Two-photon absorption arises from two-dimensional excitons

Feng Zhou; Jing Han Kua; Shunbin Lu; Wei Ji

doi:10.1364/OE.26.016093

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 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.

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.

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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)} = {(\frac{2 π}{c})}^{2} \frac{2 ω^{2}}{3 ε_{0}^{2} h^{2}} {| \sum_{q} \frac{μ_{f q} μ_{q g}}{ω_{q g} - ω} |}^{2} g (2 ω),

where 𝜔 is the angular frequency of an incident photon,

μ_{i j}

is the transition dipole moment and g is the line shape function. The 2PA coefficient is given by [27]:

α_{2} = \frac{σ^{(2)} N}{h ν},

where ℎ𝜈 (or

ℏ 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 (

α_{2}

) of monolayer TMDCs can be calculated by:

\begin{array}{l} α_{2} (h ν) = C N h ν {[\frac{E_{l o c}}{E}]}^{4} [\frac{{| μ_{q g} |}^{2}}{{(E_{A} - h ν)}^{2} + {(Γ_{A} / 2)}^{2}} + \frac{{| μ_{q g} |}^{2}}{{(E_{B} - h ν)}^{2} + {(Γ_{B} / 2)}^{2}}] \\ \times [\frac{{| μ_{f q} |}^{2} Γ_{f} / 2 π}{{(E_{f g} - 2 h ν)}^{2} + {(Γ_{f} / 2)}^{2}}] \end{array}

where

(\frac{E_{l o c}}{E}) = \frac{1}{3} (n_{0}^{2} + 2)

is the local-field correction factor;

E_{i}

and

Γ_{i}

refer to a certain energy level and its linewidth, respectively.

C

is a material-independent constant which has a value of

3.47 \times 10^{45}

in units such that

α_{2}

is in cm/MW,

N

is in cm⁻³,

h ν

,

E_{i j}

and

Γ_{i}

are in units of eV, and

μ_{i j}

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

α_{2}

-value based on band-to-band transition is considerably less than excitonic counterparts [25].

In the calculation of $μ_{q g}$ for the generated 1s-exciton (either A-exciton or B-exciton), we have $μ_{q g} = 〈 ψ_{e x c, 1 s} (r_{e}) | (- e r_{e}) | ψ_{G} (r_{e}) 〉$ , where e is the electron charge; $ψ_{e x c, 1 s} (r_{e})$ is the wave function of the excited state; and $ψ_{G} (r_{e})$ 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]:

| ψ_{n = 1, m = 0} (r) 〉 = \frac{2}{a_{B}} \sqrt{\frac{2}{π}} e^{- \frac{2 r}{a_{B}}} .

Here,

a_{B}

is 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:

| ψ_{e x c, 1 s} (r_{e}) 〉 = \int^{​} ψ_{c} (r_{e}) \cdot ψ_{n = 1, m = 0} (r_{e} - r_{h}) d r_{h} .

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

| ψ_{G} 〉

and

| ψ_{e x c, 1 s} 〉

are normalized in such a way:

〈 ψ_{e x c, 1 s} (r_{e}) ψ_{G} (r_{e}) | ψ_{e x c, 1 s} (r_{e}) ψ_{G} (r_{e}) 〉 = 1

C_{1}

and

C_{2}

are weight values and

C_{1}^{2} + C_{2}^{2} = 1

.

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

View Table | View all tables in this article

In the calculation of $μ_{f q}$ , we have

μ_{f q} = 〈 ψ_{f} (r_{e} - r_{h}) | [- e (r_{e} - r_{h})] | ψ_{e x c, 1 s} (r_{e} - r_{h}) 〉,

where

ψ_{f} (r_{e} - r_{h})

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

\begin{array}{l} ψ_{e x c, n p, m = \pm 1} (r = r_{e} - r_{h}) = \frac{1}{a} \sqrt{\frac{1}{π {(n - 1 / 2)}^{3}} \frac{(n - 2)!}{n!}} e^{- \frac{r}{(n - 1 / 2) a}} (\frac{2 r}{(n - 1 / 2) a}) \\ \times L_{n - 2}^{2} (\frac{2 r}{(n - 1 / 2) a}) e^{\pm i φ} . \end{array}

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;

L_{n - 2}^{2}

is the generalized Laguerre polynomial; and

φ

is the rotational angle in the orbital. As such, we can have

\frac{{| μ_{f q} |}^{2}}{{(E_{f g} - 2 h ν)}^{2} + {(Γ_{f} / 2)}^{2}} = \sum_{n}^{} \frac{{| 〈 W_{n} ψ_{e x c, n p, m = \pm 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}},

where W_n is the weight value with

W_{1}^{2} + W_{2}^{2} + W_{3}^{2} + \dots + W_{M A X}^{2} = 1

, (

W_{n}^{2}

= 1/MAX), and

E_{(n p) 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 MoS₂, WS₂, WSe₂ and MoSe₂. The only unknown parameter is the linewidth, $Γ_{f}$ . In Fig. 2, we plot the 2PA coefficient $α_{2}$ as functions of $Γ_{f}$ on the y-axis and laser wavelength on the x-axis.

