Issue
Natl Sci Open
Volume 2, Number 4, 2023
Special Topic: Two-dimensional Materials and Devices
Article Number 20220060
Number of page(s) 10
Section Physics
DOI https://doi.org/10.1360/nso/20220060
Published online 09 June 2023

© The Author(s) 2023. Published by China Science Publishing & Media Ltd. and EDP Sciences.

Licence Creative CommonsThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

INTRODUCTION

Excitons and their complexes (e.g., phonon replicas, biexcitons, and Fermi polarons) are elementary excitations that predominate the optical properties of a material and hence underlie the development of various emerging technological advances in photonics and optoelectronics [1]. According to the relative positions of the constituent electrons and holes in real- and momentum-spaces, excitons can be categorized into four types: real- and momentum-direct (type-I), real-direct but momentum-indirect (type-II), momentum-direct but real-indirect (type-III), and real- and momentum-indirect (type-IV), as shown in Figure 1. Type-I excitons can strongly couple to photons and show large luminescence quantum efficiency, setting a foundation for a wide variety of optoelectronic applications, such as light-emitting diodes, lasers, solar cells, and optoelectronic devices [25]. However, the rather short lifetime of real- and momentum-direct excitons strongly impedes their applications in scientific research and technological innovation where long-lived excitons are required, for example, exciton superfluid phase, exciton crystals, and exciton transistors [6,7].

thumbnail Figure 1

Real- and momentum-indirect and direct excitons. Based on the configurations of the constituent electrons and holes in real- and momentum-spaces, excitons can be divided into four types: real- and momentum-direct (type-I), real-direct but momentum-indirect (type-II), momentum-direct but real-indirect (type-III), and real- and momentum-indirect (type-IV).

Notably, in some intentionally designed systems, such as van der Waals heterostructures with a staggered band alignment [810] and quantum wells under an external electric field [11], the wave functions of the constituent electrons and holes are spatially separated, resulting in the formation of real-space indirect excitons (also referred to as interlayer excitons or spatially indirect excitons). Thanks to the spatial separation of the electrons and holes, real-space indirect excitons exhibit a much longer lifetime than the real- and momentum-direct ones [9,1214]. In the light of the long-lived real-space indirect excitons, a wide variety of captivating physical phenomena has been demonstrated, including but not limited to exciton Bose-Einstein condensation [1518], correlated excitonic insulator states [1921], and dissipationless valley exciton devices [14,22]. In addition, real-space indirect excitons harbor in-built electric dipoles and are widely tunable in applied electric fields [2,9,10,2325], representing an advantageous scenario for technological applications. Although substantial developments and progress on understanding the real-space indirect excitons have been witnessed, the studies of real-space indirect excitons have mainly focused on the momentum-bright species (i.e., type-III excitons) with electrons and holes localized in the same valley of the Brillouin zone (BZ) [9,12,14,23,2628]. However, the real- and momentum-indirect excitons (type-IV), which are expected to possess an even longer lifetime than the momentum-direct but real-indirect ones due to the dark nature in both real- and momentum-spaces, are still largely unexplored experimentally.

A suitable candidate for investigating real- and momentum-indirect excitons should meet two basic conditions simultaneously. First, it should possess multi-valleys in conduction and valence bands, endowing the possibility of momentum-indirect excitons. Second, for Bloch states at distinct valleys, the orbital compositions should be different. Consequently, the wave functions of different valleys reside at different positions in real-space, which enables the momentum-indirect excitons to be indirect in real-space as well. In particular, two-dimensional (2D) multi-valley semiconducting transition metal dichalcogenides (conduction band: Q and K valleys; valence band: K and Γ valleys) with different orbital compositions at distinct valleys provide a promising platform for real- and momentum-indirect excitons [24,25,29,30]. Despite the fact that the momentum-indirect excitons in these materials have been well uncovered [31,32], their indirect nature in real-space remains equivocal. In this work, we fabricate hexagonal boron nitride (h-BN) encapsulated dual-gate devices of a multi-valley semiconductor of bilayer MoS2. Through electric-field tunable photoluminescence (PL) spectra, we identify the out-of-plane static electric dipole and quantum-confined Stark effect of momentum-indirect exciton in bilayer MoS2, providing the smoking gun evidence of their real-space indirect characteristic and hence the existence of real- and momentum-indirect exciton. In conjunction with density functional theory (DFT) calculations, we further uncover that the observed real- and momentum-indirect exciton in bilayer MoS2 involves electron and hole respectively from K and Γ valleys, addressing the longstanding controversy of its momentum origin. As bilayer MoS2 is doped with electrons, new sets of PL peaks corresponding to real- and momentum-indirect charged excitons (namely trions) emerge below the energy of the real- and momentum-indirect neutral excitons. Remarkably, the binding energy of real- and momentum-indirect trion is giant in bilayer MoS2, twice that of the real- and momentum-direct trions in transition metal dichalcogenide systems.

