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Broadband sum frequency generation spectroscopy of dark exciton states in hBN-encapsulated monolayer WSe2

Satoshi Kusaba, Yoshiki Katagiri, Kenji Watanabe, Takashi Taniguchi, Kazuhiro Yanagi, Nobuko Naka, and Koichiro Tanaka

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Abstract

We demonstrate that broadband sum frequency generation (SFG) spectroscopy based on a partially incoherent supercontinuum light source can elucidate dark p-series excitons in monolayer WSe2 encapsulated between hexagonal boron nitride (hBN) slabs. The observed 2p exciton peak energy is a few meV higher than that predicted by the Rytova-Keldysh potential model, which is originated from the Berry phase effect. Interestingly, although the radiative relaxation of the 2p exciton is weaker, the 2p exciton peak is broader than the 1s and 2s peaks, which indicates its faster dephasing than the 1s and 2s excitons. Measuring the excitation intensity and temperature dependence, we clarified that this broader linewidth is not caused by excitation- or phonon-induced dephasing, but rather by exciton-electron scattering.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Two-dimensional (2D) transition metal dichalcogenides (TMDs) are excellent platforms for exciton physics. Their monolayers are semiconductors with direct bandgaps at two inequivalent K and K’ valleys in the first Brillouin zone [1–4] and have strong Coulomb interactions due to the reduced dielectric screening [5–11]. The electrons and holes form stable excitons with large binding energies of a few hundreds meV [5–11]. Hence, the excitons have important roles in optical properties even at room temperature. The reduced dielectric screening in monolayers also modifies the Coulomb potential shape [5–13]. This produces non-hydrogenic exciton Rydberg series [5–11] and their unique level ordering [12,13].

Recently, hexagonal boron nitride (hBN) encapsulation technique, i.e., sandwiching of monolayer TMDs between two slabs of hBN was developed [7–11,14–19]. hBN is a van der Waals layered insulator and is transparent in the visible region. This technique provides high flatness of the monolayer samples and protects them from gas adsorption. Encapsulated samples have reduced inhomogeneous excitonic linewidths and enable us to observe clear exciton Rydberg series. The s-series of the exciton Rydberg states up to 11s have been observed by photoluminescence, absorption, or photocurrent spectroscopy under a strong magnetic field [7–11]. These findings contributed greatly to the determination of exciton parameters such as exciton mass, binding energy, and dielectric parameters. On the other hand, p-series excitons, which have angular-momentum-dependent splitting induced by valley-dependent Berry phase, are expected as a pathway to detect and manipulate the quantum states and a new platform of Berry phase physics [12,15,16,20–23]. The correct identification and quantitative measurement of p-series exciton properties such as binding energy and linewidth using high-quality samples are important issues. However, p-series excitons have not been investigated sufficiently, whereas the s-series excitons have been well studied by conventional methods. This is because 1-photon absorption or emission of the p-series excitons are forbidden or partially allowed by the trigonal warping and are too weak to be observed by 1-photon absorption or emission spectroscopies [24,25].

In this study, we performed broadband sum frequency generation (SFG) spectroscopy [26] based on a partially incoherent supercontinuum light source to investigate dark p-series excitons in hBN-encapsulated monolayer WSe2. In addition to the s-series excitons observed in photoluminescence (PL) and photoluminescence excitation (PLE) spectra, 2p and 3p exciton peaks were observed at 7.0 ± 0.3 and 3.6 ± 0.7 meV lower than the 2s and 3s peaks, respectively. The 2p peak energy deviates from that of the Rytova-Keldysh potential model [5–11,27,28], which is considered to reflect the Berry phase effect [16,21–23]. Interestingly, although the radiative relaxation of the 2p excitons is weaker, the 2p exciton peak is broader than the 1s and 2s of the s-series exciton peak. Excitation intensity and temperature dependences indicate that the broader linewidth of the 2p exciton is not caused by excitation- or phonon-induced dephasing, but rather by exciton-electron scattering.

2. Broadband sum frequency generation spectroscopy

We have further developed the previous study on supercontinuum second harmonic generation (SHG) spectroscopy [26] for the case of using an incoherent supercontinuum light source and adopted this technique to detect the second order nonlinear optical response from p-series excitons. As a similar second-order nonlinear spectroscopy, scanning SHG spectroscopy [29–33] is a well-known method, where a tunable excitation light source with narrow linewidth is used similarly to 2-photon excitation PLE measurement. The SHG efficiency spectrum is measured by scanning the excitation laser wavelength as shown in Fig. 1(a). In contrast, in the case of supercontinuum SHG spectroscopy, one irradiates the sample with a broadband light pulse and obtains the second-order nonlinear signal dispersed spectrally by a spectrometer as shown in Fig. 1(b) [26]. In this condition, not only SHG but also SFG emission between various frequency pairs in the excitation laser spectrum occur simultaneously. Therefore, in this paper, we refer to this type of nonlinear spectroscopy as broadband SFG spectroscopy. Compared with the scanning method, broadband SFG spectroscopy enables us to obtain the spectrum all at once. The spectral resolution is determined by the spectrometer, whereas in the scanning method the laser linewidth determines the resolution.

 figure: Fig. 1.

Fig. 1. (a)(b) Concepts of (a) scanning SHG spectroscopy and (b) broadband SFG spectroscopy. (c) Simplified energy diagram of excitonic states, incident laser, and broadband SFG signal. Red allows show the SFG transition path.

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Let us consider the broadband SFG spectrum of exciton states in atomic layer materials. It is not necessary to consider phase matching because the thickness of atomic layers is sufficiently smaller than the coherent length. Hence, the SFG emission intensity with a frequency ω is directly proportional to the square of the second order nonlinear polarization ${\widetilde P^{(2)}}(\omega )$ given by:

$$\widetilde P_i^{(2)}(\omega ) = \frac{{{\varepsilon _0}}}{{2\pi }}\int {\widetilde \chi _{ijk}^{(2)}(\omega ;\omega ^{\prime}){{\widetilde E}_j}(\omega ^{\prime}){{\widetilde E}_k}(\omega - \omega ^{\prime})\textrm{d} \omega ^{\prime}}, $$
where $\widetilde \chi _{ijk}^{(2)}(\omega ;\omega ^{\prime})$ is the second order nonlinear susceptibility and $\widetilde E(\omega ^{\prime})$ is the excitation electric field. $\widetilde I(\omega ) = {{{{|{\widetilde E(\omega )} |}^2}} / {2Z}}$ gives a power spectrum of the excitation laser, where Z is the characteristic impedance. The indices i, j, and k refer to the components of the nonlinear polarizations and the fields. From perturbation theory, the nonlinear dielectric susceptibility is given as [33]:
$$\widetilde \chi _{ijk}^{(2)}(\omega ;\omega ^{\prime}) \propto \left\langle \textrm{g} \right|{\widehat d_i}\sum\limits_n {\frac{{|{{\psi_n}} \rangle \left\langle {{\psi_n}} \right|}}{{{\mathrm{\epsilon }_n} - \hbar \omega - i{\gamma _n}}}} {\widehat d_j}\sum\limits_v {\frac{{|{{\psi_v}} \rangle \left\langle {{\psi_v}} \right|}}{{{\mathrm{\epsilon }_v} - \hbar \omega ^{\prime} - i{\gamma _v}}}{{\widehat d}_k}} |\textrm{g} \rangle, $$
Here, $|\psi_v \rangle$ is an n-th exciton state with energy ${\mathrm{\epsilon }_n}$ and damping ${\gamma _n}$. $|\textrm{g} \rangle$ and $|{{\psi_v}} \rangle$ are the ground and virtual states, respectively, and $\widehat d$ is a dipole moment operator. The emission frequency region of interest is around the exciton energy ${\mathrm{\epsilon }_n}$. Hence, the ${({\mathrm{\epsilon }_n} - \hbar \omega - i{\gamma _n})^{ - 1}}$ term is dominant. In contrast, since the excitation photon energy $\hbar \omega ^{\prime}$ is around half of the virtual exciton energy ${\mathrm{\epsilon }_v}$ as shown in Fig. 1(c), ${({\mathrm{\epsilon }_v} - \hbar \omega ^{\prime} - i{\gamma _v})^{ - 1}}$ slowly changes. Now we treat ${({\mathrm{\epsilon }_v} - \hbar \omega ^{\prime} - i{\gamma _v})^{ - 1}}$ as a constant and then the frequency dependence of the nonlinear dielectric susceptibility is simplified as a function of ω:
$$\widetilde \chi _{ijk}^{(2)}(\omega ;\omega ^{\prime}) \propto \sum\limits_n {\frac{{{A_{nijk}}}}{{{\mathrm{\epsilon }_n} - \hbar \omega - i{\gamma _n}}}} = \widetilde \chi _{ijk}^{(2)}(\omega ), $$
where Anijk corresponds to the matrix elements:
$${A_{nijk}} \propto \sum\limits_v {\left\langle \textrm{g} \right|{{\widehat d}_i}|{{\psi_n}} \rangle \left\langle {{\psi_n}} \right|{{\widehat d}_j}|{{\psi_v}} \rangle \left\langle {{\psi_v}} \right|{{\widehat d}_k}|\textrm{g} \rangle }. $$
The three matrix elements in Eq. (4) determine the optical selection rules, which we summarize for monolayer TMDs in Section 4.2. If $|{{\mathrm{\epsilon }_m} - \hbar \omega } |\gg {\gamma _m}$, ${({\mathrm{\epsilon }_m} - \hbar \omega - i{\gamma _m})^{ - 1}}$ in Eq. (3) can be treated as a real value ${({\mathrm{\epsilon }_m} - \hbar \omega )^{ - 1}}$. Such non-resonant terms give a background $bg = \sum\nolimits_{m\textrm{:non - resonant}} {{A_{mijk}}{{({\mathrm{\epsilon }_m} - \hbar \omega )}^{ - 1}}}$ and the nonlinear dielectric susceptibility becomes:
$$\widetilde \chi _{ijk}^{(2)}(\omega ;\omega ^{\prime}) = \widetilde \chi _{ijk}^{(2)}(\omega ) = bg + \sum\limits_{n:\textrm{resonant}} {\frac{{{A_{nijk}}}}{{{\mathrm{\epsilon }_n} - \hbar \omega - i{\gamma _n}}}}, $$
where Σm:non-resonant and Σn:resonant are the summation for non-resonant and resonant excitons, respectively. Then, Eq. (1) becomes:
$$\widetilde P_i^{(2)}(\omega ) = \frac{{{\varepsilon _0}}}{{2\pi }}\widetilde \chi _{ijk}^{(2)}(\omega )\int {{{\widetilde E}_j}(\omega ^{\prime}){{\widetilde E}_k}(\omega - \omega ^{\prime})\textrm{d} \omega ^{\prime}}, $$
and the broadband SFG spectrum ${\widetilde I^{(\textrm{SFG})}}(\omega )$ is obtained as:
$$\widetilde I_i^{(\textrm{SFG})}(\omega ) \propto {|{\widetilde P_i^{(2)}(\omega )} |^2} = \frac{{\varepsilon _0^2}}{{4{\pi ^2}}}{|{\widetilde \chi_{ijk}^{(2)}(\omega )} |^2}{\left|{\int {{{\widetilde E}_j}(\omega^{\prime}){{\widetilde E}_k}(\omega - \omega^{\prime})\textrm{d} \omega^{\prime}} } \right|^2}. $$

It is not straightforward to evaluate the integral of the electric field in Eq. (7) because it needs not only the excitation power spectrum but also the phase spectrum of the excitation light. Stiehm et al. [26] evaluated the integral term by measuring the non-resonant SFG emission from hBN, which was too weak to observe in our experiment probably because of the lower excitation electric field. Here, we developed a different approach to treating the integral term. We measured the auto-correlation of the excitation laser, which suggests that our excitation laser is partially incoherent probably due to the noise-seeded stimulated Raman scattering in the nonlinear fiber [34,35] (see Section 3 in Supplement 1). In the case of incoherent excitation light of which electric field is treated as a Gaussian random process, the square of the electric field integral in Eq. (7) should be replaced by a power integral as (see Section 4 in Supplement 1) [36]:

$${\left|{\int {{{\widetilde E}_j}(\omega^{\prime}){{\widetilde E}_k}(\omega - \omega^{\prime})d \omega^{\prime}} } \right|^2} \propto \int {{{\widetilde I}_j}(\omega ^{\prime}){{\widetilde I}_k}(\omega - \omega ^{\prime})\textrm{d} \omega ^{\prime}}. $$
In addition, we examined the effect of the excitation power spectrum by measuring the broadband SFG spectrum in a β-BaB2O4 (BBO) crystal under irradiation with the excitation pulses spectrally limited by an edge filter (see Section 5 in Supplement 1). Our results show that a fraction of the excitation spectrum with a width Δω of a few tens meV around ω/2 contributes to the SFG signal at a frequency ω. Hence, assuming a slowly varying excitation spectrum, the integral term is naively replaced as:
$$\int_{{{(\omega - \Delta \omega )} / 2}}^{{{(\omega + \Delta \omega )} / 2}} {{{\widetilde I}_j}(\omega ^{\prime}){{\widetilde I}_k}(\omega - \omega ^{\prime})\textrm{d} \omega ^{\prime}} \cong {\widetilde I_j}\left( {\frac{\omega }{2}} \right){\widetilde I_k}\left( {\frac{\omega }{2}} \right)\Delta \omega. $$
Finally, the broadband SFG spectrum ${\widetilde I^{(\textrm{SFG})}}(\omega )$ is approximately given as:
$$\widetilde I_i^{(SFG)}(\omega ) \propto {|{\widetilde \chi_{ijk}^{(2)}(\omega )} |^2}{\widetilde I_j}\left( {\frac{\omega }{2}} \right){\widetilde I_k}\left( {\frac{\omega }{2}} \right)\Delta \omega. $$
This formula indicates that, if the excitation spectrum $\widetilde I(\omega ^{\prime})$ is slowly varying, the broadband SFG spectrum is insignificantly affected by the excitation spectrum. This is one advantage of using such an incoherent light source for broadband SFG spectroscopy. We confirmed that the excitation spectrum is slowly varying (see Fig. S4(a) in Supplement 1) and, in this study, we simply assumed that the broadband SFG spectrum is proportional to the squared nonlinear susceptibility ${|{\widetilde \chi_{ijk}^{(2)}(\omega )} |^2}$. We note that Δω is much larger than the typical excitation linewidth of scanning SHG spectroscopy of ∼1 meV [29,31,32]. Hence, it is reasonable to refer to the nonlinear emission as SFG rather than as SHG.