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

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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.

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

With the increase in the bandwidth $Γ_{f}$ , higher order np-excitons make more contributions towards the 2PA process and hence, decrease and blue-shift the maximal $α_{2}$ -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 $h v ∕ E g$ ∼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 $Γ_{f}$ = 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 $α_{2}$ -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].

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 $α_{2}$ -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 $α_{2}$ -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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	$\| Ψ_{G} (r_{e})$	$\| Ψ_{c} (r_{e})$
MoS₂	$C_{1} \| Ψ_{4, 2, - 2}^{M o} (r_{e}) + C_{2} \| Ψ_{3, 1, - 1}^{S} (r_{e})$	$C_{1} \| Ψ_{4, 2, 0}^{M o} (r_{e}) + C_{2} \| Ψ_{3, 1, 1}^{S} (r_{e})$
WS₂	$C_{1} \| Ψ_{5, 2, - 2}^{W} (r_{e}) + C_{2} \| Ψ_{3, 1, - 1}^{S} (r_{e})$	$C_{1} \| Ψ_{5, 2, 0}^{W} (r_{e}) + C_{2} \| Ψ_{3, 1, 1}^{S} (r_{e})$
WSe₂	$C_{1} \| Ψ_{5, 2, - 2}^{W} (r_{e}) + C_{2} \| Ψ_{4, 1, - 1}^{S e} (r_{e})$	$C_{1} \| Ψ_{5, 2, 0}^{W} (r_{e}) + C_{2} \| Ψ_{4, 1, 1}^{S e} (r_{e})$
MoSe₂	$C_{1} \| Ψ_{4, 2, - 2}^{M o} (r_{e}) + C_{2} \| Ψ_{4, 1, - 1}^{S e} (r_{e})$	$C_{1} \| Ψ_{4, 2, 0}^{M o} (r_{e}) + C_{2} \| Ψ_{4, 1, 1}^{S e} (r_{e})$

	MoS₂	WS₂	WSe₂	MoSe₂
Density of Active Atoms (N)	1.84 × 10²² cm⁻³	1.88 × 10²² cm⁻³	1.53 × 10²² cm⁻³	1.52 × 10²² cm⁻³
Lattice Constant, *a₁ or a₂*	3.16 Å	3.16 Å	3.29 Å	3.29 Å
Monolayer Thickness, L	6.3 Å	6.14 Å	7.0 Å	7.0 Å
Refractive Index, n₀	1.84	1.82	1.84	2.10
Relative Permittivity, ϵ_r	3.40	3.32	3.39	4.63
Band Gap Energy, E_g	2.7 eV	2.9 eV	2.53 eV	2.33 eV
Binding Energy *E_b*	0.897 eV	0.830eV	0.790 eV	0.751 eV
Effective Mass	0.195 m_e	0.171m_e	0.171 m_e	0.204 m_e
Effective Bohr Radius, $a_{B}$	9.3 Å	10.3 Å	10.5 Å	10.1 Å
A Exciton Binding Energy, E_A	1.9 eV	2.078 eV	1.74 eV	1.667 eV
B Exciton Binding Energy, E_B	2.06 eV	2.475 eV	2.162 eV	1.885 eV
ns or np-Excitons Binding Energies, E_b(n)	ns: $\frac{0.224}{{(n - 1 / 2)}^{2}}$ np: $\frac{0.448}{{(n - 1 / 2)}^{2}}$	ns: $\frac{0.207}{{(n - 1 / 2)}^{2}}$ np: $\frac{0.401}{{(n - 1 / 2)}^{2}}$	ns: $\frac{0.207}{{(n - 1 / 2)}^{2}}$ np: $\frac{0.395}{{(n - 1 / 2)}^{2}}$	ns: $\frac{0.187}{{(n - 1 / 2)}^{2}}$ np: $\frac{0.375}{{(n - 1 / 2)}^{2}}$
$Γ_{A} = Γ_{B}$	0.06 eV	0.06 eV	0.06 eV	0.06 eV
Orbitals	Mo: 4d S: 3p	W: 5d S: 3p	W: 5d Se: 4p	Mo: 4d Se: 4p
$Δ_{s o}$	0.16 eV	0.397 eV	0.422 eV	0.218 eV