RESULTS AND DISCUSSION

High-quality, h-BN-encapsulated dual-gate bilayer MoS2 devices (as schematically shown in Figure 2A) are fabricated by a van der Waals mediated dry-transfer method (please see Section I in Supplementary Information (SI) for more details) with few-layer graphene (FLG) as top and bottom gate electrodes. The dual-gate configuration enables us to independently tune the carrier density n0 and out-of-plane electric field Fz. Here n0=(CbVbg+CtVtg)/e and Fz=(CbVbgCtVtg)/2ε0εB, where e is the elementary charge, ε0 denotes the vacuum permittivity, εB is the out-of-plane dielectric constant of bilayer MoS2, Cb(Vbg) and Ct(Vtg) are the geometrical capacitances per area (applied voltages) for the bottom and top gates, respectively (details in Section IV in SI).

thumbnail Figure 2

(A) Schematic of dual-gate h-BN encapsulated bilayer MoS2 device. (B) PL spectrum of device D1 at Fz = −0.074 V/nm. Note that the range of 1.45–1.60 eV is magnified by 10 times for clarity. (C) Contour plot of the PL spectra of device D1 as a function of photon energy (bottom axis) and Fz (left axis). The doping density remains unchanged. (D) First-order energy derivative of (C). Real- and momentum-indirect excitons are labelled as RMX1–3 in the sequence of decreasing emission energy. Gray (green) dashed lines in (C) and (D) represent linear-fits of RMX1–3 (IX1,2). The peak at around 1.99 eV that is insusceptible to Fz corresponds to the Raman G peak of graphene.

Figure 2B depicts the PL spectrum of device D1 at Fz = −0.074 V/nm. Unless otherwise specified, the data presented in the main text are taken from the high-quality device D1 in a high vacuum at 10 K with a continuous wave optical excitation at ~2.33 eV (532 nm). In addition to the well-known momentum-direct excitons, including both the real-direct transition at around 1.92 eV (labeled as XA) and real-indirect emissions at around 1.95/2.02 eV (dubbed as IX1/IX2), three momentum-indirect excitons in the energy range of 1.50–1.58 eV (marked as RMX1, RMX2 and RMX3, in the sequence of decreasing emission energy) can be unequivocally observed. Note that the existence of two momentum-direct but real-indirect exciton species of IX1 and IX2 can be ascribed to the fact that the external electric field breaks the layer degeneracy of band structure [23]. Figure 2C presents the color contour of PL spectra against the applied out-of-plane electric field Fz. To better distinguish the fine features, we extract the first-order derivative of intensity I over photon energy E (∂I/∂E) as the function of Fz, as depicted in Figure 2D. Obviously, momentum-direct but real-indirect excitons IX1 and IX2 vary linearly with the external electric field Fz, yet have reversed slops, evidencing their opposite out-of-plane static electric dipole moments. Via linear fitting (green dashed lines in Figure 2D and section IX in SI), we extracted the out-of-plane electric dipole moments of IX1 and IX2: μz(IX1) = (0.526 ±0.009)e∙nm and μz(IX2) = −(0.530 ±0.004)e∙nm, in good agreement with the previous results [23] and our theoretical calculations (±0.578 e∙nm, as discussed in Section VII in SI). It is noteworthy that when an enough high electric field Fz is applied, the shifts of IX1/IX2 deviate from a simple linear Stark shift (Figure S5). This can be understood as the strong coupling between IX1/IX2 and real- and momentum-direct B excitons, as demonstrated by recent experimental measurements [33] and theoretical calculations [34].

