Issue 
Natl Sci Open
Volume 2, Number 4, 2023
Special Topic: Twodimensional 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 
RESEARCH ARTICLE
Real and momentumindirect neutral and charged excitons in a multivalley semiconductor
^{1}
Beijing National Laboratory for Condensed Matter Physics, Key Laboratory for Nanoscale Physics and Devices, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
^{2}
School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China
^{3}
Ningbo Institute of Materials Technology & Engineering, Chinese Academy of Sciences, Ningbo 315201, China
^{4}
Research Center for Functional Materials, National Institute for Materials Science, 11 Namiki, Tsukuba 3050044, Japan
^{5}
International Center for Materials Nanoarchitectonics, National Institute for Materials Science, 11 Namiki, Tsukuba 3050044, Japan
^{6}
Department of Electronics and Nanoengineering, Aalto University, Tietotie 3, Espoo FI02150, Finland
^{7}
Quantum Technology Finland (QTF) Centre of Excellence, Department of Applied Physics, Aalto University, Aalto FI00076, Finland
^{8}
Beijing Key Laboratory for Nanomaterials and Nanodevices, Beijing 100190, China
^{9}
Songshan Lake Materials Laboratory, Dongguan 523808, China
^{*} Corresponding authors (emails: sxdu@iphy.ac.cn (Shixuan Du); luojun.du@iphy.ac.cn (Luojun Du); gyzhang@iphy.ac.cn (Guangyu Zhang))
Received:
30
September
2022
Revised:
2
January
2023
Accepted:
16
January
2023
Excitons dominate the photonic and optoelectronic properties of a material. Although significant advancements exist in understanding various types of excitons, progress on excitons that are indirect in both real and momentumspaces is still limited. Here, we demonstrate the real and momentumindirect neutral and charged excitons (including their phonon replicas) in a multivalley semiconductor of bilayer MoS_{2}, by performing electricfield/dopingdensity dependent photoluminescence. Together with firstprinciples calculations, we uncover that the observed real and momentumindirect exciton involves electron/hole from K/Γ valley, solving the longstanding controversy of its momentum origin. Remarkably, the binding energy of real and momentumindirect charged exciton is extremely large (i.e., ~59 meV), more than twice that of real and momentumdirect charged exciton (i.e., ~24 meV). The giant binding energy, along with the electrical tunability and long lifetime, endows real and momentumindirect excitons an emerging platform to study manybody physics and to illuminate developments in photonics and optoelectronics.
Key words: excitons / real and momentumindirect exciton / giant binding energy / electrical tunability / multivalley semiconductor
© The Author(s) 2023. Published by China Science Publishing & Media Ltd. and EDP Sciences.
This 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 momentumspaces, excitons can be categorized into four types: real and momentumdirect (typeI), realdirect but momentumindirect (typeII), momentumdirect but realindirect (typeIII), and real and momentumindirect (typeIV), as shown in Figure 1. TypeI excitons can strongly couple to photons and show large luminescence quantum efficiency, setting a foundation for a wide variety of optoelectronic applications, such as lightemitting diodes, lasers, solar cells, and optoelectronic devices [2–5]. However, the rather short lifetime of real and momentumdirect excitons strongly impedes their applications in scientific research and technological innovation where longlived excitons are required, for example, exciton superfluid phase, exciton crystals, and exciton transistors [6,7].
Figure 1 Real and momentumindirect and direct excitons. Based on the configurations of the constituent electrons and holes in real and momentumspaces, excitons can be divided into four types: real and momentumdirect (typeI), realdirect but momentumindirect (typeII), momentumdirect but realindirect (typeIII), and real and momentumindirect (typeIV). 