3. Experiments

Here we briefly explain our samples and experimental setups. The details are described in Sections 1 and 2 in Supplement 1. We prepared hBN-encapsulated monolayer WSe2 samples by dry transfer of mechanically exfoliated monolayer WSe2 and thin layer hBN crystals. The monolayer WSe2 was characterized by optical contrast and photoluminescence measurements. The sample was set on the cold finger of a Gifford-McMahon cooler and cooled down to 16 K. We used a commercial picosecond supercontinuum white laser (Fianium WhiteLase, total power 4 W, 40 MHz, 0.52∼3.1 eV) as an excitation light source. In PL and PLE measurements, we used a home-built subtractive-mode double monochromator to facilitate an excitation with a narrow linewidth of ∼1 nm (∼3 meV). All PL and PLE spectra were measured by using unpolarized excitation light. In broadband SFG spectroscopy, excitation was performed with broadband near-infrared (NIR) light (0.52∼1.18 eV) cut out by a set of a dichroic mirror and edge filters. Typical excitation power was 40 mW. The excitation light was focused on the sample by reflective microscope objective lens (Newport 50105-02, x15, infinite BFL, N.A. 0.4). The emission from the sample was collected by the same objective and detected by a grating spectrograph (HORIBA Jobin Yvon, TRIAX320) equipped with a liquid nitrogen cooled charge-coupled device camera (CCD3000). The typical spectral resolution of the spectrograph was ∼0.3 nm (∼1 meV).

4. Results and discussions

4.1 PL and PLE spectra

Figure 2(a) shows the crystal structure of monolayer TMDs. They have honeycomb lattice structures similar to that of graphene but without inversion symmetry. The armchair and zigzag directions are shown by arrows in Fig. 2(a). Figure 2(b) shows the first Brillouin zone of monolayer TMDs. In monolayer TMDs, both conduction and valence bands have valleys at K and K’ points. Due to the broken inversion symmetry, these two valleys are inequivalent and distinguishable. Hence, there are two types of direct excitons; one type of the exciton consists of an electron and a hole at K valley, and the other type consists of an electron and a hole at K’ valley. In this paper, we refer to them as a K valley exciton and a K’ valley exciton, respectively. Figure 2(c) shows the schematic energy diagram of K valley excitons with 1s, 2p, and 2s orbitals in monolayer WSe2. Here we label the two kinds of p orbitals with different angular momenta +ħ and −ħ as p+ and p−, respectively. As shown in Fig. 2(c), s-series excitons are dipole-active and can be accessed by PL and 1-photon PLE spectroscopy. In contrast, the p+ orbital of K valley exciton and p− of K’ valley exciton are partially allowed by the trigonal warping [24,25], but the transitions are considerably weak. The other two types of the p-series excitons are forbidden. Hence, in this paper, we treat the p-series excitons as dark states.

 figure: Fig. 2.

Fig. 2. (a) Top view of crystal structure of monolayer TMD. Red and green arrows indicate armchair and zigzag directions. (b) Schematics of the first Brillouin zone in monolayer TMD. (c) Schematics of energy diagram for K valley excitons in monolayer WSe2. Allowed PL, SFG and 2P-PL processes are shown by arrows. Red and blue arrows indicate transitions with one right- and left-hand circularly polarized photon, respectively. Green arrow indicates transition with out-of-plane polarized PL emission. Black wiggly arrows represent relaxation processes. Green dashed lines represent virtual states. Energy separations between levels are not to scale. (d) PL spectrum in hBN encapsulated monolayer WSe2 under excitation with photon energy of 1.97 eV at T=16 K. The peak labeled BX is originating from biexciton. (e) PLE spectrum observed at dark exciton peak (1.66 eV). (f) Emission spectrum under unpolarized NIR excitation with photon energy of 0.52∼1.18 eV. Inset shows power P dependence of emission intensity I of 1s, A and 2s peaks. Solid lines represent fitting results using a power law Pα. (g)(h) Polarization-resolved emission spectra under NIR excitation with polarization parallel to (g) armchair and (h) zigzag directions.

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Figure 2(d) shows a PL spectrum obtained at T=16 K under band-to-band excitation with a photon energy of 1.97 eV. We observed two peaks at 1.698 and 1.827 eV originating from the 1s and 2s exciton states. Figure 2(e) shows a PLE spectrum. Here, as shown in Fig. 2(c), we monitored the emission at 1.659 eV from the spin-forbidden dark (or often called “grey”) exciton with out-of-plane polarized dipole moment [17,18], which is shown in Fig. S1(d) in Supplement 1, in order to avoid the resonant Raman scattering effect [37]. In addition to the 1s and 2s peaks at 1.698 and 1.828 eV, a peak corresponding to the 3s state was observed at 1.850 eV. In Table 1, we summarized the fit results obtained using Lorentzian functions.

Tables Icon

Table 1. Fitting results obtained from PL, PLE, and broadband SFG spectra at T=16 K shown in Figs. 2(d) and (e), and Fig. 3, and the Rytova-Keldysh potential (RKP) model calculation results. For the RKP model calculation, we used the reduced mass μ=0.20m0 and obtained fit parameters Ebg=1.8660 ± 0.0004 eV, κ=4.46 ± 0.05, and r0=4.25 ± 0.07 nm.

View Table

The binding energies of s-series excitons have been explained using the Rytova-Keldysh potential, which describes the Coulomb interaction of 2D carriers confined between two dielectric slabs [5,7–10,27,28]. The Rytova-Keldysh potential is expressed as a function of the electron-hole distance r as:

$${V_{\textrm{RK}}}(r) ={-} \frac{{{e^2}}}{{8{\varepsilon _0}{r_0}}}\left[ {{H_0}\left( {\frac{{\kappa r}}{{{r_0}}}} \right) - {Y_0}\left( {\frac{{\kappa r}}{{{r_0}}}} \right)} \right], $$
where H0 and Y0 are the zeroth-order Struve and Neuman functions, respectively. κ is the dielectric constant of the surrounding material (hBN). ${r_0} = 2\pi {\chi _{2D}}$ is the screening length, where ${\chi _{2D}}$ is the 2D polarizability of monolayer WSe2. The screening effect of residual carriers in monolayer WSe2 is included by the parameter r0; if the number of residual carriers increases, the screening effect is enhanced and r0 becomes larger. We carried out numerical calculation of exciton binding energies for various r0 and κ values, and then determined the fitting parameters r0, κ and the bandgap energy Ebg to best reproduce the observed exciton peak energies. The 1s, 2s, and 3s exciton peak energies obtained by the PLE measurement yielded r0=4.25 ± 0.07 nm and κ=4.46 ± 0.05. These values are in close agreement with those in previous reports [7–9]. Using the obtained parameters, the Rytova-Keldysh potential model also predicts that the 2p (3p) exciton peak exists at ∼11 meV (∼3 meV) below the 2s (3s) exciton peak.

