based on band-to-band transition is considerably less than excitonic counterparts [25].

In the calculation of μqgfor the generated 1s-exciton (either A-exciton or B-exciton), we have μqg=ψexc,1s(re)|(ere)|ψG(re), where e is the electron charge; ψexc,1s(re) is the wave function of the excited state; and ψG(re)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]:

 |ψn=1,m=0(r)=2aB2πe2raB.
Here, aBis the effective Bohr radius. The excited-state wave function is given as a convolution integral of the Bloch wave function of an electron in the conduction band multiplied by the 2D hydrogen wave function, as shown below:
|ψexc,1s(re)=ψc(re)·ψn=1, m=0(rerh)  drh.
Here, Bloch wave functions for the valence band (or the ground state) and the conduction band are listed in Table 1. It should be noticed that both |ψGand |ψexc,1sare normalized in such a way: ψexc,1s(re)ψG(re)|ψexc,1s(re)ψG(re)=1 C1 and C2 are weight values andC12+C22=1.

Tables Icon

Table 1. Bloch wave functions for the valence and conduction bands

In the calculation of μfq, we have

μfq=ψf(rerh)|[e(rerh)]|ψexc,1s(rerh),
where ψf(rerh)  is the final-state wave function in which we should take consideration of many higher-lying p-excitons. For such a p-exciton, its 2D wave function is given by
 ψexc,  np, m=±1(r=rerh)= 1a1π(n1/2)3(n2)!n!er(n1/2)a (2r(n1/2)a)×Ln22(2r(n1/2)a)e±iφ.
Here, 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; Ln22 is the generalized Laguerre polynomial; and φ is the rotational angle in the orbital. As such, we can have
|μfq|2(Efg2hν)2+(Γf/2)2=n|Wn ψexc,np, m=±1(rerh)|[e(rerh)]|ψexc,1s(rerh)|2(E(np)g2hν)2+(Γf/2)2,
where Wn is the weight value with W12+W22+W32++WMAX2  =1, (Wn2 = 1/MAX), and E(np)g=EgEb(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 MoS2, WS2, WSe2 and MoSe2. The only unknown parameter is the linewidth, Γf . In Fig. 2, we plot the 2PA coefficient α2 as functions of Γf on the y-axis and laser wavelength on the x-axis.

Tables Icon

Table 2. Parameters used in the calculation of 2PA coefficients.

 figure: Fig. 2

Fig. 2 2PA coefficient α2 (cm/MW) as functions of laser wavelength (x-axis) and Γf (y-axis) for monolayer (a) MoS2, (b) WS2, (c) WSe2, and (d) MoSe2.

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As Γf 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) α2-values vary in the range from 0.05 to 55 cm/MW in the spectral region of interest; (ii) maximal α2-values decrease and 2PA spectra become broadening; (iii) maximal α2-values are blue-shifted; and (iv) selenium-based TMDCs exhibit greater 2PA coefficients than sulfur-based TMDCs.

With the increase in the bandwidth Γf, higher order np-excitons make more contributions towards the 2PA process and hence, decrease and blue-shift the maximal α2-values. The fact that the 2PA coefficients of monolayer WSe2 and MoSe2 are larger than those of MoS2 and WS2 is attributed to the smaller binding energies of monolayer WSe2, and MoSe2, which result in stronger intra-excitonic transitions. Therefore, monolayer WSe2 and MoSe2 should have greater potential for 2PA-involved photonic devices.

In our previous study [25], we determined the wavelength-dependent 2PA coefficients of monolayer MoS2 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 hvEg ∼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 α2.

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 MoS2, WS2, WSe2 and MoSe2 are calculated by taking Γf = 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 α2-value predicted by our model at 780 nm for monolayer WS2 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 MoS2 [39], and with the data at 1040 nm for monolayer WS2 [42].

 figure: Fig. 3

Fig. 3 Comparison between our model and experimental data for monolayer TMDCs. Our calculated 2PA spectra for monolayer MoS2, WS2, WSe2, and MoSe2 are shown by the solid blue line, the dash pink line, the dash orange line and the solid green line, respectively. Experimental results are shown by symbols (stars, circles, triangles, diamonds and squares).

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In the case of 1-3 layers of WS2 at 1030 nm [40] and films of multilayer WS2 at 1040 nm [42], the calculated α2-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 α2-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.