Remarkably, the three momentum-indirect excitons RMX1-3 in bilayer MoS2 also vary linearly with the applied out-of-plane electric field Fz (gray dashed lines in Figure 2D), evidencing the quantum-confined Stark effect and their indirect nature in real-space. This demonstrates that RMX1-3 belong to real- and momentum-indirect type-IV exciton. To further confirm their indirect nature in both real- and momentum-spaces of RMX1-3, we perform DFT calculations with the Perdew-Burke-Ernzerhof generalized gradient approximation for exchange-correlation interaction (Section VII in SI). Figure 3A shows the orbital-resolved band structure of bilayer MoS2. For the valence band, its maximum is located at the center of the first BZ (i.e., Γ point). For the conduction band, there are two critical points (i.e., K and Q valleys) whose energies are almost degenerate. As a consequence, Q-Γ and K-Γ transitions are the two possible configurations of momentum-indirect excitons in bilayer MoS2. Since it is difficult for conventional techniques to directly distinguish the two transitions, the exact origin of momentum-indirect excitons in bilayer MoS2 is highly contentious. Some indicate the momentum-indirect exciton in bilayer MoS2 to be K-Γ transition [31,35], while the others reveal that it should be Q-Γ transition [3638]. To solve the longstanding controversy, we derive the real-space distribution of the spin-up wavefunctions at valence band Γ, and conduction band K/Q (Figures 3B–3D). For spin-down wavefunctions, the real-space distribution can be evidently acquired by time-reversal symmetry. Obviously, the spin-up wavefunction at valence band Γ is symmetrically distributed in both layers, and shows 100% interlayer hybridization (Figure 3B). In stark contrast, the spin-up wavefunction at conduction band K is only distributed in the lower layer and hence fully layer-polarized (Figure 3C). While the spin-up wavefunction at the conduction band Q shows strong delocalization, it still has a slight layer polarization (Figure 3D). Because of the various real-space distributions and interlayer hybridization at diverse valleys, the equivalent positions in real-space of wave functions at conduction band K/Q, and valence band Γ should be quite different. Quantitatively, the equivalent positions of wave functions in real-space, defined as rz=+r|φ(r)|2dr, are rz=−0.5t (0.5t), −0.08t (0.08t) and 0 for spin-up (spin-down) wavefunctions at conduction band K, conduction band Q, and valence band Γ, respectively. Here, |φ(r)|2 denotes the probability density of wavefunction φ(r) at real-space position r, and t = 0.615 nm represents the interlayer distance [39]. The origin point is set at the midpoint of the constituent two layers. Consequently, both the two possible momentum-indirect transitions are real-space indirect and possess nonvanishing out-of-plane static electric dipole moments: μz(K-Γ)=e[rz(Γ)rz(K)]=±0.5et=±0.308enm for K-Γ excitons and μz(Q-Γ)=e[rz(Γ)rz(Q)]=±0.08et=±0.049enm for Q-Γ excitons, as shown in Figure 3E. On applying an external electric field Fz, the exciton energies would vary linearly due to the quantum-confined Stark effect:

thumbnail Figure 3

(A) Orbital-resolved projected band structure of bilayer MoS2 considering the spin-orbit coupling. (B), (C), (D) Distribution of real-part of spin-up wavefunctions in real-space at valence band Γ, conduction band K, and conduction band Q, respectively. (E) Possible transition configuration for K-Γ (purple wave) and Q-Γ (cyan wave). The red (blue) curve denotes conduction band K (Q). The grey curve denotes valence band Γ. The electric dipole moment of each transition is labelled. rz represents the equivalent position of the wavefunction at each valley as discussed in the main text. (F) Experimental results of the photon energy of RMX1 (blue hollows), RMX2 (green hollows) and RMX3 (red hollows) in device D1 as a function of Fz extracted from Figure 2C. Orange, purple, black solid lines and corresponding shadow regions denote the linear-fit of RMX1/2/3 with a fixed slope μz(K-Γ) obtained from DFT calculations. The energy difference between RMX1 and RMX2 (RMX3) is ~22 meV (46 meV).