4.2 Broadband SFG spectrum

The selection rules of resonant SFG processes can be obtained by evaluating the matrix elements in Eq. (4). From the symmetry analysis using the group theory (see Section 6 in Supplement 1 for details), the following transition paths including three electric-dipole transitions are allowed as resonant SFG processes [24,25,33].

$$|\textrm{g} \rangle \to {|{\textrm{p}_\textrm{K}^\textrm{ + }} \rangle _{\textrm{virtual}}} \to |{{\textrm{s}_\textrm{K}}} \rangle \to |\textrm{g} \rangle, $$
$$|\textrm{g} \rangle \to {|{\textrm{p}_{\textrm{K}^{\prime}}^\textrm{ - }} \rangle _{\textrm{virtual}}} \to |{{\textrm{s}_{\textrm{K}^{\prime}}}} \rangle \to |\textrm{g} \rangle, $$
$$|\textrm{g} \rangle \to {|{{\textrm{s}_\textrm{K}}} \rangle _{\textrm{virtual}}} \to |{\textrm{p}_\textrm{K}^\textrm{ + }} \rangle \to |\textrm{g} \rangle, $$
$$|\textrm{g} \rangle \to {|{{\textrm{s}_{\textrm{K}^{\prime}}}} \rangle _{\textrm{virtual}}} \to |{\textrm{p}_{\textrm{K}^{\prime}}^\textrm{ - }} \rangle \to |\textrm{g} \rangle, $$
where $|\textrm{g} \rangle$, $|\textrm{s} \rangle$ and $|{{\textrm{p}^ \pm }} \rangle$ are the ground, s-series, and p-series exciton states, and the label K and K’ indicate the valley indices, respectively. The subscript “virtual” indicates the virtual states. As shown in Fig. 2(c), the p-series excitons are slightly split by the Berry phase effect in a reversed order for K and K’ valleys [15,21–23,38], that is, the 2p+ orbital of the K valley exciton $|{\textrm{p}_\textrm{K}^\textrm{ + }} \rangle$ and 2p− orbital of the K’ valley exciton $|{\textrm{p}_{\textrm{K}^{\prime}}^\textrm{ - }} \rangle$ locate higher, whereas the other two states, $|{\textrm{p}_\textrm{K}^\textrm{ - }} \rangle$ and $|{\textrm{p}_{\textrm{K}^{\prime}}^\textrm{ + }} \rangle$, locate lower in energy. Therefore, the selection rules in Eqs. (12–15) indicate that the broadband SFG spectroscopy allows access to the upper branch of the 2p excitons.

In order to investigate the p-series excitons, we irradiated the sample with 0.52∼1.30 eV supercontinuum light and obtained a nonlinear emission spectrum as shown in Fig. 2(f). In addition to the 1s and 2s exciton peaks observed in the PL and PLE spectra (Figs. 2(d) and (e)), a new peak labeled A was observed at 7 meV lower than the 2s peak. Besides, a peak labeled B with broad linewidth was observed approximately at the position of the 3s peak in the PLE spectrum. The excitation power dependences of the emission intensity at 1s, 2s, and peak A positions are shown in the inset of Fig. 2(f). The emission peaks exhibited almost quadratic dependence. Therefore, these peaks are ascribed to a second-order nonlinear process, i.e., resonant SFG emission and/or 2-photon excitation photoluminescence (2P-PL) as shown in Fig. 2(c). Comparing the experimental data with the calculation results using the Rytova-Keldysh potential model explained above, we attribute peak A to resonant SFG emission from 2p excitons and assume that peak B includes not only the 3s but also 3p exciton components.

In order to confirm the above assignments, we investigated the polarization selection rules. In monolayer TMDs, PL from excitons is weakly polarized along the excitation laser polarization at low temperature due to the valley coherence [19,33,39] and is independent of the crystal orientation. In contrast, the SFG emission reflects the D3h crystal symmetry [33,40,41]. The polarization selection rules for the SFG process resonant to the p-series excitons can be obtained by considering the matrix elements contained in Eq. (4). In Eq. (4), the following matrix elements are nonzero:

$${M_{ -{+} + }} = \left\langle g \right|{\widehat d_ - }|{p_\textrm{K}^ + } \rangle \left\langle {p_\textrm{K}^ + } \right|{\widehat d_ + }|{{s_\textrm{K}}} \rangle \left\langle {{s_\textrm{K}}} \right|{\widehat d_ + }|g \rangle, $$
$${M_{ +{-} - }} = \left\langle g \right|{\widehat d_ + }|{p_{\textrm{K}^{\prime}}^ - } \rangle \left\langle {p_{\textrm{K}^{\prime}}^ - } \right|{\widehat d_ - }|{{s_{\textrm{K}^{\prime}}}} \rangle \left\langle {{s_{\textrm{K}^{\prime}}}} \right|{\widehat d_ - }|g \rangle, $$
where the indices $+$ and $-$ indicate the right-hand and left-hand circular polarizations, respectively. The dipole operators are defined as:
$${\widehat d_ + } = \frac{{{{\widehat d}_x} + i{{\widehat d}_y}}}{{\sqrt 2 }}, $$
$${\widehat d_ - } = \frac{{{{\widehat d}_x} - i{{\widehat d}_y}}}{{\sqrt 2 }}. $$
Here, we define the x and y axes along the armchair and zigzag directions, respectively, as shown in Fig. 2(a). The monolayer crystal is symmetric under the mirror operation R with respect to the xz plane. Each state relevant to Eqs. (16) and (17) transforms by the operation R as:
$$|g \rangle \to |g \rangle, $$
$$|{{s_\textrm{K}}} \rangle \to |{{s_{\textrm{K}^{\prime}}}} \rangle, $$
$$|{p_\textrm{K}^ + } \rangle \to |{p_{\textrm{K}^{\prime}}^ - } \rangle, $$
and from the definitions, the dipole operators transform as:
$${\widehat d_ + } \to {\widehat d_ - }, $$
$${\widehat d_ - } \to {\widehat d_ + }, $$
which yield:
$$\begin{aligned} {M_{ -{+} + }} &= \left\langle \textrm{g} \right|{\widehat d_ - }|{p_\textrm{K}^ + } \rangle \left\langle {p_\textrm{K}^ + } \right|{\widehat d_ + }|{{s_\textrm{K}}} \rangle \left\langle {{s_\textrm{K}}} \right|{\widehat d_ + }|\textrm{g} \rangle \\ &= \left\langle \textrm{g} \right|{R^\dagger }R{\widehat d_ - }{R^\dagger }R|{p_\textrm{K}^ + } \rangle \left\langle {p_\textrm{K}^ + } \right|{R^\dagger }R{\widehat d_ + }{R^\dagger }R|{{s_\textrm{K}}} \rangle \left\langle {{s_\textrm{K}}} \right|{R^\dagger }R{\widehat d_ + }{R^\dagger }R|\textrm{g} \rangle \\ &= \left\langle \textrm{g} \right|{\widehat d_ + }|{p_{\textrm{K}^{\prime}}^ - } \rangle \left\langle {p_{\textrm{K}^{\prime}}^ - } \right|{\widehat d_ - }|{{s_{\textrm{K}^{\prime}}}} \rangle \left\langle {{s_{\textrm{K}^{\prime}}}} \right|{\widehat d_ - }|\textrm{g} \rangle = {M_{ +{-} - }} = M. \end{aligned}$$