References and links

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2. V. I. Klimov, S. A. Ivanov, J. Nanda, M. Achermann, I. Bezel, J. A. McGuire, and A. Piryatinski, “Single-exciton optical gain in semiconductor nanocrystals,” Nature 447(7143), 441–446 (2007). [CrossRef]   [PubMed]  

3. F. Wang, G. Dukovic, L. E. Brus, and T. F. Heinz, “The optical resonances in carbon nanotubes arise from excitons,” Science 308(5723), 838–841 (2005). [CrossRef]   [PubMed]  

4. T. Takagahara, “Excitonic optical nonlinearity and exciton dynamics in semiconductor quantum dots,” Phys. Rev. B Condens. Matter 36(17), 9293–9296 (1987). [CrossRef]   [PubMed]  

5. P. Chen, C. Piermarocchi, and L. J. Sham, “Control of exciton dynamics in nanodots for quantum operations,” Phys. Rev. Lett. 87(6), 067401 (2001). [CrossRef]   [PubMed]  

6. X. Li, Y. Wu, D. Steel, D. Gammon, T. H. Stievater, D. S. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “An all-optical quantum gate in a semiconductor quantum dot,” Science 301(5634), 809–811 (2003). [CrossRef]   [PubMed]  

7. R. D. Schaller, V. M. Agranovich, and V. I. Klimov, “High-efficiency carrier multiplication through direct photogeneration of multi-excitons via virtual single-exciton states,” Nat. Phys. 1(3), 189–194 (2005). [CrossRef]  

8. J. S. Ross, P. Klement, A. M. Jones, N. J. Ghimire, J. Yan, D. G. Mandrus, T. Taniguchi, K. Watanabe, K. Kitamura, W. Yao, D. H. Cobden, and X. Xu, “Electrically tunable excitonic light-emitting diodes based on monolayer WSe2 p-n junctions,” Nat. Nanotechnol. 9(4), 268–272 (2014). [CrossRef]   [PubMed]  

9. G. Grancini, M. Maiuri, D. Fazzi, A. Petrozza, H.-J. Egelhaaf, D. Brida, G. Cerullo, and G. Lanzani, “Hot exciton dissociation in polymer solar cells,” Nat. Mater. 12(1), 29–33 (2013). [CrossRef]   [PubMed]  

10. W. Brütting, Physics of Organic Semiconductors (John Wiley & Sons, 2006) pp.349–369, chap.12.

11. L. Yang, J. Deslippe, C. H. Park, M. L. Cohen, and S. G. Louie, “Excitonic effects on the optical response of graphene and bilayer graphene,” Phys. Rev. Lett. 103(18), 186802 (2009). [CrossRef]   [PubMed]  

12. K. F. Mak and J. Shan, “Photonics and optoelectronics of 2D semiconductor transition metal dichalcogenides,” Nat. Photonics 10(4), 216–226 (2016). [CrossRef]  

13. Q. H. Wang, K. Kalantar-Zadeh, A. Kis, J. N. Coleman, and M. S. Strano, “Electronics and optoelectronics of two-dimensional transition metal dichalcogenides,” Nat. Nanotechnol. 7(11), 699–712 (2012). [CrossRef]   [PubMed]  

14. A. Miyata, A. Mitioglu, P. Plochocka, O. Portugall, J. T.-W. Wang, S. D. Stranks, H. J. Snaith, and R. J. Nicholas, “Direct measurement of the exciton binding energy and effective masses for charge carriers in organic-inorganic tri-halide perovskites,” Nat. Phys. 11(7), 582–587 (2015). [CrossRef]  

15. X. Wang, A. M. Jones, K. L. Seyler, V. Tran, Y. Jia, H. Zhao, H. Wang, L. Yang, X. Xu, and F. Xia, “Highly anisotropic and robust excitons in monolayer black phosphorus,” Nat. Nanotechnol. 10(6), 517–521 (2015). [CrossRef]   [PubMed]  

16. A. Ramasubramaniam, “Large excitonic effects in monolayers of molybdenum and tungsten dichalcogenides,” Phys. Rev. B 86(11), 115409 (2012). [CrossRef]  

17. H.-P. Komsa and A. V. Krasheninnikov, “Effects of confinement and environment on the electronic structure and exciton binding energy of MoS2 from first principles,” Phys. Rev. B 86(24), 241201 (2012). [CrossRef]  

18. T. C. Berkelbach, M. S. Hybertsen, and D. R. Reichman, “Theory of neutral and charged excitons in monolayer transition metal dichalcogenides,” Phys. Rev. B 88(4), 045318 (2013). [CrossRef]  