	$\| Ψ_{G} (r_{e})$	$\| Ψ_{c} (r_{e})$
MoS₂	$C_{1} \| Ψ_{4, 2, - 2}^{M o} (r_{e}) + C_{2} \| Ψ_{3, 1, - 1}^{S} (r_{e})$	$C_{1} \| Ψ_{4, 2, 0}^{M o} (r_{e}) + C_{2} \| Ψ_{3, 1, 1}^{S} (r_{e})$
WS₂	$C_{1} \| Ψ_{5, 2, - 2}^{W} (r_{e}) + C_{2} \| Ψ_{3, 1, - 1}^{S} (r_{e})$	$C_{1} \| Ψ_{5, 2, 0}^{W} (r_{e}) + C_{2} \| Ψ_{3, 1, 1}^{S} (r_{e})$
WSe₂	$C_{1} \| Ψ_{5, 2, - 2}^{W} (r_{e}) + C_{2} \| Ψ_{4, 1, - 1}^{S e} (r_{e})$	$C_{1} \| Ψ_{5, 2, 0}^{W} (r_{e}) + C_{2} \| Ψ_{4, 1, 1}^{S e} (r_{e})$
MoSe₂	$C_{1} \| Ψ_{4, 2, - 2}^{M o} (r_{e}) + C_{2} \| Ψ_{4, 1, - 1}^{S e} (r_{e})$	$C_{1} \| Ψ_{4, 2, 0}^{M o} (r_{e}) + C_{2} \| Ψ_{4, 1, 1}^{S e} (r_{e})$

	MoS₂	WS₂	WSe₂	MoSe₂
Density of Active Atoms (N)	1.84 × 10²² cm⁻³	1.88 × 10²² cm⁻³	1.53 × 10²² cm⁻³	1.52 × 10²² cm⁻³
Lattice Constant, *a₁ or a₂*	3.16 Å	3.16 Å	3.29 Å	3.29 Å
Monolayer Thickness, L	6.3 Å	6.14 Å	7.0 Å	7.0 Å
Refractive Index, n₀	1.84	1.82	1.84	2.10
Relative Permittivity, ϵ_r	3.40	3.32	3.39	4.63
Band Gap Energy, E_g	2.7 eV	2.9 eV	2.53 eV	2.33 eV
Binding Energy *E_b*	0.897 eV	0.830eV	0.790 eV	0.751 eV
Effective Mass	0.195 m_e	0.171m_e	0.171 m_e	0.204 m_e
Effective Bohr Radius, $a_{B}$	9.3 Å	10.3 Å	10.5 Å	10.1 Å
A Exciton Binding Energy, E_A	1.9 eV	2.078 eV	1.74 eV	1.667 eV
B Exciton Binding Energy, E_B	2.06 eV	2.475 eV	2.162 eV	1.885 eV
ns or np-Excitons Binding Energies, E_b(n)	ns: $\frac{0.224}{{(n - 1 / 2)}^{2}}$ np: $\frac{0.448}{{(n - 1 / 2)}^{2}}$	ns: $\frac{0.207}{{(n - 1 / 2)}^{2}}$ np: $\frac{0.401}{{(n - 1 / 2)}^{2}}$	ns: $\frac{0.207}{{(n - 1 / 2)}^{2}}$ np: $\frac{0.395}{{(n - 1 / 2)}^{2}}$	ns: $\frac{0.187}{{(n - 1 / 2)}^{2}}$ np: $\frac{0.375}{{(n - 1 / 2)}^{2}}$
$Γ_{A} = Γ_{B}$	0.06 eV	0.06 eV	0.06 eV	0.06 eV
Orbitals	Mo: 4d S: 3p	W: 5d S: 3p	W: 5d Se: 4p	Mo: 4d Se: 4p
$Δ_{s o}$	0.16 eV	0.397 eV	0.422 eV	0.218 eV

Two-photon absorption arises from two-dimensional excitons

Abstract

1. Introduction

2. Theoretical calculation

3. Results and discussion

4. Conclusion

Funding

References and links

Cited By

Figures (3)

Tables (2)

Equations (8)

Optics Express