When the incident electric field with linear polarization is expressed as:

$$\begin{aligned} {\textbf E}(\theta )&= E\left( {\begin{array}{{c}} {\cos \theta }\\ {\sin \theta } \end{array}} \right) = E({\cos \theta {{\textbf e}_x} + \sin \theta {{\textbf e}_y}} )\\ &= \frac{E}{{\sqrt 2 }}({{e^{ - i\theta }}{{\textbf e}_ + } + {e^{i\theta }}{{\textbf e}_ - }} )= {E_ + }{{\textbf e}_ + } + {E_ - }{{\textbf e}_ - }, \end{aligned}$$
where θ is the angle between the polarization and the armchair direction, the nonlinear polarization is given as:
$$\begin{aligned} {{\textbf P}^{(2)}} &\propto ME_ + ^2{{\textbf e}_ - } + ME_ - ^2{{\textbf e}_ + }\\ &= \frac{{M{E^2}}}{2}({{e^{ - 2i\theta }}{{\textbf e}_ - } + {e^{2i\theta }}{{\textbf e}_ + }} )\\ &= \frac{{M{E^2}}}{2}[{\cos ( - 2\theta ){{\textbf e}_x} + \sin ( - 2\theta ){{\textbf e}_y}} ]\\ &= \frac{{M{E^2}}}{2}\left( {\begin{array}{{c}} {\cos ( - 2\theta )}\\ {\sin ( - 2\theta )} \end{array}} \right). \end{aligned}$$
In particular, if the incident field is polarized along the armchair (θ=0°) or zigzag (θ=30°) directions, P(2) and thus the SFG emission have a polarization parallel or perpendicular to that of excitation, respectively. Performing similar calculations leads to the same polarization selection rules for the SFG in resonance with the s-series excitons.

The polarization-resolved spectra are shown in Figs. 2(g) and (h). Under excitation with polarization along the armchair direction (Fig. 2(g)), each peak has parallel polarization. In contrast, when the excitation is polarized along the zigzag direction (Fig. 2(h)), peak A is polarized perpendicularly, being consistent with the above polarization selection rules for SFG, which means that peak A arises mainly from resonant SFG emission. Conversely, the 2s peak still has a polarization parallel to the incident polarization, which indicates that the 2s peak dominantly includes 2P-PL.

4.3 Spectral analysis

For further analysis, we performed fits of the A, 2s, and B peaks in the broadband SFG spectrum. Because the spectrum includes not only coherent SFG emission but also incoherent 2P-PL signal, i.e., the PL from excitons created by relaxation of 2-photon excited electron-hole pairs (see Fig. 2(c)), we used the sum of each component:

$$I(\omega ) = {I_{\textrm{SFG}}}(\omega ) + {I_{\textrm{2P - PL}}}(\omega ), $$
where the first and second terms are given as Eq. (5) and the sum of Lorentzian functions, respectively. According to the selection rules, the s-series excitons contribute to both SFG and 2P-PL terms, whereas the p-series excitons contribute only to the SFG term. In addition, both SFG and 2P-PL terms should include non-resonant background terms. Here, for simplicity, we assumed that 1) the s-series excitons’ emission is dominated by 2P-PL and the SFG emission from s-series excitons is insignificant. 2) The 2P-PL background is approximately zero because it is as small as the emission spectra under the SFG forbidden condition as shown by blue curves in Fig. 2(g). 3) The SFG background $bg = \sum\nolimits_{m\textrm{:non - resonant}} {{A_{mijk}}{{({\mathrm{\epsilon }_m} - \hbar \omega )}^{ - 1}}}$ is approximately given as a linear function of frequency ω. Then, we obtained the following fit function for excited excitons (2p, 2s, 3p, 3s) higher than the ground-state exciton (1s):
$${I_{\textrm{fit}}}(\omega ) = {\left|{a + b\omega + \sum\limits_{k = \textrm{2p,3p}} {\frac{{\sqrt {{A_k}} }}{{{\mathrm{\epsilon }_k} - \hbar \omega - i{\gamma_k}}}} } \right|^2} + \sum\limits_{k = \textrm{2s,3s}} {\frac{{{A_k}}}{{{{({\mathrm{\epsilon }_k} - \hbar \omega )}^2} + \gamma _k^2}}}, $$
where a, b, Ak, ${\mathrm{\epsilon }_k}$, and ${\gamma _k}$ are real fit parameters.

The black curve in Fig. 3(a) shows the fit result together with the 1s peak using a single Lorentzian function. Both results are in good agreement with the measured spectra (the thick light red curves). The real and imaginary parts of each SFG component are shown in Figs. 3(b) and (c), respectively. The real part of the background is expected to have a positive slope, because it is derived from the shoulders of nonlinear susceptibility ${\tilde{\chi} ^{(2)}} \propto {(\mathrm{\epsilon } - \hbar \omega )^{ - 1}}$ for the 1s exciton, higher exciton states, free electron-hole pairs, and the excitons corresponding to the second valence band. The fit result is consistent with this expectation. In Figs. 3(a) and (b), the background function and the SFG spectrum are extrapolated to lower and higher frequencies and shown by the dotted curves. Note that we linearly extrapolated the background to the lower energy side. The deviation between the calculation and experimental result is probably due to the shoulder of the 1s exciton PL peak. On the higher energy side, we set the SFG background so that the calculated emission spectrum matches the experimental data. We can see a similar shape to the real part of ${(\mathrm{\epsilon } - \hbar \omega - i\gamma )^{ - 1}}$ around 1.87 eV. This background is considered to originate from the excitons at higher states than 3p and the band edge.

 figure: Fig. 3.

Fig. 3. (a) Broadband SFG spectrum (light red curve) and fit result using Eq. (29) (black curve). Orange, blue and red curves show the 2s, 3s and SFG components, respectively. The excitation light was unpolarized. (b) Real parts and (c) imaginary parts of 1s, 2p, 2s and non-resonant background components of ${\tilde{\chi }^{(2 )}}$. Extrapolation are shown as dotted curves.

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 figure: Fig. 4.

Fig. 4. (a)-(c) Schematics of crystal structure and excitation laser polarization. (d)-(l) Detection angle dependence of emission intensity at 1s, A and 2s peaks. Red curves show the fit result using a fit function of $I = F_1{\cos ^2}({\theta + {\theta_0}} )+ F_2$. Green and blue curves show the cosine square and constant components, respectively. Orange arrows show directions of excitation polarization.