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References

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  1. V. M. Agranovich and V. Ginzburg, Crystal optics with spatial dispersion, and excitons (Springer Science & Business Media, 2013) pp. 271–300, chap. 5, Vol. 42.
  2. V. I. Klimov, S. A. Ivanov, J. Nanda, M. Achermann, I. Bezel, J. A. McGuire, and A. Piryatinski, “Single-exciton optical gain in semiconductor nanocrystals,” Nature 447(7143), 441–446 (2007).
    [Crossref] [PubMed]
  3. F. Wang, G. Dukovic, L. E. Brus, and T. F. Heinz, “The optical resonances in carbon nanotubes arise from excitons,” Science 308(5723), 838–841 (2005).
    [Crossref] [PubMed]
  4. T. Takagahara, “Excitonic optical nonlinearity and exciton dynamics in semiconductor quantum dots,” Phys. Rev. B Condens. Matter 36(17), 9293–9296 (1987).
    [Crossref] [PubMed]
  5. P. Chen, C. Piermarocchi, and L. J. Sham, “Control of exciton dynamics in nanodots for quantum operations,” Phys. Rev. Lett. 87(6), 067401 (2001).
    [Crossref] [PubMed]
  6. X. Li, Y. Wu, D. Steel, D. Gammon, T. H. Stievater, D. S. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “An all-optical quantum gate in a semiconductor quantum dot,” Science 301(5634), 809–811 (2003).
    [Crossref] [PubMed]
  7. R. D. Schaller, V. M. Agranovich, and V. I. Klimov, “High-efficiency carrier multiplication through direct photogeneration of multi-excitons via virtual single-exciton states,” Nat. Phys. 1(3), 189–194 (2005).
    [Crossref]
  8. J. S. Ross, P. Klement, A. M. Jones, N. J. Ghimire, J. Yan, D. G. Mandrus, T. Taniguchi, K. Watanabe, K. Kitamura, W. Yao, D. H. Cobden, and X. Xu, “Electrically tunable excitonic light-emitting diodes based on monolayer WSe2 p-n junctions,” Nat. Nanotechnol. 9(4), 268–272 (2014).
    [Crossref] [PubMed]
  9. G. Grancini, M. Maiuri, D. Fazzi, A. Petrozza, H.-J. Egelhaaf, D. Brida, G. Cerullo, and G. Lanzani, “Hot exciton dissociation in polymer solar cells,” Nat. Mater. 12(1), 29–33 (2013).
    [Crossref] [PubMed]
  10. W. Brütting, Physics of Organic Semiconductors (John Wiley & Sons, 2006) pp.349–369, chap.12.
  11. L. Yang, J. Deslippe, C. H. Park, M. L. Cohen, and S. G. Louie, “Excitonic effects on the optical response of graphene and bilayer graphene,” Phys. Rev. Lett. 103(18), 186802 (2009).
    [Crossref] [PubMed]
  12. K. F. Mak and J. Shan, “Photonics and optoelectronics of 2D semiconductor transition metal dichalcogenides,” Nat. Photonics 10(4), 216–226 (2016).
    [Crossref]
  13. Q. H. Wang, K. Kalantar-Zadeh, A. Kis, J. N. Coleman, and M. S. Strano, “Electronics and optoelectronics of two-dimensional transition metal dichalcogenides,” Nat. Nanotechnol. 7(11), 699–712 (2012).
    [Crossref] [PubMed]
  14. A. Miyata, A. Mitioglu, P. Plochocka, O. Portugall, J. T.-W. Wang, S. D. Stranks, H. J. Snaith, and R. J. Nicholas, “Direct measurement of the exciton binding energy and effective masses for charge carriers in organic-inorganic tri-halide perovskites,” Nat. Phys. 11(7), 582–587 (2015).
    [Crossref]
  15. X. Wang, A. M. Jones, K. L. Seyler, V. Tran, Y. Jia, H. Zhao, H. Wang, L. Yang, X. Xu, and F. Xia, “Highly anisotropic and robust excitons in monolayer black phosphorus,” Nat. Nanotechnol. 10(6), 517–521 (2015).
    [Crossref] [PubMed]
  16. A. Ramasubramaniam, “Large excitonic effects in monolayers of molybdenum and tungsten dichalcogenides,” Phys. Rev. B 86(11), 115409 (2012).
    [Crossref]
  17. H.-P. Komsa and A. V. Krasheninnikov, “Effects of confinement and environment on the electronic structure and exciton binding energy of MoS2 from first principles,” Phys. Rev. B 86(24), 241201 (2012).
    [Crossref]
  18. T. C. Berkelbach, M. S. Hybertsen, and D. R. Reichman, “Theory of neutral and charged excitons in monolayer transition metal dichalcogenides,” Phys. Rev. B 88(4), 045318 (2013).
    [Crossref]
  19. D. Y. Qiu, F. H. da Jornada, and S. G. Louie, “Optical spectrum of MoS2: many-body effects and diversity of exciton states,” Phys. Rev. Lett. 111(21), 216805 (2013).
    [Crossref] [PubMed]
  20. T. Cheiwchanchamnangij and W. R. L. Lambrecht, “Quasiparticle band structure calculation of monolayer, bilayer, and bulk MoS2,” Phys. Rev. B 85(20), 205302 (2012).
    [Crossref]
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  39. Y. Li, N. Dong, S. Zhang, X. Zhang, Y. Feng, K. Wang, L. Zhang, and J. Wang, “Giant two‐photon absorption in monolayer MoS2,” Laser Photonics Rev. 9(4), 427–434 (2015).
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2017 (1)

2016 (3)

2015 (10)

A. Miyata, A. Mitioglu, P. Plochocka, O. Portugall, J. T.-W. Wang, S. D. Stranks, H. J. Snaith, and R. J. Nicholas, “Direct measurement of the exciton binding energy and effective masses for charge carriers in organic-inorganic tri-halide perovskites,” Nat. Phys. 11(7), 582–587 (2015).
[Crossref]