E = E 0 μ z F z , (1) where E0 is the exciton energy at zero electric field. Importantly, the experiment results of RMX1-3 (hollows in Figure 3F) can be described perfectly by Eq. (1) using the out-of-plane static electric dipole moment of K-Γ transition (i.e.,μz(K-Γ) = 0.308e·nm), as shown by solid lines in Figure 3F (please refer to Section VI in SI for more information). This clearly manifests that the constituent electrons/holes of real- and momentum-indirect RMX1-3 originate from K/Γ valley in the BZ, solving the longstanding debate on momentum origin. It is noteworthy that the energy difference between RMX1 and RMX2 (RMX3) is ~22 meV (~46 meV), which coincides with the energy of phonon mode TA(K) or ZA(K) [2TA(K) or 2ZA(K)] [40]. Thus, the RMX1 transition is probably a kind of zero-phonon line which may be activated by defect [25], and RMX2 and RMX3 should correspond to the one-phonon and two-phonon replicas of RMX1, respectively. Note that two-phonon replica RMX3 is too weak to be distinguished at zero electric field, but exhibits an intensity comparable to or even stronger than that of RMX1 at high electric fields (Figures 2C and 2D). This may indicate a largely enhanced electron-phonon coupling by electric fields and deserves further studies.

Finally, we study the doping-density dependent responses of real- and momentum-indirect excitons in bilayer MoS2. Figures 4A and 4B respectively depict the contour plot and line-cuts of PL spectra as a function of doping density. Here the out-of-plane electric field Fz is fixed at zero. Remarkably, when bilayer MoS2 is doped with electrons, new sets of PL peaks emerge below the energy of the neutral real- and momentum-indirect RMX1 and RMX2, which can be fitted to two new peaks with Lorentzian function (Figure 4B) and correspond to real- and momentum-indirect trions (labeled as RMT1 and RMT2, respectively). This is in close resemblance to the well-studied real- and momentum-direct exciton XA, which is tuned into charged exciton (XA-) upon electrostatic doping [41]. Note that at Fz=0, only RMX1 and RMX2 can be probed (Figures 2C and 2D). Figure 4C plots the energies of RMT1/RMT2 and RMX1/RMX2 , as well as XA/XA-, as a function of doping density. Notably, the binding energy of real- and momentum-indirect RMT1 (RMT2), i.e., the energy difference between RMX1 (RMX2) and RMT1 (RMT2), can reach up to ~59 meV (~57 meV), which is more than twice that of real- and momentum-direct XA- (~24 meV) in bilayer MoS2 and also larger than the state-of-the-art results of previously reported trion in 2D semiconductors [4245].

thumbnail Figure 4

(A) Contour plot of the PL spectra of device D1 as a function of photon energy (bottom axis) and n0 (left axis). The out-of-plane electric field Fz is fixed at zero. (B) PL spectra at diverse n0. Cyan and green peaks represent Lorentz-fit peaks for RMX1&2 and RMT1&2, respectively. Note that the range of 1.45–1.60 eV in both (A) and (B) are magnified by 10 times for clarity. Offset is set in (B) for better resolution. (C) Lorentz-fitted results extracted from (B) of XA (red hollows), XA- (carmine hollows), RMX1 (green triangles), RMX2 (blue cubes), RMT1 (light green triangles) and RMT2 (light blue cubes). The binding energy of XA-, RMT1 and RMT2 is approximately 24, 59 and 57 meV, respectively.

CONCLUSION

In short, we demonstrate the real- and momentum-indirect neutral/charged excitons (including their phonon replicas) in a multi-valley semiconductor of bilayer MoS2 by the combination of electric-field/doping-density tunable PL measurements and first-principles calculations. Because of the sizable in-built electric dipoles, real- and momentum-indirect excitons present quantum-confined Stark effect and are widely tunable in applied electric fields. Moreover, the Coulomb interaction between real- and momentum-indirect excitons and free carriers is astonishingly strong in bilayer MoS2, endowing the giant binding energy of real- and momentum-indirect trion of ~59 meV, more than twice that of real- and momentum-direct trion (i.e., ~24 meV). Our work not only fulfills the knowledge on real- and momentum-indirect neutral and charged excitons, but also sheds light on the understanding and engineering of many-body physics and optoelectronics based on multi-valley semiconductors.

Data availability

All data needed to evaluate the conclusions are presented in the paper and/or the Supplementary Information. Additional data related to this paper may be requested from the authors.