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Figures 4(a)-(c) schematize the configurations of the incident polarization in polarization-resolved measurement with respect to the crystal lattice. The spectral measurement and decomposition of the peak by fitting were performed for each spectrum obtained with the detection polarization angle φ varied in steps of approximately 10 degrees. Figures 4(d)-(l) show polar plots of the 1s, peak A, and 2s emission intensities as functions of the detection polarization angle under linearly polarized excitation. According to the polarization selection rules, the SFG, polarized PL, and unpolarized PL components show the angle dependence of ${F_{\textrm{SFG}}}{\cos ^2}(\varphi + 2\theta )$, ${F_{\textrm{PLVC}}}{\cos ^2}(\varphi - \theta )$, and a constant ${F_{\textrm{NPPL}}}$, respectively. The sum of these three components effectively has an angle dependence of ${F_1}{\cos ^2}(\varphi + {\theta _0}) + {F_2}$, with three independent parameters F1, F2, and θ0. Red curves in Figs. 4(d)-(l) show the fit results using ${F_1}{\cos ^2}(\varphi + {\theta _0}) + {F_2}$. Green and blue curves show the first and second terms of the fitting function, i.e., the cosine square and constant components, respectively. We can see that, as the excitation polarization is rotated clockwise, the polarization of peak A emission (Figs. 4(g)-(i)) is rotated counterclockwise with approximately twice the rotation angle of the excitation polarization. This is consistent with the polarization selection rules for SFG described by Eq. (27). In addition, the emission always exhibited almost perfect linear polarization, which means that peak A effectively includes no 2P-PL components but SFG resonant to the 2p excitons. In contrast, the 2s emission is less perfectly polarized along the excitation polarization (Figs. 4(j)-(l)). This indicates that the 2s emission consists of mainly unpolarized 2P-PL and collinearly polarized 2P-PL owing to the valley coherence [19,33,39]. The polarization dependence of the 1s emission (Figs. 4(d)-(f)) was similar to that of peak A, but the degree of polarization $DOP = {{({I_\parallel } - {I_ \bot })} / {({I_\parallel } + {I_ \bot })}}$, where ${I_\parallel }$ and ${I_ \bot }$ are the emission intensity of parallel and cross polarization, respectively, was smaller, particularly when the incident polarization was tilted from the armchair direction. Hence, the 1s emission includes not only the SFG but also 2P-PL components.

According to the polarization selection rules, the proportion of SFG rSFG can be calculated as (see Section 10 in Supplement 1 for the detail):

$${r_{\textrm{SFG}}} = \frac{{DO{P^{(\textrm{Arm}\textrm{.})}} - DO{P^{(\textrm{Zig}{.})}}}}{2}. $$
where $DO{P^{(\textrm{Arm}\textrm{.})}}$ and $DO{P^{(\textrm{Zig}\textrm{.})}}$ are the degrees of linear polarization under excitation with polarization along the armchair and zigzag directions, respectively. Performing fits of polarization-dependence shown in Figs. 2(g) and (h), we estimated rSFG=56.8 ± 0.4%, 96.7 ± 1.1%, and 18.8 ± 2.0%, respectively, for the 1s, 2p, and 2s peaks.

4.4 Peak energy and linewidth of p-series excitons

In Table 1, we summarized the parameters of each exciton peak obtained from the fit and numerical calculation using the Rytova-Keldysh potential. We can see that the observed 2p and 3p exciton energies are 4.1 ± 0.4 meV higher and 1.0 ± 0.4 meV lower than the calculated ones, respectively. Because our Rytova-Keldysh potential model does not include the Berry phase effect, we attributed these deviations to the valley-dependent Berry phase [13,16,21–23]. According to Srivastava et al. [22], the perturbation Hamiltonian due to the Berry phase is approximately given as:

$${V^I} = \frac{{|{{\Omega _0}} |}}{2}\frac{{2\pi {e^2}}}{{S\varepsilon }}\left( {\frac{1}{2}{{|{{\textbf k} - {\textbf k}^{\prime}} |}^2} + i{\tau_\textrm{v}}\frac{{{\textbf k}^{\prime} \times {\textbf k}}}{{|{{\textbf k} - {\textbf k}^{\prime}} |}}} \right), $$
where |Ω0| is the Berry curvature at the K and K’ points, S is the quantization area, ɛ is the dielectric constant, k and k’ are the Bloch wavevectors and τv=±1 is the valley index. The first term gives rise to redshifts of both p+ and p− exciton states, whereas the second term causes them to split. As the splitting energy is larger than the redshift, the upper side 2p exciton (2p+ of K valley exciton or 2p− of K’ valley exciton), which can be accessed by the resonant SFG process, should blueshift from the original energy calculated by the Rytova-Keldysh potential model. The deviation of the 2p exciton energy between our experiment and calculation is consistent with this qualitative estimation. In contrast, the deviation of the 3p exciton energy was different in sign from the estimation. Actually, the s-series exciton energies are also affected by the Berry phase, but this effect was not included in our analysis, which could affect the determinations of the parameters. This is one possible reason why the signs of the deviation from the Rytova-Keldysh potential model were opposite between the 2p and 3p excitons.

In Table 1, we listed the relative intensity of each exciton peak. We estimated the intensity of the SFG component by utilizing the fit results of the polarization-resolved measurement (Figs. 2(g) and (h)). Interestingly, the SFG signal of the 2p excitons is approximately two orders of magnitude weaker than that of the 1s excitons, whereas the squares of the leading matrix elements ${\left|{\left\langle g \right|{{\widehat d}_i}|{2p} \rangle \left\langle {2p} \right|{{\widehat d}_j}|{1s} \rangle \left\langle {1s} \right|{{\widehat d}_k}|g \rangle } \right|^2}$ and ${\left|{\left\langle g \right|{{\widehat d}_i}|{1s} \rangle \left\langle {1s} \right|{{\widehat d}_j}|{2p} \rangle \left\langle {2p} \right|{{\widehat d}_k}|g \rangle } \right|^2}$ for each SFG process are the same. This difference is probably because of other virtual states also contributing to the matrix elements.

Our fit results show that the 1s and 2s excitons have a similar linewidth, which is consistent with the previous study [42]. In contrast, the linewidth 8.5 ± 0.1 meV of the 2p exciton peak is larger than that of the 1s and 2s exciton peaks (3.8 and 5.0 ± 0.1 meV, respectively, in the broadband SFG spectrum). This suggests that the dephasing of the 2p excitons is faster than that of the 1s and 2s excitons whereas the radiative relaxation of the 2p excitons is slower [24]. The broader linewidth of the 2p exciton was reproduced in another sample as shown in Fig. S7 in the Supplement 1. We confirmed that the linewidth does not change with the excitation power reduced by one order of magnitude (see Fig. S13 in Supplement 1). Hence, the broader linewidth is not caused by excitation-induced dephasing [43].

4.5 Temperature dependence of broadband SFG spectrum

Figure 5(a) shows the broadband SFG spectra measured at various temperatures. The peak B, corresponding to the 3s and 3p excitons, disappeared around 100 K, and the 2s peak became weaker above ∼150 K. Conversely, the 1s and peak A were observed at all temperatures up to 294 K. Figure 5(b) shows the temperature dependence of exciton energies estimated by a global fit of the PL and broadband SFG spectra (see also Sections 14 and 15 in the Supplement 1 for the details). As the temperature increases, all peaks including peaks A and B redshifted, which confirms that peaks A and B are derived from excitonic states in monolayer WSe2 rather than from other signals, such as the emission from substrates. We remark here that elucidation of the energy position of the 2p exciton state in addition to that of the 1s and 2s exciton states opens an unprecedented possibility to evaluate the temperature dependence of material parameters up to room temperature. Figure 5(c) shows the temperature dependence of energy differences between the 2s, 2p, and 1s excitons. Both the 1s-2s and 1s-2p energy differences increase as the temperature increases. This is considered to be due to the temperature evolution of the Berry phase, the dielectric parameters κ, and/or r0.

 figure: Fig. 5.

Fig. 5. (a) Broadband SFG spectra of excitons in hBN-encapsulated monolayer WSe2 at various temperatures. Inset shows image plot. (b)(c) Temperature dependence of (b) peak energy positions and (c) energy differences from the 1s peak. (d) Temperature dependence of linewidth. Lines show linear fitting results for data points below 120 K. (e) Differences between measured linewidths and linear fit results shown in (d). The error bars in (b)-(d) indicate the global fit errors.