Notably, in some intentionally designed systems, such as van der Waals heterostructures with a staggered band alignment [8–10] 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 realspace indirect excitons (also referred to as interlayer excitons or spatially indirect excitons). Thanks to the spatial separation of the electrons and holes, realspace indirect excitons exhibit a much longer lifetime than the real and momentumdirect ones [9,12–14]. In the light of the longlived realspace indirect excitons, a wide variety of captivating physical phenomena has been demonstrated, including but not limited to exciton BoseEinstein condensation [15–18], correlated excitonic insulator states [19–21], and dissipationless valley exciton devices [14,22]. In addition, realspace indirect excitons harbor inbuilt electric dipoles and are widely tunable in applied electric fields [2,9,10,23–25], representing an advantageous scenario for technological applications. Although substantial developments and progress on understanding the realspace indirect excitons have been witnessed, the studies of realspace indirect excitons have mainly focused on the momentumbright species (i.e., typeIII excitons) with electrons and holes localized in the same valley of the Brillouin zone (BZ) [9,12,14,23,26–28]. However, the real and momentumindirect excitons (typeIV), which are expected to possess an even longer lifetime than the momentumdirect but realindirect ones due to the dark nature in both real and momentumspaces, are still largely unexplored experimentally.
A suitable candidate for investigating real and momentumindirect excitons should meet two basic conditions simultaneously. First, it should possess multivalleys in conduction and valence bands, endowing the possibility of momentumindirect 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 realspace, which enables the momentumindirect excitons to be indirect in realspace as well. In particular, twodimensional (2D) multivalley 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 momentumindirect excitons [24,25,29,30]. Despite the fact that the momentumindirect excitons in these materials have been well uncovered [31,32], their indirect nature in realspace remains equivocal. In this work, we fabricate hexagonal boron nitride (hBN) encapsulated dualgate devices of a multivalley semiconductor of bilayer MoS_{2}. Through electricfield tunable photoluminescence (PL) spectra, we identify the outofplane static electric dipole and quantumconfined Stark effect of momentumindirect exciton in bilayer MoS_{2}, providing the smoking gun evidence of their realspace indirect characteristic and hence the existence of real and momentumindirect exciton. In conjunction with density functional theory (DFT) calculations, we further uncover that the observed real and momentumindirect exciton in bilayer MoS_{2} involves electron and hole respectively from K and Γ valleys, addressing the longstanding controversy of its momentum origin. As bilayer MoS_{2} is doped with electrons, new sets of PL peaks corresponding to real and momentumindirect charged excitons (namely trions) emerge below the energy of the real and momentumindirect neutral excitons. Remarkably, the binding energy of real and momentumindirect trion is giant in bilayer MoS_{2}, twice that of the real and momentumdirect trions in transition metal dichalcogenide systems.
RESULTS AND DISCUSSION
Highquality, hBNencapsulated dualgate bilayer MoS_{2} devices (as schematically shown in Figure 2A) are fabricated by a van der Waals mediated drytransfer method (please see Section I in Supplementary Information (SI) for more details) with fewlayer graphene (FLG) as top and bottom gate electrodes. The dualgate configuration enables us to independently tune the carrier density n_{0} and outofplane electric field F_{z}. Here ${n}_{0}=({C}_{b}{V}_{bg}+{C}_{t}{V}_{tg})/e$ and ${F}_{z}=({C}_{b}{V}_{bg}{C}_{t}{V}_{tg})/2{\epsilon}_{0}{\epsilon}_{B}$, where e is the elementary charge, ${\epsilon}_{0}$ denotes the vacuum permittivity, ${\epsilon}_{B}$ is the outofplane dielectric constant of bilayer MoS_{2}, ${C}_{b}({V}_{bg})$ and ${C}_{t}\left({V}_{tg}\right)$ are the geometrical capacitances per area (applied voltages) for the bottom and top gates, respectively (details in Section IV in SI).