X. Wang, A. M. Jones, K. L. Seyler, V. Tran, Y. Jia, H. Zhao, H. Wang, L. Yang, X. Xu, and F. Xia, “Highly anisotropic and robust excitons in monolayer black phosphorus,” Nat. Nanotechnol. 10(6), 517–521 (2015).
[Crossref] [PubMed]

H. Zhang, Y. Ma, Y. Wan, X. Rong, Z. Xie, W. Wang, and L. Dai, “Measuring the refractive index of highly crystalline monolayer MoS2 with high confidence,” Sci. Rep. 5(1), 8440 (2015).
[Crossref] [PubMed]

A. Kormányos, G. Burkard, M. Gmitra, J. Fabian, V. Zólyomi, N. D. Drummond, and V. Fal’ko, “k●p theory for two-dimensional transition metal dichalcogenide semiconductors,” 2D Materials 2(2), 022001 (2015).
[Crossref]

A. T. Hanbicki, M. Currie, G. Kioseoglou, A. L. Friedman, and B. T. Jonker, “Measurement of high exciton binding energy in the monolayer transition-metal dichalcogenides WS2 and WSe2,” Solid State Commun. 203, 16–20 (2015).
[Crossref]

J. H. Jiang and S. John, “Photonic architectures for equilibrium high-temperature Bose-Einstein condensation in dichalcogenide monolayers,” Sci. Rep. 4(1), 7432 (2015).
[Crossref] [PubMed]

G. Wang, I. C. Gerber, L. Bouet, D. Lagarde, A. Balocchi, M. Vidal, and B. Urbaszek, “Exciton states in monolayer MoSe2: impact on interband transitions,” 2D Materials 2(4), 045005 (2015).
[Crossref]

X. Zheng, Y. Zhang, R. Chen, X. Cheng, Z. Xu, and T. Jiang, “Z-scan measurement of the nonlinear refractive index of monolayer WS2.,” Opt. Express 23(12), 15616–15623 (2015).
[Crossref] [PubMed]

Y. Li, N. Dong, S. Zhang, X. Zhang, Y. Feng, K. Wang, L. Zhang, and J. Wang, “Giant two‐photon absorption in monolayer MoS2,” Laser Photonics Rev. 9(4), 427–434 (2015).
[Crossref]

S. Zhang, N. Dong, N. McEvoy, M. O’Brien, S. Winters, N. C. Berner, C. Yim, Y. Li, X. Zhang, Z. Chen, L. Zhang, G. S. Duesberg, and J. Wang, “Direct observation of degenerate two-photon absorption and its saturation in WS2 and MoS2 monolayer and few-layer films,” ACS Nano 9(7), 7142–7150 (2015).
[Crossref] [PubMed]

2014 (6)

Z. Ye, T. Cao, K. O’Brien, H. Zhu, X. Yin, Y. Wang, S. G. Louie, and X. Zhang, “Probing excitonic dark states in single-layer tungsten disulphide,” Nature 513(7517), 214–218 (2014).
[Crossref] [PubMed]

D. Wickramaratne, F. Zahid, and R. K. Lake, “Electronic and thermoelectric properties of few-layer transition metal dichalcogenides,” J. Chem. Phys. 140(12), 124710 (2014).
[Crossref] [PubMed]

R. J. Wu, M. L. Odlyzko, and K. A. Mkhoyan, “Determining the thickness of atomically thin MoS2 and WS2 in the TEM,” Ultramicroscopy 147, 8–20 (2014).
[Crossref] [PubMed]

A. Chernikov, T. C. Berkelbach, H. M. Hill, A. Rigosi, Y. Li, O. B. Aslan, D. R. Reichman, M. S. Hybertsen, and T. F. Heinz, “Exciton binding energy and nonhydrogenic Rydberg series in monolayer WS2.,” Phys. Rev. Lett. 113(7), 076802 (2014).
[Crossref] [PubMed]

D. Kozawa, R. Kumar, A. Carvalho, K. Kumar Amara, W. Zhao, S. Wang, M. Toh, R. M. Ribeiro, A. H. Castro Neto, K. Matsuda, and G. Eda, “Photocarrier relaxation pathway in two-dimensional semiconducting transition metal dichalcogenides,” Nat. Commun. 5(1), 4543 (2014).
[Crossref] [PubMed]

J. S. Ross, P. Klement, A. M. Jones, N. J. Ghimire, J. Yan, D. G. Mandrus, T. Taniguchi, K. Watanabe, K. Kitamura, W. Yao, D. H. Cobden, and X. Xu, “Electrically tunable excitonic light-emitting diodes based on monolayer WSe2 p-n junctions,” Nat. Nanotechnol. 9(4), 268–272 (2014).
[Crossref] [PubMed]

2013 (4)

G. Grancini, M. Maiuri, D. Fazzi, A. Petrozza, H.-J. Egelhaaf, D. Brida, G. Cerullo, and G. Lanzani, “Hot exciton dissociation in polymer solar cells,” Nat. Mater. 12(1), 29–33 (2013).
[Crossref] [PubMed]