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Figure 5(d) shows the temperature dependence of the 1s, 2p, and 2s linewidths obtained by the global fits of both PL and broadband SFG spectra. Below ∼100 K, the 2p linewidth was broader than that of the 1s and 2s excitons. With increasing temperature, all peaks broadened. In the low temperature region, the evolution of the linewidth was found to follow that of acoustic phonon scattering and is simply described by a linear function of temperature T [43]:

$$2\gamma = 2{\gamma _0} + {c_1}T = 2{\gamma _{\textrm{ac}}}, $$

Here, 2γ0 is the residual linewidth at 0 K. For the 2p exciton, we obtained 2γ0=7.91 ± 0.07 meV, which is much larger than that of the 1s and 2s excitons of 2γ0=3.12 ± 0.01 and 2.45 ± 0.03 meV, respectively. c1 corresponds to the strength of the acoustic phonon coupling. The value c1=19.1 μeV/K for the 2p exciton is smaller than c1=25.2 ± 0.2 μeV/K for the 1s exciton and c1=65.1 ± 0.7 μeV/K for the 2s exciton. These results mean that the 2p exciton has a smaller coupling strength with the acoustic phonons than the 1s and 2s excitons.

Above ∼100 K, the linewidth deviated from the linear fit result. Figure 5(e) shows the differences between the measured linewidths and linear fit results. These deviations are considered to originate from the optical phonon scattering activated at relatively high temperatures. The magnitude of the deviation was similar among the respective excitons, which indicates that the 2p exciton has a similar coupling strength with the optical phonons as the 1s and 2s excitons.

Now we consider the effect of exciton-phonon scattering on the 2p exciton linewidth. Generally, phonon-induced dephasing is caused by three processes: phonon absorption, emission, and Raman scattering process consisting of the absorption of a phonon and the emission of another phonon [44]. The rate of each process is proportional to the occupation number of phonons $N = {[{\textrm{exp} ({{{\hbar \Omega } / {{k_\textrm{B}}T}}} )- 1} ]^{ - 1}}$, $N + 1$, and $N(N + 1)$, respectively. Here, ħΩ is the corresponding phonon energy. At T=0 K, N becomes zero and the phonon absorption and Raman scattering processes disappear. Hence, the larger 2γ0 value of the 2p exciton than that of the 1s and 2s excitons indicates that the phonon absorption or Raman scattering processes do not cause the broader 2p linewidth at low temperatures.

Meanwhile, the residual linewidth 2γ0 at T=0 K could still include the contribution of the phonon-emission process. However, our results also indicate that the phonon coupling strength of the 2p exciton is not larger than that of the 1s and 2s excitons. Hence, it is considered that the phonon-emission rate of the 2p exciton is similar to that of the 1s and 2s excitons and does not contribute to the broader 2p linewidth.

From the above discussion, we concluded that the broader 2p linewidth at low temperature is not caused by the phonon-induced dephasing. In addition, as we already mentioned in the previous section, the excitation-intensity dependence shows that the linewidth of the 2p exciton is not broadened by the excitation-induced dephasing. Here, we propose that the broader 2p linewidth is originating from the exciton-electron scattering process. It is well known that monolayer WSe2 samples without electrical gating are typically negatively doped. In our samples, negative trion peaks were observed in the PL spectrum (see Figure S1(d) in the Supplement 1). A recent theoretical study on monolayer TMDs [45] shows that the 2p exciton can form stable negative trion states with trion binding energy of a few tens of meV, which suggests the existence of strong interaction between 2p excitons and residual electrons. The present monolayer samples are not gated and the carrier density cannot be controlled. We expect that future experiments to be performed on gated samples clarify the roles of the exciton-electron scattering in 2p exciton dephasing quantitatively.

5. Summary

We have developed broadband SFG spectroscopy in hBN-encapsulated monolayer WSe2 using the partially incoherent supercontinuum light source, and derived approximate expressions for the nonlinear susceptibility. We observed clear p-series excitons in the broadband SFG spectra which cannot be detected by PL or 1-photon PLE measurement. By separating contributions of SFG and 2P-PL signals using spectral and polarization information, we could evaluate exciton energies and linewidths up to the 3p/3s exciton states. The 2p exciton energy was found to slightly deviate from that predicted by the Rytova-Keldysh potential model, which indicates the emergence of the Berry phase effect. Interestingly, the 2p excitons showed slower radiative relaxation while exhibited a broader linewidth than the 1s and 2s excitons. Based on the dependence on excitation intensity and temperature, we elucidated the excitation- or phonon-induced dephasing mechanisms and proposed the exciton-electron scattering as a cause for the broadening. Our study demonstrates that the broadband SFG spectroscopy based on partially incoherent supercontinuum light is a simple and powerful tool to explore the dark states in 2D materials.

Funding

Japan Society for the Promotion of Science (JP17H06124, JP19K21849, JP20H00354, JP20H02573, JP21H05017); Japan Science and Technology Agency (JPMJCR17I5, JPMJMI17F2); Ministry of Education, Culture, Sports, Science and Technology (JPMXS0118067634).

Acknowledgments

We thank Prof. K. Matsuda, Dr. K. Shinokita and Mr. H. Nishidome for helpful advices on sample preparation. We also thank Dr. M. Takahata and Dr. K. Uchida for fruitful discussions.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

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Supplementary Material (1)

NameDescription
Supplement 1       Experimental setups, additional data and discussions

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (5)
Fig. 1.
Fig. 1. (a)(b) Concepts of (a) scanning SHG spectroscopy and (b) broadband SFG spectroscopy. (c) Simplified energy diagram of excitonic states, incident laser, and broadband SFG signal. Red allows show the SFG transition path.

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Fig. 2.
Fig. 2. (a) Top view of crystal structure of monolayer TMD. Red and green arrows indicate armchair and zigzag directions. (b) Schematics of the first Brillouin zone in monolayer TMD. (c) Schematics of energy diagram for K valley excitons in monolayer WSe2. Allowed PL, SFG and 2P-PL processes are shown by arrows. Red and blue arrows indicate transitions with one right- and left-hand circularly polarized photon, respectively. Green arrow indicates transition with out-of-plane polarized PL emission. Black wiggly arrows represent relaxation processes. Green dashed lines represent virtual states. Energy separations between levels are not to scale. (d) PL spectrum in hBN encapsulated monolayer WSe2 under excitation with photon energy of 1.97 eV at T=16 K. The peak labeled BX is originating from biexciton. (e) PLE spectrum observed at dark exciton peak (1.66 eV). (f) Emission spectrum under unpolarized NIR excitation with photon energy of 0.52∼1.18 eV. Inset shows power P dependence of emission intensity I of 1s, A and 2s peaks. Solid lines represent fitting results using a power law Pα. (g)(h) Polarization-resolved emission spectra under NIR excitation with polarization parallel to (g) armchair and (h) zigzag directions.

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Fig. 3.
Fig. 3. (a) Broadband SFG spectrum (light red curve) and fit result using Eq. (29) (black curve). Orange, blue and red curves show the 2s, 3s and SFG components, respectively. The excitation light was unpolarized. (b) Real parts and (c) imaginary parts of 1s, 2p, 2s and non-resonant background components of ${\tilde{\chi }^{(2 )}}$ . Extrapolation are shown as dotted curves.