Figure 2 (A) Schematic of dualgate hBN encapsulated bilayer MoS_{2} device. (B) PL spectrum of device D1 at F_{z} = −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 F_{z} (left axis). The doping density remains unchanged. (D) Firstorder energy derivative of (C). Real and momentumindirect excitons are labelled as RMX_{1–3} in the sequence of decreasing emission energy. Gray (green) dashed lines in (C) and (D) represent linearfits of RMX_{1–3} (IX_{1,2}). The peak at around 1.99 eV that is insusceptible to F_{z} corresponds to the Raman G peak of graphene. 
Figure 2B depicts the PL spectrum of device D1 at F_{z} = −0.074 V/nm. Unless otherwise specified, the data presented in the main text are taken from the highquality 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 wellknown momentumdirect excitons, including both the realdirect transition at around 1.92 eV (labeled as X_{A}) and realindirect emissions at around 1.95/2.02 eV (dubbed as IX_{1}/IX_{2}), three momentumindirect excitons in the energy range of 1.50–1.58 eV (marked as RMX_{1}, RMX_{2} and RMX_{3}, in the sequence of decreasing emission energy) can be unequivocally observed. Note that the existence of two momentumdirect but realindirect exciton species of IX_{1} and IX_{2} 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 outofplane electric field F_{z}. To better distinguish the fine features, we extract the firstorder derivative of intensity I over photon energy E (∂I/∂E) as the function of F_{z}, as depicted in Figure 2D. Obviously, momentumdirect but realindirect excitons IX_{1} and IX_{2} vary linearly with the external electric field F_{z}, yet have reversed slops, evidencing their opposite outofplane static electric dipole moments. Via linear fitting (green dashed lines in Figure 2D and section IX in SI), we extracted the outofplane electric dipole moments of IX_{1} and IX_{2}: μ_{z}(IX_{1}) = (0.526 ±0.009)e∙nm and μ_{z}(IX_{2}) = −(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 F_{z} is applied, the shifts of IX_{1}/IX_{2} deviate from a simple linear Stark shift (Figure S5). This can be understood as the strong coupling between IX_{1}/IX_{2} and real and momentumdirect B excitons, as demonstrated by recent experimental measurements [33] and theoretical calculations [34].
Remarkably, the three momentumindirect excitons RMX_{13} in bilayer MoS_{2} also vary linearly with the applied outofplane electric field F_{z} (gray dashed lines in Figure 2D), evidencing the quantumconfined Stark effect and their indirect nature in realspace. This demonstrates that RMX_{13} belong to real and momentumindirect typeIV exciton. To further confirm their indirect nature in both real and momentumspaces of RMX_{13}, we perform DFT calculations with the PerdewBurkeErnzerhof generalized gradient approximation for exchangecorrelation interaction (Section VII in SI). Figure 3A shows the orbitalresolved band structure of bilayer MoS_{2}. 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, QΓ and KΓ transitions are the two possible configurations of momentumindirect excitons in bilayer MoS_{2}. Since it is difficult for conventional techniques to directly distinguish the two transitions, the exact origin of momentumindirect excitons in bilayer MoS_{2} is highly contentious. Some indicate the momentumindirect exciton in bilayer MoS_{2} to be KΓ transition [31,35], while the others reveal that it should be QΓ transition [36–38]. To solve the longstanding controversy, we derive the realspace distribution of the spinup wavefunctions at valence band Γ, and conduction band K/Q (Figures 3B–3D). For spindown wavefunctions, the realspace distribution can be evidently acquired by timereversal symmetry. Obviously, the spinup wavefunction at valence band Γ is symmetrically distributed in both layers, and shows 100% interlayer hybridization (Figure 3B). In stark contrast, the spinup wavefunction at conduction band K is only distributed in the lower layer and hence fully layerpolarized (Figure 3C). While the spinup wavefunction at the conduction band Q shows strong delocalization, it still has a slight layer polarization (Figure 3D). Because of the various realspace distributions and interlayer hybridization at diverse valleys, the equivalent positions in realspace of wave functions at conduction band K/Q, and valence band Γ should be quite different. Quantitatively, the equivalent positions of wave functions in realspace, defined as ${r}_{z}={{\displaystyle \int}}_{\infty}^{+\infty}r{\left\phi (r)\right}^{2}\text{d}r$, are r_{z}=−0.5t (0.5t), −0.08t (0.08t) and 0 for spinup (spindown) wavefunctions at conduction band K, conduction band Q, and valence band Γ, respectively. Here, ${\left\phi (r)\right}^{2}$ denotes the probability density of wavefunction $\phi (r)$ at realspace 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 momentumindirect transitions are realspace indirect and possess nonvanishing outofplane static electric dipole moments: ${\mu}_{z}(\text{K\Gamma})=e\cdot [{r}_{z}(\text{\Gamma}){r}_{z}(\text{K})]=\pm 0.5e\cdot t=\pm 0.308e\cdot \text{nm}$ for KΓ excitons and ${\mu}_{z}(\text{Q\Gamma})=e\cdot [{r}_{z}(\text{\Gamma}){r}_{z}(\text{Q})]=\pm 0.08e\cdot t=\pm 0.049e\cdot \text{nm}$ for QΓ excitons, as shown in Figure 3E. On applying an external electric field F_{z}, the exciton energies would vary linearly due to the quantumconfined Stark effect:
Figure 3 (A) Orbitalresolved projected band structure of bilayer MoS_{2} considering the spinorbit coupling. (B), (C), (D) Distribution of realpart of spinup wavefunctions in realspace at valence band Γ, conduction band K, and conduction band Q, respectively. (E) Possible transition configuration for KΓ (purple wave) and QΓ (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. r_{z} represents the equivalent position of the wavefunction at each valley as discussed in the main text. (F) Experimental results of the photon energy of RMX_{1} (blue hollows), RMX_{2} (green hollows) and RMX_{3} (red hollows) in device D1 as a function of F_{z} extracted from Figure 2C. Orange, purple, black solid lines and corresponding shadow regions denote the linearfit of RMX_{1/2/3} with a fixed slope μ_{z}(KΓ) obtained from DFT calculations. The energy difference between RMX_{1} and RMX_{2} (RMX_{3}) is ~22 meV (46 meV). 
Finally, we study the dopingdensity dependent responses of real and momentumindirect excitons in bilayer MoS_{2}. Figures 4A and 4B respectively depict the contour plot and linecuts of PL spectra as a function of doping density. Here the outofplane electric field F_{z} is fixed at zero. Remarkably, when bilayer MoS_{2} is doped with electrons, new sets of PL peaks emerge below the energy of the neutral real and momentumindirect RMX_{1} and RMX_{2}, which can be fitted to two new peaks with Lorentzian function (Figure 4B) and correspond to real and momentumindirect trions (labeled as RMT_{1} and RMT_{2}, respectively). This is in close resemblance to the wellstudied real and momentumdirect exciton X_{A}, which is tuned into charged exciton (X_{A}^{}) upon electrostatic doping [41]. Note that at F_{z}=0, only RMX_{1} and RMX_{2} can be probed (Figures 2C and 2D). Figure 4C plots the energies of RMT_{1}/RMT_{2} and RMX_{1}/RMX_{2} , as well as X_{A}/X_{A}^{}, as a function of doping density. Notably, the binding energy of real and momentumindirect RMT_{1} (RMT_{2}), i.e., the energy difference between RMX_{1} (RMX_{2}) and RMT_{1} (RMT_{2}), can reach up to ~59 meV (~57 meV), which is more than twice that of real and momentumdirect X_{A}^{} (~24 meV) in bilayer MoS_{2} and also larger than the stateoftheart results of previously reported trion in 2D semiconductors [42–45].