Funding

This work was supported by the National Natural Science Foundation of China (NSFC) (12274447, 61888102, 11834017, 61734001, and 12074412), the National Key Research and Development Program (2021YFA1202900 and 2021YFA1400502), the Strategic Priority Research Program of Chinese Academy of Sciences (XDB30000000), and the Key-Area Research and Development Program of Guangdong Province (2020B0101340001).

Author contributions

G.Z. and L.D. supervised this work; L.D. and Z.H. conceived the project and designed the experiments; Z.H. fabricated the devices with the assistance from J.T., Y.C., X.Z., Y.P., X.L. and Y.Z., and carried out the optical measurements with the help of F.W; Z.H. conducted AFM measurements with the help of L.L., Y.Y. and Y.J; Y.L. under the supervision of S.D. and T.B. conducted the DFT calculations; K.W. and T.T. provided high-quality h-BN crystals; Z.H. and L.D. analyzed the data; Y.Z. , L.L., W.Y., D.S. and Z.S. helped with data analysis; Z.H., Y.L., L.D. and G.Z. co-wrote the manuscript. All authors discussed the results and commented on the paper.

Conflict of interest

The authors declare no conflict of interest.

Supplementary information

The Supporting Information is available free of charge at https://doi.org/10.1360/nso/20220060. The supporting materials are published as submitted, without typesetting or editing. The responsibility for scientific accuracy and content remains entirely with the authors.

References

All Figures

thumbnail Figure 1

Real- and momentum-indirect and direct excitons. Based on the configurations of the constituent electrons and holes in real- and momentum-spaces, excitons can be divided into four types: real- and momentum-direct (type-I), real-direct but momentum-indirect (type-II), momentum-direct but real-indirect (type-III), and real- and momentum-indirect (type-IV).

In the text
thumbnail Figure 2

(A) Schematic of dual-gate h-BN encapsulated bilayer MoS2 device. (B) PL spectrum of device D1 at Fz = −0.074 V/nm. Note that the range of 1.45–1.60 eV is magnified by 10 times for clarity. (C) Contour plot of the PL spectra of device D1 as a function of photon energy (bottom axis) and Fz (left axis). The doping density remains unchanged. (D) First-order energy derivative of (C). Real- and momentum-indirect excitons are labelled as RMX1–3 in the sequence of decreasing emission energy. Gray (green) dashed lines in (C) and (D) represent linear-fits of RMX1–3 (IX1,2). The peak at around 1.99 eV that is insusceptible to Fz corresponds to the Raman G peak of graphene.

In the text
thumbnail Figure 3

(A) Orbital-resolved projected band structure of bilayer MoS2 considering the spin-orbit coupling. (B), (C), (D) Distribution of real-part of spin-up wavefunctions in real-space at valence band Γ, conduction band K, and conduction band Q, respectively. (E) Possible transition configuration for K-Γ (purple wave) and Q-Γ (cyan wave). The red (blue) curve denotes conduction band K (Q). The grey curve denotes valence band Γ. The electric dipole moment of each transition is labelled. rz represents the equivalent position of the wavefunction at each valley as discussed in the main text. (F) Experimental results of the photon energy of RMX1 (blue hollows), RMX2 (green hollows) and RMX3 (red hollows) in device D1 as a function of Fz extracted from Figure 2C. Orange, purple, black solid lines and corresponding shadow regions denote the linear-fit of RMX1/2/3 with a fixed slope μz(K-Γ) obtained from DFT calculations. The energy difference between RMX1 and RMX2 (RMX3) is ~22 meV (46 meV).

In the text
thumbnail Figure 4

(A) Contour plot of the PL spectra of device D1 as a function of photon energy (bottom axis) and n0 (left axis). The out-of-plane electric field Fz is fixed at zero. (B) PL spectra at diverse n0. Cyan and green peaks represent Lorentz-fit peaks for RMX1&2 and RMT1&2, respectively. Note that the range of 1.45–1.60 eV in both (A) and (B) are magnified by 10 times for clarity. Offset is set in (B) for better resolution. (C) Lorentz-fitted results extracted from (B) of XA (red hollows), XA- (carmine hollows), RMX1 (green triangles), RMX2 (blue cubes), RMT1 (light green triangles) and RMT2 (light blue cubes). The binding energy of XA-, RMT1 and RMT2 is approximately 24, 59 and 57 meV, respectively.

In the text

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