D. Y. Qiu, F. H. da Jornada, and S. G. Louie, “Optical spectrum of MoS2: many-body effects and diversity of exciton states,” Phys. Rev. Lett. 111(21), 216805 (2013).
[Crossref] [PubMed]

T. C. Berkelbach, M. S. Hybertsen, and D. R. Reichman, “Theory of neutral and charged excitons in monolayer transition metal dichalcogenides,” Phys. Rev. B 88(4), 045318 (2013).
[Crossref]

D. Y. Qiu, F. H. da Jornada, and S. G. Louie, “Optical spectrum of MoS2: many-body effects and diversity of exciton states,” Phys. Rev. Lett. 111(21), 216805 (2013).
[Crossref] [PubMed]

2012 (5)

T. Cheiwchanchamnangij and W. R. L. Lambrecht, “Quasiparticle band structure calculation of monolayer, bilayer, and bulk MoS2,” Phys. Rev. B 85(20), 205302 (2012).
[Crossref]

A. Ramasubramaniam, “Large excitonic effects in monolayers of molybdenum and tungsten dichalcogenides,” Phys. Rev. B 86(11), 115409 (2012).
[Crossref]

H.-P. Komsa and A. V. Krasheninnikov, “Effects of confinement and environment on the electronic structure and exciton binding energy of MoS2 from first principles,” Phys. Rev. B 86(24), 241201 (2012).
[Crossref]

Q. H. Wang, K. Kalantar-Zadeh, A. Kis, J. N. Coleman, and M. S. Strano, “Electronics and optoelectronics of two-dimensional transition metal dichalcogenides,” Nat. Nanotechnol. 7(11), 699–712 (2012).
[Crossref] [PubMed]

T. Cheiwchanchamnangij and W. R. Lambrecht, “Quasiparticle band structure calculation of monolayer, bilayer, and bulk MoS2,” Phys. Rev. B 85(20), 205302 (2012).
[Crossref]

2009 (1)

L. Yang, J. Deslippe, C. H. Park, M. L. Cohen, and S. G. Louie, “Excitonic effects on the optical response of graphene and bilayer graphene,” Phys. Rev. Lett. 103(18), 186802 (2009).
[Crossref] [PubMed]

2007 (1)

V. I. Klimov, S. A. Ivanov, J. Nanda, M. Achermann, I. Bezel, J. A. McGuire, and A. Piryatinski, “Single-exciton optical gain in semiconductor nanocrystals,” Nature 447(7143), 441–446 (2007).
[Crossref] [PubMed]

2005 (2)

F. Wang, G. Dukovic, L. E. Brus, and T. F. Heinz, “The optical resonances in carbon nanotubes arise from excitons,” Science 308(5723), 838–841 (2005).
[Crossref] [PubMed]

R. D. Schaller, V. M. Agranovich, and V. I. Klimov, “High-efficiency carrier multiplication through direct photogeneration of multi-excitons via virtual single-exciton states,” Nat. Phys. 1(3), 189–194 (2005).
[Crossref]

2003 (1)

X. Li, Y. Wu, D. Steel, D. Gammon, T. H. Stievater, D. S. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “An all-optical quantum gate in a semiconductor quantum dot,” Science 301(5634), 809–811 (2003).
[Crossref] [PubMed]

2002 (1)

D. G. W. Parfitt and M. E. Portnoi, “The two-dimensional hydrogen atom revisited,” JOMP 43(10), 4681–4691 (2002).

2001 (1)

P. Chen, C. Piermarocchi, and L. J. Sham, “Control of exciton dynamics in nanodots for quantum operations,” Phys. Rev. Lett. 87(6), 067401 (2001).
[Crossref] [PubMed]

1987 (1)

T. Takagahara, “Excitonic optical nonlinearity and exciton dynamics in semiconductor quantum dots,” Phys. Rev. B Condens. Matter 36(17), 9293–9296 (1987).
[Crossref] [PubMed]

1985 (1)

1981 (1)

M. A. Kramer and R. W. Boyd, “Three-photon absorption in Nd-doped yttrium aluminum garnet,” Phys. Rev. B 23(3), 986–991 (1981).
[Crossref]

Achermann, M.

V. I. Klimov, S. A. Ivanov, J. Nanda, M. Achermann, I. Bezel, J. A. McGuire, and A. Piryatinski, “Single-exciton optical gain in semiconductor nanocrystals,” Nature 447(7143), 441–446 (2007).
[Crossref] [PubMed]

Agranovich, V. M.

R. D. Schaller, V. M. Agranovich, and V. I. Klimov, “High-efficiency carrier multiplication through direct photogeneration of multi-excitons via virtual single-exciton states,” Nat. Phys. 1(3), 189–194 (2005).
[Crossref]

Anghel, S.

Aslan, O. B.

A. Chernikov, T. C. Berkelbach, H. M. Hill, A. Rigosi, Y. Li, O. B. Aslan, D. R. Reichman, M. S. Hybertsen, and T. F. Heinz, “Exciton binding energy and nonhydrogenic Rydberg series in monolayer WS2.,” Phys. Rev. Lett. 113(7), 076802 (2014).
[Crossref] [PubMed]

Balocchi, A.