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Fig. 4.
Fig. 4. (a)-(c) Schematics of crystal structure and excitation laser polarization. (d)-(l) Detection angle dependence of emission intensity at 1s, A and 2s peaks. Red curves show the fit result using a fit function of $I = F_1{\cos ^2}({\theta + {\theta_0}} )+ F_2$ . Green and blue curves show the cosine square and constant components, respectively. Orange arrows show directions of excitation polarization.

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Fig. 5.
Fig. 5. (a) Broadband SFG spectra of excitons in hBN-encapsulated monolayer WSe2 at various temperatures. Inset shows image plot. (b)(c) Temperature dependence of (b) peak energy positions and (c) energy differences from the 1s peak. (d) Temperature dependence of linewidth. Lines show linear fitting results for data points below 120 K. (e) Differences between measured linewidths and linear fit results shown in (d). The error bars in (b)-(d) indicate the global fit errors.

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Tables (1)

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Table 1. Fitting results obtained from PL, PLE, and broadband SFG spectra at T=16 K shown in Figs. 2(d) and (e), and Fig. 3, and the Rytova-Keldysh potential (RKP) model calculation results. For the RKP model calculation, we used the reduced mass μ=0.20m0 and obtained fit parameters Ebg=1.8660 ± 0.0004 eV, κ=4.46 ± 0.05, and r0=4.25 ± 0.07 nm.

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Equations (32)

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P ~ i ( 2 ) ( ω ) = ε 0 2 π ∫ χ ~ i j k ( 2 ) ( ω ; ω ′ ) E ~ j ( ω ′ ) E ~ k ( ω − ω ′ ) d ω ′ ,
χ ~ i j k ( 2 ) ( ω ; ω ′ ) ∝ ⟨ g | d ^ i ∑ n | ψ n ⟩ ⟨ ψ n | ϵ n − ℏ ω − i γ n d ^ j ∑ v | ψ v ⟩ ⟨ ψ v | ϵ v − ℏ ω ′ − i γ v d ^ k | g ⟩ ,
χ ~ i j k ( 2 ) ( ω ; ω ′ ) ∝ ∑ n A n i j k ϵ n − ℏ ω − i γ n = χ ~ i j k ( 2 ) ( ω ) ,
A n i j k ∝ ∑ v ⟨ g | d ^ i | ψ n ⟩ ⟨ ψ n | d ^ j | ψ v ⟩ ⟨ ψ v | d ^ k | g ⟩ .
χ ~ i j k ( 2 ) ( ω ; ω ′ ) = χ ~ i j k ( 2 ) ( ω ) = b g + ∑ n : resonant A n i j k ϵ n − ℏ ω − i γ n ,
P ~ i ( 2 ) ( ω ) = ε 0 2 π χ ~ i j k ( 2 ) ( ω ) ∫ E ~ j ( ω ′ ) E ~ k ( ω − ω ′ ) d ω ′ ,
I ~ i ( SFG ) ( ω ) ∝ | P ~ i ( 2 ) ( ω ) | 2 = ε 0 2 4 π 2 | χ ~ i j k ( 2 ) ( ω ) | 2 | ∫ E ~ j ( ω ′ ) E ~ k ( ω − ω ′ ) d ω ′ | 2 .
| ∫ E ~ j ( ω ′ ) E ~ k ( ω − ω ′ ) d ω ′ | 2 ∝ ∫ I ~ j ( ω ′ ) I ~ k ( ω − ω ′ ) d ω ′ .
∫ ( ω − Δ ω ) / 2 ( ω + Δ ω ) / 2 I ~ j ( ω ′ ) I ~ k ( ω − ω ′ ) d ω ′ ≅ I ~ j ( ω 2 ) I ~ k ( ω 2 ) Δ ω .
I ~ i ( S F G ) ( ω ) ∝ | χ ~ i j k ( 2 ) ( ω ) | 2 I ~ j ( ω 2 ) I ~ k ( ω 2 ) Δ ω .
V RK ( r ) = − e 2 8 ε 0 r 0 [ H 0 ( κ r r 0 ) − Y 0 ( κ r r 0 ) ] ,
| g ⟩ → | p K  +  ⟩ virtual → | s K ⟩ → | g ⟩ ,
| g ⟩ → | p K ′  -  ⟩ virtual → | s K ′ ⟩ → | g ⟩ ,
| g ⟩ → | s K ⟩ virtual → | p K  +  ⟩ → | g ⟩ ,
| g ⟩ → | s K ′ ⟩ virtual → | p K ′  -  ⟩ → | g ⟩ ,
M − + + = ⟨ g | d ^ − | p K + ⟩ ⟨ p K + | d ^ + | s K ⟩ ⟨ s K | d ^ + | g ⟩ ,
M + − − = ⟨ g | d ^ + | p K ′ − ⟩ ⟨ p K ′ − | d ^ − | s K ′ ⟩ ⟨ s K ′ | d ^ − | g ⟩ ,
d ^ + = d ^ x + i d ^ y 2 ,
d ^ − = d ^ x − i d ^ y 2 .
| g ⟩ → | g ⟩ ,
| s K ⟩ → | s K ′ ⟩ ,
| p K + ⟩ → | p K ′ − ⟩ ,
d ^ + → d ^ − ,
d ^ − → d ^ + ,
M − + + = ⟨ g | d ^ − | p K + ⟩ ⟨ p K + | d ^ + | s K ⟩ ⟨ s K | d ^ + | g ⟩ = ⟨ g | R † R d ^ − R † R | p K + ⟩ ⟨ p K + | R † R d ^ + R † R | s K ⟩ ⟨ s K | R † R d ^ + R † R | g ⟩ = ⟨ g | d ^ + | p K ′ − ⟩ ⟨ p K ′ − | d ^ − | s K ′ ⟩ ⟨ s K ′ | d ^ − | g ⟩ = M + − − = M .
E ( θ ) = E ( cos ⁡ θ sin ⁡ θ ) = E ( cos ⁡ θ e x + sin ⁡ θ e y ) = E 2 ( e − i θ e + + e i θ e − ) = E + e + + E − e − ,
P ( 2 ) ∝ M E + 2 e − + M E − 2 e + = M E 2 2 ( e − 2 i θ e − + e 2 i θ e + ) = M E 2 2 [ cos ⁡ ( − 2 θ ) e x + sin ⁡ ( − 2 θ ) e y ] = M E 2 2 ( cos ⁡ ( − 2 θ ) sin ⁡ ( − 2 θ ) ) .
I ( ω ) = I SFG ( ω ) + I 2P - PL ( ω ) ,
I fit ( ω ) = | a + b ω + ∑ k = 2p,3p A k ϵ k − ℏ ω − i γ k | 2 + ∑ k = 2s,3s A k ( ϵ k − ℏ ω ) 2 + γ k 2 ,
r SFG = D O P ( Arm . ) − D O P ( Zig . ) 2 .
V I = | Ω 0 | 2 2 π e 2 S ε ( 1 2 | k − k ′ | 2 + i τ v k ′ × k | k − k ′ | ) ,
2 γ = 2 γ 0 + c 1 T = 2 γ ac ,
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