Figure 4 (A) Contour plot of the PL spectra of device D1 as a function of photon energy (bottom axis) and n_{0} (left axis). The outofplane electric field F_{z} is fixed at zero. (B) PL spectra at diverse n_{0}. Cyan and green peaks represent Lorentzfit peaks for RMX_{1&2} and RMT_{1&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) Lorentzfitted results extracted from (B) of X_{A} (red hollows), X_{A}^{} (carmine hollows), RMX_{1} (green triangles), RMX_{2} (blue cubes), RMT_{1} (light green triangles) and RMT_{2} (light blue cubes). The binding energy of X_{A}^{}, RMT_{1} and RMT_{2} is approximately 24, 59 and 57 meV, respectively. 
CONCLUSION
In short, we demonstrate the real and momentumindirect neutral/charged excitons (including their phonon replicas) in a multivalley semiconductor of bilayer MoS_{2} by the combination of electricfield/dopingdensity tunable PL measurements and firstprinciples calculations. Because of the sizable inbuilt electric dipoles, real and momentumindirect excitons present quantumconfined Stark effect and are widely tunable in applied electric fields. Moreover, the Coulomb interaction between real and momentumindirect excitons and free carriers is astonishingly strong in bilayer MoS_{2}, endowing the giant binding energy of real and momentumindirect trion of ~59 meV, more than twice that of real and momentumdirect trion (i.e., ~24 meV). Our work not only fulfills the knowledge on real and momentumindirect neutral and charged excitons, but also sheds light on the understanding and engineering of manybody physics and optoelectronics based on multivalley 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 KeyArea 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 highquality hBN 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. cowrote 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.
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All Figures
Figure 1 Real and momentumindirect and direct excitons. Based on the configurations of the constituent electrons and holes in real and momentumspaces, excitons can be divided into four types: real and momentumdirect (typeI), realdirect but momentumindirect (typeII), momentumdirect but realindirect (typeIII), and real and momentumindirect (typeIV). 

In the text 
Figure 2 (A) Schematic of dualgate hBN encapsulated bilayer MoS_{2} device. (B) PL spectrum of device D1 at F_{z} = −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 F_{z} (left axis). The doping density remains unchanged. (D) Firstorder energy derivative of (C). Real and momentumindirect excitons are labelled as RMX_{1–3} in the sequence of decreasing emission energy. Gray (green) dashed lines in (C) and (D) represent linearfits of RMX_{1–3} (IX_{1,2}). The peak at around 1.99 eV that is insusceptible to F_{z} corresponds to the Raman G peak of graphene. 

In the text 
Figure 3 (A) Orbitalresolved projected band structure of bilayer MoS_{2} considering the spinorbit coupling. (B), (C), (D) Distribution of realpart of spinup wavefunctions in realspace at valence band Γ, conduction band K, and conduction band Q, respectively. (E) Possible transition configuration for KΓ (purple wave) and QΓ (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. r_{z} represents the equivalent position of the wavefunction at each valley as discussed in the main text. (F) Experimental results of the photon energy of RMX_{1} (blue hollows), RMX_{2} (green hollows) and RMX_{3} (red hollows) in device D1 as a function of F_{z} extracted from Figure 2C. Orange, purple, black solid lines and corresponding shadow regions denote the linearfit of RMX_{1/2/3} with a fixed slope μ_{z}(KΓ) obtained from DFT calculations. The energy difference between RMX_{1} and RMX_{2} (RMX_{3}) is ~22 meV (46 meV). 

In the text 
Figure 4 (A) Contour plot of the PL spectra of device D1 as a function of photon energy (bottom axis) and n_{0} (left axis). The outofplane electric field F_{z} is fixed at zero. (B) PL spectra at diverse n_{0}. Cyan and green peaks represent Lorentzfit peaks for RMX_{1&2} and RMT_{1&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) Lorentzfitted results extracted from (B) of X_{A} (red hollows), X_{A}^{} (carmine hollows), RMX_{1} (green triangles), RMX_{2} (blue cubes), RMT_{1} (light green triangles) and RMT_{2} (light blue cubes). The binding energy of X_{A}^{}, RMT_{1} and RMT_{2} is approximately 24, 59 and 57 meV, respectively. 

In the text 
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