G. Wang, I. C. Gerber, L. Bouet, D. Lagarde, A. Balocchi, M. Vidal, and B. Urbaszek, “Exciton states in monolayer MoSe2: impact on interband transitions,” 2D Materials 2(4), 045005 (2015).
[Crossref]

Berkelbach, T. C.

A. Chernikov, T. C. Berkelbach, H. M. Hill, A. Rigosi, Y. Li, O. B. Aslan, D. R. Reichman, M. S. Hybertsen, and T. F. Heinz, “Exciton binding energy and nonhydrogenic Rydberg series in monolayer WS2.,” Phys. Rev. Lett. 113(7), 076802 (2014).
[Crossref] [PubMed]

T. C. Berkelbach, M. S. Hybertsen, and D. R. Reichman, “Theory of neutral and charged excitons in monolayer transition metal dichalcogenides,” Phys. Rev. B 88(4), 045318 (2013).
[Crossref]

Berner, N. C.

S. Zhang, N. Dong, N. McEvoy, M. O’Brien, S. Winters, N. C. Berner, C. Yim, Y. Li, X. Zhang, Z. Chen, L. Zhang, G. S. Duesberg, and J. Wang, “Direct observation of degenerate two-photon absorption and its saturation in WS2 and MoS2 monolayer and few-layer films,” ACS Nano 9(7), 7142–7150 (2015).
[Crossref] [PubMed]

Bezel, I.

V. I. Klimov, S. A. Ivanov, J. Nanda, M. Achermann, I. Bezel, J. A. McGuire, and A. Piryatinski, “Single-exciton optical gain in semiconductor nanocrystals,” Nature 447(7143), 441–446 (2007).
[Crossref] [PubMed]

Bikorimana, S.

Bouet, L.

G. Wang, I. C. Gerber, L. Bouet, D. Lagarde, A. Balocchi, M. Vidal, and B. Urbaszek, “Exciton states in monolayer MoSe2: impact on interband transitions,” 2D Materials 2(4), 045005 (2015).
[Crossref]

Boyd, R. W.

M. A. Kramer and R. W. Boyd, “Three-photon absorption in Nd-doped yttrium aluminum garnet,” Phys. Rev. B 23(3), 986–991 (1981).
[Crossref]

Brida, D.

G. Grancini, M. Maiuri, D. Fazzi, A. Petrozza, H.-J. Egelhaaf, D. Brida, G. Cerullo, and G. Lanzani, “Hot exciton dissociation in polymer solar cells,” Nat. Mater. 12(1), 29–33 (2013).
[Crossref] [PubMed]

Brus, L. E.

F. Wang, G. Dukovic, L. E. Brus, and T. F. Heinz, “The optical resonances in carbon nanotubes arise from excitons,” Science 308(5723), 838–841 (2005).
[Crossref] [PubMed]

Burkard, G.

A. Kormányos, G. Burkard, M. Gmitra, J. Fabian, V. Zólyomi, N. D. Drummond, and V. Fal’ko, “k●p theory for two-dimensional transition metal dichalcogenide semiconductors,” 2D Materials 2(2), 022001 (2015).
[Crossref]

Cao, T.

Z. Ye, T. Cao, K. O’Brien, H. Zhu, X. Yin, Y. Wang, S. G. Louie, and X. Zhang, “Probing excitonic dark states in single-layer tungsten disulphide,” Nature 513(7517), 214–218 (2014).
[Crossref] [PubMed]

Carvalho, A.

D. Kozawa, R. Kumar, A. Carvalho, K. Kumar Amara, W. Zhao, S. Wang, M. Toh, R. M. Ribeiro, A. H. Castro Neto, K. Matsuda, and G. Eda, “Photocarrier relaxation pathway in two-dimensional semiconducting transition metal dichalcogenides,” Nat. Commun. 5(1), 4543 (2014).
[Crossref] [PubMed]

Castro Neto, A. H.

D. Kozawa, R. Kumar, A. Carvalho, K. Kumar Amara, W. Zhao, S. Wang, M. Toh, R. M. Ribeiro, A. H. Castro Neto, K. Matsuda, and G. Eda, “Photocarrier relaxation pathway in two-dimensional semiconducting transition metal dichalcogenides,” Nat. Commun. 5(1), 4543 (2014).
[Crossref] [PubMed]

Cerullo, G.

G. Grancini, M. Maiuri, D. Fazzi, A. Petrozza, H.-J. Egelhaaf, D. Brida, G. Cerullo, and G. Lanzani, “Hot exciton dissociation in polymer solar cells,” Nat. Mater. 12(1), 29–33 (2013).
[Crossref] [PubMed]

Cheiwchanchamnangij, T.

T. Cheiwchanchamnangij and W. R. L. Lambrecht, “Quasiparticle band structure calculation of monolayer, bilayer, and bulk MoS2,” Phys. Rev. B 85(20), 205302 (2012).
[Crossref]

T. Cheiwchanchamnangij and W. R. Lambrecht, “Quasiparticle band structure calculation of monolayer, bilayer, and bulk MoS2,” Phys. Rev. B 85(20), 205302 (2012).
[Crossref]

Chen, P.

P. Chen, C. Piermarocchi, and L. J. Sham, “Control of exciton dynamics in nanodots for quantum operations,” Phys. Rev. Lett. 87(6), 067401 (2001).
[Crossref] [PubMed]

Chen, R.

Chen, Z.

S. Zhang, N. Dong, N. McEvoy, M. O’Brien, S. Winters, N. C. Berner, C. Yim, Y. Li, X. Zhang, Z. Chen, L. Zhang, G. S. Duesberg, and J. Wang, “Direct observation of degenerate two-photon absorption and its saturation in WS2 and MoS2 monolayer and few-layer films,” ACS Nano 9(7), 7142–7150 (2015).
[Crossref] [PubMed]

Cheng, X.

Chernikov, A.

A. Chernikov, T. C. Berkelbach, H. M. Hill, A. Rigosi, Y. Li, O. B. Aslan, D. R. Reichman, M. S. Hybertsen, and T. F. Heinz, “Exciton binding energy and nonhydrogenic Rydberg series in monolayer WS2.,” Phys. Rev. Lett. 113(7), 076802 (2014).
[Crossref] [PubMed]

Cobden, D. H.

J. S. Ross, P. Klement, A. M. Jones, N. J. Ghimire, J. Yan, D. G. Mandrus, T. Taniguchi, K. Watanabe, K. Kitamura, W. Yao, D. H. Cobden, and X. Xu, “Electrically tunable excitonic light-emitting diodes based on monolayer WSe2 p-n junctions,” Nat. Nanotechnol. 9(4), 268–272 (2014).
[Crossref] [PubMed]

Cohen, M. L.

L. Yang, J. Deslippe, C. H. Park, M. L. Cohen, and S. G. Louie, “Excitonic effects on the optical response of graphene and bilayer graphene,” Phys. Rev. Lett. 103(18), 186802 (2009).
[Crossref] [PubMed]

Coleman, J. N.

Q. H. Wang, K. Kalantar-Zadeh, A. Kis, J. N. Coleman, and M. S. Strano, “Electronics and optoelectronics of two-dimensional transition metal dichalcogenides,” Nat. Nanotechnol. 7(11), 699–712 (2012).
[Crossref] [PubMed]

Cui, Y.

Currie, M.

A. T. Hanbicki, M. Currie, G. Kioseoglou, A. L. Friedman, and B. T. Jonker, “Measurement of high exciton binding energy in the monolayer transition-metal dichalcogenides WS2 and WSe2,” Solid State Commun. 203, 16–20 (2015).
[Crossref]

da Jornada, F. H.

D. Y. Qiu, F. H. da Jornada, and S. G. Louie, “Optical spectrum of MoS2: many-body effects and diversity of exciton states,” Phys. Rev. Lett. 111(21), 216805 (2013).
[Crossref] [PubMed]

D. Y. Qiu, F. H. da Jornada, and S. G. Louie, “Optical spectrum of MoS2: many-body effects and diversity of exciton states,” Phys. Rev. Lett. 111(21), 216805 (2013).
[Crossref] [PubMed]

Dai, L.

H. Zhang, Y. Ma, Y. Wan, X. Rong, Z. Xie, W. Wang, and L. Dai, “Measuring the refractive index of highly crystalline monolayer MoS2 with high confidence,” Sci. Rep. 5(1), 8440 (2015).
[Crossref] [PubMed]

Deslippe, J.

L. Yang, J. Deslippe, C. H. Park, M. L. Cohen, and S. G. Louie, “Excitonic effects on the optical response of graphene and bilayer graphene,” Phys. Rev. Lett. 103(18), 186802 (2009).
[Crossref] [PubMed]

Dong, N.

N. Dong, Y. Li, S. Zhang, N. McEvoy, X. Zhang, Y. Cui, L. Zhang, G. S. Duesberg, and J. Wang, “Dispersion of nonlinear refractive index in layered WS2 and WSe2 semiconductor films induced by two-photon absorption,” Opt. Lett. 41(17), 3936–3939 (2016).
[Crossref] [PubMed]

Y. Li, N. Dong, S. Zhang, X. Zhang, Y. Feng, K. Wang, L. Zhang, and J. Wang, “Giant two‐photon absorption in monolayer MoS2,” Laser Photonics Rev. 9(4), 427–434 (2015).
[Crossref]

S. Zhang, N. Dong, N. McEvoy, M. O’Brien, S. Winters, N. C. Berner, C. Yim, Y. Li, X. Zhang, Z. Chen, L. Zhang, G. S. Duesberg, and J. Wang, “Direct observation of degenerate two-photon absorption and its saturation in WS2 and MoS2 monolayer and few-layer films,” ACS Nano 9(7), 7142–7150 (2015).
[Crossref] [PubMed]

Dorsinville, R.

Drummond, N. D.

A. Kormányos, G. Burkard, M. Gmitra, J. Fabian, V. Zólyomi, N. D. Drummond, and V. Fal’ko, “k●p theory for two-dimensional transition metal dichalcogenide semiconductors,” 2D Materials 2(2), 022001 (2015).
[Crossref]

Duesberg, G. S.

N. Dong, Y. Li, S. Zhang, N. McEvoy, X. Zhang, Y. Cui, L. Zhang, G. S. Duesberg, and J. Wang, “Dispersion of nonlinear refractive index in layered WS2 and WSe2 semiconductor films induced by two-photon absorption,” Opt. Lett. 41(17), 3936–3939 (2016).
[Crossref] [PubMed]

S. Zhang, N. Dong, N. McEvoy, M. O’Brien, S. Winters, N. C. Berner, C. Yim, Y. Li, X. Zhang, Z. Chen, L. Zhang, G. S. Duesberg, and J. Wang, “Direct observation of degenerate two-photon absorption and its saturation in WS2 and MoS2 monolayer and few-layer films,” ACS Nano 9(7), 7142–7150 (2015).
[Crossref] [PubMed]

Dukovic, G.

F. Wang, G. Dukovic, L. E. Brus, and T. F. Heinz, “The optical resonances in carbon nanotubes arise from excitons,” Science 308(5723), 838–841 (2005).
[Crossref] [PubMed]

Eda, G.

D. Kozawa, R. Kumar, A. Carvalho, K. Kumar Amara, W. Zhao, S. Wang, M. Toh, R. M. Ribeiro, A. H. Castro Neto, K. Matsuda, and G. Eda, “Photocarrier relaxation pathway in two-dimensional semiconducting transition metal dichalcogenides,” Nat. Commun. 5(1), 4543 (2014).
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Egelhaaf, H.-J.

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H. Zhang, Y. Ma, Y. Wan, X. Rong, Z. Xie, W. Wang, and L. Dai, “Measuring the refractive index of highly crystalline monolayer MoS2 with high confidence,” Sci. Rep. 5(1), 8440 (2015).
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Figures (3)

Fig. 1
Fig. 1 (a) Energy vs. optical absorption and (b) energy diagram in the momentum space near K point for a monolayer TMDC. The dashed lines represent the energy levels of excitons, and the blue curves show the valence bands (VBs) and conduction band (CB). Δ s o is the energy split due to the spin-orbital interaction. The auburn arrows show the 2PA process.
Fig. 2
Fig. 2 2PA coefficient α 2 (cm/MW) as functions of laser wavelength (x-axis) and Γ f (y-axis) for monolayer (a) MoS2, (b) WS2, (c) WSe2, and (d) MoSe2.
Fig. 3
Fig. 3 Comparison between our model and experimental data for monolayer TMDCs. Our calculated 2PA spectra for monolayer MoS2, WS2, WSe2, and MoSe2 are shown by the solid blue line, the dash pink line, the dash orange line and the solid green line, respectively. Experimental results are shown by symbols (stars, circles, triangles, diamonds and squares).

Tables (2)

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Table 1 Bloch wave functions for the valence and conduction bands

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Table 2 Parameters used in the calculation of 2PA coefficients.

Equations (8)

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σ ( 2 ) = ( 2 π c ) 2 2 ω 2 3 ε 0 2 h 2 | q μ f q μ q g ω q g ω | 2 g ( 2 ω ) ,
α 2 = σ ( 2 ) N h ν ,
  α 2 ( h ν ) = C N h ν [ E l o c E ] 4 [ | μ q g | 2 ( E A h ν ) 2 + ( Γ A / 2 ) 2 +   | μ q g | 2 ( E B h ν ) 2 + ( Γ B / 2 ) 2 ] × [ | μ f q | 2 Γ f / 2 π ( E f g 2 h ν ) 2 + ( Γ f / 2 ) 2 ]
  | ψ n = 1 , m = 0 ( r ) = 2 a B 2 π e 2 r a B .
| ψ e x c , 1 s ( r e ) = ψ c ( r e ) · ψ n = 1 ,   m = 0 ( r e r h )     d r h .
μ f q = ψ f ( r e r h ) | [ e ( r e r h ) ] | ψ e x c , 1 s ( r e r h ) ,
  ψ e x c ,     n p ,   m = ± 1 ( r = r e r h ) =   1 a 1 π ( n 1 / 2 ) 3 ( n 2 ) ! n ! e r ( n 1 / 2 ) a   ( 2 r ( n 1 / 2 ) a ) × L n 2 2 ( 2 r ( n 1 / 2 ) a ) e ± i φ .
| μ f q | 2 ( E f g 2 h ν ) 2 + ( Γ f / 2 ) 2 = n | W n   ψ e x c , n p ,   m = ± 1 ( r e r h ) | [ e ( r e r h ) ] | ψ e x c , 1 s ( r e r h ) | 2 ( E ( n p ) g 2 h ν ) 2 + ( Γ f / 2 ) 2 ,

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